Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy
The Geological Society of London Books Editorial Committee Chief Editor
Bob Pankhurst (UK) Society Books Editors
John Gregory (UK) Jim Griffiths (UK) John Howe (UK) Howard Johnson (UK) Rick Law (USA) Phil Leat (UK) Nick Robins (UK) Randell Stephenson (UK) Society Books Advisors
Eric Buffetaut (France) Jonathan Craig (Italy) Tom McCann (Germany) Mario Parise (Italy) Satish-Kumar (Japan) Gonzalo Veiga (Argentina) Maarten de Wit (South Africa)
Geological Society books refereeing procedures The Society makes every effort to ensure that the scientific and production quality of its books matches that of its journals. Since 1997, all book proposals have been refereed by specialist reviewers as well as by the Society’s Books Editorial Committee. If the referees identify weaknesses in the proposal, these must be addressed before the proposal is accepted. Once the book is accepted, the Society Book Editors ensure that the volume editors follow strict guidelines on refereeing and quality control. We insist that individual papers can only be accepted after satisfactory review by two independent referees. The questions on the review forms are similar to those for Journal of the Geological Society. The referees’ forms and comments must be available to the Society’s Book Editors on request. Although many of the books result from meetings, the editors are expected to commission papers that were not presented at the meeting to ensure that the book provides a balanced coverage of the subject. Being accepted for presentation at the meeting does not guarantee inclusion in the book. More information about submitting a proposal and producing a book for the Society can be found on its web site: www.geolsoc.org.uk. It is recommended that reference to all or part of this book should be made in one of the following ways: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) 2011. Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360. Rutter, E. H., Mecklenburgh, J. & Brodie, K. H. 2011. Rock mechanics constraints on mid-crustal, low-viscosity flow beneath Tibet. In: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 329– 336.
GEOLOGICAL SOCIETY SPECIAL PUBLICATION NO. 360
Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy
EDITED BY
DAVID J. PRIOR University of Otago, New Zealand
ERNEST H. RUTTER University of Manchester, UK
and DANIEL J. TATHAM University of Liverpool, UK
2011 Published by The Geological Society London
THE GEOLOGICAL SOCIETY The Geological Society of London (GSL) was founded in 1807. It is the oldest national geological society in the world and the largest in Europe. It was incorporated under Royal Charter in 1825 and is Registered Charity 210161. The Society is the UK national learned and professional society for geology with a worldwide Fellowship (FGS) of over 10 000. The Society has the power to confer Chartered status on suitably qualified Fellows, and about 2000 of the Fellowship carry the title (CGeol). Chartered Geologists may also obtain the equivalent European title, European Geologist (EurGeol). One fifth of the Society’s fellowship resides outside the UK. To find out more about the Society, log on to www.geolsoc.org.uk. The Geological Society Publishing House (Bath, UK) produces the Society’s international journals and books, and acts as European distributor for selected publications of the American Association of Petroleum Geologists (AAPG), the Indonesian Petroleum Association (IPA), the Geological Society of America (GSA), the Society for Sedimentary Geology (SEPM) and the Geologists’ Association (GA). Joint marketing agreements ensure that GSL Fellows may purchase these societies’ publications at a discount. The Society’s online bookshop (accessible from www.geolsoc. org.uk) offers secure book purchasing with your credit or debit card. To find out about joining the Society and benefiting from substantial discounts on publications of GSL and other societies worldwide, consult www.geolsoc.org.uk, or contact the Fellowship Department at: The Geological Society, Burlington House, Piccadilly, London W1J 0BG: Tel. þ 44 (0)20 7434 9944; Fax þ 44 (0)20 7439 8975; E-mail:
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Contents Dedication to Martin Casey (1948–2008)
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PRIOR, D. J., RUTTER, E. H. & TATHAM, D. J. Deformation mechanisms, rheology and tectonics: microstructures, mechanics and anisotropy: introduction
1
Lattice Preferred Orientations and Anisotropy LLOYD, G. E., BUTLER, R. W. H., CASEY, M., TATHAM, D. J. & MAINPRICE, D. Constraints on the seismic properties of the middle and lower continental crust
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DEMPSEY, E. D., PRIOR, D. J., MARIANI, E., TOY, V. G. & TATHAM, D. J. Mica-controlled anisotropy within mid-to-upper crustal mylonites: an EBSD study of mica fabrics in the Alpine Fault Zone, New Zealand
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LLOYD, G. E., HALLIDAY, J. M., BUTLER, R. W. H., CASEY, M., KENDALL, J.-M., WOOKEY, J. & MAINPRICE, D. From crystal to crustal: petrofabric-derived seismic modelling of regional tectonics
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DI´AZ-AZPIROZ, M., LLOYD, G. E. & FERNA´NDEZ, C. Deformation mechanisms of plagioclase and seismic anisotropy of the Acebuches metabasites (SW Iberian massif)
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WATANABE, T., SHIRASUGI, Y., YANO, H. & MICHIBAYASHI, K. Seismic velocity in antigorite-bearing serpentinite mylonites
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WALLIS, S. R., KOBAYASHI, H., NISHII, A., MIZUKAMI, T. & SETO, Y. Obliteration of olivine crystallographic preferred orientation patterns in subduction-related antigorite-bearing mantle peridotite: an example from the Higashi-Akaishi body, SW Japan
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WASSMANN, S., STO¨CKHERT, B. & TREPMANN, C. A. Dissolution precipitation creep versus crystalline plasticity in high-pressure metamorphic serpentinites
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MORALES, L. F. G., MAINPRICE, D., LLOYD, G. E. & LAW, R. D. Crystal fabric development and slip systems in a quartz mylonite: an approach via transmission electron microscopy and viscoplastic self-consistent modelling
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MAINPRICE, D., HIELSCHER, R. & SCHAEBEN, H. Calculating anisotropic physical properties from texture data using the MTEX open-source package
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Microstructures, Mechanisms and Rheology AUSTIN, N. J. The microstructural and rheological evolution of shear zones
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SKEMER, P., SUNDBERG, M., HIRTH, G. & COOPER, R. Torsion experiments on coarse-grained dunite: implications for microstructural evolution when diffusion creep is suppressed
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FARLA, R. J. M., FITZ GERALD, J. D., KOKKONEN, H., HALFPENNY, A., FAUL, U. H. & JACKSON, I. Slip-system and EBSD analysis on compressively deformed fine-grained polycrystalline olivine
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HILDYARD, R. C., PRIOR, D. J., MARIANI, E. & FAULKNER, D. R. Characterization of microstructures and interpretation of flow mechanisms in naturally deformed, fine-grained anhydrite by means of EBSD analysis
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PEARCE, M. A. & WHEELER, J. Grain growth and the lifetime of diffusion creep deformation
257
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HOBBS, B. E. & ORD, A. Microstructures in deforming–reactive systems
273
IACOPINI, D., FRASSI, C., CAROSI, R. & MONTOMOLI, C. Biases in three-dimensional vorticity analysis using porphyroclast system: limits and application to natural examples
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BERTON, J. R., DURNEY, D. W. & WHEELER, J. Diffusion-creep modelling of fibrous pressure shadows II: influence of inclusion size and interface roughness
319
RUTTER, E. H., MECKLENBURGH, J. & BRODIE, K. H. Rock mechanics constraints on mid-crustal low-viscosity flow beneath Tibet
329
Index
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Fig 1. Martin Casey on location (a) at the Lac d’Emosson, NW Alps in the mid 1980s, the era of ‘heroic fieldwork’; (b) at Knockan Crag, NW Scotland in the mid 2000s, rediscovering a passion for mylonites; and (c) teaching an undergraduate class on the Moine Thrust.
Martin Casey (1948 – 2008) Martin Casey, who died on the 10 June 2008 after a short illness, was a structural geologist of world renown. He led numerical approaches to understanding rock deformation and the development of tectonic structures, pioneering the quantitative analysis of rock fabrics, textures and folds. These researches now underpin diverse areas of geology, such as quantifying the strength of rocks in the crust which leads to calibrations of the seismic properties of the minerals in continents and the prediction of the fine-scale structure of hydrocarbon reservoirs. This Special Publication is dedicated to his memory not only for his fundamental contribution to science but also as a mentor, colleague and friend. Martin was born on the 3 September 1948 into Lancastrian farming stock, and brought up at White Goat Farm outside the village of Whalley. His secondary education at the Royal Grammar School in Clitheroe was marked by a growing flair for mathematics and an extra-curricular passion for high-speed hill-walking. His younger brother and cousin both talk of chasing down distant walkers on a Snowdonia path or youthful races up a Lakeland fell. This instilled in Martin a lifelong love of mountains and geology and an intensely competitive spirit. He left school in 1966 with the Weeks Memorial Exhibition prize for excellent A-levels. After a brief flirtation with engineering at ICI on Teeside, he went to Churchill College, Cambridge to read Natural Sciences. There he met Jane Wild, lifelong confidante and soon his wife. After graduation in 1970 they moved to London. Keen to find an area of geology within which to apply his mathematical skills, Martin went to Imperial College first for MSc studies and then on to a PhD. As fellow graduate John Tipper (now at Freiburg, Switzerland) remembers, Martin quickly sold off his palaeontology textbooks, clearing the shelves for a career in structural geology. Once at IC, Martin settled into PhD life occupying one of the legendary cubicles that housed graduate students in the Royal School of Mines. The early 1970s at IC were especially eventful, with John Ramsay building a group of talented young geologists who pushed the boundaries of rock deformation studies. Fellow inmate of the time Stan White recalls the continuous discussions and banter on subjects ranging from the merits (or otherwise) of touring Australian cricket teams to the esoterica of deformation of quartz crystals. Stan remembers long nights at the Queen’s Arms, a favourite watering hole close to the department,
developing the theoretical basis for textural and microstructural analysis. Martin completed his PhD thesis on the computer modelling of geological structures and took up a Natural Environment Research Council postdoctoral fellowship in Leeds in 1976, joining the newly installed court of John Ramsay. It did not last long, as Ramsay moved to ETH Zurich with Martin following a couple of years later. At Leeds Martin had pioneered X-ray methods for the measurement of crystallographic orientations of deformed minerals with Andrew Siddans. The technology also went to Zurich and, in collaboration with Stefan Schmid, Martin set upon a remarkable period of research. Stefan records Martin’s arrival, resplendent in formal shirt and jacket but wearing mountain boots and sporting a rucksack full of computer cards (the medium by which Martin, and others of the time, loaded programs). It must have been a big rucksack. The early 1980s in Zurich found Martin developing and applying new microstructural methods. With Stefan Schmid, he ran a pioneering set of experiments on samples of deformed limestone and quartzite using their new X-ray texture goniometer. As part of this work he programmed an efficient algorithm for the quantitative texture analysis of low-symmetry minerals using X-ray pole figure data and the orientation distribution function (Casey 1981). As Dave Mainprice recalls, this was a very difficult task requiring a major effort on Martin’s part; he was justifiably proud of this code. As was typical of him, he freely distributed the source code and his hand-written manual to all those genuinely interested. The use of this texture analysis package by Martin and collaborators at ETH greatly advanced the understanding of quartz and calcite deformation mechanisms in shear zones (e.g. Schmid et al. 1981a, b), a very controversial subject at the time. Together, Martin and Stefan collected deformed quartzites from different types of structures and carefully measured the microstructures and crystal fabrics, thereby calibrating them to known deformation states. This work has underpinned much subsequent work on quantitative microstructural geology and its application to larger-scale dynamics. It is characteristic of Martin that the seminal paper with Stefan was published not in a highly cited journal but in a monograph, albeit in honour of legendary rock deformer Mervyn Paterson (Schmid & Casey 1986). As Stefan comments – ‘He stood, as is typical for him, as the mastermind of that kind of analysis – in the background’.
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The other really influential microstructure publication from the time was developed in collaboration with Rick Law and Rob Knipe (both then at Leeds), applying microstructural methods to understand the evolution of quartz mylonites from the Moine Thrust Zone (Law et al. 1986). Again the paper appeared in a journal outside the scope of conventional citation indexes, but has had enormous influence manifest by the pilgrimages made by numerous researchers to the key outcrops at the Stack of Glencoul in northwest Scotland. A critical part of the research involved Rick Law visiting Zurich through the summer of 1984 for an extended period of collaboration with Martin. By then Martin and Jane had a young family, daughters Helen and Gwen. As ever, Martin generously offered the hospitality of the family apartment: a grand place with fantastic views over Zurich. Rick was invited en famille – and arrived in July – to the warm welcome of Martin and the complete surprise of the rest of the Caseys. This uncoordinated hospitality was a regular occurrence of the time. With all this activity in microstructural geology, it would be wrong to think that through the early years at Zurich Martin was stuck in the laboratory. Being in John Ramsay’s group made field research, especially in the Helvetics, obligatory. Martin was a very capable and careful observer of structures in the field but had a slightly off-hand attitude to the business of actually recording data, in marked contrast to his careful laboratory work and computer programming. Zurich colleague Dorothee Dietrich records that: ‘on the outcrop he preferred to rest and smile, often a bit sarcastically, whilst we ran around observing and measuring’. The field was a place to inspire new theoretical understanding, commonly applied by research students that Martin cosupervised. This type of collaboration led to one of his favourite publications from the time with Peter Huggenberger (Casey & Huggenberger 1985) on numerical modelling of folds formed during shearing, a study inspired by the structures of the Morcles Nappe. Through the early 1980s, Martin also took a keen interest in conflict resolution. It was a time when thrust tectonics research, especially in Britain, was at its height. Martin cast a cynical eye on the proceedings at various Tectonic Studies Group meetings, where half the presentations might be on various duplex structures from around the world, deriding the participants as ‘the Ramping Club’. In contrast, Zurich was the home of true structural geology where rocks folded and careful kinematic research still survived. Young British researchers (the author included) were taken under Martin’s wing and shown parts of the Alps – especially the Helvetics – where proper deformation could be found. It was a time for, as he put it, ‘heroic
fieldwork’, and vigorous romps up to high mountain vantage points to observe another spectacular structural panorama. This of course led to huge arguments on the interpretation of all the geology, especially the significance of thrust faults; this was always Martin’s intention (and great fun). A kind of common ground was reached in which folds and strain could be related to the same deformation as thrusting, ideas that Martin wrote up with Dorothee Dietrich (Dietrich & Casey 1989). Into the 1990s Martin applied his petrofabric research with Dave Mainprice at Montpellier to explore the seismic responses of deformed rocks (Mainprice & Casey 1990). He also co-advised research students in Ramsay’s group and embedded his careful quantitative approaches in a new generation of scientists. In addition, he provided critical input into experimental rock deformation, working closely with Dave Olgaard on some of the first torsion experiments (Casey et al. 1998). Research using the X-ray texture goniometer continued (e.g. Mainprice et al. 1993), reinvigorating Martin’s studies of calcite microstructures (e.g. Bruhn & Casey 1997; Khazanehdari et al. 1998). He always had a soft spot for this, his ‘ancienne carrie`re de marbre’ (a typically deliberate Casey mistranslation of ‘old marble quarry’, a common feature shown on maps of the French Alps). Martin’s career in Zurich eventually led to conferment of Privatdozent (equivalent to a British D.Sc.) in 1991, but still allied to a post reliant on the continued patronage of John Ramsay. All changed with Ramsay’s retirement. His replacement in the structural geology chair at Zurich, Jean-Pierre Burg, was unable to find a way of continuing Martin’s contract; he therefore moved to Leeds as a senior research assistant in 1997 which initially led to collaboration with Rob Knipe and his Rock Deformation Research group. Martin was unprepared to enter the intensely entrepreneurial world of UK geoscience, however, nor was he inclined to compromise his own research ambitions to suit the needs of company sponsors. It was a difficult transition but did lead to research collaboration with Quentin Fisher, at the time a fellow RDR member. And so Martin moved into industry-led research and developed new numerical approaches for the investigation of compaction in sandstones, important for understanding hydrocarbon reservoir properties (Fisher et al. 1999, 2003). He developed research collaborations with the author (Casey & Butler 2004) after years of talking about the idea while gazing upon folds in the Helvetics; with Cindy Ebinger on continental extensional tectonics and magma emplacement (Ebinger & Casey 2001); with Geoff Lloyd and the author on seismic anisotropy and deformation fabrics (Tatham et al. 2008; Lloyd et al. 2010); and with Geoff and
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Mike Kendall on the seismic response of deformed sediments (Valcke et al. 2006). The result was a string of important papers, again with Martin taking an undeserved back-seat. But his research standing was recognized by visiting professorships at the Ecole Normale Supe´rieure in Nancy, France, and the University of Go¨ttingen, Germany. Before he left Zurich, Martin had been gently jibed by Jean-Pierre Burg that true structural geologists have all worked in the Himalayas. When the chance came in 1999 to join the author and colleagues on a research trip to Nanga Parbat, he jumped at it. Martin loved Pakistan, the friendly people, the chaotic-yet-functional roads and the spectacular structural geology on the hillsides above the Indus. He mused, in typical droll fashion: ‘Hmmmm, Himalayas (pause) – only one outcrop (longer pause) – all of it!’ Whilst the Nanga Parbat work continued Martin’s research on mylonites and folding that he had followed his entire career (e.g. Casey 1980), it reflected also his increasing interest in diverse larger-scale tectonic features. This interest saw Martin become an increasingly important figure in the newly constituted Institute of Geophysics and Tectonics in Leeds. Despite international status and a citation record equivalent to that of most of the professors in the then Earth Sciences department at Leeds, Martin was offered only temporary teaching appointments. He patched together a career out of various shortterm teaching contracts: a stressful enough existence for a young post-doc and frustrating, to say the least, for an established international scientist. Thus, it is indeed curious that neither Zurich nor Leeds were able to appoint Martin to the permanent faculty – something that would have given him the necessary stability to develop more of his research and supervise his own string of PhD students (a research and training approach at which Martin excelled). As Stan White commented on learning of Martin’s death: ‘he had a brilliant mind but never really had a proper turn’. If Martin’s research was arguably never truly appreciated by the universities that employed him, his teaching and pastoral interactions with students definitely were. That this should be so is not at all obvious. Martin had a passion for numerical approaches that he attempted to instil in an increasingly mathematically challenged student population. In this regard he would regularly quote lines from the theme to the cult TV cowboy show Rawhide ‘don’t try to understand them, just rope ‘em up and brand them’ (applied to students). This of course belied his true beliefs; he was enormously patient and would carefully explain an especially challenging concept to anyone who was interested. Able students found the approach stimulating and
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many were inspired to take higher degrees that they had originally thought beyond them. Even the less-interested student still appreciated the care with which Martin would go through a problem. He became a kind of intellectual mascot for the younger generation and an increasingly rotund and white-haired Father Christmas at seasonal student parties (albeit a not entirely respectable one). Despite his increasing weight, Martin remained physically competitive. While in Zurich, during the close season for Alpine fieldwork, he would keep himself fit by joining ambitious aerobics classes and midday ‘Kondi’ sessions. It wasn’t simply about fitness and personal-trainer appreciation: encouraging unsuspecting visitors and other staff at ETH to his classes provided a ready lunch group for geological and cultural debates in the nearby Tannenbar. At Leeds, Martin’s approach to exercise revolved around the squash court and many a lean and hungry student would be run off their feet. This was great exercise for the student, who emerged well-beaten from the Sports Hall. It was rather less so for the smug senior man who, possessing the great racquet skills and tactical nous, had remained largely static in the centre of the court. The move to Leeds saw Martin take up residence in Otley and cement relationships that had been started during various visits through the 1990s. For a proud Lancastrian, setting up home in deepest Yorkshire (which he called ‘the occupied eastern slopes’) was not too traumatic. He struck up an instant friendship with his neighbour Ron Smith and became a regular member of the quiz teams in the Junction, his favourite pub. Of course he took great delight in winning, especially against teams containing other academics from Leeds. He also embarked on a series of ambitious and chaotic home-improvement plans. Central to these was the regeneration of his kitchen into a high-tech environment from which he could entertain visitors (not that he had any expertise in cooking). His first soire´e required careful and patient assistance and a meticulous list of instructions. At the top of the list was written ‘failing to plan is planning to fail’. It was a one-liner he recited on almost every available occasion. Unfortunately, away from the kitchen, he singly failed to apply it. His office in Leeds, for example, was regularly on the verge of condemnation as a fire risk. All manner of draft manuscripts interleaved between crisp packets, journals, trays of rock specimens, exam scripts, class lists and geological maps were piled high across desk, floor and half-hidden mounds of defunct computers. Accounts of Martin’s career and shared experiences can only scratch the surface. He had a disarming way of admitting his human frailties and many of his constant streams of anecdotes commonly had himself as the anti-hero. One particular
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favourite of the author’s is Martin’s account of locking himself out of his room in the Cathedral Hill Hotel, San Francisco at a meeting of the American Geophysical Union, after failing to distinguish the door to his en-suite bathroom from that to the corridor. He found himself on the fifth floor in the early hours of the morning, completely naked without even a pot-plant for cover and faced with the only course of action: a descent to the hotel lobby for a new key. Not many other people would share such a shocking experience with the world! But for all of the stories, Martin did keep himself to himself. He valued his own opinions and rarely asked for help. It seems likely that Martin truly liked to solve all problems and meet all life’s challenges himself. Perhaps this reflected his intense competitive spirit. Some may have found this offhand or even arrogant but he was also enormously kind. Although Martin above all liked to win, he did not actually like other people to lose –he was really competing with himself. For those of us who knew him (and Martin left hundreds of friends all over the world) it is hard to accept that he is gone. We half-expect him to appear inappropriately at a prominent door in the front of a conference auditorium, as he did in life, fashionably late for a presentation, only to look slightly disapproving and retreat to a less conspicuous entrance. Future generations of students will miss having his quiet encouragement and insightful discussion. We are all the poorer for his passing. Rob Butler, with notes from Dorothee Dietrich, Nigel Harris, Geoff Lloyd, Stefan Schmid, Ron Smith, Tracy Rushmer and Stan White.
References Bruhn, D. F. & Casey, M. 1997. Texture development in experimentally deformed two-phase aggregates of calcite and anhydrite. Journal of Structural Geology, 19, 909–925. Casey, M. 1980. Mechanics of shear zones in isotropic dilatant materials. Journal of Structural Geology, 2, 143– 147. Casey, M. 1981. Numerical analysis of X-ray texture data: an implementation in FORTRAN allowing triclinic or axial specimen symmetry and most crystal symmetries. Tectonophysics, 78, 51–64. Casey, M. & Huggenberger, P. 1985. Numerical modelling of finite-amplitude similar folds developing under general deformation histories. Journal of Structural Geology, 7, 103–114. Casey, M. & Butler, R. W. H. 2004. Modelling approaches to understanding fold development: implications for hydrocarbon reservoirs. Marine and Petroleum Geology, 21, 933– 946. Casey, M., Kunze, K. & Olgaard, D. L. 1998. Texture of Solnhofen limestone deformed to high strains in torsion. Journal of Structural Geology, 20, 255–267.
Dietrich, D. & Casey, M. 1989. A new tectonic model of the Helvetic nappes. In: Coward, M. P., Dietrich, D. & Ries, A. C. (eds) Alpine Tectonics. Geological Society, London, Special Publications, 45, 47–63. Ebinger, C. J. & Casey, M. 2001. Continental break-up in magmatic provinces: an Ethiopian example. Geology, 29, 527– 530. Fisher, Q. J., Casey, M., Clennell, M. B. & Knipe, R. J. 1999. Mechanical compaction of deeply buried sandstones of the North Sea. Marine and Petroleum Geology, 16, 605 –618. Fisher, Q. J., Casey, M., Harris, S. D. & Knipe, R. J. 2003. Fluid-flow properties of faults in sandstone: the importance of temperature history. Geology, 31, 965–968. Khazanehdari, J., Rutter, E. H., Casey, M. & Burlini, L. 1998. The role of crystallographic fabric in the generation of seismic anisotropy and reflectivity of high strain zones in calcite rocks. Journal of Structural Geology, 20, 293– 299. Law, R. D., Casey, M. & Knipe, R. J. 1986. Kinematic and tectonic significance of microstructures and crystallographic fabrics within quartz mylonites from the Assynt and Eriboll regions of the Moine Thrust Zone, NW Scotland. Transactions of the Royal Society of Edinburgh: Earth Sciences, 77, 99– 125. Lloyd, G. E., Butler, R. W. H., Casey, M. & Mainprice, D. 2010. Deformation fabrics, mica and seismic properties of the continental crust. Earth and Planetary Sciences Letters, 288, 320–328. Mainprice, D. & Casey, M. 1990. The complete seismic properties of quartz mylonites with typical fabrics: relationship to kinematics and temperature. Geophysical Journal, 103, 599–608. Mainprice, D., Bouchez, J.-L., Casey, M. & Dervin, P. 1993. Quantitative texture analysis of naturally deformed anhydrite by neutron diffraction texture goniometry. Journal of Structural Geology, 15, 793–804. Schmid, S. M. & Casey, M. 1986. Complete fabric analysis of some commonly observed quartz c-axis patterns. In: Hobbs, B. E. & Heard, H. C. (eds) Deformation of Rocks: Laboratory Studies. American Geophysical Union, Washington, Geophysical Monograph, 36, 263–286. Schmid, S. M., Casey, M. & Starkey, J. 1981a. An illustration of the advantages of a complete texture analysis described by the orientation distribution function (ODF) using quartz pole figure data. Tectonophysics, 78, 101– 117. Schmid, S. M., Casey, M. & Starkey, J. 1981b. The microfabric of calcite tectonites from the Helvetic nappes (Swiss Alps). In: McClay, K. R. & Price, N. J. (eds) Thrust and Nappe Tectonics. Geological Society, London, Special Publications, 9, 151–158. Tatham, D. J., Lloyd, G. E., Butler, R. W. H. & Casey, M. 2008. Amphibole and lower crustal seismic properties. Earth and Planetary Science Letters, 267, 118–128. Valcke, S. L., Casey, M., Lloyd, G. E., Kendall, J. M. & Fisher, Q. J. 2006. Lattice preferred orientation and seismic anisotropy in sedimentary rocks. Geophysical Journal International, 166, 652– 666.
Deformation mechanisms, rheology and tectonics: microstructures, mechanics and anisotropy: introduction DAVID J. PRIOR1*, ERNEST H. RUTTER2 & DANIEL J. TATHAM3 1
Department of Geology, University of Otago, Dunedin, New Zealand
2
Rock Deformation Laboratory, School of Earth, Atmospheric and Environmental Sciences, University of Manchester, Oxford Road, Manchester M13 9PL, UK 3
School of Earth and Ocean Sciences, University of Liverpool, Liverpool L69 3GP, UK *Corresponding author (e-mail:
[email protected])
This special publication of the Geological Society of London presents recent advances in the study of deformation mechanisms and rheology and their application to tectonics. We have subdivided the papers into two themed sections. The inference of deformation processes, conditions and rheology at depth in active tectonic settings is of fundamental importance to a quantitative geodynamic understanding of deformation in the Earth. The papers in the section on Lattice Preferred Orientations and Anisotropy are extremely important as they underpin our ability to make such geodynamic interpretations from global seismic data. These papers reflect the growing emphasis on the determination of elastic properties from microstructures, from which acoustic properties can be computed for comparison with in situ seismic measurements. The component of the microstructure that receives most attention is the lattice preferred orientation (LPO), otherwise known as the crystallographic preferred orientation (CPO) or the texture (the term used in material science and metallurgy). The papers include new LPO measurements (made almost exclusively by the relatively new technique of electron backscatter diffraction or EBSD), exploration of the significance of these data for seismic properties of both the crust and the mantle and modelling of LPO generation. An invited contribution from Mainprice and colleagues introduces a computational toolbox to help researchers calculate anisotropic physical properties from their LPO data. Rock microstructures evolve during deformation and rock physical properties, including both elastic properties and creep rheology, evolve with the microstructures as a function of strain and time. The section on Microstructures, Mechanisms and Rheology reflects the fundamental importance of understanding microstructural evolution to our ability to estimate deformation processes and conditions from recovered samples or geophysical data and to our modelling of tectonics. An invited contribution from Austin focuses on some of the
key issues from the last few decades: how different mechanisms (grain size sensitive and grain size insensitive) compete and interact to control the evolution of grain size and LPO. Many of the other papers touch on these issues and make use of combinations of laboratory experiments, field studies and computational methods to explore the controls on microstructural evolution and to relate microstructural evolution to rheology and largescale tectonic processes. It is clear from this collection of papers that resolution of the controls on microstructural evolution in rocks remains at the cutting edge of Earth sciences.
Lattice preferred orientations and anisotropy The majority of published work on LPO relates to a relatively limited number of minerals, in particular olivine, quartz and calcite. There are probably a number of reasons for this, not least the fact that key components of the LPO of these minerals can be measured using a universal stage mounted on an optical microscope (technology that has been widely available for many decades). We have a good empirical understanding of the controls on LPO evolution in these minerals from extensive measurements of well-constrained rock samples (especially for quartz) and laboratory experiments (especially for calcite and olivine). Modelling of LPO evolution has also had significant developments in the last few decades and now does a good job of simulating LPOs in monomineralic aggregates undergoing plastic deformation. The chief difficulty in applying this understanding of LPO development is that real rocks are not generally monomineralic. Although nearly pure dunites, quartzites and marbles do exist in rock outcrops, we have to consider that the length scales sampled by seismic waves that are detected in broadband studies used to generate seismic anisotropy data are hundreds of metres to
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 1 –5. DOI: 10.1144/SP360.1 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
2
D. J. PRIOR ET AL.
tens of kilometres. Occurrences of pure quartzites in the crust certainly do not span these length scales, and it is probably true that rock volumes sampled by these seismic waves will be polymineralic. A recent trend in using LPOs to constrain anisotropic properties is therefore to consider the contributions of all of the mineral phases in a rock and all of the different rocks in a volume commensurate with the volume sampled by a seismic investigation. This approach has been made much easier by the development of the EBSD technique, enabling measurements of the full crystallographic orientation of most mineral phases and is an approach that links many of the papers in this section. Lloyd, Butler et al. explore the contribution to seismic anisotropy of a range of minerals important in the continental crust through construction of ‘fabric recipes’: essentially a synthesis of the fabric that relates to a particular mineralogy from the common LPOs of the component minerals. The fabric recipes enable the anisotropy signature of simple crustal-scale structures and compositions to be explored. The paper reviews the singlecrystal anisotropies of minerals important in the continental crust and explores which seismic parameters are more and less sensitive to mineralogy and strain symmetry. Mica is identified as potentially the most important mineral in controlling crustal anisotropy. Mica is particularly important from the perspective of crustal anisotropy as single-crystal anisotropies are high; moreover, micas are commonly aligned along the basal planes, giving rise to a strong aggregate anisotropy. Regional schist belts with aligned micas and large sedimentary basin systems with aligned clays, or micas grown mimetically on clays, will be significant components of the crustal anisotropy signal. Dempsey et al. present measurements from samples in the Alpine Fault Zone of New Zealand which show that, even in small proportions (20%), aligned mica will dominate the seismic anisotropy of quartz –feldspar – mica rocks. Additionally, this paper shows that mica LPOs can yield kinematic information. The calculated anisotropies match laboratory measurements and geophysical data are integrated into a qualitative large-scale model for crustal anisotropy of the Southern Alps. Modelling the influence of rock composition, deformation kinematics and structural geometry is taken to the next level by Lloyd, Halliday et al. who measure the LPOs of all volumetrically significant minerals in samples from the Nanga Parbat–Haramosh massif in the Himalayas and use these data to generate synthetic seismograms that explore simple structural geometries and resulting seismic anisotropy. These analyses show that
observed seismic anisotropies will be highly sensitive to foliation orientation. Plagioclase is probably the single most important mineral in the continental crust, particularly the middle and lower crust. Plagioclase can have very high single-crystal anisotropy, depending upon its composition. The three earlier papers in this section all measure plagioclase fabrics and, in all cases, there is little LPO and thus little contribution to sample anisotropy. Dı´az-Aspiroz et al. explore in more depth the potential role of plagioclase in crustal anisotropy. They measure plagioclase and hornblende LPOs from metabasites from the Variscan massif of southwest Spain. These data show that LPO development in plagioclase corresponds to inferred deformation mechanisms, with dislocation creep and twinning leading to strong LPOs and grain-boundary sliding leading to significantly weaker LPOs. Where plagioclase has a strong LPO it contributes to seismic anisotropy even though hornblende (also a highly anisotropic mineral) is also strongly aligned. For several decades we have considered olivine LPOs as perhaps the most important contribution to seismic anisotropy in the Earth’s upper mantle. Like the crust, the mantle is unlikely to be monomineralic and due consideration is now being given to other phases. Serpentine minerals are clearly of relevance given the potential for their formation from olivine and orthopyroxene in hydrating mineral reactions. Serpentine minerals are likely to be of significance to the rheology, microstructure and elastic physical properties of oceanic lithosphere in a variety of tectonic settings. Antigorite is a high-pressure serpentine mineral that is thought to have particular significance for subduction complexes. Watanabe et al. measure the ultrasonic velocities and LPOs of two antigorite-bearing samples from the Happo ultramafic complex in Japan. The data show that antigorite could give rise to high seismic anisotropies. Neither homogeneous strain (Voigt) nor homogeneous stress (Reuss) averaging recreate the velocity data from the LPO acceptably, and more sophisticated models which incorporate the microstructure need to be considered. Wallis et al. pick up on a previously unexplored aspect of the importance of serpentine minerals. They studied samples from an exhumed section of forearc mantle that had been partially serpentinized. Samples with less than 1% antigorite have a strong olivine LPO, with a-axes perpendicular to lineation in the foliation plane and b-axes perpendicular to foliation. Samples with more than 10% antigorite have weaker LPOs and are predicted to generate a different seismic signature, with lower anisotropy. The weakening of LPO is interpreted as the
INTRODUCTION
result of sliding and rotation of the olivine grains accommodated by internal deformation and grainboundary slip of antigorite. The significance is that olivine LPO may not be the important factor controlling mantle wedge seismic anisotropy. Wassmann et al. provide detailed microstructural data and interpret the deformation mechanisms in metamorphic serpentinites from the Zermatt – Saas zone in the Alps. The paper does not address LPO development directly, but is included in this section as a complement to the two previous papers. This paper focuses on the complexity of rock microstructures including the development of schistosity, layering and folding involving antigorite. It is inferred that the dominant deformation mechanisms are diffusion creep mechanisms rather than dislocation creep, with significant implications for rheology in a subduction setting. Modelling has been a key aspect of understanding LPO for nearly 40 years. It is rare that modelling relates to individual well-constrained samples. A quartz vein mylonite from Torridon has already been the subject of a number of detailed publications and Morales et al. use the existing data, together with new transmission electron microscopy (TEM) observations and some new LPO measurements to constrain viscoplastic selfconsistent (VPSC) models for LPO development. In this very detailed study, an up-strain sequence of LPO and TEM data places constraints on the relative activity of slip systems in quartz, constraints that may be translated to less wellconstrained scenarios. The growing research community with an interest in LPO and anisotropy owe a great debt to a small group of individuals who have dedicated significant time and effort to produce software for calculating and displaying data and have made these programs freely available. Martin Casey was one of the earliest proponents of this approach. Most of the papers in the first section of this volume use the open-access software packages written by David Mainprice to plot orientation data, calculate orientation statistics and calculate and visualize polycrystal elastic properties. David has facilitated a significant broadening of research activity in petrofabrics and this community is very grateful to him for this service. We are very pleased that he accepted our invitation to publish the next step in open-access software in this volume. Mainprice et al. presents a MatLab toolbox to help researchers calculate and manipulate anisotropic physical property data from LPOs. The paper clearly outlines the tensor mathematics important in such calculations and explains how to use the software package for a variety of tasks. We are sure that this paper alone will be of significant value to a large number of researchers.
3
Microstructures, mechanisms and rheology Microstructures play a key role in studies of deformed rock. They are the link between laboratory experiments and natural deformation; data from a laboratory experiment that produces a microstructure that we can recognize in natural tectonites has clear relevance to processes in nature. Microstructures also link deformation mechanisms and rheology; the microscale physical and chemical processes that facilitate microstructural change include the processes that will control the rheology. We have realized for a long time that microstructure, deformation mechanisms and rheology are intimately linked. The papers in this volume provide the next increment in our understanding of these links. When we examine naturally deformed rocks, microstructures are our principal link to deformation conditions such as stress magnitude and to the deformation mechanisms and rheology during active deformation. The quantitative microstructural measure that has had most significance over the last four decades is the grain size. The correlation of smaller grain sizes with larger flow stress magnitudes during crystal plastic deformation is broadly supported by laboratory data and observations in nature. Palaeopiezometers, which link recrystallized grain sizes and subgrain sizes to stress magnitudes, have therefore become important tools even though there is no complete agreement as to the way physical processes interact to control grain size during deformation. Nick Austin has recently played a key role in this area, most particularly in developing the concept of palaeowattmeters. The invited contribution by Austin explores the processes that control microstructural and rheological evolution with a particular focus on the attainment of steady-state grain sizes and LPO. He shows that the different approaches to understanding grain size have similar stress sensitivities but predict different grain-size magnitudes and temperature sensitivities. Evolution of LPO strength is less well understood. He concludes that we need to take a more holistic approach to quantification of microstructure as individual components, such as grain size and LPO, are not necessarily independent. One traditional limitation of laboratory deformation experiments has been that strains comparable to those in natural tectonites were difficult to achieve. In the last decade, torsion experiments in the Paterson apparatus have enabled creep to shear strains of 3 or much more in a wide range of rocks. Such high-strain experiments in olivine have been limited to fine grain sizes, where grain-size-sensitive mechanisms accommodate a significant proportion of the strain. Skemer et al. present the first high-strain data from coarse
4
D. J. PRIOR ET AL.
dunites. They explain the modified experimental protocols needed to conduct these experiments and present the results from experiments to shear strains of 3.5 on natural dunites with a pre-existing LPO. They demonstrate that microstructures and LPO can be inherited to high strain, something not predicted by experiments on fine-grained olivine. Given that our qualitative and quantitative understanding of microstructure and LPO generation during dislocation creep is dependent upon understanding the behaviour of individual slip systems as a function of grain orientation, it is alarming that there are few studies that try to correlate grain orientation and dislocation distribution in deformed rocks. Farla et al. provide new data from experimentally deformed, natural and synthetic dunites. They compare the dislocation densities from decorated grains with Schmid-factor, LPO and misorientation analyses of EBSD orientation data. Most grains are interpreted to deform by slip on several slip systems and the pattern of dislocation densities observed is rather more complex than we expect from the simple models used to predict LPO generation and evolution. They suggest that heterogeneous stress distributions may provide an explanation for these observations. Hildyard et al. present microstructural and LPO data from a suite of naturally deformed anhydrite rocks. Quantitative microstructural data from coarse-grained anhydrites show a correlation of grain shapes and an intense alignment of one crystal axis, suggesting that mechanisms other than dislocation creep can generate intense LPOs. In finer-grained samples LPO strength correlates inversely with grain size, supporting the idea that grain-boundary sliding, which accommodates more strain at finer grain sizes, weakens LPOs. When grain-size-sensitive mechanisms accommodate significant strain, the rate of grain growth relative to the strain rate will control the microstructural and rheological evolution. Pearce & Wheeler explore this through a modelling approach to define the timescale, controlled by grain growth, in which diffusion creep accommodates significant deformation. Their models predict that significant strain can be accommodated in the diffusion creep field in both plagioclase and olivine, although there will be an upper temperature limit above which rapid grain growth will shut down diffusion creep. They take the idea upscale in exploring the role of diffusion creep in a mantle plume where fine-grained olivine has been formed on passing through the spinel-olivine phase transformation. The driving forces for chemical changes, including phase changes, and microstructural changes can be formalized using thermodynamics. Although static equilibrium thermodynamics is in common
use in geology, understanding deformation really requires application of non-hydrostatic thermodynamics. Hobbs & Ord contribute to this at a fundamental level by considering the interaction of deformation and mineral reactions and conclude that shape fabrics should reflect the deformation rate tensor rather than the finite strain tensor. This paper generated considerable discussion between the authors and the referees – all with considerable expertise in this area. There is clearly no full agreement as to the form of this type of analysis and hopefully this paper will provide stimulation and a stepping stone to further development. Most deformed rocks are mechanically and microstructurally heterogeneous. Porphyroclasts are a common manifestation of this and are important as they have been used to interpret shear senses, strain symmetry and rheology. Iacopini et al. consider the errors associated with vorticity analysis based on porphyroclast geometry. They provide a protocol for testing of the data that enable the underlying assumptions of the vorticity analysis to be assessed, and conclude that ambiguities in analysis are most likely in deformation with a large flattening component. Rigid porphyroclasts often develop ‘pressure shadows’. Analysis of these has been important in constraining deformation kinematic history and diffusive mass-transfer processes associated with deformation. Berton et al. use a numerical model of diffusion creep to simulate the growth of fibrous pressure shadows around a rigid pyrite grain. They show that pyrite size and the diffusional conductance of the pyrite –matrix interface are important controls on pressure shadow development, and that inclusion geometry is not. Upscaling our understanding of rheology to help solve tectonic problems must always be a key goal for researchers interested in rock deformation. Many geodynamic models use inappropriate rheological models, with little consideration to the impact of their choices. Rutter et al. take a new look at the geodynamics of Tibet, where geophysical data have been used to suggest detachment across a mid-crustal low-viscosity layer; it has been suggested that the layer involves partial melt. In this paper laboratory-determined rheologies from partially molten granitic systems are used in a geodynamic model that reproduces the geophysical constraints. Use of the appropriate rheological data lends weight to the inference of the partially molten layer and provides the basis for a more sophisticated understanding of this large-scale system. This volume highlights some of the cutting edges of research into deformation mechanisms, rheology and tectonics at the start of a new decade. One major theme that emerges is that interpreting structure,
INTRODUCTION
dynamics and physical properties of the Earth’s interior requires an understanding of deformation mechanisms, microstructure and LPO development in polyphase rocks. Work on all scales, from the microstructural analysis of deformed rocks to the interpretation of whole-Earth geophysical data,
5
needs to consider that polyphase rocks will dominate in the crust and the mantle. Dave Prior Ernie Rutter & Dan Tatham March 2011
Constraints on the seismic properties of the middle and lower continental crust G. E. LLOYD1*, R. W. H. BUTLER2, M. CASEY1†, D. J. TATHAM1,4 & D. MAINPRICE3 1
Institute of Geophysics and Tectonics, School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK
2
Geology and Petroleum Geology, School of Geosciences, University of Aberdeen, Meston Building, King’s College, Aberdeen AB24 3UE, UK
3
Ge´osciences Montpellier, CNRS & Universite´ Montpellier 2, 34095 Montpellier, France 4
School of Earth and Ocean Sciences, University of Liverpool, Liverpool L69 3GP, UK *Corresponding author (e-mail:
[email protected])
Abstract: For the past two decades geodetic measurements have quantified surface displacement fields for the continents, illustrating a general complexity. However, the linkage of geodetically defined displacements in the continents to mantle flow and plate tectonics demands understanding of ductile deformations in the middle and lower continental crust. Advances in seismic anisotropy studies are beginning to allow such work, especially in the Himalaya and Tibet, using passive seismological experiments (e.g. teleseismic receiver functions and records from local earthquakes). Although there is general agreement that measured seismic anisotropy in the middle and lower crust reflects bulk mineral alignment (i.e. crystallographic preferred orientation, CPO), there is a need to calibrate the seismic response to deformation structures and their kinematics. Here, we take on this challenge by deducing the seismic properties of typical mid- and lower-crustal rocks that have experienced ductile deformation through quantitative measures of CPO in samples from appropriate outcrops. The effective database of CPO and hence seismic properties can be expanded by a modelling approach that utilizes ‘rock recipes’ derived from the as-measured individual mineral CPOs combined in varying modal proportions. In addition, different deformation fabrics may be diagnostic of specific deformation kinematics that can serve to constrain interpretations of seismic anisotropy data from the continental crust. Thus, the use of ‘fabric recipes’ based on subsets of individual rock fabric CPO allows the effect of different fabrics (e.g. foliations) to be investigated and interpreted from their seismic response. A key issue is the possible discrimination between continental crustal deformation models with strongly localized simple-shear (ductile fault) fabrics from more distributed (‘pure-shear’) crustal flow. The results of our combined rock and fabric-recipe modelling suggest that the seismic properties of the middle and lower crust depend on deformation state and orientation as well as composition, while reliable interpretation of seismic survey data should incorporate as many seismic properties as possible.
Geodetic measurements quantifying displacement fields for the Earth’s surface illustrate a general complexity that questions the existence of a simple relationship to mantle flow and plate tectonics (e.g. Fouch & Rondenay 2006; Caporali et al. 2009; Thatcher 2009). Linking geodetically defined continental displacements to mantle flow and plate tectonics demands understanding of ductile deformation in the middle and lower continental crust, for example, the discrimination between deformation models with strongly localized
†
‘simple-shear’ fabrics (e.g. Burg 1999) from those involving more distributed ‘pure-shear’ crustal flow (e.g. Butler et al. 2002). Studies of the in situ seismic characteristics of the middle and lower continental crust should lead to enhanced understanding of ductile deformation in these regions. However, until recently the continental crust was considered too heterogeneous mineralogically and tectonically to reflect coherent patterns of, for example, seismic anisotropy. Today, seismic anisotropy is now not only recognized regularly in the continental crust
Deceased 2008.
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 7 –32. DOI: 10.1144/SP360.2 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
8
G. E. LLOYD ET AL. stereographic projection
crystal form
Vp km/s
c
10.00
quartz
5.31
z
z r
m r z -a am
m
7.0 6.0
7.02
c
m a -a r z r
m a
calcite
v
5.61
7.76
orthoclase feldspar
c
5.30
b
plagioclase feldspar muscovite mica
c b
8.16
b
a
5.10 AVp = 46.2%
c
6.13
e
M a
M
b
c
biotite mica hornblende amphibole
.00
10.00
120.00
33.62
30.0 10.0
.00
a
8.20
c o M a
c
o M
.00 120.00
b
b
.00 120.00 0.36
8.0 7.0
70.0 50.0 30.0 10.0
.00
.00
10.00
120.00 110.0 90.0 70.0 50.0 30.0 10.0
113.82
.00
.00
10.00
120.00
6.01
7.5 6.5
c
b
010
110
30.69
7.89
a AVp = 27.1%
c
.00
0.15
z
b y a
50.0 30.0 10.0
5.60
011
x
53.24
10.00
7.0 6.0
a
c
8.0 6.0
b
e
z
augite pyroxene
.00
72.09
AVp = 32.9%
forsterite olivine
50.0 10.0
0.47
Mm a p o
.00
30.0 10.0
0.80
.00
10.00
001 111 010
6.92
b y
100
c
AVp = 26.2%
cz
9.77
001 101 021
c
010 b
19.31
b
.00 10.00 9.0 8.0
0.52 17.95
10.0 AVp = 24.3%
.00
.00
10.00
a
8.88
a AVp = 0.3%
120.00 0.72
8.85
a
120.00
7.65
a
a
10.0 .00
y a
n n d n n n d a n n n dn
120.00 0.24
b a
a
110
9.0 8.0 7.0
9.01
110
d
0.02
10.00
7.81
garnet
6.0
1.12
AVp = 28.9%
Fig. 1.
.00 120.00
7.50
AVp = 34.5%
x a
.00 10.00
a
c
x
40.0
7.0 6.0
c b
0.02
59.98 AVp = 32.1%
m m a y x
120.00
43.17
7.0
m r v a v a m
c
m m a a m a
Vs1P
10.0 AVp = 27.7%
c r vv
AVs %
0.00
.00
.00
SEISMIC PROPERTIES OF CONTINENTAL CRUST
but is also used increasingly to image zones of deformation (e.g. Meltzer et al. 2001; Shapiro et al. 2004; Sherrington et al. 2004; Moschetti et al. 2010). Seismic anisotropy studies (e.g. in the Himalaya and Tibet) are therefore beginning to investigate middle and lower continental crustal deformation using passive seismological experiments, such as teleseismic receiver functions (e.g. Ozacar & Zandt 2004) and records from local earthquakes (e.g. Schulte-Pelkum et al. 2005). Such studies claim that it is possible to distinguish different types and magnitudes of deformation via seismic anisotropy. If correct, there is much to be gained from seismic (anisotropy) analysis of the (ductile) continental crust, provided the source of the anisotropy is rigorously constrained. Although there is general agreement that measured seismic anisotropy in the middle and lower continental crust reflects bulk mineral crystallographic preferred orientation (CPO), there is a need to calibrate the seismic response to deformation structures and their kinematics (e.g. Okaya & Christensen 2002; Mahan 2006; Meissner et al. 2006). Almost all of the common rock-forming minerals in the continental crust exhibit significant seismic velocity anisotropy as single crystals (Barruol & Mainprice 1993; Ji et al. 2002). These are illustrated in Figure 1 where the individual seismic properties for each mineral have been contoured using the same scale ranges to emphasize relative values and distributions. Plotting the seismic properties in this way indicates which minerals are likely to control each seismic property (e.g. mafic minerals increase compressional wave velocity Vp while most minerals have similar seismic anisotropy AVs except for the micas, which have very high maximum anisotropy). However, for many minerals (including quartz and feldspars) crystal symmetry creates geometrically complex seismic responses that, with small degrees of misorientation within polycrystalline aggregates, interfere destructively to produce nearisotropic behaviour (Barruol & Mainprice 1993; Tatham et al. 2008; Lloyd et al. 2009). Key
9
exceptions are monoclinic mica and amphibole minerals (e.g. biotite and hornblende) that commonly align both crystallographically and dimensionally to create a single slow direction for seismic transmission. Thus, the seismic properties measured for the (ductile) middle and lower continental crust are usually attributed to the CPO of mica (e.g. Kern & Wenk 1990; Barruol & Mainprice 1993; Nishizawa & Yoshino 2001; Shapiro et al. 2004; Mahan 2006; Meissner et al. 2006; Lloyd et al. 2009) or amphibole (e.g. Siegesmund et al. 1989; Barruol & Mainprice 1993; Rudnick & Fountain 1995; Kitamura 2006; Meissner et al. 2006; Barberini et al. 2007; Tatham et al. 2008). It should also be mentioned that other microstructural parameters (e.g. grain shape fabric and variations in spatial distribution of mineral phases, grainboundary properties, porosity, etc.) may contribute in particular to seismic anisotropy (e.g. Wendt et al. 2003). Furthermore, in the (brittle) upper crust the effects of open fractures (e.g. Crampin 1981; Kendall et al. 2007), sedimentary layering (e.g. Vernik & Liu 1997; Valke et al. 2006; Kendall et al. 2007) and/or grain-scale effects (e.g. Hall et al. 2008) must be considered in addition to CPO. According to recent compilations (e.g. Rudnick & Fountain 1995; Rudnick & Gao 2003), increasing average P-wave seismic velocities with depth indicates increasing proportions of mafic lithologies and increasing metamorphic grade (Fig. 2). Superposition of single-crystal Vp values for common rockforming minerals (e.g. Fig. 1) confirms this general trend. Thus, the lower continental crust (i.e. below c. 20 –25 km depth) is expected to be lithologically diverse but dominated by granulite facies mafic lithologies with an average composition approaching that of primitive basalt, although amphibolite facies may be significant where high water fluxes occur and felsic-to-intermediate lithologies can also be important locally (Rudnick & Fountain 1995). In contrast, the average middle crust (i.e. between 10 –15 and 20– 25 km depth), where Pwave velocities are too low to be explained by
Fig. 1. Principal seismic properties of elastically anisotropic single crystals of major rock-forming minerals relative to mineral form and crystallography (Vp: compressional wave velocities; AVs: percentage shear-wave splitting; Vs1P: polarization direction of fastest shear wave). The seismic property for each mineral has been contoured using the same scale range to emphasize relative values and distributions (i.e. Vp, minimum 5.10 km/s for plagioclase and maximum 9.77 km/s for olivine; AVs, minimum 0% and maximum 113.82% for biotite). For example, blues and reds indicate relatively high or low values respectively (NB contour lines are drawn and numbered only for the precise range of that property for that mineral). The seismic properties were derived using the Mainprice (1990) suite of programs (i.e. Pfch5, Anis and VpGC) and appropriate mineral single-crystal elastic constants: quartz (McSkimin et al. 1965), calcite (Dandekar 1968), orthoclase (Aleksandrov et al. 1974), plagioclase (Aleksandrov et al. 1974), muscovite (Vaughn & Guggenheim 1986), biotite (Aleksandrov & Ryzhova 1961), hornblende (Aleksandrov & Ryzhova 1961), augite (Aleksandrov & Ryzhova 1961), garnet (Bonczar et al. 1977) and olivine (Abramson et al. 1997).
10
G. E. LLOYD ET AL.
olivine
garnet
augite
muscovite
hornblende
biotite
plagioclase
calcite
8
orthoclase
9
quartz
Single crystal Vp range (km/s)
10
7
LsC LC LMC MC
6
5
Common rock-forming minerals
Fig. 2. Summary of the average P-wave velocities compiled by Rudnick & Fountain (1995) for the middle (MC), lower middle (LMC), lower (LC) and lowest lower (LsC) continental crust. Also shown are the ranges (black columns) and approximate averages (white circles) of single-crystal Vp values for common rock-forming minerals (see Fig. 1). The latter can be compared to the values suggested by Rudnick & Fountain (1995) to indicate the likely minerals present and hence responsible for the seismic properties in each crustal region.
dominantly mafic lithologies, is considered to consist of a mixture of mafic, intermediate and felsic amphibolite facies gneisses (e.g. Rudnick & Fountain 1995; Rudnick & Gao 2003). Suggestions that micas control the seismic properties of the deeper continental crust (e.g. Meltzer & Christensen 2001; Meltzer et al. 2001; Takanashi et al. 2001; Chlupacova et al. 2003; Shapiro et al. 2004; Mahan 2006; Meissner et al. 2006) are difficult to reconcile with the view that there is little evidence for metapelite-dominated layers in these regions (e.g. Rudnick & Fountain 1995). Nevertheless, muscovite Vp values are clearly compatible with those expected for the lower crust rather than the middle crust (Fig. 2). In contrast, biotite values are typically too low and are more appropriate to the middle crust. It is possible that high metamorphic grade former metapelites that now comprise kyanite and/or sillimanite could also explain lower crustal Vp values; both of these minerals have high seismic velocities as well as large seismic anisotropy (e.g. Ji et al. 2002), but they probably represent ,10% of the lower crust in general (Rudnick & Fountain 1995). While single-crystal Vp characteristics of common rock-forming minerals provide initial constraints on the composition and nature of the ductile continental crust (i.e. Figs 1 & 2), actual lithologies are polygranular and usually polymineralic in reality. Thus, the Vp properties need to be derived from the whole-rock petrofabrics (i.e. the sum of the CPO for the individual constituent minerals) to provide more rigorous constraints (e.g. Barruol & Mainprice 1993). For example,
combining the appropriate single-crystal Vp characteristics shown in Figure 2 indicates that granitic (i.e. quartz + feldspars + micas), amphibolitic (i.e. hornblende + plagioclase + micas) and mafic (i.e. +amphibole + pyroxene + plagioclase + garnet) compositions are responsible for the Vp properties of the middle crust, lower –middle and lower crust and lower/lowermost ductile continental crust, respectively. However, this approach omits consideration of seismic anisotropy, which typically reflects the strength of individual mineral and whole-rock CPO. In general, CPOs form in response to deformation and their detailed distributions can reflect and hence distinguish the kinematic reference frame (e.g. Schmid & Casey 1986; Passchier & Trouw 2005). Investigation of the complete whole-rock petrofabric-derived seismic properties should therefore lead to a better understanding of ductile deformation in the middle and lower continental crust, which ultimately should help in linking geodetically defined continental displacements to mantle flow and plate tectonics. The present contribution takes up this challenge by providing constraints on the seismic properties of the ductile middle and lower continental crust via quantitative measures of CPO in appropriate natural representative samples.
Methodology Seismic property determination The velocity (Vp, Vs1, Vs2), anisotropy (AVp, AVs) and polarization (Vs1P) of seismic compressional (p) and shear (s) waves depend on the threedimensional elastic ‘stiffness’ (Cij) of rocks, which varies with crystal direction in the constituent minerals (e.g. Babuska & Cara 1991). Bulk CPO and hence seismic properties are therefore determined by: individual mineral crystallography (symmetry, etc.); whole-rock mineralogy (composition, modal proportions); deformation mechanisms (e.g. crystal slip); deformation kinematics (deformation type, magnitude and rate); and geological environment (pressure, temperature, fluids, etc.). The influence of these variables is considered here via ‘end-member’ rock types that span typical middle and lower continental crustal compositions (e.g. Rudnick & Fountain 1995; Rudnick & Gao 2003), namely felsic (i.e. +quartz + feldspar + mica) and mafic (i.e. +amphibole + pyroxene + plagioclase + mica). Typical samples were analysed via electron backscattered diffraction (EBSD) in the scanning electron microscope (SEM) to measure their CPOs, from which their seismic properties were derived following conventional procedures (e.g. Mainprice 1990; Mainprice & Humbert 1994; Lloyd & Kendall 2005).
SEISMIC PROPERTIES OF CONTINENTAL CRUST
Samples When modelling the seismic response of the continents, we are aware that outcrop lithologies that once resided at deep crustal levels may not be representative of the lower crust in situ. Nevertheless, the samples used in this study are considered to be representative of many of the lithologies that make up the ductile middle and lower continental crust (e.g. Rudnick & Fountain 1995; Rudnick & Gao 2003). Garnet-bearing lower crustal lithologies have however been omitted from our analysis due to the isotropic seismic properties and relatively high density of garnet which predictably act to dilute seismic anisotropy and increase seismic (P-wave) velocity (e.g. Figs 1 & 2), respectively, with increasing garnet content (e.g. Brown et al. 2009) even if the garnet exhibits (rare) CPO. In addition, we consider orthopyroxene (specifically hypersthene) to be representative of pyroxenes generally due to the similarity in crystal structure and hence seismic properties of orthopyroxenes and clinopyroxenes (see Barruol & Mainprice 1993 or Ji et al. 2002 for specific examples). The four samples used in this study are described as follows. (1)
(2)
(3)
Felsic orthogneiss (Nanga Parbat, west Himalaya; Butler et al. 2002). This rock comprises quartz –orthoclase –plagioclase –biotite – muscovite. Such orthogneisses are derived largely from a granodioritic protolith composed of feldspar, quartz and biotite, with subordinate muscovite. Overall, this composition is considered to represent a good analogue for much of the Earth’s ductile hydrous middle continental crust. Furthermore, mica-bearing tectonites frequently exhibit S –C fabrics, which allow the impact of different textural development on seismic properties to be investigated (Lloyd et al. 2009). ‘Banded’ amphibolite (south Harris, northwest Scotland; Lapworth et al. 2002). This rock comprises hornblende– plagioclase– quartz. The compositions of different ‘bands’ (e.g. quartz–plagioclase, quartz–plagioclase– hornblende, plagioclase–quartz–hornblende, hornblende–plagioclase–quartz, hornblende– plagioclase and hornblende only) are considered to be representative of a range of hydrous ductile middle-to-lower continental crustal lithologies. Sheared mafic dyke (Badcall, northwest Scotland; Tatham & Casey 2007; Tatham et al. 2008). This rock comprises hornblende – plagioclase –quartz and is therefore apparently compositionally similar to sample 2. However, it represents a suite of specimens (1–9; note specimen 1 contains also
(4)
11
appreciable clinopyroxene that presumably reflects the original igneous composition) exhibiting progressively increasing deformation. It is therefore used to study the impact of deformation on seismic properties. It is considered to be representative of the hydrous ductile upper– mid lower continental crust. Pyroxene granulite (Kerala, south India; Prasannakumar & Lloyd 2007, 2010). This rock comprises hypersthene–plagioclase – quartz–(biotite). It is considered to be representative of the anhydrous (hydrous where micaceous) ductile lower continental crust.
Results Composition and texture The effective database of whole-rock CPO and hence seismic properties can be expanded by a modelling approach that utilizes ‘rock recipes’ based on the mineral modal proportions determined via SEM-EBSD analysis (e.g. Tatham 2008; Tatham et al. 2008; Lloyd et al. 2009). In this approach, the modal content of one mineral is varied progressively from 0 to 100% while the other phases are varied according to their original modal proportions. The seismic properties are then calculated per ‘recipe’. The recipes considered here are based on nine specimens comprising hornblende, plagioclase and quartz collected from sample 3 above, the sheared mafic dyke rock (Tatham & Casey 2007; Tatham 2008). The CPO (not shown here – see Tatham et al. 2008) of the individual minerals in these specimens was determined via SEM-EBSD, from which the whole-rock seismic properties per specimen were determined in the conventional manner (see above). In general, the specimens exhibit a trend in seismic properties from an almost isotropic aggregate (specimen 1) to a strong and ordered pattern of orthorhombic symmetry (specimens 2– 9) with increasing strain that accords with the finite strain axes and indicates the dominant role of hornblende in controlling these properties (e.g. Tatham et al. 2008). However, the roles played by all three constituent minerals can be investigated, both within and across the range of specimens from sample 3, via a rock-recipe modelling approach as follows. Rock-recipe modelling. As the specimens of sheared mafic dyke rock typically comprise only three mineral phases (i.e. hornblende, plagioclase and quartz), modelled variations in their compositions can be represented by simple ternary plots (see Fig. 3). Each plot is based on the interpolation of monomineralic aggregate elastic properties for the specimen over all permutations of modal fractions,
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Fig. 3. Ternary rock-recipe plots for mafic dyke samples 1 –9 showing the sensitivity of various seismic properties to variations in modal composition for different shear strains (g) and ellipticity (R) of the finite strain ellipse in a three-phase aggregate system comprising hornblende, plagioclase and quartz. The shaded areas of the legend ternary plots indicate the regions of confident extrapolation of data due to potential differences in the behaviour of polymineralic and monomineralic rocks (see text for discussion). Black dots indicate the actual modal composition of each sample. (a) Maximum in Vp; (b) AVp; (c) AVs.
including monomineralic, in each mineral phase. However, the elastic properties of each mineral phase are representative of their deformation
behaviour in a polyphase aggregate with the specific relative modal volume fractions of the original specimen. In practice, both the strain distribution
SEISMIC PROPERTIES OF CONTINENTAL CRUST
in the original specimen and its partitioning into specific mineral phases are likely to be such that the microstructure, petrofabric and hence elastic properties of the constituent mineral phase fractions are not representative of the behaviour of each phase for a monomineralic rock of that phase at the strain observed in the bulk specimen. For example, deformation of an aggregate comprising 60% hornblende, 30% plagioclase and 10% quartz (similar to that of specimens 19; see Fig. 3) seems to preferentially partition strain into the hornblende phase as the plagioclase and quartz phases exhibit poorly developed CPO (see Tatham 2008; Tatham et al. 2008). Deformation of approximately monomineralic aggregates of either quartz or plagioclase to the same bulk strain would almost certainly involve the development of strong CPO (e.g. Marshall & McLaren 1977a, b; Lister & Dornsiepen 1982; Olsen & Kohlstedt 1984, 1985; Mainprice et al. 1986; Kruhl 1987). The range of relative modal fractions (i.e. rock recipes) for which the ternary plots (Fig. 3) can be applied is therefore likely limited. From results of low-law calculations in a selection of two-phase aggregates, Handy (1994) postulated that the presence of .10% of a weak phase is necessary for that phase to govern the bulk strength of the aggregate. In detail, this prediction depends on a number of factors including temperature, strain rate and the strength contrast between adjacent phases. It is therefore suggested that the rock-recipe modelling results are accurate only between 10 and 90% in each phase for each possible two-phase system of the three-phase aggregate (i.e. the shaded areas indicated in the legend ternary plots in Fig. 3). The ternary rock-recipe plots reveal the sensitivity of various seismic properties to variations in modal composition for different deformation states (e.g. shear strain and ellipticity of the finite strain ellipse). It is clear that the maxima in Vp, AVp and AVs are most strongly dependent upon hornblende modal fraction. For example, there are increases of up to 1 km/s in Vp (Fig. 3a), 5% in AVp (Fig. 3b) and 8% in AVs between end-member plagioclase and hornblende compositions. Figure 3a, b also indicates that both AVp and AVs tend to increase with increasing strain, but only up to a shear strain of c. 10. Beyond this value there tends to be little increase in either anisotropy. This behaviour suggests that seismic anisotropy cannot increase indefinitely with deformation, but saturates at a certain value. In general, Vp values show little variation with increasing strain (Fig. 3a). In contrast, the impact of changing quartz modal fraction on these seismic properties is minor. For example, from a starting aggregate of 50% hornblende and 50% plagioclase, progressively increasing the quartz fraction up to 100% yields only a 2% increase overall in AVp (Fig. 3b) and a
13
2% increase in AVs at 50% quartz, returning to no increase at 100% quartz (Fig. 3c). The change in the maximum in Vp is also minimal (Fig. 3a). Previous investigations into the relative effects of component phases on seismic properties have also highlighted the dominant role of mafic components (e.g. hornblende) compared to felsic components (plagioclase and quartz), with the latter often acting as ‘dilutants’ (e.g. Christensen & Fountain 1975; Fountain & Christensen 1989; Tatham et al. 2008; Lloyd et al. 2009). Rock-recipe modelling approaches indicate that the seismic properties of (hydrous) mafic rocks are most likely to be controlled by amphibole (e.g. Tatham et al. 2008). In contrast, for felsic rocks, micas (in particular biotite) are the controlling mineral phase (e.g. Lloyd et al. 2009, 2011). Other minerals in both lithologies, such as plagioclase and quartz but also pyroxene (see below), act as dilutants and tend to reduce seismic properties (particularly anisotropy) although some (e.g. plagioclase and pyroxene) may increase velocities (see below). Fabric-recipe modelling. Many rocks may comprise different deformation fabrics (e.g. S–C foliations). Use of fabric recipes based on subsets of individual rock fabric CPOs therefore allows the impact of different fabrics on seismic properties to be investigated. In this recipe modelling approach, CPOs are measured (e.g. via SEM-EBSD) for each fabric element and also for the whole rock, and are then varied in a similar manner to rock recipes. The elastic properties of each fabric recipe are then used to calculate the impact of different proportions of each fabric on the whole-rock seismic properties. For example, Lloyd et al. (2009, 2011) have shown that the typical transverse isotropy characteristics associated with individual mica fabric elements can disappear in whole rocks comprising S–C type foliations.
Foliation and azimuthal effects Seismic modelling. Rock and fabric-recipe modelling suggest that foliation, and in particular its orientation, exerts a significant impact on seismic properties (see also Lloyd et al. 2011). To investigate this impact, a seismic model was constructed representing a 10 km slice of deformed lower continental crust (Fig. 4). This thickness of crust was chosen to provide practical values of shear-wave splitting (dVs). The model was populated with the elastic stiffness properties of the nine specimens of mafic dyke rock using a common rock-recipe composition of 60% hornblende, 30% plagioclase and 10% quartz, which represents the average composition of these specimens.
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Fig. 4. Schematic representation of the seismic model used to investigate the impact of foliation orientation on different seismic properties (see Figs 5 & 6).
As discussed previously, the elastic properties of rock sample 3 vary due to the increasing strain exhibited by specimens 1–9 (e.g. Fig. 3). One axis of the model can therefore be scaled according to increasing strain, represented as either shear strain (g) or the length of the major axis of the strain ellipse (S1, where S1 ¼ 1 defines an undeformed unit circle). Representing the variation in seismic properties with respect to either g or S1 means that the plots can be used to consider either simple or pure-shear-dominated deformations, respectively. The other axis of the model is scaled according to the orientation of sample foliation relative to propagating seismic waves. For the purposes of this model it is assumed that the waves propagate vertically through the 10 km of crust and can effectively be considered as teleseismic waves (Fig. 4). Sample foliation is rotated from horizontal (08) to vertical (908) in both clockwise and anticlockwise directions about either the X or Y tectonic axes. In the former configuration, the X or lineation axis remains unchanged and horizontal while the foliation rotates from horizontal to vertical. In the latter configuration, both the lineation and foliation rotate between horizontal and vertical. The variation in seismic properties with strain and foliation orientation in the model is determined in three stages (e.g. Fig. 4). Firstly, the bulk elastic stiffness matrices (Cij) for each specimen of deformed mafic dyke rock are interpolated between their pre-determined finite shear strains
(g) or the associated strain ellipse long axes (S1) to describe the variation of elastic stiffness with strain. The interpolated grid is discrete but of high resolution, with interpolations at increments of 0.1 for both g and S1. Secondly, the interpolated strain scale of Cij is incrementally rotated from 08 to 908 either clockwise or anticlockwise with respect to either the X or Y tectonic axes. Again, the interpolated scale of petrofabric rotation is discrete but of high resolution, with increments of +58. Thirdly, the values of Vp and dVs are calculated for each node of the interpolated grids, from which contoured plots are constructed (e.g. Fig. 4). Note that the calculated P-wave velocities refer specifically to vertically propagating waves and hence are not necessarily the maximum in P-wave velocities. In addition, the fast shear-wave polarization orientation (Vs1P) is also determined, but for a more sparsely populated grid necessary for display purposes. Foliation orientation. Results of seismic modelling of the impact of foliation orientation on seismic properties (Vp and dVs) for different strains (i.e. ‘simple’ and ‘pure’ shears) are shown in Figure 5. In general, all relationships indicate a crude bilateral symmetry about the line of zero rotation (horizontal foliation). Deviations from perfect bilateral symmetry are due to a combination of the slightly nonorthorhombic nature of the aggregate Cij, which probably reflects complex interactions between constituent phases with different seismic properties and symmetries and other natural variations (e.g. microstructure, CPO, etc.). Apparently anomalous steps or jumps in the distribution of seismic properties are an amplification effect of specimen variations. Discontinuities in the smoothly varying distributions mark points of specimen control, while smooth variations denote regions of interpolation of physical properties between specimens. Similarly, complex patterns of ‘peaks’ and ‘cusps’, particularly within central bands adjacent to the symmetry axis, are due to local effects around specimen control points. The peaks and cusps are of comparatively low relief and are considered to be insignificant with respect to the overall trends indicated. The impact of each of these irregularities would be reduced by employing a greater specimen density. If they are overlooked, a number of
Fig. 5. Seismic modelling of the effects of foliation orientation and deformation on P-wave velocity (Vp) and shear-wave splitting (dVs) for vertically propagating (‘teleseismic’) waves through 10 km of crust populated by the samples of mafic dyke rock (see Fig. 4). For each pair of diagrams, deformation is represented in terms of shear strain (g) and the length of the major axis of the strain ellipse (S1, where S1 ¼ 1 defines an undeformed unit circle). (a) Variation in Vp for rotation about the X tectonic axis. (b) Variation in Vp for rotation about the Y tectonic axis. (c) Variation in dVs for rotation about the X tectonic axis. (d) Variation in dVs for rotation about the Y tectonic axis. See text for discussion.
SEISMIC PROPERTIES OF CONTINENTAL CRUST
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significant trends in the seismic properties can be recognized as follows (see Fig. 5). (1)
(2)
(3)
(4)
The bilateral symmetries recognized in all plots suggest that the direction of fabric rotation (i.e. clockwise or anticlockwise) about the kinematic X or Y axes does not affect the pattern or magnitude of seismic velocity or anisotropy to any significant degree. Furthermore, there are notable similarities between the velocity and anisotropy distributions irrespective of whether the foliation is rotated about either the X or Y axes. There is a general decrease in the magnitude of seismic velocity and anisotropy with increasing strain for subhorizontal foliation orientations. This contrasts with the tendency for velocity and anisotropy values to increase with strain for steep to vertical foliations orientations. The greatest rate of change occurs for values 0 ≤ g ≤ 10 or 1 ≤ S1 ≤3, beyond which increasing strain results in negligible changes in either velocity or anisotropy. This observation supports the suggestion made earlier that both petrofabric and hence petrophysical properties become saturated by g c. 10 (i.e. S1 –3). Somewhat steeper gradients in the change in Vp and dVs with strain are observed for values 0 ≤ g ≤ 0.4 or 1 ≤ S1 ≤ 1.2. These behaviours suggest that small initial strains imposed upon initially isotropic protoliths can have dramatic effects upon the seismic properties. The change from dominantly decreasing to increasing trends of Vp and dVs values with strain occurs for foliation rotations about X of 20–408. Similar behaviour is shown by Vp for rotations about Y. For dVs the change occurs closer to 608 due to details in the AVs distribution, whereby low values of anisotropy occupy a greater proportion of the XZ plane compared to the YZ plane; this leads to a comparatively wider band of low dVs values about subhorizontal foliations. It therefore follows that, for a given strain, a move from subhorizontal to subvertical foliations is associated with an overall increase in Vp and dVs values.
Based on these seismic modelling results, interpolation of the petrophysical properties of a discrete strain-calibrated microstructurally characterized and compositionally normalized suite of specimens allows the evaluation of seismic properties against finite strains (both simple and pure shears) and foliation orientation in that material, across a continuum. A similar conclusion, again based on petrofabric-derived seismic modelling, is reported
elsewhere in this volume for mica-dominated felsic rocks (Lloyd et al. 2011). It therefore appears that CPO observations and Vp and dVs (sic AVs) distributions are potentially useful proxies for each other in regional-scale crustal geodynamic investigations and models. Azimuthal considerations. In the previous section it was shown that accurate interpretation of seismic wave propagation through foliated rocks depends on knowledge of the source and propagation direction of the seismic waves relative to the foliation orientation. The propagation direction was always vertical in the seismic model described above, implying either teleseismic waves or possibly local seismic events. In nature, source locations are likely to be variable and hence propagation directions also vary in their orientations (i.e. azimuths and plunges). For example, assuming a constant steeply dipping foliation, waves from a teleseismic source would propagate (sub-) parallel to foliation while waves from a remote (i.e. ‘wideangle’) source would propagate (sub-) normal to foliation. The same seismic array would therefore detect very different seismic properties. This simple example emphasizes the need to also consider CPO-derived polarization/birefringence (i.e. Vs1P) effects in the seismic models. Analysis of Vs1P behaviour for the seismic model (Fig. 4) reveals distributions similar to those of Vp and dVs (Fig. 6). The vertical expression of shear-wave polarization becomes more apparent with increasing strain and as the foliation rotates towards the vertical. Furthermore, as indicated in the previous section, Vs1P is also associated with an increasing component of the total AVs as the foliation rotates towards parallelism with the vertical wave propagation direction. However, the direction of rotation does produce some differences. For rotations about X, the direction of polarization is ‘parallel’ to the strain axis of the plots which effectively defines the orientation of X (Fig. 6a). In contrast, for rotations about Y, the direction of polarization is ‘normal’ to the strain axis of the plots which effectively defines the orientation of Y (Fig. 6b). These results are perhaps intuitive and reflect the development of an increasingly welldefined foliation with strain, together with the increasing parallelism of that foliation with the seismic ray-path. The Vs1P polarization plane is consistently parallel to the plane of petrofabric foliation (Fig. 6 inserts), which in turn is parallel to the girdle of greatest shear-wave anisotropy. The seismic modelling described here illustrates the significance of azimuth in the interpretation of seismic properties. A similar conclusion, again based on petrofabric-derived seismic modelling, is reported elsewhere in this volume for
SEISMIC PROPERTIES OF CONTINENTAL CRUST x
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Fig. 6. Relationship between the fast shear-wave polarization direction (Vs1P), deformation and CPO for a vertical ray (i.e. teleseismic wave) passing through a 10 km section of ‘crust’ populated with the elastic properties and estimated shear strain (Tatham & Casey 2007) of the mafic dyke samples 1 –9. Inserts to the right of each plot illustrate the CPO orientation in terms of the finite strain ellipsoid (i.e. X ≥ Y ≥ Z ). Polarization direction and intensity is indicated by the line lengths at each nodal point; points with no ticks exhibit no surface expression of shear-wave polarization. (a) Rotation of CPO about the X tectonic axis with respect to shear strain (g) and length of major axis of the strain ellipse (above and below, respectively). (b) Rotation of CPO about the Y tectonic axis with respect to shear strain (g) and length of major axis of the strain ellipse (above and below, respectively).
mica-dominated felsic rocks (Lloyd et al. 2011). However, in that contribution, it is shown that rocks comprising multiple (mica-defined) foliations exert complicated responses on Vs1P orientations and magnitudes, varying with the relative proportions of the different foliations (see also Lloyd et al. 2009). In particular, the VS1P orientation does not necessarily reflect variations in kinematics indicated by changing foliation proportions. It therefore appears that CPO observations and seismic property distributions are potentially useful proxies for each other, but only in regional-scale crustal geodynamic investigations and models involving a single and/or dominant foliation. In regions comprising multiple foliations, which are likely to generate variations in orientation and magnitude of both AVs and VS1P, ‘maps’ of variations in these properties with depth need careful (geological) consideration and interpretation.
CPO and deformation CPO develops during ductile deformation via dislocation creep on crystal slip systems that are mainly temperature dependent, but are also sensitive to deformation type (e.g. Nicolas & Poirier 1976; Passchier & Trouw 2005). Complex natural deformations are usually simplified (e.g. plane strain, pure and simple shear, flattening, constriction, etc.) and represented on diagrams such as Flinn plots (e.g. Flinn 1956; see below and Figs 7 & 8). However, these deformations may not always be distinguished via the seismic properties of the dominant minerals, as the following examples indicate. Figure 7a is a Flinn diagram illustrating the variation in quartz c- and a-axes CPO with deformation type and magnitude (Lister & Hobbs 1980). Note the tendency for dispersion rather than concentration
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Fig. 7. Left: idealized Flinn plot indicating the expected variation in quartz c- and a-axes CPO with different types and magnitude of deformation (Lister & Dornsiepen 1982). Note the tendency for dispersion rather than concentration of crystal axes. Right: Flinn plot representation of examples of natural quartz c- and a-axes CPO and the resulting Vp and AVs seismic properties (data from G. E. Lloyd archive).
into unique directions of the crystal axes. These dispersions are responsible for the ‘diluting’ effect of quartz on whole-rock CPO. Simply stated, the elastic anisotropy due to minerals such as quartz (which do not form ‘quasi-single-crystal’ CPO) is distributed rather than concentrated such that both the whole-rock velocity and anisotropy of seismic waves are reduced. This effect is indicated clearly in Figure 7b, which is a Flinn plot representation of the Vp and AVs seismic properties for different quartz CPOs due to different deformations. The approach illustrated in Figure 7 for quartz can be extrapolated to the seismically important mica and amphibole minerals as follows. In general, natural CPO data for either mica or amphibole minerals are lacking when compared to that available for quartz. However, it is possible to model the CPO expected for different deformations due to the simple relationship between mica/amphibole crystal shape and crystal axes. In effect, the shape of the mineral grains accurately reflects the direction of the crystal axes. Both mineral groups therefore tend to form quasi-single-crystal CPO during deformation (e.g. Tatham et al. 2008; Lloyd et al. 2009), involving the operation of relatively
few and simple crystal slip systems. These systems are (001) ,110. and/or (001)[100] for micas and (100)[001], (010)[001] or (hk0)[001] for amphiboles (e.g. Nicolas & Poirier 1976). Furthermore, due to the simple relationship between mineral grain shape and crystal axes orientations, both mineral groups may also develop similar CPO via rigid-body rotation rather than crystal slip (e.g. Meissner et al. 2006; Dı´az-Azpiroz et al. 2007; Tatham et al. 2008). For flattening type deformations (i.e. where the Flinn K parameter approaches zero such that the tectonic axes have the relationship X Y ≫ Z, leading to S-type tectonites), both micas and amphiboles are expected to form strongly foliated rock fabrics. Consequently, their CPOs are characterized by uniform distributions of a(001)/b(010) and b(010)/c(001), respectively, with the mica c[001] axis and the amphibole a(100) axis defining the foliation normal (Fig. 8). These CPOs lead to so-called vertical transverse isotropy or VTI seismic distributions for Vp and AVs, with maximum values in both symmetrically distributed parallel to foliation. The minimum in Vp is normal to foliation in both mineral groups. However, the minimum in AVs is
Fig. 8. Idealized ‘Flinn-type’ plots of (a) biotite and (b) hornblende CPO and their derived seismic properties for ‘flattening’, ‘simple-shear’ and ‘constriction’ deformations. Note: 1. biotite-controlled Vp and AVs are sensitive only to foliation due to mica VTI symmetry and therefore can distinguish only constriction deformation; 2. hornblendecontrolled Vp and AVs are sensitive to foliation and lineation due to ‘orthorhombic’ symmetry and can therefore distinguish different deformations.
SEISMIC PROPERTIES OF CONTINENTAL CRUST CPO density low
Z
seismic contours
b(010)
c[001] PS1
Y
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AVs
X>Y>Z simple shear LS-tectonite
Flattening K=0
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AV s
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1]
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)
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oblate ellipsoid X Y>>Z flattening L-tectonite Vp
AVs
PS1
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Y
Z
b(010)
c[001]
X>Y>Z simple shear LS-tectonite
c[
00
a(100) Y
X
1]
X
seismic contours
high
X/Y
(b)
X
0) 10
Vp
a(
Flattening K=0
X
Y
Z
a(100) X
oblate ellipsoid X Y>>Z flattening L-tectonite Vp
Fig. 8.
AV s
Z
PS1 Y
AVs
Constriction K= Pl an K es = tra 1 in
Vp prolate ellipsoid X>>Y Z constriction L-tectonite
b(
01
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0)
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20
G. E. LLOYD ET AL.
normal to foliation only for amphiboles; for micas, it defines a small circle inclined to the foliation normal. For both minerals, the orientation of Vs1P is radial for waves propagating subvertical to foliation but becomes circumferential for waves propagating subparallel to foliation. For plane strain deformations (i.e. K 1 and X . Y . Z, leading to LS-type tectonites) micas and amphiboles are expected to form both foliated and lineated rock fabrics. Consequently, their CPOs are characterized by quasi-single-crystal distributions in which (100) and [001] are parallel to lineation and foliation normal for micas, respectively, but this relationship is reversed for amphiboles (Fig. 8). In both mineral groups, (010) also lies within the foliation. Thus, as both (100) and (010) lie within the foliation and due to the VTI characteristics of the single crystal (Fig. 1), the seismic properties for micas exhibit exactly the same distributions as for the previous case (Fig. 8). It is therefore not possible to distinguish between deformations defined by 0 , K 1 on the basis of mica seismic properties. However, amphibole single crystal do not exhibit VTI (Fig. 1); the CPO-based seismic properties are therefore different from the previous case and can distinguish both foliation (with the foliation normal defined by the minimum in both Vp and AVs) and lineation (defined by the albeit slight maximum in both Vp and AVs). The orientations of Vs1P for amphiboles are parallel to the YZ plane for waves propagating subvertical to foliation and circumferential for waves propagating subparallel to foliation (i.e. there is no indication of lineation). For constrictional deformations (i.e. K 1 and X ≫ Y Z, leading to L-type tectonites) micas and amphiboles are expected to form strongly lineated rock fabrics. Consequently, their CPOs are characterized by strong concentrations of the principal crystal slip directions (i.e. (100) in micas and [001] in amphiboles) parallel to the tectonic extension (X ) direction (Fig. 8). However, due to the fact that foliation development is impeded, neither [001] of micas nor (100) of amphiboles form orientation clusters but tend to form, with (010) in both minerals, great circle CPO distributions normal to X. The resulting CPO-based seismic property distributions are therefore different to previous cases for both mineral groups (Fig. 8). They both show maximum in Vp parallel to X and a very diffuse spread of low Vp velocities parallel-to-oblique to the foliation. In contrast, it is the minimum in AVs that aligns parallel with X for both mineral groups, while the foliation plane is defined by a narrow great circle of intermediate values for micas and a more diffuse great circle of alternating low-tointermediate values for amphiboles. The orientation of Vs1P for micas is consistently parallel to XY for
all propagation directions subparallel to this plane; for all other propagation directions it is typically circumferential. For amphiboles, the orientation of Vs1P is parallel to either XZ or YZ for all propagation directions within these planes, leading to a complex interference pattern for vertically propagating (i.e. teleseismic) waves. For all other propagation directions, Vs1P is oriented circumferentially. Figure 8 illustrates that it may be difficult (if not impossible) to distinguish seismically between plane strain and flattening type deformations in micaceous (i.e. felsic) rocks due to the similarity of their CPO-based seismic property distributions, which results from the VTI characteristics of mica single crystals. However, constrictional type deformations, which are likely to be rarer in nature, can be easily recognized via their distinctive CPO-based seismic property distributions. In contrast, there is a difference between CPO-based seismic property distributions for plane strain and flattening deformations in amphibolitic (i.e. mafic) rocks, although this may often be subtle, while constrictional deformations yield very different distributions. These observations suggest that seismic property distributions can be used as proxies for deformation types and hence are potentially able to distinguish geodynamic behaviour in mafic lithologies dominated by amphibole. Figure 8 also emphasizes the significant impact foliation development has on influencing the (observed) seismic properties, supporting observations presented above (e.g. Figs 5 & 6). Indeed, it is the absence of foliation that results in the distinctive seismic responses of amphiboles and particularly micas for constrictional deformations.
Modelling seismic properties of ductile continental crust This section considers the implications of the results presented above for the explanation and interpretation of the seismic properties of the ductile continental crust using the CPO-derived seismic properties (specifically Vp, AVp and AVs) of the four rock samples described previously. It is emphasized that the controlling variables constraining the seismic properties appear to be: (1) the single-crystal elastic constants of the individual rock-forming minerals as reflected in the modal composition; (2) CPO; (3) foliation development and orientation; and (4) deformation type.
Compressional wave velocities Rudnick & Fountain (1995) suggested the following Vp ‘stratigraphy’ for the ductile continental crust
SEISMIC PROPERTIES OF CONTINENTAL CRUST 1. Vp (max & min) km/s 0
4
5
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7
8 0
4
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8
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AVp 60
60
Lowermost crust Lower crust Lower middle crust Middle crust
% Modal content biotite
40
80
100
100 4
5
6
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8 0
2
4
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8
10
12
14
1. Vp (max & min) km/s 4
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8 0
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2
4
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8
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100
% Modal content biotite
Micas - biotite (pyroxene granulite)
% Modal content orthopyroxene
80
8 0
0
20
40
80
100 5
6
7
8 0
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5
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8 0
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AVp AVs
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% Modal content orthopyroxene
7
60
% Modal content biotite
6
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AVp
(d)
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5
14
100
0
AVp
4
12
Amphiboles - hornblende
Lowermost crust Lower crust Lower middle crust Middle crust
100
10
80
4
20
80
8
60
AVs
60
6
AVs
16
20
40
4
Lowermost crust Lower crust Lower middle crust Middle crust
0
2. AVp and AVs % 2
60
Micas - biotite (felsic gneiss)
(c)
8 0
% Modal content hornblende
20
AVs
80
0
0
Lowermost crust Lower crust Lower middle crust Middle crust
20
% Modal content biotite
(b)
2. AVp and AVs % 2
% Modal content hornblende
(a)
21
100
Orthopyroxene - Hypersthene
Fig. 9. Constraints on the seismic properties of the middle and lower ductile continental crust based on CPO and rock-recipe modelling of: (a) felsic orthogneiss (biotite variable modal content); (b) biotite-bearing pyroxene granulite (biotite variable modal content); (c) mafic dyke (hornblende variable modal content); and (d) pyroxene granulite (hypersthene biotite variable modal content). All samples are deformed. Also shown are the typical Vp ranges (broken lines) for different parts of the middle-to-lower crust as suggested by Rudnick & Fountain (1999). See text for discussion.
(e.g. Fig. 9): (1) middle crust, 6.20 –6.50 km/s; (2) lower middle crust, 6.50– 6.85 km/s; (3) lower crust, 6.85–6.95 km/s; and (4) lowermost lower crust, 7.15 –7.25 km/s. Using the felsic orthogneiss, mafic dyke and pyroxene granulite as ‘rock standards’ for the ductile continental crust, the concept of rock recipes can be used to determine the impact of varying modal composition on CPOderived Vp values and, in particular, the roles played by mica (biotite), amphibole (hornblende) and, to a lesser extent, pyroxene (augite). It is clear from Figure 9a that biotite (or other micas) cannot be responsible for the Vp values recognized in the ductile continental crust. This is not surprising given the single-crystal Vp values for biotite (Fig. 1). Although muscovite has a somewhat higher single-crystal Vp range than biotite, this is unlikely to be capable of increasing values sufficiently to explain Vp values, particularly in the middle crust. This situation becomes even
more problematical when the impact of foliation orientation is included. Based on observations described above (i.e. Figs 5, 6 & 8), regions of the ductile continental crust dominated by (sub-) horizontal layering (i.e. foliation) should exhibit the minimum Vp values available through micas when imaged via near-vertical (teleseismic) ray-paths (see e.g. Dı´az-Azpiroz et al. 2011). Such situations demand wide-angled seismic surveys. It appears that neither mica content nor deformation state can account for the Vp values observed in the middle and lower continental crust, which therefore must be due to other minerals. From Figure 1, the most likely candidates are quartz and particularly feldspars, unless the rocks also consist of a mafic mineral. However, increasing the wholerock Vp is not simply a matter of adding common rock-forming minerals that possess relatively high Vp as a rock-recipe model for the pyroxene granulite sample indicates (Fig. 9b). Although this rock
22
G. E. LLOYD ET AL.
comprises only minor biotite, its Vp values only satisfy those expected for the middle crust for the actual rock composition. Increasing the biotite content gradually decreases the maximum in Vp but significantly decreases the minimum in Vp, such that most values of Vp fall below those expected for the middle crust. Amphibole is clearly capable of explaining Vp values throughout the whole ductile continental crust (Fig. 9c). Indeed, it does not matter whether layering (i.e. foliation) is flat lying or steep, Vp values are still well within limits. Rocks with very high amphibole contents could account for the Vp values observed in the deepest regions of the continental crust (i.e. in excess of 7 km/s). If this interpretation is correct, it implies that these regions are also hydrous. However, mafic anhydrous minerals (and in particular pyroxene) can contribute even more strongly to Vp values due to their singlecrystal characteristics (Fig. 1) and are capable of achieving velocities up to 8 km/s in overtly pyroxene-dominated lithologies (Fig. 9d). It is therefore not possible to reconcile between amphibole and pyroxene as the specific cause of Vp values observed in the ductile continental crust, except where velocities exceed perhaps 7.5 km/s. Such high velocities might also be explained by increasing garnet content, although with concomitant decrease in anisotropy (e.g. Brown et al. 2009).
P- and S-waves anisotropy Unlike velocity, seismic anisotropy (both AVp and AVs) is not dependent directly upon depth as it tends to reflect CPO development and hence magnitude of deformation rather than mineralogy per se. Nevertheless, it is clear that micas are the most seismically anisotropic of the common rock-forming minerals, with maximum AVp and AVs values (for biotite) of up to 64 and 114% respectively (see Fig. 1). This suggests that felsic rocks with well-aligned micas should be the most anisotropic. The sample of felsic orthogneiss (Fig. 9a) supports this suggestion, showing rapid increase in AVp and particularly AVs with increasing biotite modal content. Similar behaviour is also shown by progressively increasing the initial minor biotite content in the pyroxene granulite sample (Fig. 9b). Indeed, although mica (specifically biotite) contents in excess of c. 30% are unusual (except perhaps in slates), even normal modal contents of say 10–25% are more than capable of contributing significant anisotropy. In contrast, amphibole modal contents of 10–25% appear to be capable of contributing only relatively low values of AVp and AVs (Fig. 9c). However, for modal amphibole contents above c. 35%, both AVp and AVs increase rapidly and approach similar
values to those due to mica (biotite) for .60% amphibole. As amphibole modal contents of c. 60% may not be unusual naturally, amphibole can therefore be regarded as a potential source of AVp and AVs in the ductile continental crust. The contribution of pyroxenes to anisotropy, as indicated by the pyroxene granulite sample, appears to be opposite to both micas and amphiboles. For this particular sample, AVp remains approximately constant with increasing pyroxene content but AVs decreases progressively and eventually converges on the AVp value (Fig. 9d). As this sample is deformed and the pyroxene exhibits a relatively strong CPO, it therefore seems that pyroxene content may have little or no impact on AVp and a negative impact on AVs. This behaviour is perhaps explained by recalling that pyroxene exhibits significantly less single-crystal anisotropy, particularly in AVs, compared to other common rock-forming minerals (e.g. Fig. 1). For example, for augite AVp ¼ 24.3% and AVs ¼ 18.0% but for hornblende AVp ¼ 27.1% and AVs ¼ 30.7%. This suggests that pyroxene-dominated lithologies are not as anisotropic as their amphibolitic counterparts, which is likely to have significant implications for the seismic anisotropy of the lower continental crust. Finally in this section, it should be recognized that AVs is critically azimuthally dependent, whereas AVp has no azimuthal dependence and merely represents the absolute difference between the maximum and minimum in Vp. For example, shear waves propagating normal to foliation exhibit significantly lower AVs than the maximum possible (e.g. Fig. 5), which argues that crustal regions dominated by (sub-) horizontal layering (i.e. much of the lower continental crust according to Rudnick & Fountain 1995) should exhibit relatively low AVs values irrespective of mineralogy (e.g. Tatham 2008; Lloyd et al. 2011; Dı´az-Azpiroz et al. 2011). It is therefore important to incorporate also propagation direction (i.e. AVs combined with Vs1P) when considering and/or interpreting AVs from continental regions. The samples described in this section were considered simply in terms of their bulk AVp and AVs characteristics as functions of composition alone (e.g. Fig. 9). In the next section, two of these samples – the ‘banded’ amphibolite and the pyroxene granulite – are considered as microcosms of layered continental crustal sections, so-called ‘seismic stratigraphy modelling’ in which case the azimuthal dependence of AVs can be incorporated.
Seismic stratigraphy modelling The sample of ‘banded’ amphibolite used to construct Figure 9c comprises hornblende, plagioclase
SEISMIC PROPERTIES OF CONTINENTAL CRUST
and quartz. However, these minerals have combined together in different proportions to form alternating bands of different compositions. In effect, this layered structure mimics the compositional changes from felsic to mafic lithologies expected with increasing depth through the ductile continental crust (e.g. Rudnick & Fountain 1999), although admittedly not in actual sequence in the sample. By defining and analysing separately individual layers/lithologies, their CPOs can be used to derive their particular seismic properties from which a ‘seismic stratigraphy’ for Vp, AVp and AVs can be constructed and used to model the seismic response of the middle and lower continental crust (Fig. 10). Although not layered, a similar modelling approach to that used for the banded amphibolite can be applied to the pyroxene granulite based on rock-recipe variations in modal pyroxene, plagioclase and biotite compositions. The seismic stratigraphy that can be constructed from this sample can be used to model (Fig. 11) the seismic response of not only mafic and felsic lower continental crust but also potential anhydrous and hydrous variants (a) H-P-Q %
5
7
6
0
8
plagioclasequartz
Middle-to-lower crust (‘banded’ amphibolite). Figure 10 is a potential CPO-derived Vp, AVp and AVs seismic stratigraphy for a middle-to-lower continental crustal section. In terms of Vp (Fig. 10a), values increase progressively as composition changes from felsic (i.e. plagioclase –quartz dominated) to mafic (i.e. hornblende–plagioclase dominated) and eventually to ultramafic (i.e. hornblende dominated). These values encompass the range of velocities suggested as being typical of the middle and lower crust (e.g. Rudnick & Fountain 1999) and are also in close agreement with the results obtained from rock-recipe modelling of the sheared mafic dyke sample (Fig. 9c). In terms of anisotropy (Fig. 10b), both AVp and AVs values are relatively low for felsic (i.e. plagioclase –quartz dominated) compositions, with AVp noticeably smaller than AVs. With an increasing mafic (i.e. hornblende) component, AVp increases initially only slowly while AVs decreases slightly
felsic
P/S-wave Anisotropy (%) 2
6
4 S
P
P
S
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H-P-Q %
8-49-43
P
Maxima in AVp & AVs per compositional band
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E-W
plagioclase-quartzhornblende
31-38-31
12
N-S
middle crust
8-49-43
(i.e. pyroxene granulite or pyroxenite, as opposed to biotitic anorthosite granulite or biotitic pyroxene granulite).
(b)
P-wave Velocity (km/s) 4
23
31-38-31 S P
hornblende
92-5-3
40-32-28
P
S
54-35-11 E-W
‘banded’ amphibolite composition (scaled to hornblende)
Lowermost crust
Middle crust
Lower crust Lower middle crust
Vp range per compositional band
lower-middle crust
hornblendeplagioclase
E-W
lower crust
54-35-11
P
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lowest crust
hornblendeplagioclase-quartz
‘banded’ amphibolite composition (scaled to hornblende)
40-32-28
‘composional depth’
S
AVp E-W
mafic
92-5-3
P
4
5
6
7
8
0
2
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6
8
10
S
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16
Fig. 10. Constraints on the seismic properties of the middle and lower ductile continental crust based on CPO and rock-recipe modelling of individual compositional bands in a deformed ‘banded’ amphibolite, taken to simulate a middle to lower continental crust compositional profile (i.e. felsic to mafic compositions). Also shown (grey) are the results from Figure 9c of ‘rock-recipe’ modelling for amphibole (hornblende) content in the mafic dyke sample and typical Vp ranges for different parts of the middle-lower crust as suggested by Rudnick & Fountain (1999). See text for discussion. (a) P-wave velocity; (b) P- and S-waves anisotropy.
24
G. E. LLOYD ET AL.
(a)
(b)
P-wave Velocity (km/s) 0
4
5
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6
P/S-wave Anisotropy (%) 2
8
Px Pl B Q %
Lowermost crust
Middle crust
pyroxenite
Lower crust Lower middle crust
80 13 3 4
middle crust lower middle crust
40
50 30 15 5
NE-SW P
S
60 26 7 7 E-W
NE-SW
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80 13 3 4
mafic
AVp
Vp range per compositional band 5
Px Pl B Q %
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Maxima in AVp & AVs per compositional band pyroxene granulite vertical NE-SW AVs & orientation of Vs1P
P
100 4
0
15 51 20 14
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lower crust
‘composional depth’
pyroxene granulite
16
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lowest crust
‘hydrous’
60 26 7 7
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biotitic pyroxene granulite
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granulite composition (scaled to pyroxene)
granulite composition (scaled to pyroxene)
15 51 20 14
8
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AVs
4
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S
10
12
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100
Fig. 11. Constraints on the seismic properties of the middle and lower ductile continental crust based on CPO and rock-recipe modelling of individual compositional bands in a deformed pyroxene granulite, taken to simulate in particular a lowermost middle to lowest continental crust compositional profile. The presence of biotite in some ‘bands’ permits not only felsic to mafic compositions but also hydrous to anhydrous conditions to be considered. Also shown (grey) are the results from Figure 9d of ‘rock-recipe’ modelling for pyroxene (hypersthene) content in this rock and typical Vp ranges for different parts of the middle-lower crust as suggested by Rudnick & Fountain (1999). See text for discussion. (a) P-wave velocity; (b) P- and S-waves anisotropy.
and becomes less than AVp for intermediate compositions (i.e. c. 1/3 mafic content). At c. 50% mafic content, AVs exceeds AVp and increases more rapidly towards ultramafic (i.e. so-called ‘hornblendite’) compositions. These results are in broad agreement with the behaviour exhibited by the mafic dyke, including the tendency for AVs , AVp for felsic compositions (Fig. 9c). Due to the azimuthal dependence of AVs, simply plotting maximum values of shear-wave splitting can be misleading (e.g. Fig. 6). This situation is exacerbated by the impact of foliation and/or compositional layering orientation (e.g. Figs 5 & 8). To emphasize this effect, the layering in the banded amphibolite sample has been considered to be oriented horizontally in a geographical sense. For a vertically propagating (i.e. teleseismic) shear wave, the value of AVs for each composition is significantly smaller (i.e. from 2% to 3%) than the maximum recorded (Fig. 10b). Indeed, for most compositions it is smaller than the azimuthally independent absolute P-wave anisotropy. Furthermore, the polarization direction of the fast (teleseismic) shear wave (i.e. Vs1P) is not constant and changes through 908
as the composition changes from felsic to intermediate, beyond which it remains constant. This behaviour of Vs1P clearly reflects the increasing influence of the amphibole (hornblende). From Figure 8b, this would indicate not only a horizontal foliation/layering (as initially defined) but also an ‘east –west’ (in the arbitrary coordinate system assumed) tectonic Y-direction and hence a ‘north– south’ oriented lineation. Thus, this banded amphibolite sample can be interpreted from the CPOderived seismic characteristics as being an LS tectonite that developed under conditions close to plane strain (i.e. K 1). Lower crust (pyroxene granulite). Figure 11 is a potential CPO-derived Vp, AVp and AVs seismic stratigraphy for a lower continental crustal section. In terms of Vp (Fig. 11a), values increase progressively as composition changes from felsic (i.e. biotite–plagioclase dominated) to mafic (i.e. pyroxene dominated) and eventually to ultramafic (i.e. so-called ‘pyroxenites’). However, the most felsic compositions (e.g. biotitic anorthosite granulite) exhibit Vp values less than those expected for even the middle crust, while compositions between
SEISMIC PROPERTIES OF CONTINENTAL CRUST
biotitic anorthosite granulite and biotitic pyroxene granulite yield Vp values typical of the middle crust (e.g. Rudnick & Fountain 1999). In general, essentially mafic compositions (i.e. biotitic pyroxene and pyroxene granulites) are required to produce typical lower crustal Vp values. Ultramafic compositions exhibit Vp values in excess of those expected in typical lower crust. In terms of anisotropy, AVp is relatively small for felsic (i.e. biotite –plagioclase dominated) compositions but AVs is much larger (Fig. 11b). As the mafic component steadily increases towards pyroxene-dominated compositions, AVp remains essentially constant but AVs reduces significantly and therefore trends towards AVp. These results are obviously in broad agreement with those shown in Figure 9c due to the fact that the same sample is involved in both. However, there are some differences because the seismic-stratigraphy compositions have been defined via rock recipes. To investigate the impact of azimuthal dependence on AVs, the sample of pyroxene granulite (which is an XZ tectonic section) is considered to have been oriented with X horizontal and Z vertical geographically (i.e. horizontal foliation and/or layering). For a vertically propagating (i.e. teleseismic) shear wave, the value of AVs for each composition considered is not only significantly smaller than the maximum recorded but it is also smaller than the equivalent and azimuthally independent absolute AVp value (Fig. 11b). The magnitude of the vertical AVs also decreases with increasing mafic (pyroxene) content, although a slight increase may be significant at the change from effectively felsic to effectively mafic granulite compositions. The polarization direction of the fast (teleseismic) shear wave (i.e. Vs1P) is essentially constant ‘northeast –southwest’ for all compositions except for the slight increase in AVs values, where it is ‘east– west’. The behaviours of the vertical (teleseismic) AVs and Vs1P clearly reflect the increasing influence of pyroxene and suggest that mafic and ultramafic lithologies with (sub-) horizontal layering may exhibit little recognizable and/or interpretable shear-wave splitting anisotropy (e.g. see Dı´azAzpiroz et al. 2011).
Discussion While recent studies suggest that crustal deformation is distinguishable via seismic profiling using velocity and/or anisotropy (e.g. Meltzer et al. 2001; Meltzer & Christensen 2001; Chlupacova et al. 2003; Brown et al. 2009; Moschetti et al. 2010), the source(s) of these seismic properties must be constrained. In the ductile continental crust, the usual source is the CPO characteristics of the
25
rocks through which the seismic waves travel. It appears to be tacitly assumed that CPO not only indicates foliation but is also mimicked by the seismic symmetry (e.g. Siegesmund et al. 1989; Okaya & Christensen 2002; Kitamura 2006; Meissner et al. 2006; Barberini et al. 2007). Thus, micas are generally regarded as controlling the seismic properties of much of the ductile continental crust (e.g. Vernik & Liu 1997; Meltzer & Christensen 2001; Nishizawa & Yoshino 2001; Mahan 2006; Valcke et al. 2006; Lloyd et al. 2009). However, there is little evidence for meta-pelitic layers in deep continental crust as increasing Vp values with depth are interpreted as indicating increasing mafic component (e.g. Rudnick & Fountain 1999; Rudnick & Gao 2003). Furthermore, the present contribution has shown (i.e. Figs 1– 8) that while relationships exist between mineralogy, CPO, deformation and seismic symmetry, the precise natures need to be established rigorously, usually for each geodynamic situation. Figures 9–11 summarized these results and initially considered the potential origins of the seismic properties of the middle and lower ductile continental crust, which form the basis for many tectonic and/or geodynamic interpretations. The simple questions to be posed/answered when considering the seismic properties of the ductile continental crust are therefore: (1) what seismic velocity and/or anisotropy is/are observed/needed; (2) what mineralogy/composition and/or CPO/ deformation state is/are observed/required; and/ or (3) what tectonic/geodynamic setting(s) explain(s) the seismic properties and/or geology observed? The rest of this discussion considers these questions based on the results presented in this study and provides some caveats for the accuracy and reliability of some current geodynamic models based on seismic profiling of the ductile (middle and lower) continental crust.
Seismic velocity (Vp) profiling The typical mica modal contents of most rocks is ,30% and alone cannot account for observed midcrustal Vp values of 6.2 –6.8 km/s, particularly if foliation/layering is (sub-) horizontal (e.g. Fig. 9a, b). Consequently, other faster common rockforming minerals (such as feldspars) must make the most significant contributions (e.g. Figs 1, 2, 10a & 11a). Micas therefore cannot be regarded as being responsible for the Vp values observed in the ductile continental crust. Furthermore, while micas are amongst the most significant and sensitive foliation-forming minerals, their VTI characteristics make them insensitive to most deformation states (e.g. Fig. 8a). Indeed, it appears that micas are capable only of distinguishing constrictional
26
G. E. LLOYD ET AL.
deformations, ironically due to the lack of foliation development. Exceptional care must therefore be taken when attempting to infer tectonic states from micaceous rocks, particularly where the orientation and/or number of any foliation(s) is/are unknown. In contrast to micas, the wide range of amphibole modal contents possible naturally (i.e. up to and exceeding 65%) could explain observed Vp values in not only the middle but also much of the lower continental crust (e.g. Figs 9c & 10a). Indeed, Vp values in excess of 7 km/s and regarded as indicative of the lowermost continental crust can be explained by hydrous rocks with very high amphibole contents (i.e. so-called ‘hornblendites’) and subvertical foliation(s). However, as with micas, amphiboles form foliation/layering relatively easily and (sub-) horizontally oriented foliation/ layering acts to reduce Vp values significantly (e.g. Figs 9c & 10a). Nevertheless, amphiboles (unlike micas) are sensitive to most deformation states due to their lack of VTI characteristics (e.g. Fig. 8b). The Vp characteristics of most of the ductile continental crust can therefore be explained by amphiboles alone, but conditions must be regarded as being hydrous. The highest Vp values recognized are provided by pyroxenes (e.g. Figs 9d & 11a). Indeed, so-called ‘pyroxenites’ are capable of exhibiting velocities of c. 8 km/s, perhaps associated more usually with upper mantle lithologies (e.g. Ben Ismail & Mainprice 1998). Even relatively low pyroxene modal contents can explain Vp values associated with the middle continental crust, while pyroxene granulite compositions readily account for most lower continental crustal values (e.g. Figs 9a & 11b). Although (sub-) horizontally oriented foliation/layering will again act to reduce Vp, values are likely to remain sufficiently high to explain most middle and especially lower crustal observations. The Vp characteristics of most of the ductile continental crust can therefore be explained by pyroxenes alone, but conditions must be regarded as being anhydrous. In summary, it is unlikely that seismic profiling observations based on Vp characteristics alone are capable of providing accurate and/or reliable interpretations of the middle or lower ductile continental crust. Both amphiboles and pyroxenes yield acceptable Vp values but micas do not and require significant contribution from other common rockforming mineral phases (e.g. feldspars).
Seismic anisotropy (AVp, AVs) profiling Both mica-dominated felsic and amphiboledominated mafic lithologies are capable of generating the levels of AVp and AVs recognized naturally (e.g. Figs 9a–c & 10). In detail, micas and amphiboles contribute differently to whole-rock
anisotropy. In felsic rocks, relatively low (i.e. ,30%) modal mica contents are offset by relatively high mica single-crystal elastic anisotropy (Fig. 1). In mafic rocks, relatively high (.50%) modal amphibole contents offset relatively low amphibole single-crystal elastic anisotropy. However, the impact of micas on both AVp and AVs appears to be more significant overall (compare Figs 9a & 9c). In contrast to micas and amphiboles, pyroxenes do not appear to contribute significantly to wholerock anisotropy (e.g. Figs 9d & 11). This behaviour is clearly related to the relatively low single-crystal elastic anisotropy of pyroxenes (Fig. 1). In summary, significant AVp and AVs values observed in the ductile middle to lower continental crust are due to either mica (felsic) and/or amphibole (mafic hydrous) lithologies. Furthermore, relatively low or intermediate values of anisotropy could also be due to pyroxene (mafic anhydrous) lithologies. It will therefore be difficult and/or impossible to distinguish between contrasting felsic and mafic compositions and/or conditions using seismic anisotropy profiling observations alone.
Seismic shear-wave splitting polarization (AVs, Vs1P) profiling AVs values measured normal to foliation and/or compositional layering are typically significantly lower than the maximum AVs values possible (e.g. Figs 5 & 6) due to the composition and intrinsic properties of the single-crystal elastic anisotropy and whole-rock CPO (e.g. Figs 1 & 8). In addition, the polarization direction of the fast shear wave (i.e. Vs1P) may also vary with shear-wave propagation direction, which similarly depends on composition, single-crystal elastic anisotropy and CPO. Thus, variations in AVs–Vs1P (e.g. Figs 10 & 11) may not necessarily reflect different kinematics, as is often assumed in many geodynamic interpretations of seismic profiling data (e.g. Ozacar & Zandt 2004; Shapiro et al. 2004; Sherrington et al. 2004; Schulte-Pelkum et al. 2005). For example, if the lower continental crust comprises layered mafic lithologies (whether amphibole or pyroxene dominated) and the layering is oriented (sub-) horizontally as is often assumed (e.g. Rudnick & Fountain 1999), it is likely to exhibit only low shear-wave splitting anisotropy and may even appear essentially isotropic irrespective of its actual composition and deformation state (e.g. see Dı´az-Azpiroz et al. 2011).
Seismic properties of ductile continental crust: an example Recently, Moschetti et al. (2010) used ambient noise tomography to argue for strong, deep
SEISMIC PROPERTIES OF CONTINENTAL CRUST
(middle-to-lower) crustal radial seismic anisotropy beneath the western USA, confined mainly to geological provinces that have undergone significant extension in the last 65 Ma. They consider that the coincidence of crustal radial anisotropy with extensional provinces suggests that the anisotropy results from the CPO of anisotropic crustal minerals caused by extensional deformation. Consequently, their observations provide support for the hypothesis that the deep crust within these regions has undergone widespread and relatively uniform strain in response to crustal thinning and extension. We now consider the results of Moschetti et al. (2010) in terms of the observations presented in this contribution. In general, the seismic results of Moschetti et al. (2010) recognize seismic anisotropy in the middle-to-lower crust and hence support our basic conclusion that the ductile continental crust can contribute significantly to the overall seismic anisotropy detected in the Earth. However, can their seismic results be interpreted geologically by the observations and relationships established in the present study? To attempt to answer this question we have incorporated the best-fit results of Moschetti et al. (2010, supplementary fig. 6a, b), which predict AVs of c. 5%, into our observations on the impact of different modal contents of mica (biotite), amphibole (hornblende) and pyroxene (hypersthene) on seismic properties (i.e. Figs 9–11). For the purposes of this comparison, these figures have been amended to include the maximum and minimum in the CPOderived Vs1 and Vs2. To comply with the analysis presented by Moschetti et al. (2010), these are termed VSV and VSH here to represent a horizontally propagating shear wave split into vertical and horizontal components (Fig. 12). According to the results presented previously (Fig. 9), felsic lithologies with mica contents of ,18% are required to explain the middle crust Vs values indicated by Moschetti et al. (2010) while their AVs of c. 5% require ,5% modal mica (Fig. 12a). However, only the latter composition can result in typical middle crustal Vp values (e.g. Rudnick & Fountain 1995). Felsic lithologies with mica contents of up to 18% require the presence of one or more seismically ‘fast’ minerals, that do not impact on anisotropy, to achieve middle crustal Vp values. The lower crustal Vs and AVs values indicated by Moschetti et al. (2010) cannot be achieved by variation in modal mica content in the felsic lithology considered here (Fig. 12a). In addition, micas cannot explain the overall results indicated by Moschetti et al. (2010) due to their VTI characteristics, which force isotropy in the radial plane for flattening and plane strain (simpleshear) deformations (e.g. Fig. 8a). Consequently, any anisotropy observed in the radial plane must
27
be due to other minerals, which may be possible for low biotite contents. While hydrous mafic lithologies with amphibole contents of 27 –72% (Fig. 10) readily explain the lower crustal Vs values indicated by Moschetti et al. (2010), even the lowest amphibole contents yield shear-wave velocities that are too high for their middle crust (Fig. 12b). Notwithstanding these comparisons, increasing amphibole content can account for the Vp values expected for the middle and lower crust (e.g. Rudnick & Fountain 1995). The AVs value of c. 5% predicted by Moschetti et al. (2010) can be satisfied by amphibole contents of 35 –40%, which are compatible with both middle and lower crustal Vp values and the Vs values predicted by Moschetti et al. (2010) for their lower crust (Fig. 12b). In addition, amphiboles exhibit radial VTI characteristics for flattening type deformations but, for deformations where K 1 (e.g. plane strain, simple shear, etc.), they exhibit slight radial anisotropy. Any anisotropy observed in the radial plane must therefore be due to either other minerals for flattening type deformations, which implies relatively low amphibole contents, or implies some type of shear deformation. Anhydrous mafic lithologies with pyroxene contents of ,18% and 17 –37% explain the Vs values of Moschetti et al. (2010) for the middle and lower crust, respectively (Fig. 12c). However, Vp values for the former are less than expected for the middle crust while for the latter Vp is only that expected for the middle crust (e.g. Rudnick & Fountain 1995). Furthermore, AVs values are considerably higher than the 5% predicted by Moschetti et al. (2010), probably due to the lack of impact of pyroxene. The ratio Vp/Vs is often used seismologically to characterize rock types and, in particular, to recognize hot rocks and/or the presence of melt. Moschetti et al. (2010) also calculated the variation in this parameter with depth in their seismic analyses but observed little change (Fig. 12 insets). In terms of the CPO-derived seismic properties, the ratio Vp/Vs can be defined either by Vp-max/ Vs2-min to give a maximum, or by Vp-min/ Vs1-max to give a minimum value. A range of Vp/Vs values can therefore be plotted with variation in, for example, modal content of a specific mineral (Fig. 12 insets). The ranges of Vp/Vs expand significantly with increasing modal content of, particularly, mica in felsic lithologies and amphibole in hydrous mafic lithologies (Fig. 12a, b insets). In contrast, the increasing modal content of pyroxene in anhydrous mafic lithologies results in a contraction of this range (Fig. 12c inset). Nevertheless, the ranges predicted from the CPO-derived seismic properties generally bracket the values
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(a)
1. Vp (max & min) km/s 3
02
7
6
5
4
2. AVp and AVs % 8 0
2
6
4
VSH
8
10
12
16
14
0
Avs
middle crust VSV
Vp-max biotite felsic gneiss Vp-min
60
100
3
2
5
4
(b)
(fe bio ls tite ic A gn V ei p ss )
Lowermost crust Lower crust Lower middle crust
Middle crust
80
6
7
Vp-max 2.00 Vs2-min 1.50 Vp-min Vs1-max 1.00 0 20
8 0
5
6
7
Vpmin
Vpmax
lower crust
Biotite% 40 60
80
8
12
10
100
14
2
8
6
4
10
16 0
14
12
2.50
Vp-max 2.00 Vs2-min
M-S6b 1.50 Vp-min Vs1-max Hornblende% 1.00 0 20 40 60 80 100
AVs
20
40
60
60 VSV
80
80 Vs- Vsmin max
100
3
4
AVp mafic dyke 5
6
7
100
16
8 0
2
4
6
8
10
% Modal content hornblende
40
80
M-S6b
2. AVp and AVs % 8 0
20 VSH
6
4
Lowermost crust Lower crust Lower middle crust Middle crust
% Modal content hornblende
VSV
2
Micas - biotite
Vp/Vs
4 VSH middle crust
60
2.50
1. Vp (max & min) km/s 03
20
40
Vp/Vs
40
Vs1-max biotite felsic gneiss Vs2-min
% Modal content biotite
b (fe iotit lsi e A cg V ne s iss )
VSV
AVs
12
16
14
100
Amphiboles - hornblende (c)
1. Vp (max & min) km/s 03
4
5
6
7
2. AVp and AVs % 8 0
2
4
6
8
10
12
14
16
0
VSH
orthopyroxene AVp
AVs
lower crust VSV
40
20
40
60
60
100
Lowermost crust Lower crust Lower middle crust Middle crust
3
4
5
6
7
2.5
1.5 Vp-min Vs1-max AVs 1.00 20
8 0
2
4
6
8
Pyroxenes - hypersthene Fig. 12.
80
Vp-max 2.0 Vs2-min
Vp/Vs
80
10
M-S6b Pyroxene% 40 60 80
12
14
100
16
100
% Modal content orthopyroxene
VSH
ax Vp-m oxene pyr ortho p-min V
20
VSV
ax Vs-m oxene pyr ortho s-min V
% Modal content orthopyroxene
middle crust
% Modal content biotite
VSH lower crust
20
SEISMIC PROPERTIES OF CONTINENTAL CRUST
suggested by Moschetti et al. (2010), except for pyroxene contents in excess of 80%. Based on the observations presented in this contribution, we can therefore explain geologically the seismic results described by Moschetti et al. (2010) in terms of a dominantly amphibolitic and hence hydrous middle/lower continental crust beneath the current western USA. A composition of c. 40% amphibole and c. 60% felsic minerals (e.g. plagioclase and quartz) is compatible with our results. Furthermore, the extensional deformation indicated is due to horizontal (simple?) shear rather than vertical flattening, accommodated by the development of CPO in the amphibole to produce the observed radial crustal anisotropy of c. 5%.
Seismic profiling strategy for the ductile continental crust The observations made and experiences gained in this contribution have led us to suggest the following strategy for in situ seismic profiling of the ductile continental crust. Essentially, there is one basic question that needs to be considered: what seismic velocities and/or anisotropies are observed and/or needed to explain a particular geodynamic setting? To answer this question it is important to make use of as much seismic information as possible as individual parameters (e.g. Vp, Vs, AVp, AVs, etc.) are sensitive to different geological constraints and hence provide information on different aspects of the geodynamical setting, as the following considerations illustrate. Relatively low mica modal contents are capable of generating strong mica CPO in felsic rocks, resulting in relatively large seismic anisotropy but relatively low P-wave velocity. Thus, anisotropic felsic rocks with typical middle crustal P-wave velocities must consist of mica(s) and seismically faster minerals. However, if any associated foliation is oriented normal to the seismic wave propagation direction, both anisotropy and velocity will be minimized. Nevertheless, due to the VTI characteristics of mica(s) (which tend to dominate seismic property distributions in three dimensions), it will be difficult to distinguish specific types of deformation (except
29
constriction) unless the other minerals are able to impart significant modifications. To generate similar seismic anisotropy in mafic rocks as micaceous felsic rocks requires large but not unrealistic amphibole modal contents, which can also account for most middle and lower crustal P-wave velocities. Such lithologies imply hydrous conditions. Again, foliation orientation is significant in terms of the velocity and particularly anisotropy detected. The non-VTI characteristics of amphibole mean that different types of deformation (e.g. flattening, simple shear, constriction) can be distinguished. The fastest P-wave velocities are associated with pyroxene-dominated lithologies which can account for velocities associated with both the lower crust and upper mantle, particularly when garnet is also present (see Brown et al. 2009). Such situations imply anhydrous conditions. However, it appears that such lithologies tend to exhibit relatively low anisotropy (whether normal or parallel to foliation), especially when garnetiferous. Although in principal the non-VTI characteristics of pyroxene mean that different types of deformation can be distinguished, in practice this may prove difficult due to the lack of detail in the seismic property distributions.
Conclusions Results of rock- and fabric-recipe modelling using representative rock samples suggest that the seismic properties of the middle and lower continental crust depend on CPO and deformation as well as composition, while reliable interpretation of seismic survey data should incorporate as many seismic properties (i.e. Vp, AVp, AVs, Vs1, Vs2, Vs1P) as possible. Crustal AVs is controlled by the dominant rock fabric-forming mineral: micas provide the most significant values, even at low concentrations, and amphiboles are significantly weaker except at higher concentrations. As lithology and anisotropy vary with depth, AVs can have a significant crustal component (e.g. steep, pervasive coaxial mica fabrics). Micas do not contribute significantly to Vp, however; observed Vp and AVs values imply
Fig. 12. Explanation of the seismic results of Moschetti et al. (2010) in terms of the potential geology of the middle-to-lower crust beneath the western USA using observations presented in this contribution (see Figs 9– 11) amended to include the CPO-derived shear-wave velocities. The black boxes in the velocity graphs represent the splitting of horizontally propagating shear waves into vertical and horizontal components, while in the anisotropy graphs they represent the best estimate AVs as reported by Moschetti et al. (2010, fig. 6a, b) for their best-fitting results, plotted in order to intercept with the respective CPO-derived values. The inset in each part compares their Vp/Vs ratio with the ranges predicted via CPO (NB minor inflection points indicate change from upper, middle and lower crust and mantle). (a) Felsic lithologies (varying modal biotite content); (b) hydrous mafic lithologies (varying modal amphibole content); and (c) anhydrous mafic lithologies (varying modal pyroxene content).
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the middle/lower crust is dominated by mafic lithologies, including ‘amphibolites’. Seismic symmetry is crucial in distinguishing strain type. Micas exhibit VTI and are unlikely to distinguish either radial deformation typical of flattening strains (i.e. X Y ≫ Z; K 0) or ductile deformation typical of simple-shear zones (i.e. X . Y . Z; K 1), where multiple fabrics may also interfere. Amphiboles exhibit ‘orthorhombic’ symmetry and distinguish potentially all common strain types, providing more sensitive tests of anisotropy-derived tectonic models (e.g. anisotropic lower crust with high Vp values can be explained by foliated amphibolites, although pyroxene granulites with only minor biotite content can achieve similar impact). These results pose significant constraints for many currently extant seismically based geodynamic models. The samples of Nanga Parbat gneisses were collected during fieldwork funded by a Royal Society research grant (RWHB). We are grateful to Tamsin Lapworth and V. Prasannakumar for the samples of ‘banded’ amphibolite and pyroxene granulite, respectively, used in this study. We thank Craig Storey and Stanislav Ulrich for their constructive reviews and Dave Prior for his editorial comments that have helped to improve the original version of this contribution. The UK NERC supported part of the SEM/EBSD facilities via Small Grant GR9/3223 (GEL, MC) and a PhD studentship to DJT.
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Takanashi, M., Nishizawa, O., Kanagawa, K. & Yasunga, K. 2001. Laboratory measurements of elastic anisotropy parameters for the exposed crustal rocks from the Hidaka Metamorphic Belt, Central Hokkaido, Japan. Geophysical Journal International, 145, 33– 47. Tatham, D. J. 2008. Towards using seismic anisotropy to interpret ductile deformation in mafic lower crust. Unpublished PhD thesis, University of Leeds, UK. Tatham, D. J. & Casey, M. 2007. Inferences from shear zone geometry: an example from the Laxfordian shear zone at Upper Badcall, Lewisian Complex, NW Scotland. In: Ries, A. C., Butler, R. W. H. & Graham, R. H. (eds) Deformation of the Continental Crust: The Legacy of Mike Coward. Geological Society, London, Special Publications 272, 47–57. Tatham, D. J., Lloyd, G. E., Butler, R. W. H. & Casey, M. 2008. Amphibole and lower crustal seismic properties. Earth & Planetary Sciences Letters, 267, 118–128. Thatcher, W. 2009. How the continents deform: the evidence from tectonic geodesy. Annual Review of Earth and Planetary Sciences, 37, 237– 262. Valcke, S. L. A., Casey, M., Lloyd, G. E., Kendall, J.-M. & Fisher, Q. 2006. Lattice preferred orientation and seismic anisotropy in sedimentary rocks Geophysical Journal International, 166, 652–666. Vaughan, M. T. & Guggenheim, S. 1986. Elasticity of muscovite and its relationship to crystal-structure. Journal of Geophysical Research, 91, 4657– 4664 Vernik, L. & Liu, X. 1997. Velocity anisotropy in shales: a petrophysical study. Geophysics, 62, 521–532. Wendt, A. S., Bayuk, I. O., Covey-Crump, S. J., Wirth, R. & Lloyd, G. E. 2003. An experimental and numerical study of the microstructural parameters contributing to the seismic anisotropy of rocks. Journal of Geophysical Research, 108, 2365. doi: 10.1029/ 2002JB001915.
Mica-controlled anisotropy within mid-to-upper crustal mylonites: an EBSD study of mica fabrics in the Alpine Fault Zone, New Zealand EDWARD D. DEMPSEY1*, DAVE J. PRIOR1, ELISABETTA MARIANI1, VIRGINIA G. TOY2 & DANIEL J. TATHAM1 1
School of Environmental Sciences, University of Liverpool, 4 Brownlow Street, Liverpool L69 3GP, UK
2
Geology Department, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand *Corresponding author (e-mail:
[email protected]) Abstract: The lattice preferred orientation (LPO) of both muscovite and biotite were measured by electron backscatter diffraction (EBSD) and these data, together with the LPOs of the other main constituent minerals, were used to produce models of the seismic velocity anisotropy of the Alpine Fault Zone. Numerical experiments examine the effects of varying modal percentages of mica within the fault rocks. These models suggest that when the mica modal proportions approach 20% in quartzofeldspathic mylonites the intrinsic seismic anisotropy of the studied fault zone is dominated by mica, with the direction of the fastest P and S wave velocities strongly dependent on the mica LPOs. The LPOs show that micas produce three distinct patterns within mylonitic fault zones: C-fabric, S-fabric and a composite S –C fabric. The asymmetry of the LPOs can be used as kinematic indicators for the deformation within mylonites. Kinematic data from the micas matches the kinematic interpretation of quartz LPOs and field data. The modelling of velocities and velocity anisotropies from sample LPOs is consistent with geophysical data from the crust under the Southern Alps. The Alpine Fault mylonites and parallel Alpine schists have intrinsic P-wave velocity anisotropies of 12% and S-wave anisotropies of 10%.
Transpressional plate boundaries are often accompanied by a highly anisotropic crustal zone where seismic waves propagate at higher velocities in preferred orientations. Seismic anisotropy information may be used to assess the extent of deformation, not only in the mantle but also in the continental crust (Tatham et al. 2008). By understanding this anisotropy, geophysicists can better constrain the nature of the deep structure of fault zones and map out the deformation therein (Mainprice 2007). At depth this anisotropy is generally ascribed to the regional alignment of minerals, often micas in mid-lower crustal settings, caused by the kinematic regime of the fault zone. The micaceous quartzofeldspathic Alpine schist is the main rock type of the Southern Alps and was formed during the Cretaceous Rangitata Orogeny (Little et al. 2002). The rock types analysed here are protomylonites, mylonites and ultramylonites derived from this schist. The Alpine Fault marks the dextral convergent boundary of the Australian and Pacific plates. It has existed in this form for the past 5 million years (Sutherland 1996; Toy et al. 2008) with little apparent change in the rate and magnitude of relative Pacific –Australian plate motion across the boundary over that time. Associated hanging wall fault rocks comprise a 1 km wide zone of
cataclasites, ultramylonites, mylonites and protomylonites (Sibson et al. 1979). From the plate vector calculations obtained for this region for the past 5 million years (Beavan et al. 1999; Cande & Stock 2004) we can assume that the mylonites we see at the surface today have been exhumed since then and are therefore representative of rock currently being deformed at depth (Norris & Cooper 2003). Strong seismic reflectors from inferred anisotropic shear-zone rocks indicate that the fault zone continues to at least 25 km with an average dip of 40 –608SE (Davey et al. 1998; Kleffmann et al. 1998). The mylonites studied for this paper have been deformed at between c. 8 and c. 25 km depth (Toy et al. 2008) with frictional melting occurring between c. 3.5 and 11 km depth (Warr & van der Pluijm 2005; Warr et al. 2007). The Alpine Fault Zone (AFZ) is an active plate boundary on which large earthquakes (Mw . 7.6) occur approximately every 300 years and, as the last event was in 1717AD, the fault is in the late stages of its seismic cycle (Sutherland et al. 2007). It is therefore of fundamental importance to understand the kinematics and seismic properties of this fault zone. The Alpine Fault mylonites give us an opportunity to study fault rocks which are comparable to those believed to be present but not exhumed within this
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 33– 47. DOI: 10.1144/SP360.3 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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and other crustal-scale fault zones. Previous studies of lattice preferred orientations (LPO) fabrics within the Alpine Fault mylonites have focused on quartz (Toy et al. 2008) and, while this is important, a complete picture of textures within the polyphase shear zone can only be gathered through understanding the deformation behaviour of other important constituent minerals such as micas. Due to their weak basal planes and highly anisotropic nature, phyllosilicate minerals play an important role in the deformation processes and seismic properties of ductile fault rocks (e.g. Byerlee 1978; Kronenberg et al. 1990; Mares & Kronenberg 1993). The predominant phyllosilicate minerals in the Alpine Fault mylonites are muscovite and biotite. These micas are present as both a finegrained matrix and as larger (1–2 mm) porphyroclasts. Mica porphyroclasts may be used as kinematic indicators in the field, through examination of mica streaks that are formed due to movement along the displacement vector (Toy et al. 2008) and mica fish. However, these only comprise a small proportion of the mica present within the fault rocks of the Alpine Fault Zone. When examined optically, most of the matrix mica shows remarkably consistent foliation-parallel to subparallel orientation. This paper uses electron backscatter diffraction (EBSD) to obtain the LPOs of mica crystals in the Alpine Fault Zone mylonites. These data are then used to evaluate mica LPOs as kinematic indicators and their effect on the seismic properties of the fault zone in relation to other minerals, namely quartz and feldspar. Based on these data, a kinematic and seismic profile of the Alpine Fault Zone is produced.
Methods The eight samples used in this study (Table 1) are part of a much larger sample suite collected over
two field seasons in the AFZ. These samples were collected from localities that vary in distance from and along strike of the fault trace. Samples selected were deemed to be typical of the AFZ mylonites: these are thinly banded micaceous quartzofeldspathic mylonites, ultramylonites and protomylonites derived from the quartzofeldspathic Alpine schist. Successful and reliable acquisition of EBSD data on micas is generally challenging (Prior et al. 2009). In particular, when examined in a foliation normal section, micas may index with difficulty due to very weak electron backscatter diffraction patterns. This may be due to the platy habit of the mineral and its ductility, which compromise the quality of the final polishing as they may deform during the polishing process. However, recent work on micas using EBSD was carried out by Lloyd et al. (2009). These authors gained EBSD data from thin sections cut perpendicular to foliation (and therefore to mica cleavage) and parallel to the shear direction. The data collected were then utilized successfully for LPO analyses. The method employed by Lloyd et al. (2009) could not be replicated for this study as adequate EBSD data could not be obtained from the AFZ mylonite samples cut perpendicular to foliation. This problem was overcome by polishing foliationparallel surfaces, looking down upon the basal planes of the mica. Samples prepared in this orientation provided a good final polish and a larger surface area within individual crystals to examine, and yielded a high indexing during EBSD data acquisition. Samples were cut into thin chips sub-parallel to foliation and polished using 500, 800 and 1200 grade SiC paper followed by 6, 3 then 1 mm diamond paste for 1 h on each medium. After this 0.3 mm alumina paste was used for 20 min, and finally the samples were polished using 0.25 mm colloidal silica. Samples were then given a thin carbon coat to prevent charging effects and were
Table 1. Sample list including data obtained from EBSD analysis Sample number
Lithology
Distance from fault
Shear direction
J index
Max Vp (km/s)
AVs (%)
Max Vs1 (km/s)
RCE7 RCE6 BCE4 BCE5 BCE8 BCE9 VCE5 VCE6
Mylonite Mylonite Ultramylonite Ultramylonite Protomylonite Protomylonite Mylonite Mylonite
c. 500 m c. 430 m c. 60 m c. 130 m c. 920 m c. 1000 m c. 620 m c. 350 m
34/056 31/072 34/090 24/098 10/042 8/016 45/125 40/128
10.53 7.91 8.15 9.23 10.59 8.40 6.75 10.15
7.00 7.50 6.82 7.59 7.70 7.15 6.84 7.69
70.06 49.19 59.45 50.95 52.71 63.34 55.85 54.42
4.32 4.59 4.17 4.62 4.66 4.27 4.10 4.70
RCE, Rough Creek; BCE, Bullock Creek; VCE, Vine Creek; Vp, compressional wave velocity; AVs, shear-wave anisotropy; Vs1, fast shear-wave velocity.
MICA FABRICS AND ANISOTROPY FROM EBSD
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Fig. 1. (a) Phase map, red: biotite, blue: muscovite, yellow: quartz and green: feldspar. (b) Orientation map with colours representing various orientations of crystals based on Euler angles.
analysed in the CamScan X500 Crystal Probe# and a Phillips XL30# scanning electron microscopes (SEMs) using the Oxford Instruments/HKL Technology Channel5# automated EBSD system. Operating conditions of 20 keV accelerating voltage, 25 mm working distance, 208 specimen tilt to the electron beam and 3–40 nA beam current were used. EBSD data were collected using a set grid, with step sizes of 5 or 8 mm depending on grain size and area to be covered.
The Channel5# post-processing software allows the generation of colour-coded EBSD maps displaying crystallographic orientations and phases (Fig. 1), as well as stereographic point and contour plots of the crystallographic orientations. The phase and orientation maps were processed with extrapolation of data into non-indexed pixels with 8 and 7 neighbouring pixels. For the generation of stereographic projections for LPO and kinematic analyses, one orientation data point (pixel) per
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Fig. 2.
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MICA FABRICS AND ANISOTROPY FROM EBSD
grain was used so that results were not biased towards large grains containing many data points (Fig. 2). Conversely, for modelling of seismic anisotropy, all data points available were used as the area-weighted data are more representative of the volumetric proportions of orientation components in the rock (Fig. 2). Reorientation of the samples in a virtual chamber, available in the software, allowed us to examine the data orientated foliationparallel (Fig. 2) in a standard kinematic reference frame and reoriented back into its original field orientation. Seismic properties of the mylonites were modelled from the LPOs of each mineral type using the programs AnisCh5 and VpG (Mainprice 1990). Results are displayed by foliationparallel section (Fig. 2) in a kinematic reference frame and in the original field orientation for comparison to the fault zone geometry.
Testing the reliability of EBSD applied to micas As anticipated previously, the acquisition of microstructural data from mica using EBSD can be problematic. During the development of sample preparation and analytical methods, we posed the following key question. Does EBSD only index favourably orientated mica crystals (sub-parallel to basal plane), ignoring those at higher angles to the polished surface? To answer this question, samples from the Alpine Fault mylonites were analysed at various orientations; surfaces analysed were cut parallel to foliation (08), 108, 208, 308, 458 and 608 to foliation. Analysis of these results showed that the orientation of the mica fabric remains consistent regardless of the angle the sample sectioned; however, the greater the angle the lower the % indexing and higher the % misindexing (Fig. 3). This suggests, when carried out on a surface parallel to foliation, EBSD was capable of analysing all the mica present within the mylonites (in these samples the micas are rarely orientated .308(from foliation). In analyses of polished surfaces between 458 and 608 to foliation, the indexing % greatly decreased and misindexing problems increased. This indicates that sections through mica crystals at ,458 to their basal plane may be analysed with confidence, while higher-angle sections may be problematic. A further consideration which must be made is that, generally, in foliated
37
rocks, mica fabrics (textures) are strong and therefore any errors involved with measuring these fabrics using EBSD might be too small to be significant. Despite the different preparation methods used, our results are comparable to those obtained by Lloyd et al. (2009) indicating that errors involved in the measurements may be unimportant given the strength of mica LPOs. We however suggest that, for samples which are not characterized by such strong fabrics, mica orientation data may be collected confidently using EBSD on three orthogonal polished sections from the same sample. This would allow all mica orientations to be measured and reliable mica LPOs would be obtained.
Results Mica LPO fabrics of the Alpine Fault and their use as kinematic indicators The stereographic projections, used for LPO data display, represent the k001l, k100l and k010l crystallographic directions. As expected from inspection of the macroscopic fabric, the k001l directions in both muscovite and biotite produce a strong cluster in all samples analysed while the k100l and k010l directions produce well-defined girdles. However, poles to the basal planes of the mica do not always cluster around the pole to foliation. The k001l clusters are centred up to 308 off the pole to foliation with faint tails due to locally distorted crystals. Mica LPOs obtained using EBSD in this paper are displayed in two orientations. Firstly, they are displayed looking down upon the foliation (foliation parallel section) with north to the top of the figure (the data were acquired in this orientation to overcome polishing and indexing issues as discussed previously). In this reference frame, the deviation of the k001l cluster from the pole to foliation is clearly visible (Fig. 3). The direction in which the cluster is deflected from the pole to foliation is interpreted as the shear direction. Note that many Alpine Fault mylonites do not contain lineations or have lineations that are not related to the transport direction (Toy 2008; Toy et al. 2008), so that ‘reconstruction’ of shear direction from LPOs is a valuable tool. The second reference frame is the standard kinematic ref frame, where foliation is the XY plane and the lineation (or reconstructed lineation from LPO
Fig. 2. The first column displays the orientations of the k001l direction of micas within the eight samples of Alpine Fault mylonite. These samples are all viewed looking down upon the foliation with north to the top. The interpreted shear direction for each of these samples is represented by a small arrow. The second, third and fourth columns contain the modelled seismic anisotropy of the same eight samples. It is noted that the LPO clustering defines the orientation of the anisotropy but their density does not have any obvious control upon the maximum P- or S1-velocities.
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Fig. 3. Plots of the k001l direction gathered from the same sample cut at different angles to the foliation. The plots have been rotated into a common reference frame where the ZX plane is foliation. They show the orientation of k001l in relation to the foliation and the number of points indexed with a mean angular difference of less ,1.
asymmetry) is parallel to X. This reference frame provides the easiest comparison to structures and microstructures that are conventionally viewed in the profile plane. Viewed in this way it is clear that mica LPOs form two distinct fabrics (Figs 4 & 5). In samples where there is no macroscopically visible S –C fabric the k100l and k010l directions occupy the C plane of a type II S –C fabric as described by Lister & Snoke (1984) (Fig. 4), forming a girdle at an angle of up to 20p (from the foliation (the S plane). In samples with a macroscopically penetrative extensional S –C fabric the majority of ,(100(. and ,(010(. directions lie within the S plane of an S–C –C0 ’ fabric as described by Berthe´ et al. (1979) (Fig. 5). The C0 -plane is faintly visible in these samples but does contain few mica crystals. Although one fabric is usually dominant, the second fabric is often faintly visible as in sample VCE3 where the fabric is dominated by mica occupying the S-plane but a weak C0 -plane fabric is visible due to the tails of the S-plane micas being aligned with the C0 -plane (Fig. 6a). Relating the mica LPOs to the microstructure in this way enhances their use as kinematic indicators. This can be carried out first examining the mica LPOs in a kinematic reference frame where the foliation is the XY plane and X is the lineation. This allows the fabric type to be identified as either S fabric, C fabric or a composite S –C –C0 fabric
(Fig. 6a). The second stage involves rotation of the sample (and thus the fabrics) into its original field orientation. Again the k100l and k010l directions form a girdle representing the great circle of an Splane, a C-plane, a C0 -plane or a composite. When the best-fit great circles to the S- and C-plane girdles identified are plotted, the shear direction will lie at 908 to their intersection upon the foliation plane (Fig. 6b) and within the foliation plane; both the plunge and plunge direction of the shear direction can therefore be determined (Fig. 6c). Shear directions from the Alpine Fault mylonites can be obtained in the field using various lineations and macroscopic fabrics. The interpretation of these shear directions is complicated by the fact they are strung out in a girdle along the fault (Norris & Cooper 2003; Toy et al. 2008) (Fig. 6d). Shear directions within the Alpine Fault Zone mylonites may also be obtained from quartz fabrics. Shear directions from the same samples as those analysed for mica LPOs were obtained from quartz c-axis LPOs using the methodology of Toy et al. (2008). Shear directions obtained using quartz and mica data are consistent from the same sample, with never more than a few degrees of discrepancy (Fig. 6c). Mica fabrics may be viewed in a third reference frame, the geographic reference frame, which allows the reconstruction of fault geometry based on initially oriented hand specimens. This is useful for the analysis of regional tectonics and
MICA FABRICS AND ANISOTROPY FROM EBSD
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Fig. 4. LPO of muscovite showing the k100l and k010l direction creating a girdle occupying the C-plane of a type II S–C fabric of Lister & Snoke (1984) where foliation parallel layers of micas allow layer-parallel extension and layer-perpendicular shortening through slip upon their (001) basal plane. Z is the pole to foliation and X is the principal kinematic lineation.
Fig. 5. LPO of muscovite showing the k100l and k010l directions creating a girdle predominantly occupying the S-plane of an extensional S–C– C0 fabric of Berthe´ et al. (1979). Note the girdles broaden or deflect close to the edge of the hemisphere. This is due to the indexing of the ‘tails’ of mica occupying the C0 -plane. Z is the pole to foliation and X is the principal kinematic lineation.
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has been used here to display the seismic properties of the fault zone based on LPOs.
Seismic anisotropy modelling The seismic properties of the sampled mylonites were modelled on data collected using EBSD. This technique is based on that set out by Mainprice et al. (2000). The FORTRAN program put forward by Mainprice et al. (2000) was used for this study. This program models seismic anisotropy by coupling LPO data with the density of the constituent minerals and their elastic properties. The elastic properties of the samples were derived from the Voigt –Reuss–Hill approximation, an averaging scheme by which anisotropic single-crystal elastic constants can be converted into isotropic polycrystalline elastic moduli (Chung & Beussem 1967). The technique for creating rock recipes followed that of Tatham et al. (2008) where the elastic matrices of each mineral are merged to create a bulk anisotropy for the sample with the modal percentage of each mineral defined by the user. The seismic properties of the material are modelled using the elastic properties of quartz (McSkimin et al. 1965), feldspar (Aleksandrov et al. 1974), muscovite (Vaughan & Guggenheim 1986) and biotite (Aleksandrov & Ryzhova 1961). Using EBSD it is possible to couple these properties with the LPO data and analyse the seismic properties of the bulk material in both a kinematic and geographic reference frame. The modelling process allows the input of various modal percentages of the different phases. This leads to the generation of models of quartz, mica and feldspar in various quantities. As in Lloyd et al. (2009), the seismic properties displayed here are the compressional wave velocity (Vp km/s) and anisotropy (AVp %), shear-wave splitting anisotropy (AVs%) and the polarization of the fastest shear wave (Vs1p). The pure mica model. To understand the effect of mica upon the anisotropy of mylonitic rocks it is
Fig. 6. (a) LPO of mica from the same sample as Fig. 4 with the S and C0 planes highlighted in red and green respectively. (b) Stereographic projection of the S (red) and C0 (green) planes in relation to the macroscopic foliation (black). The apparent shear direction (red box) lies upon the foliation plane 908 from its intersection with the S and C0 planes. (c) Stereographic projection of the shear directions calculated from mica fabrics (black) and quartz fabrics (red). Note the quartz-obtained shear directions lie within a few degrees of the mica-obtained shear directions from the same sample (within green loops in diagram). Shear directions from mica streaks and shear band fabrics from Toy et al. (2008).
MICA FABRICS AND ANISOTROPY FROM EBSD
first important to look at the seismic properties of the mica itself in relation to its LPOs. Upon examination, it is clear that the high-velocity directions are parallel to the basal planes as mapped out using EBSD. This strongly aligned LPO results in very high seismic anisotropy: over 50% in most cases when examining mica independent of other phases. The pure quartz model. Anisotropy of the quartz fabrics from the Alpine Fault Zone mylonites was modelled because of the strength and consistency of the fabrics and because quartz can account for .40% of the bulk rock. The seismic properties of Alpine Fault quartz were modelled in accordance with the c-axis profile across the fault zone as described in Toy et al. (2008). This profile is based on the idea that fabrics demonstrate characteristics of a decrease in strain and depth of formation with increasing distance from the fault trace. Closest to the fault, within the ultramylonites, a Y-maxima fabric is observed. Further from the fault within the mylonites the fabric is an asymmetric single girdle; at maximum distance from the fault, within the protomylonites, symmetric cross-girdles are found. The LPOs of these various quartz fabrics were coupled with the elastic properties of quartz, creating a seismic profile of the fault zone based on quartz alone (Fig. 7). These data are represented in a standard kinematic reference frame. The pattern of anisotropy shown by a Y-maxima quartz fabric is similar to that of a single quartz crystal as described by Tatham et al. (2008) with 12.7% P-wave anisotropy and 15.51% AVs. For a single girdle it is noted that the three zones of high Vp anisotropy begin to merge but, within the cross-girdle fabric, there is two merged zones of merged high Vp anisotropy. Here P-wave anisotropy has dropped to 8.2% with AVs at 10.32%. The AVs pattern also changes with fabric, with the Y-maxima fabric again producing the zones of higher anisotropy (orientated between the three zones of high Vp anisotropy), but the highest anisotropy is found around the circumference of the projection. For a single girdle the zones of higher anisotropy are divided by three girdles of lower anisotropy. The cross-girdle fabric produces four zones of higher AVs (up to 9.17%) anisotropy with the lower zones coincidental with the high Vp zones (up to 10.6% anisotropy). The various quartz fabrics also produce distinct patterns of maximum Vs1 which approximate to a mirror image of the maximum Vp patterns. Vs1 shows a decrease in maximum velocity moving away from the fault; for Vp however, the lowest maximum velocities are found within the mylonites rather than the protomylonites. The fastest velocities of both are found within Y-maxima quartz fabric ultramylonites. The maximum values of Vp, AVs and Vs1
41
all lie on or close to the foliation plane and regularly orthogonal (or close to orthogonal) in relation to the lineation. Multiphase model. The next phase of this study was to examine how the mica anisotropy modelled above influenced the bulk rock anisotropy, including all constituent minerals. We therefore obtained the LPO data for quartz, feldspar, biotite and muscovite. These various LPOs were combined to create a model for the bulk seismic anisotropy of the Alpine Fault Zone mylonites. The modal proportions of the constituent minerals were varied to examine the corresponding change in velocities and anisotropies. Two separate tests were carried out; the first test was to examine the effect of mica anisotropy on a pure quartz matrix. Six models were run containing 0 –50% mica (Fig. 8). For the second test, biotite and muscovite were considered to be of equal proportion (as were quartz and feldspar) to reflect the natural rock type of the AFZ. Six models containing 0– 50% mica were again run (Fig. 9). Our results show that once mica reaches a modal percentage of 20% or over, its seismic properties mask those of other minerals and thus dominate those of the bulk rock. Seismic anisotropy viewed in relation to fault geometry. By rotating samples back into their original field orientation and analysing their seismic properties in this orientation, it is possible to relate these properties to the regional tectonic context of the Alpine Fault Zone itself. The main foliation of the AFZ mylonites runs parallel to the active fault plane with only local variations due to folding or rotation. To relate the seismic data to the fault zone, all eight samples were combined to create one bulk sample to represent the mylonites of the fault zone with a composite S –C –C0 fabric (Fig. 10). This representative bulk sample contains 20% mica, 40% quartz and 40% feldspar. When modelled, the plane of high velocities and anisotropy lies parallel to the foliation and thus the fault. Velocities are not consistent throughout the girdle, with distinct clusters of higher velocities seen which approximate to the maximum velocities defined by the quartz fabric. There is also a difference between the directions of the highest velocities depending on which type of wave is examined. The orientations of the highest velocities can be related directly to structures found throughout the fault zone. S1-waves appear to have their highest velocities close to the dip and strike lines of the fault, whereas the velocities of P-waves are highest oblique to strike of the fault, at approximately 908 to the mean Alpine Fault mylonitic lineation within the foliation plane (as commonly observed within the quartz fabric) and parallel to the average S–C0 intersection
42
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Fig. 7. Seismic anisotropy for the three main quartz fabrics found within the Alpine Fault Zone where Z is pole to foliation and X is the principal kinematic lineation. Y-maxima fabrics are located nearest the fault, with cross-girdles being the most distal. This shows that, in relation to quartz, the highest anisotropy is found nearest the fault with AVs and Vs1 maximum velocities and % anisotropy falling with distance from the fault. For Vp velocities and anisotropies, the values are again highest nearest the fault but the lowest values are found within single-girdle quartz fabric.
lineation. In addition to this, the modelled P-wave and S1-wave velocities are very different, with the maximum S1-wave velocities only 60% of the maximum P-wave velocities. Note that the Alpine Fault mylonites blend eastwards into 10 km thickness of Alpine schists, with the foliation of the mylonites merging with that of the Alpine schist, and they have similar mica content. Christensen & Okaya (2007) measured 6% compressional wave anisotropy, maximum P-wave velocities of 5.8– 6.15 km/s and maximum S-wave velocities of 3.1–4.1 km/s within a zone just east of the Alpine
Fault, consistent with laboratory-measured velocities of the protolith schists. There is therefore likely to be a 10 km thick zone east of and parallel to the fault with high seismic velocity anisotropy (Fig. 10).
Discussion The results of this analysis show that within mylonitic zones, biotite and muscovite LPOs can be used as an additional tool in constraining the kinematics of
MICA FABRICS AND ANISOTROPY FROM EBSD
43
such zones. The study shows that micas have an LPO constrained by the regional kinematics. Two distinct fabrics have been identified using EBSD: C-plane micas and S-plane micas. In each sample the S, C0 and C planes may be visible; however, they will in general be preferentially orientated within one of these planes. Samples which exhibit micas marking out the C-Plane are interpreted as high-strain subplanar laminated mylonites with tightly packed layer-dependent C-planes exploiting the weakly bonded basal plane of micas. Those containing predominantly S-plane micas are interpreted as S–C –C0 mylonites with regularly spaced local high-strain shear zones. These fabrics can then in turn be used to infer shear directions. The shear directions, obtained using this method, are consistent with those measured directly in the field (where possible) and those inferred from quartz LPO fabrics. The evidence from seismic modelling based on LPOs in Figures 8 and 9 suggests that once a modal percentage of 20% mica is present it dominates the bulk seismic properties of the rock. It must however be noted that the role of feldspar LPOs within these samples may be important as, with the presence of 40% feldspar, 40% quartz and 20% mica, the minimum s1 velocities show a marked reduction. This effect manifests itself in the AVs where there is no change in anisotropy from 10–20%. Based on our results it is viable to suggest that, at modal percentages even as low as 10% mica in a quartzofeldspathic matrix, the mica LPOs have a significant influence on anisotropy. The 20% threshold identified, where the crystal orientation of micas becomes the controlling factor on anisotropy, is in agreement with the results presented in Lloyd et al. (2009) obtained from S–C tectonites from Nanga Parbat. It should be noted that values of maximum P- and S-wave velocities and percentage anisotropy modelled in Lloyd et al. (2009) are lower than those observed from Alpine Fault samples. This is due to the fact that Alpine Fault micas are concentrated close (,308) to the foliation whereas the Nanga Parbat micas are reported to be occupying multiple planes at slightly larger angles
Fig. 8. Effect of varying the mica content within a quartz matrix: note that by 20% mica, a faint girdle of higher velocities and anisotropy has formed upon the XY plane (foliation). Samples with 10% mica do show an increase in Vp and AVs anisotropy, but 10% mica content does not effect the anisotropy of Vs1. The maximum and minimum velocities increase/decrease at the same rate for Vp; however for Vs1 maximum velocity does not increase at the same rate as the decrease of minimum velocity.
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to the foliation. The Alpine Fault Zone mylonite sequence does have variable mica content, which varies from 15 to 20% of the bulk rock. It is therefore assumed that the LPO of biotite and muscovite within the Alpine Fault Zone has a very strong control over the seismic anisotropy of the ductile region of the fault zone at depths greater than 10 km where fracturing should be annealed. The observed mica LPO fabrics combined with the LPOs of the other phases can be combined to create a model of the seismic behaviour of the fault rock. However, it must be noted that this model defines these properties of the bulk rock based solely on crystal preferred orientations and does not account for interference from other variables such as pore pressure, fracture density, grain shape or phase boundary interference, which can alter the amount and orientation of anisotropy and velocities. Modelling of both explosion and onshore–offshore seismic data (Davey et al. 1998; Stern & McBride 1998) suggest a region of low velocity (5.9–6.2 km/s) overlying a lower crustal layer of higher velocity (7 km/s) (Fig. 11), again consistent with measurements described by Christensen & Okaya (2007). This low-velocity zone deep in the Pacific plate extends upwards through the Alpine Fault Zone. Upon entering the fault zone, the velocity decreases to c. 5.8 km/s at depth and ,5.5 km/s at depths less than 10 km. Very low velocities near the surface have been ascribed to open fracturing and the formation of fault gouge, which may reduce the velocity of the rock (fault gouge in particular may reduce velocities by up to 20%; Eberhart-Phillips et al. 1995). Deeper in the fault the fault gouge and open fracturing are absent so the reduction in velocity has been ascribed to high pore pressure (Stern & McBride 1998). At elevated pore pressures it is suggested that P-wave velocities may be reduced by up to 11% (Jones & Nur 1982). As stated before, the modelling of seismic velocities from LPOs carried out for this work does not account for fracturing and high fluid pressure, and yielded maximum P-wave velocities of 6.19 km/s, P-wave anisotropy of 11.8%, maximum S-wave velocity of 3.8 km/s and AVs of 10.35%.
Fig. 9. Effect on varying the mica content within a quartz matrix: note that by 20% mica, a girdle of higher velocities and anisotropy has formed upon the XY plane (foliation). 20% mica shows a small increase of max Vs1 but a distinct increase in the rate at which min Vs1 velocity decreases. This feature coincides with no change of the AVs % anisotropy between 10– 20% mica. However, at this volume of mica, the orientation of the higher velocities and % anisotropy is defined by the mica LPOs.
MICA FABRICS AND ANISOTROPY FROM EBSD
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Fig. 10. Bulk anisotropy and velocities for the eight samples studied using modal percentages typical of the Alpine Fault mylonites. This model is representative of mylonites .10 km deep where the effects of open fracturing are absent. Due to the strong mica fabrics, girdles of high anisotropy and high velocities are parallel to the fault line. The highest S1-wave velocities are located close to the strike and dip lines of the fault and the highest P-wave velocities are within the fault-parallel foliation and perpendicular to the mean mylonitic lineation.
These results are generally consistent with those obtained and described by Davey et al. (1998) and Stern & McBride (1998) (Fig. 11). These authors also identify a drop in velocity from c. 6.2 to c. 5.8 km/s at depths between 10 and 25 km, which they ascribe to enhanced pore fluid pressures. This reduction in velocity would require a drop in our LPO modelled P-wave velocity of 6.3%, which falls within the estimate of seismic velocity changes (,11% drop), due to the effect of high pore fluid pressures as suggested by Jones & Nur (1982). Davey et al. (1998) and Stern & McBride (1998) also observe and describe a reduction in
velocity in the upper 10 km of the fault zone. Measured velocities of 5.5 km/s would require a reduction in our LPO-modelled P-wave velocity of 11.15%. This 11.15% reduction is within the ,20% drop in velocities that Eberhart-Phillips et al. (1995) suggest may occur in the presence of fracturing and fault gouge. It therefore may be reasonable to infer that while at depths of 10 – 25 km velocities may be reduced by the presence of high pore fluid pressure, at shallow levels within the fault zone (0 –10 km) velocities decrease as a result of fracturing and the formation of fault gouge Again it should be emphasized that the LPO
Fig. 11. Velocity profile of the Alpine Fault Zone showing P-wave velocities (km/s) from Davey et al. (1998) based on data from Stern & McBride (1998).
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modelling does not take into account the effect of grain shape or phase changes interference. To refine more accurately the link between LPO anisotropy and the large-scale anisotropy of the fault zone, we suggest two areas of further research. Further work should include expanding upon the current dataset of mica LPOs from the Alpine Fault Zone (and comparable fault zones worldwide). Secondly, the brittle nature of the fault zone (in particular fracture density and orientation and their effect on seismic velocities) needs to be investigated in much more detail.
Conclusions For this study, LPOs of mica from samples collected along traverses in the Alpine Fault Zone mylonites were obtained using EBSD. Our results suggest the following. † EBSD analysis of mica is possible and is improved in highly foliated samples by working from foliation parallel sections. † Within mylonitic zones, mica produces three distinct LPOs that correspond to localized S-plane and C-plane fabrics and a regional composite S–C fabric. † Kinematic analysis of theses LPOs produces shear directions consistent with other methods of kinematic analysis, most particularly quartz LPOs from the same samples. † The LPOs of mica dominate the bulk anisotropy of the samples once there is c. 20% mica present, but also have a strong influence at lower percentages. † The Alpine Fault mylonites and neighbouring Alpine schists are a 10 km thick unit that has an intrinsic P-wave velocity anisotropy of 12% and S-wave velocity anisotropy of 10%. Fast velocities are contained in the foliation plane, dipping parallel to the fault.
References Aleksandrov, K. S. & Ryzhova, T. V. 1961. The elastic properties of rock-forming minerals, I: pyroxenes and amphiboles. Izvestiva Academy of Sciences USSR, Geophysics Series, 9, 1339– 1344 (in Russian). Aleksandrov, K. S., Alchikov, V. V., Belikov, B. P., Zaslavskii, B. I. & Krupnyi, A. I. 1974. Velocities of elastic waves in minerals at atmospheric pressure and increasing of precision of elastic constants by means of EVM. Izvestiya Academy of Sciences USSR, Geophysics Series, 10, 15–24 (in Russian). Beavan, J., Moore, M. et al. 1999. Crustal deformation during 1994– 1998 due to oblique continental collision in the central Southern Alps, New Zealand, and implications for seismic potential of the Alpine
fault. Journal of Geophysical Research, 104, 25 233–25 255. Berthe´, D., Choukroune, P. & Jegouzo, P. 1979. Orthogneiss, mylonite and non-coaxial deformation of granites: the example of the South Amorican Shear Zone. Journal of Structural Geology, 1, 31–42. Byerlee, J. D. 1978. Friction of rocks. Pure and Applied Geophysics, 116, 615– 626. Cande, S. C. & Stock, J. 2004. Pacific-AntarcticaAustralia motion and the formation of the Macquarie Plate. Geophysical Journal International, 157, 399–414. Christensen, N. I. & Okaya, D. 2007. Compressional and shear wave velocities in South Island NZ rocks and their application to the interpretation of seismological models of the New Zealand crust. In: Okaya, D., Stern, T. & Davey, F. (eds) A Continental Plate Boundary: Tectonics of South Island. New Zealand, AGU Geophysical Monograph, Washington, 175, 125–158. Chung, D. H. & Beussem, W. R. 1967. The Voigt-ReussHill approximation and elastic moduli of polycrystalline MgO, CaF2, b-ZnS, ZnSe, and CdTe. Journal of Applied Physics, 38, 2535– 2540. Davey, F. J., Henyey, T. et al. 1998. Preliminary results from a geophysical study across a modern continent– continent collisional plate boundary—the Southern Alps, New Zealand. Tectonophysics, 288, 221– 235. Eberhart-Phillips, D., Stanley, W. D., Rodriguez, B. D. & Lutter, W. J. 1995. Surface seismic and electrical methods to detect fluids related to faulting. Journal of Geophysical Research, 100, 12 919–12 936. Jones, T. D. & Nur, A. 1982. Seismic velocity and anisotropy in mylonites and the reflectivity of deep crustal fault zones. Geology, 10, 260–263. Kleffmann, S., Davey, F., Melhuish, A., Okaya, D., Stern, T. & SIGHT TEAM 1998. Crustal structure in the central South Island, New Zealand, from the Lake Pukaki seismic experiment. New Zealand Journal of Geology and Geophysics, 41, 39– 49. Kronenberg, A. K., Kirby, S. H. & Pinkston, J. 1990. Basal slip and mechanical anisotropy of biotite. Journal of Geophysical Research, 95, 19 257– 19 278. Lister, G. S. & Snoke, A. W. 1984. S-C mylonites. Journal of Structural Geology, 6, 617– 638. Little, T. A., Holcombe, R. J. & Ilg, B. R. 2002. Kinematics of oblique collision and ramping inferred from microstructures and strain in middle crustal rocks, central Southern Alps, New Zealand. Journal of Structural Geology, 24, 219–239. Lloyd, G. E., Butler, R. W. H., Casey, M. & Mainprice, D. 2009. Mica, deformation fabrics and the seismic properties of the continental crust. Earth and Planetary Science Letters, 288, 320 –328. Mainprice, D. 1990. An efficient Fortran program to calculate seismic anisotropy from the lattice preferred orientation of minerals. Computers and Geosciences, 16, 385– 393. Mainprice, D. 2007. Seismic anisotropy of the deep Earth from a mineral and rock physics perspective. In: Schubert, G. (ed.) Treatise in Geophysics, Elsevier, Oxford, 2, 437– 492. Mainprice, D., Barruol, G. & Ben Ismail, W. 2000. The Seismic Anisotropy of the Earth’s Mantle: from Single
MICA FABRICS AND ANISOTROPY FROM EBSD Crystal to Polycrystal. In: Karato, S. I., Forte, A. M., Lierberman, R. C., Masters, G. & Stixrude, L. (eds) Earth’s Deep Interior; Mineral Physics and Tomography from the Atomic to the Global Scale. AGU, Washington, Geophysics Monographs, 117, 237– 264. Mares, V. M. & Kronenberg, A. K. 1993. Experimental deformation of muscovite. Journal of Structural Geology, 15, 1061–1075. McSkimin, H. J., Andreatch, P. Jr. & Thurston, R. N. 1965. Elastic moduli of quartz versus hydrostatic pressure at 258C and 2195.88C. Journal of Applied Physics, 36, 1624–1632. Norris, R. J. & Cooper, A. F. 2003. Very high strains recorded in mylonites along the Alpine Fault, New Zealand: implications for the deep structure of plate boundary faults. Journal of Structural Geology, 20, 2141–2157. Prior, D. J., Mariani, E. & Wheeler, J. 2009. EBSD in the Earth Sciences: applications, common practice and challenges. In: Schwatz, A. J., Kumar, M., Adams, B. L. & Field, D. P. (eds) Electron Backscatter Diffraction in Materials Science. 2nd edn. Springer, Berlin, 345– 360. Sibson, R. H., White, S. H. & Atkinson, B. K. 1979. Fault rock distribution and structure within the Alpine Fault zone: a preliminary account. Royal Society of New Zealand Bulletin, 18, 55–65. Stern, T. A. & McBride, J. 1998. Seismic exploration of continental strike–slip zones. Tectonophysics, 286, 63– 78. Sutherland, R. 1996. Transpressional development of the Australia-Pacific boundary through southern South Island, New Zealand: constraints form
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Miocene–Pliocene sediments, Waiho-1 borehole, South Westland. New Zealand Journal of Geology and Geophysics, 39, 251–264. Sutherland, R., Eberhart-Phillips, D. et al. 2007. Do great earthquakes occur on the Alpine fault in central South Island, New Zealand? In: Okaya, D., Stern, T. & Davey, F. (eds) A Continental Plate Boundary: Tectonics at South Island, New Zealand. American Geophysical Union, Washington, Geophysical Monograph Series, 175, 235–251. Tatham, D. J., Lloyd, G. E., Butler, R. W. H. & Casey, M. 2008. Amphibole and lower crustal seismic properties. Earth and Planetary Science Letters, 267, 118– 128. Toy, V. G. 2008. Rheology of the Alpine Fault Mylonite Zone: deformation processes at and below the base of the seismogenic zone in a major plate boundary structure. PhD thesis, University of Otago. Toy, V. G., Prior, D. J. & Norris, R. J. 2008. Quartz fabrics in the Alpine Fault mylonites: influence of preexisting preferred orientations on fabric development during progressive uplift. Journal of Structural Geology, 30, 602–621. Vaughan, M. T. & Guggenheim, S. 1986. Elasticity of muscovite and its relationship to crystal structure. Journal of Geophysical Research, 91, 4657– 4664. Warr, L. N. & van der Pluijm, B. A. 2005. Crystal fractionation in the friction melts of seismic faults (Alpine Fault, New Zealand). Tectonophysics, 402, 111– 124. Warr, L. N., van der Pluijm, B. A. & Tourscher, S. 2007. The age and depth of exhumed friction melts along the Alpine fault, New Zealand. Geology, 35, 603– 606.
From crystal to crustal: petrofabric-derived seismic modelling of regional tectonics G. E. LLOYD1*, J. M. HALLIDAY1, R. W. H. BUTLER2, M. CASEY1†, J.-M. KENDALL3, J. WOOKEY3 & D. MAINPRICE4 1
Institute of Geophysics and Tectonics, School of Earth & Environment, University of Leeds, Leeds LS2 9JT, UK
2
Geology and Petroleum Geology, School of Geosciences, University of Aberdeen, Meston Building, King’s College, Aberdeen AB24 3UE, UK 3
Department of Earth Sciences, University of Bristol, Bristol BS8 1RJ, UK 4
Ge´osciences Montpellier, CNRS & Universite´ Montpellier 2, 34095 Montpellier, France *Corresponding author (e-mail:
[email protected])
Abstract: The Nanga Parbat Massif (NPM), Pakistan Himalaya, is an exhumed tract of Indian continental crust and represents an area of active crustal thickening and exhumation. While the most effective way to study the NPM at depth is through seismic imaging, interpretation depends upon knowledge of the seismic properties of the rocks. Gneissic, ‘mylonitic’ and cataclastic rocks emplaced at the surface were sampled as proxies for lithologies and fabrics currently accommodating deformation at depth. Mineral crystallographic preferred orientations (CPO) were measured via scanning electron microscope (SEM)/electron backscatter diffraction (EBSD), from which three-dimensional (3D) elastic constants, seismic velocities and anisotropies were predicted. Micas make the main contribution to sample anisotropy. Background gneisses have highest anisotropy (up to 10.4% shear-wave splitting, AVs) compared with samples exhibiting localized deformations (e.g. ‘mylonite’, 4.7% AVs; cataclasite, 1% AVs). Thus, mylonitic shear zones may be characterized by regions of low anisotropy compared to their wall rocks. CPOderived sample elastic constants were used to construct seismic models of NPM tectonics, through which P-, S- and converted waves were ray-traced. Foliation orientation has dramatic effects on these waves. The seismic models suggest dominantly pure-shear tectonics for the NPM involving horizontal compression and vertical stretching, modified by localized ductile and brittle (‘simple’) shear deformations.
Knowledge of tectonic and geodynamic processes occurring within the continental crust, particularly at deeper levels, is traditionally obtained via detailed geological analysis in both the field and subsequently in the laboratory of tracts of former middle and lower crust now exposed at the surface. Although relevant to past processes, this approach provides only guides to what might be happening today in the modern middle and lower continental crust. In situ techniques involving remote sensing of the current state of the middle and lower continental crust have therefore recently become increasingly popular and useful. The potentially most useful of these techniques is provided by seismic waves, as both natural and controlled source seismology can be used to image tectonic structures occurring at depth (e.g. Burlini et al. 1998; Khananehdari et al. 1998). However, such approaches demand that the control †
geology exerts on the seismic properties is fully understood. For example, as seismic resolution depends on the magnitude and spatial extent of seismic anisotropy exhibited by the geology, there are limits to the size of structures that can be observed. Shear zones, which are typically narrower than the seismic wavelength, are consequently not normally visible using standard seismic survey methods such as multi-azimuth wide-angle controlledsource seismic surveys. However, it has been suggested (e.g. Guest et al. 1993; Lloyd & Kendall 2005) that vertical resolution can be improved further using knowledge of reflection coefficients and mode conversions at (lithological) interface boundaries, which demands detailed knowledge of the geological control on the seismology. Large-scale continental structures are typically exhumed or emplaced on localized zones of
Deceased 2008.
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 49– 78. DOI: 10.1144/SP360.4 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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Fig. 1.
G. E. LLOYD ET AL.
FROM CRYSTAL TO CRUSTAL
high-strain deformation that may persist to great depths, perhaps through the entire crust, and form both simple and complex linked three-dimensional arrays (e.g. Sibson 1977; Ramsay 1980; Coward 1994). It has long been known that the deformation mechanisms and processes responsible for emplacement and/or exhumation are frequently preserved in the grain-scale microstructures observed in surface outcrops (e.g. Nicolas & Poirier 1976; Passchier & Trouw 1996). More recently, it has been realized that these microstructures can also be used to estimate a variety of petrophysical properties and, in particular, their seismic characteristics (e.g. Babuska & Cara 1991; Kocks et al. 1998). Ductile shear zones in lower continental crust contain structures at the scale of the crystal lattice, grains and lithological layering and larger. The seismic anisotropy of such zones is a function of all these structures. Characterization of this seismic anisotropy necessitates information on the various scales and a method of integrating the effects to obtain the bulk properties (e.g. Christensen 1984; Ben Ismail & Mainprice 1998). In contrast, concentrated shear in upper levels of the continental crust often occurs along cataclastic zones, the seismic properties of which are determined by the fracture density or porosity of the fault rocks (e.g. Crampin 1981). This contribution concentrates on the petrofabric or crystal lattice preferred orientation (CPO) determination of seismic properties (e.g. Kern 1982; Siegesmund et al. 1989; Barruol & Mainprice 1993; Burlini & Kern 1994; Barruol & Kern 1996). It has long been recognized that CPO is a major cause of seismic anisotropy (Hess 1964; Christensen 1971). The question posed is therefore whether or not it can be expected to be able to distinguish between different structural, tectonic and geodynamic configurations (e.g. shear and fault zones) at depth in crystalline rocks by measurement of their seismic properties. The Nanga Parbat Massif (NPM), Pakistan Himalaya (Fig. 1), is considered an ideal example of the possibility of using seismic observations to determine whether its emplacement was/is accommodated solely on localized fault and shear zones via essentially simple-shear deformation (e.g. Burg 1999), or whether bulk pure-shear deformation within the massif as a whole played/ plays a significant role (e.g. Butler et al. 2002). CPO data from characteristic rock samples, measured via scanning electron microscope (SEM) electron backscattered diffraction (EBSD), are used to determine the overall elastic and seismic properties of rocks from the NPM. These can then
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be used to populate various seismic models to investigate the regional geodynamic setting.
Regional geology and sample descriptions Regional geology The NPM (Fig. 1a) lies within the Indo –Asian continental collision zone (Fig. 1b), which forms the major current example of active continental collision and compressive intra-continental tectonics. It represents an exhumed tract of Indian continental crust and, as an area of active crustal thickening and exhumation, offers insights into how localized thrust faults exposed at the Earth’s surface couple with more distributed strain at depth. The NPM exhibits a remarkably high topography (.8 km) and forms an elongate, upright, north–south antiform with c. 40 km wavelength that probably involves the whole crust (Zeitler et al. 2001; Butler et al. 2002). It comprises exhumed young (,2 Ma) Indian continental granitic and metasedimentary gneisses and granitic orthogneisses basement and calc-silicate and marble metasedimentary cover. However, there is a distinct geochemical, metamorphic and tectonic regime relative to the surrounding terrain, which suggests an anomalous structure beneath the massif (George et al. 1993; Zeitler et al. 1993, 2001). The massif can be defined by the shape of the contact, known as the Main Mantle Thrust (MMT), between Indian continental gneisses with the overlying Kohistan arc rocks (Fig. 1c, d). The entire complex was subducted and metamorphosed during collision to amphibolite grade typified by a quartz-muscovitebiotite-plagioclase assemblage (Zeitler et al. 2001). Peak metamorphic conditions towards the west of the massif, in the vicinity of the sampling region (see below), are thought to have been c. 7– 11 kbars and 550–700 8C (i.e. 25– 40 km depth), in contrast to c. 7– 14 kbars and 675– 800 8C (i.e. 40–50 km depth) on the eastern margin of the massif (Poage et al. 2000). The NPM has been, and continues to be, exhumed on a series of ductile shear and brittle fault zones, resulting in at least 22 km of erosion during emplacement (Butler et al. 2000; see Fig. 1c, d). It is assumed that the overall tectonic regime has not changed significantly over the last few million years (Zeitler et al. 2001), with geochronological data suggesting that exhumation is exposing rocks that are representative of the deformation kinematics currently operating at depth
Fig. 1. The Nanga Parbat Massif, Pakistan Himalaya. (a) General view of the Nanga Parbat Massif. (b) General location, with the Nanga Parbat Massif region indicated by the white box. (c) Main geological features of the Nanga Parbat Massif region (after Butler et al. 2002), with the location of the study area indicated. (d) Schematic cross-section of the Nanga Parbat Massif (after Butler et al. 2002); note location of the study area.
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Fig. 2.
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(Zeitler et al. 1993). The central part of the massif is bound by two primary shear zones (the Liachar and Rupal), although current exhumation is achieved on a cataclastic fault zone termed either the Liachar Thrust (Butler & Prior 1988) or the Raikhot Fault (Zeitler et al. 2001) which emplaces gneisses onto Quaternary gravels (e.g. Fig. 2a– c). The 2 km wide Liachar Shear Zone occurs in the hanging wall of this structure (Fig. 1d), which led Butler & Prior (1988) to suggest that the surface cataclastic faults pass downwards into a ductile simple-shear zone. However, an alternative model (Butler et al. 2002) argues for more distributed deformation at depth, partitioned between broadly simple shear in the ductile shear and cataclastic fault zones and broadly pure shear in the rest of the hanging wall. Various models currently exist to explain the kinematics of the exhumation and emplacement of the NPM. Nevertheless, as an area of active crustal thickening and exhumation, the NPM offers insight into how localized thrust faults at the Earth’s surface couples with more distributed strain at depth (Butler et al. 2002). It has long been recognized that seismogenic faulting often passes into aseismic creep with depth (e.g. Sibson 1977). However, the kinematics and distribution of strain at depth remains controversial (Butler et al. 2002). While simple-shear deformation is localized into relatively narrow tracts of non-coaxial strain, anastomosing discrete tracts could produce a thick zone of macroscopically distributed strain (e.g. Burg 1999). Alternatively, distributed strain at depth could be accommodated by pure-shear subvertical stretching (e.g. Butler et al. 2002). The latter model is one of volume conservation: layers parallel to the principle axes of strain do not rotate and experience no shear strain, while layers perpendicular to compression are shortened and thickened (e.g. Twiss & Moores 1992). Butler et al. (2002) suggest that the bulk deformation within the crust at Nanga Parbat may be described as heterogeneous vertical stretching; it is only when strain gradients become pronounced that simple shear dominates. If this view is correct, a combination of pure and simple shear is responsible for the exhumation of the NPM. Evidence for the mechanism(s) of exhumation of the NPM should be preserved within the rock record. For example, when a rock is deformed, one of the fundamental changes observed is a re-orientation of the crystal components, usually into a direction defined by the stress field, to form
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CPO. It is well known that natural deformation mechanisms in rocks are responsible for inducing anisotropy of the inherent petrophysical properties (Babuska & Cara 1991; Wendt et al. 2003). As the majority of rock-forming minerals are elastically anisotropic, seismic anisotropy (i.e. variation of seismic velocities with direction) is an intrinsic property of most minerals and rocks. Seismic anisotropy should therefore be present at all scales in the Earth and hence can be used to infer information on structures induced by large-scale geodynamic processes at depth (e.g. Davis et al. 1997; Shapiro et al. 2004). This contribution uses CPO measured from rocks sampled from appropriate parts of the NPM to estimate their seismic properties and, in particular, their anisotropy in order to predict the most likely mechanism(s) of exhumation.
Sample descriptions The most recent phase of exhumation of the NPM has occurred close to the surface on the Liachar Thrust/Raikhot Fault (Butler & Prior 1988; Zeitler et al. 2001). In the hanging wall of this structure there is a high-strain zone known as the Liachar Shear Zone (Butler & Prior 1988), which emplaces augen gneisses of the Indian continental crust onto the Kohistan Arc rocks (Fig. 2a–c). Although deformation of these gneisses (as indicated by varying degrees of lithological layering and grainsize reduction) effectively increases towards the Liachar Thrust, such deformation is not related to the thrusting. Rather, it is related to the formation of the ductile Liachar Shear Zone that developed previously at deeper levels, which is now being exhumed on the brittle Liachar Thrust. A suite of rocks has been collected from the Liachar Shear Zone in the Tato Ridge area above the Liachar Thrust (Figs 1d & 2a– c). Two specific samples (B1 and B2) from this suite are considered in this contribution (Fig. 3). Both are augen orthogneisses and contain the same quartz + plagioclase + orthoclase + biotite + muscovite mineralogy, although in varying proportions, while sample B2 also contains minor garnet. Sample B1 (Fig. 3a) contains relict clasts of the original igneous protolith, with a large range in clast and grain sizes. The clasts are mainly of plagioclase and orthoclase (although there are also regions of biotite + muscovite), while quartz has behaved in a ductile manner. The segregation of each of these phases, particularly the micas,
Fig. 2. Tato– Raikhot–Bulder study region, Nanga Parbat Massif. (a) Geological map; note the specific locations of samples B1 and B2 along the sampling traverse. (b) General view of the Tato Road section from where the samples were collected; note steeply dipping foliation. (c) Schematic cross-section of the Tato Road region; note general locations of samples B1 and B2.
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Fig. 3. SEM back-scattered electron (BSE) atomic number (Z) contrast images of the samples of Nanga Parbat orthogneisses used in this study, illustrating the textural variations between the samples. Minerals indicated: Q, quartz; K, orthoclase; P, plagioclase; B, biotite; M, white mica; G, garnet; and C, calcite. (a) Sample B1, representing the ‘protolith’: note relatively coarser grain size and metamorphic layering or banding. (b) Sample B2m, developed from the ‘protolith’: note relatively finer scale shear-zone foliation defined by all minerals except porphyroblastic garnet. (c) Sample B2c, developed from the ‘protolith’ and/or the shear zone: cataclasite; note variation in size and composition of individual fragments, set in an orthoclase cement seal cut by a late calcite vein.
imparts a crude lithological layering. Sample B2 comprises two distinctive subsamples. Half of the sample (B2m) is considered to represent the ultimate development of lithological layering and grain-size reduction due to ductile shearing (Fig. 3b). All minerals appear extended in the foliation except for garnet, which occurs as porphyroblasts. In the other half of the sample (B2c), the lithological layering due to ductile shearing has been completely destroyed by the effects of the brittle deformation associated with the Liachar Trust (Fig. 3c). However, a weak ‘foliation’ due to variations in fragment sizes is apparent within a cement comprising very-fine-grained orthoclase. The kinematic indicators associated with the microstructures of samples B1, B2m and B2c, whether
ductile or brittle, suggest a top-to-the-northwest sense of movement, consistent with the large-scale geometry of the NPM (e.g. Figs 1d & 2b; see Butler et al. 2002 for details). The varying degrees of lithological layering, grain-size reduction of quartz and feldspars by ductile processes and other microstructures in the augen gneiss samples (e.g. Fig. 3a, b) attest to different degrees of deformation under (at least) lower amphibolite facies temperatures. The samples are therefore considered to indicate an increasing strain gradient within the Liachar Shear Zone, partly responsible for the exhumation and emplacement of the NPM at depth (e.g. Figs 1d & 2c). As sample B1 was collected from the periphery of the Liachar Shear Zone, it is considered to represent
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the ‘background’ deformation state of the Indian continental gneisses protolith. Sample B2m represents the maximum expression of the (increasing) strain gradient associated with the shear zones involved in the exhumation of the NPM. In contrast, differences in orthoclase composition between relict gneissic fragments and cement within sample B2c suggest that faulting was either associated with and/or succeeded by fluid flow that precipitated orthoclase in void spaces created by fracturing, resulting in a cemented cataclasite. Such processes are typical of low temperatures and hence relatively near-surface deformation. Cataclasized sample B2c is therefore considered to represent the brittle fault zones responsible for the exhumation/emplacement of Nanga Parbat at shallower crustal levels, a manifestation of shallow-level strain partitioning (e.g. Figs 1d & 2c). This sample is obviously not typical of the fault zones, including those currently active (e.g. the Liachar Thrust), during their movement episodes when fracturing dominates the microstructure and hence petrophysical properties. Rather, the orthoclase cemented cataclasite is typical of the fossilized fault structures as they now occur. However, the volume and distribution of orthoclase cement provides an estimate of fault zone porosity and hence fracture density during such movement episodes.
Methodology Factors that determine the seismic properties of rocks include (e.g. Mainprice & Nicolas 1989; Babuska & Cara 1991): mineralogy and lithology, layering, grain shape, crack and fracture arrays and CPO. The latter is particularly significant because single-crystal seismic velocities of minerals vary with crystal symmetry and direction due to variations in the elastic properties of the crystals. Consequently, deformation processes such as dislocation creep that lead to CPO of anisotropic minerals in deformed rocks must also impact upon the elastic stiffness matrix and hence seismic properties of the rock aggregate (e.g. Lloyd et al. 2011). Crack and fracture arrays (including porosity) are only likely to be significant, and therefore impact on seismic properties, at shallow depths (e.g. Crampin 1981). As samples B1 and B2m are from either the background gneisses or the shear zones developed at depth, it is therefore expected that their seismic properties (and hence those of most of the NPM) vary mainly with CPO. However, the cataclastic fault zone (i.e. sample B2c) developed at shallow depths, where it was subsequently cemented by orthoclase feldspar. The feldspar has acted to seal the fault zone and thereby
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removed the influence of fractures on the seismic properties. The methodology used to estimate seismic properties from CPO follows standard procedure (e.g. Mainprice 1990; Barruol & Mainprice 1993; Mainprice & Humbert 1994; Lloyd & Kendall 2005). The crystal orientation of each mineral grain in a sample is measured automatically via SEM/EBSD (e.g. Prior et al. 1999) to determine the individual mineral CPO. The single-crystal elastic properties for each mineral CPO are rotated into the sample reference frame, such that the elastic parameters of the polycrystal are derived by integration over all possible orientations in the 3D orientation distribution function (e.g. Bunge 1982; Ben Ismail & Mainprice 1998). Due to stress/strain compatibility assumptions, three different averaging schemes are possible (Crosson & Lin 1971). The constant strain or Voigt (V) average (Voigt 1928) and constant stress or Reuss (R) average (Reuss 1929) provide upper and lower bounds. However, the arithmetic mean or Hill (H) average (Hill 1952) of V and R is often taken as the best estimate of the elastic parameters as it is observed to give results close to experimental values (e.g. Bunge et al. 2000). The individual mineral CPO measurements are then combined in their correct mineral proportions, from which the whole-rock seismic properties are estimated via the Christoffel equation (e.g. Babuska & Cara 1991; Kendall 2000). The seismic properties of interest include the compressional wave (Vp) and shear waves (Vs1, Vs2) phase velocity distributions in threedimensions, as well as the degree of shear-wave splitting for a given direction. The latter is represented as either the absolute difference in shearwave velocities (i.e. dVs ¼ Vs1 – Vs2) or the shear-wave anisotropy, which is conventionally defined (e.g. Mainprice & Silver 1993): AVs% = 100(Vs1 − Vs2 )/[(Vs1 + Vs2 )0.5] The absolute anisotropy of Vp, Vs1 and Vs2 can also be calculated by substituting their appropriate maximum and minimum values for Vs1 and Vs2 into this equation. Shear-wave splitting analyses of real data estimate the degree of splitting and orientation of the fast shear wave for a given ray direction (e.g. Kendall 2000). Thus, the polarizations (or birefringence) of the fast shear waves (Vs1P) are also calculated from the whole-rock seismic properties via the Christoffel equation (Mainprice 1990). In practice, all constituent minerals contribute to the overall seismic properties of a rock aggregate depending on their single-crystal elastic parameters, volume fraction and CPO. Thus, the aggregate
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seismic properties may be different from any of the individual mineral properties in rocks where one mineral does not dominate in any particular manner (i.e. modally, elastically and/or strength of CPO). Indeed, there must be a significant degree of crystal alignment (most typically produced by dislocation creep due to tectonic stresses in deformed rocks) of at least one major mineral phase to produce seismic anisotropy, as randomly oriented crystals must generate an isotropic bulk rock. Where one mineral does dominate, the aggregate seismic properties are likely to reflect the characteristics of that mineral. For example, micas are one of the most anisotropic of the common rock-forming minerals, exhibiting a vertical transverse isotropy (VTI) parallel to the crystal c axis (i.e. normal to the mineral cleavage plane). Depending on the modal content and/or CPO strength, micas are therefore likely to contribute most to the seismic anisotropy of polymineralic rock aggregates in which they occur (e.g. Lloyd et al. 2009, 2011).
CPO-derived seismic properties CPO distributions CPO data are conventionally represented in the tectonic or kinematic reference frame (XYZ, where X ≥ Y ≥ Z ), usually from samples cut parallel to the XZ plane. However, seismic data are viewed in the natural or geographical reference frame (north–south/east –west). The SEM-EBSD CPO data measured from XZ sections of samples B1 and B2 have therefore been rotated into the geographical reference frame by means of two (different) rotations per sample as follows: B1 is firstly rotated by 222.58 about an axis plunging 908 towards 0008 (i.e. vertical) and secondly by 258 about 08/0678; and B2 is firstly rotated by 2408 about 908/0008 and secondly by 508 about 08/0908. The biotite and, to a lesser extent, muscovite CPO for samples B1 and B2m clearly indicate the ductile foliations present in these rocks (Fig. 4a, b).
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In the former, the foliation is steeply NNE-dipping and strikes ENE –WSW while in the latter it is moderately south-dipping and east–west striking. The non-mica phases exhibit less distinct CPO patterns, possibly reflecting local constraints on CPO development imposed by neighbouring grains. The CPOs for sample B2c are different and do not indicate a foliation (Fig. 4c). Rather, they show tendencies for small/great circle distributions centred on either maxima or minima that plunge consistently moderately towards the southwest.
Seismic properties The seismic properties mimic the CPO distributions, particularly for biotite, in terms of the foliation in samples B1 and B2m (Fig. 4a, b). However, the steep c. northeast-plunging stretching lineation characteristic of the ‘background’ gneisses is not reflected in the seismic properties of sample B1 and the moderately c. southeast-plunging stretching lineation characteristic of the ductile shear zones is only indicated by the maximum in AVs for sample B2m. These results support the suggestion (e.g. Mahan 2006; Lloyd et al. 2009, 2011) that, when present, micas (in this case biotite) control the seismic properties of the rock, particularly anisotropy. Furthermore, because micas indicate mainly foliation and cannot recognize lineation, it is not possible to differentiate between flattening and plane-strain (e.g. simple-shear) type deformations. Thus, the seismic properties for samples B1 and B2m could have arisen from similar or very different deformations. Sample B2c is therefore interesting because its seismic properties (Fig. 3c) are indicative of a non-foliation-forming deformation, such as expected for constriction. The actual magnitudes of the seismic properties for the three samples are also interesting (Fig. 4). In terms of Vp their velocities are generally similar, reflecting their similar compositions. However, in detail, the ranges in Vp do vary between samples; B1 shows the absolute minimum (5.7 km/s) and
Fig. 4. SEM/EBSD-measured CPO and CPO-derived seismic properties plotted in the geographical reference frame (north– south/east–west) using the Mainprice (2003) suite of programs. All CPO lower hemisphere (except plagioclase, which is both upper and lower hemisphere), contoured in multiples of the uniform distribution (m.u.d.) as indicated (dotted line, 0.5 m.u.d.) with maximum and minimum values indicated by black squares and open circles, respectively. For quartz, the a(11–20), m(10–10), c(0001), r (1 –101) and z(01–11) poles are plotted. For all other minerals the respective a(100), b(010) and c(001) poles are plotted. The seismic properties plotted are: the compressional (Vp) and shear (Vs1, Vs2) waves velocities; the azimuthal-dependent shear-wave anisotropy (either AVs or dVs); and the polarization direction (birefringence) of the fast shear wave (Vs1P), indicated by the black/white lines for the wave propagation direction as indicated by the plunge/azimuth position on the stereogram. (a) Sample B1: note the steep north-dipping ‘background’ foliation, principally defined by biotite which is mimicked by the seismic properties. (b) Sample B2m: note the moderate south-dipping shear-zone foliation, principally defined by biotite which is mimicked by the seismic properties. (c) Sample B2c: note the generally weak fabrics, although a definite tendency for each mineral to show small/great circle dispersions about a moderated southwest-plunging axis, which are mimicked by the seismic properties.
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B2m the absolute maximum (6.3 km/s). These variations in Vp impact on the AVp values per sample, which is a maximum for B1 (8.8%) and a minimum for B2c (1.3%), with B2m intermediate (4.4%). Equivalent behaviour is shown by AVs with sample B1 exhibiting the shear-wave splitting of 10.43% (equivalent to a dVs of 0.35 km/s), while sample B2m has an AVs of 4.69% (dVs ¼ 0.16 km/s) and sample B2c has an AVs of 1.03% (dVs ¼ 0.04 km/s). Thus, the ductile Liachar simple-shear zone exhibits significantly lower seismic anisotropy than the ‘background’ gneisses of the NPM. Similar behaviour, whereby a shear zone is less anisotropic than its wall rocks, has been observed by Michibayashi & Mainprice (2004) and Michibayashi et al. (2006), being attributed in both cases to the presence and significance of pre-existing mechanical anisotropy on shear-zone development. If correct, this suggests that the ‘background’ gneisses of the NPM exhibit a significant deformation fabric in terms of the seismic properties. Although a brittle fault zone, sample B2c is almost anisotropic. This is presumably due to the dispersion of the various CPO along small and great circles about the moderately southwest-plunging ‘rotation’ axis and the orthoclase cement seal, which has acted to remove any potential crackinduced anisotropy. Finally in this section, in terms of polarization the fast shear waves (i.e. Vs1P) for samples B1 and B2m are polarized parallel to the foliation when they propagate within the foliation (Fig. 4a, b); the behaviour of vertically propagating (e.g. teleseismic) waves (Vs1Pv) is therefore different in both samples. For sample B1, which is characterized by a steeply dipping ENE –WSE foliation, Vs1Pv is polarized ENE –WSW and has an AVs value close to the maximum. In contrast, for sample B2m, which is characterized by a moderately dipping east– west foliation, Vs1Pv is polarized northeast– southwest and has an AVs value of less than half the maximum for this sample. Sample B2c exhibits similar behaviour to sample B2m (Fig. 4c).
CPO-derived seismic modelling The determination of seismic properties via CPO provides elastic stiffness matrices for the rock aggregates involved. Such matrices can be used to populate seismic models with realistic rock properties. This approach is used to construct several seismic models populated with the CPO-derived elastic properties of the samples described previously in order to investigate the impact of these properties (sic rocks) on the seismic characteristics of the NPM. In these models, sample B1 is considered to represent the background deformation
state of the Indian continental gneisses protolith away from the localized high-strain zones, whether ductile or brittle. In other words, it is considered to represent the rock fabric that comprises the bulk of the NPM. In contrast, samples B2m and B2c are considered to represent the maximum expressions of the localized ductile shear and brittle fault zones, respectively, responsible potentially for the (near-surface) exhumation of the NPM. The seismic models erected investigated the following parameters: (1) effect of mineralogy, foliation orientation and deformation (partitioning); (2) wave propagation; (3) seismic reflection coefficients; (4) controlled source surveys; and (5) deformation determination from reflection coefficients and mode conversions. In the seismic models the aggregate elastic constants derived for each sample via CPO analysis are converted into a density-normalized format. Only two elastic constants are required for isotropic media but at least nine elastic constants must be specified (e.g. Babuska & Cara 1991) for any other form of anisotropy. A computer program (Atrak, Guest & Kendall 1993) is then used to construct the models based on the aggregate elastic constants, assuming the geometry of the NPM. This program is capable of: (1) deriving slowness surfaces and reflection coefficients between adjacent lithologies; (2) tracking compressional and shearwave ray-paths in the geological situations envisaged where the effects of seismic anisotropy may be important; (3) providing estimates of the effect of crustal anisotropy on either passive or teleseismic data; and (4) generating three-component (3C) synthetic seismograms from the ray-traced models, using the reflection coefficients and mode conversions derived earlier to aid interpretations.
Effects of mineralogy, foliation and deformation Mineralogy. Although CPO relationships between individual mineral phases in polymineralic rocks are often complex (e.g. Fig. 4), it is possible to determine the impact of each mineral phase on the bulk rock seismic properties. This is achieved by recognizing that individual mineral CPOs represent a ‘recipe book’ from which rocks of different modal proportions but with the same measured CPO can be constructed (e.g. Tatham et al. 2008; Lloyd et al. 2009). A single mineral phase can therefore be chosen and its modal proportion varied between 0 and 100%, with the other minerals left to comprise the residual composition in their relative (i.e. measured) modal proportions. Using this ‘recipe’ approach, the impact of modal proportion on the seismic anisotropy of samples B1, B2m and B2c has been considered (Fig. 5). It
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Fig. 5. Rock-recipe modelling of the effect of composition on P- and S-waves anisotropy (AVp and AVs respectively) for samples B1, B2m and B2c. Note different anisotropy scales per plot. The line ornament per mineral is the same for each plot (see key, top left plot).
is clear that micas, and especially biotite, make the main contribution to the bulk seismic anisotropy (see also Takanashi et al. 2001; Chlupacova et al. 2003). Due to their generally weak CPO in these samples, feldspars and quartz impose a diluting effect on the anisotropy introduced by micas. The absence of CPO in any of these minerals in sample B2c means that the cataclastic fault zone exhibits very little anisotropy (Fig. 5). Foliation orientation. The results of CPO and seismic property analysis of samples collected from the NPM (Fig. 4) suggest that the orientation and strength of foliation play crucial roles in the extent of shear-wave splitting (see also Lloyd et al. 2011). From the previous section, such effects are likely to be exacerbated as the phyllosilicate content increases. To investigate the impact of (mica-controlled) foliation orientation, a simple elastic model was therefore designed (Fig. 6a). Although model width can be arbitrary (in the present models, the X and Y dimensions were set at 10 and +5 km, respectively), depth was set at Z ¼ 50 km because the NPM is believed to
involve the whole crust which is estimated to be c. 48 km thick (Butler et al. 2002). The model was populated with the elastic constants of sample B1, considered as representative of the early fabric state of the deformed NPM orthogneisses. The effect of foliation orientation was investigated by rotating the elastic constants of sample B1 such that the foliation changed progressively from vertical (i.e. 08) to horizontal (i.e. 908). Figure 6b illustrates the end-member foliation configurations. Finally, a shear-wave source was positioned near the base of the model within an isotropic layer to avoid complications associated with sources in anisotropic media. Figure 6c shows the travel times versus offset plots of the shear waves for the end-member horizontal and vertical foliations situations (Fig. 6b). The original shear wave is clearly split into two shear waves for both cases. However, there are obvious differences in the travel times and offsets for the two foliation orientations. This behaviour is shown more clearly by considering the magnitude of the shear-wave splitting for each increment of foliation orientation (Fig. 6d). From 0–458 there is
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Fig. 6. Seismic modelling of the effect of foliation orientation on shear-wave splitting characteristics. (a) Seismic model: the ‘crust’ is populated with the elastic properties of sample B1, while the seismic source region is isotropic. (b) End-member foliation configurations. Left: horizontal (908); right: vertical (08). (c) Fast (solid lines) and slow (broken lines) shear-wave travel-time plots for end-member configurations. Upper: horizontal (908); lower: vertical (08). (d) Amount of shear-wave splitting in terms of foliation angle (08, vertical; 908, horizontal) for vertically propagating shear waves.
little variation, but beyond 458 there is a trend of steadily decreasing shear-wave splitting to a minimum of ,0.15 s at 908. There is therefore a difference of c. 1.3 s between the maximum and minimum shear-wave splitting for vertical and horizontal foliation orientations. Although the impact of foliation orientation on shear-wave splitting is perhaps intuitive, the maximum splitting actually occurs at 108 to the vertical (Fig. 6d). This specific behaviour reflects the detail present in both the CPO distributions
(Fig. 4a) and the plunge of the lineation within the foliation plane. The splitting observed therefore depends on CPO, foliation orientation and ray geometry, which suggests that small-scale petrofabric observations can be effectively used as proxies to investigate regional-scale geodynamics in the crust (see below). Indeed, as the potential for c. 1.3 s of shear-wave splitting due to CPO is larger than the average SKS splitting observed (e.g. Silver 1996), this suggests that the crust can make a significant contribution to SKS splitting.
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Furthermore, values of 1.3 s splitting would be relatively easy to distinguish in local events or teleseismic P– S conversions at the Moho, and highlights the importance in considering such factors in both global and local seismic surveys. Deformation partitioning. Deformation partitioning (e.g. between pure and simple shear) can occur on all scales (e.g. Twiss & Moores 1992; Passchier & Trouw 1996). To investigate the effect of deformation partitioning on seismic properties, assuming similar rock properties rotated into different relative orientations (e.g. vertical or pure-shear and inclined or simple-shear components), a 40 km deep elastic model was designed in which the lowest 4 km represents a basal isotropic layer containing a shearwave source located at 38 km depth (Fig. 7a). Initially, the remaining 36 km of the model was populated with the elastic properties of sample B2m rotated into the vertical to represent 100% horizontal pure shear. Next, a shallowly dipping 2 km thick simple-shear zone (i.e. comparable in orientation and thickness to the Liachar Shear Zone in the NPM) was incorporated using the elastic properties of sample B2m rotated into the appropriate orientation (Fig. 7b). However, the relative proportions of pure and simple shear in the model can be varied progressively by increasing shear-zone width. The travel-time-offset behaviours for the two end-member configurations (i.e. 100% pure-shear and 100% simple-shear elastic properties) indicate considerable shear-wave splitting for both models and a slight increase in travel times for the simpleshear model (Fig. 7c, d). In detail, the degree of shear-wave splitting measured for each increment of increasing shear-zone width shows a negative (almost linear) relationship with the amount of simple shear present in the model (Fig. 7e). More shear-wave splitting is generated by the 100% pureshear model compared to the 100% simple-shear model, which again emphasizes the influence of steep foliations on shear-wave splitting (i.e. Fig. 6d). Furthermore, the pure-shear model increases velocity, as shown by the c. 0.4 s difference between the arrivals of the S1-waves between the two models (Fig. 7c, d). However, the difference in shear-wave splitting is only c. 0.6 s due to the rocks having similar seismic properties. Greater differences in seismic properties between the constituent rocks are therefore expected to induce larger differences in shear-wave splitting. The results of these pure- and simple-shear models (Fig. 7) suggest that it may be possible seismically to not only recognize deformation partitioning but also to distinguish between different styles of deformation at depth. However, there are certain provisos as follows: (1) the geology is
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well-constrained; (2) the CPO characteristics are pervasive over sufficient distances depending on seismic resolution; (3) the measurements of shearwave splitting are well constrained; and, crucially, (4) there is sufficient contrast between host and shear-zone rocks in terms of foliation, fabric and structural relationships (e.g. Lloyd et al. 2011).
Seismic wave propagation It is known that seismic wave propagation is more complex in anisotropic compared to isotropic media, while waveform effects due to anisotropy may be dramatic and unexpected (e.g. Guest & Kendall 1993). Wave fronts are no longer spherical and the directions of particle motion, rays and wave-front normal are generally not aligned. However, the primary effect of wave propagation from isotropic to anisotropic media is shear-wave splitting or seismic anisotropy (e.g. Crampin 1981), resulting in the separation of shear waves into two orthogonal quasi-shear waves (qSH and qSV). The results of the previous sections indicate that shear-wave splitting is to be expected in many of the samples from the NPM. To investigate the potential effects of (shear) wave propagation through the NPM, the program SLWVEL (Guest & Kendall 1993) was used to calculate the slowness surfaces (including polarization vectors and group velocity surfaces) for samples B1, B2m and B2c based on their Voight-ReussHill (VRH)-averaged elastic constants via numerical decomposition of the Christoffel equation into eigenvalues (phase velocities) and eigenvectors (displacements). Slowness is the reciprocal of phase velocity; a slowness surface is an envelope that encompasses all of the slowness vectors and therefore provides a description of elastic anisotropy (e.g. Lloyd & Kendall 2005). Slowness curves are directly analogous to CPO-derived velocity pole figures (e.g. Fig. 4). In sample B1, both foliation-parallel and -normal sections exhibit variable slowness and velocity, with off-axis shear-wave splitting illustrated by the presence of two shear-wave surfaces resulting in anisotropic behaviour and almost orthorhombic symmetry (Fig. 8a). In contrast, sample B2m is almost isotropic with only slight splitting and almost constant slowness and velocity (Fig. 8c). Sample B2c is clearly isotropic and exhibits no shear-wave splitting and constant slowness and velocity seismically (Fig. 8b).
Seismic reflection coefficients Up to six new reflected and transmitted secondary waves may be generated when a ray encounters an interface between two different media, including
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Volume of shear zone (%)
Fig. 7. Seismic modelling of the effect of deformation partitioning on seismic anisotropy, assuming similar rock properties rotated into different relative orientations (e.g. vertical or pure-shear and inclined or simple-shear components). (a) Seismic model with vertical foliation defined by sample B2m properties rotated into the vertical orientation. (b) Seismic model with vertical foliation cut by a shallow-dipping 2 km thick simple-shear zone defined by sample B2m properties rotated into the appropriate orientation. (c) Travel-time plot for seismic model comprising 100% vertical foliation. (d) Travel-time plot for seismic model comprising 100% shear zone. (e) Variation in amount of shear-wave splitting with variation in simple-shear zone content for vertically propagating shear waves.
both fast and slow shear waves. A unique feature of such reflected-converted waves in anisotropic media is that they have energy at normal incidence due to a difference in group and phase velocity produced by low-symmetry and anisotropy, which never
exists in isotropic cases (e.g. Lloyd & Kendall 2005). As an example of this phenomenon, the seismic reflection coefficients derived from the VRH elastic constants were calculated for P-wave velocities at
Horizontal slowness cross-section
Horizontal velocity cross-section
S1
0.2
P
0
-0.2
P 4
0.4
S1 S2
S2
S1 S2
2
0
-2
S1
0.2
P
0
2
0
-0.2
-2
-0.4
-4
-4
-0.4
0.6 0.4 0.2 x1 slowness (s/km)
Horizontal slowness cross-section
(b)
0
B2-cat
0.3
2 4 x1 velocity (km/s)
6
Horizontal velocity cross-section
-0.6
0.4 0.2 x1 slowness (s/km)
0
(c)
S
x2 slowness (s/km)
x2 velocity (km/s)
0
2
0
-0.1
-2
-0.2
-4
0
0.1 0.2 0.3 x1 slowness (s/km)
0
P 4
P
0
S1 S2
2
0
-2
-0.1
-4 -0.2 -6
-6
-0.3
0.1
6
S2 S1
0.2
P
4
P
6
Horizontal slowness Horizontal velocity cross-section cross-section B2-myl
S
0.1
2 4 x1 velocity (km/s)
0.3
6
0.2
-6 0
0.6
x2 velocity (km/s)
0
FROM CRYSTAL TO CRUSTAL
-6
-0.6
x2 slowness (s/km)
P
4
S2 x2 velocity (km/s)
x2 slowness (s/km)
0.4
6
Vertical velocity cross-section
6
x2 velocity (km/s)
0.6
Vertical slowness cross-section
0.6
x2 slowness (s/km)
(a)
4 2 x1 velocity (km/s)
6
-0.3 0
0.1 0.2 x1 slowness (s/km)
0.3
0
6 2 4 x1 velocity (km/s)
63
Fig. 8. Examples of seismic slowness and velocity surfaces and polarization directions (arrows) for P- and S-waves (occurrence of two S-waves indicates shear-wave splitting). (a) Sample B1. Left: foliation (XY ) parallel or ‘horizontal’ sections; right: foliation (XZ ) normal or ‘vertical’ sections. (b) Sample B2c foliation (XY ) parallel sections. (c) Sample B2m foliation (XY ) parallel sections.
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G. E. LLOYD ET AL.
an imagined interface between samples B2m and B2c. Energy (as defined by the total displacement ratio) clearly exists for normal incidence P-waves and P-to-P conversions, but not for P-to-S wave conversions (Fig. 9). However, from 15– 538 incidence angles, energy associated with P-to-S conversions is greater than that associated with P-to-P conversions. This behaviour implies that converted phases are likely to contain information about the properties of the interface boundary (see also Lloyd & Kendall 2005). Such clear azimuthal variation in reflections suggests also that multi-azimuth wideangle reflection data can be used to study sense of deformation in deep-rooted deformation zones. Furthermore, the amplitude of reflected shear waves is sensitive to anisotropy (Helbig 1993, 1994). The ability to calculate reflection and transmission coefficients at boundaries between media with different material properties can be used to aid the interpretation of seismic datasets (e.g. Guest et al. 1993). The veracity of such interpretations depends critically upon knowledge of the seismic properties of the rocks involved. These properties can be determined from seismic datasets, laboratory techniques or, as in this contribution,
Total displacement ratio
sagittal plane Z
from CPO. However, to properly gauge the effects of anisotropy on seismic wave propagation, isotropic ‘control models’ are needed for each sample constructed from its VRH-averaged elastic constants. Models using the isotropic elastic constants only consider the impedance contrast in a layer and at a boundary, and generally yield consistently lower P- and S-wave velocities than those observed in the actual (i.e. anisotropic) cases. This aspect is considered in the next section.
Seismic reflection coefficients and controlled source surveys The models described so far have considered only shear-wave splitting where a CPO is persistent over a large region. Typical shear-zone widths will not be detected in such seismic surveys, as they are too narrow to have a significant effect on shearwave splitting. However, it has been suggested that reflection coefficients yield greater vertical resolution than shear-wave splitting at boundaries between anisotropic media (Guest et al. 1993; Lloyd & Kendall 2005). An elastic model was therefore designed to investigate the effects of anisotropy
X
Tectonic reference frame critical angle: phase & waveform changes (analytical problems)
1.0
P-waves wide angle reflection surveys
normal seismic reflection data angles
0.5
normal incidence (nb energy)
PP>PS PS>PP P-to-S waves
PP>PS
Phase
0.0
0 180
10
20
30
40
50
60
70
80
90
0
–180 Incident phase angle (degs) Fig. 9. Example of seismic modelling of reflection coefficients and mode conversions for P-waves at an interface between samples B2m and B2c in terms of energy (total displacement) and incidence wave angle (08, vertical; 908, horizontal). For normal incidence (i.e. teleseismic waves), energy exists for P-waves and P-to-P conversions but not for P-to-S conversions. From 15– 538 incidence angles, P-to-S conversions energy is greater P-to-P conversions energy. Note also the significant increase in energies for wide-angle (.708) reflections.
FROM CRYSTAL TO CRUSTAL
on the reflection, transmission and conversion coefficients at various angles of incidence for boundaries between the background rock of the NPM, represented by the elastic properties of sample B1, and a cross-cutting shear zone (Fig. 10a). For the purposes of this model, the shear zone was populated with the elastic properties of another sample (Gn7) that can be proven in the field to have been derived from the protolith as represented by sample B1 (e.g. Butler et al. 2002). For control purposes, an isotropic version of the model was also constructed. Two interfaces are present in the model, with rays shot from the surface passing from and interacting with B1-to-Gn7 and Gn7-to-B1. Reflected waves and mode conversions therefore originate at both interfaces (Fig. 10a). The travel-time plots for the anisotropic cases are very similar for all azimuths and indicate that fast and slow shear waves (i.e. shear-wave splitting) occur for both interfaces in the anisotropic models (Fig. 10b). In contrast, the isotropic control models for both wall-rock-toshear-zone and shear-zone-to-wall-rock interfaces reveal relatively simple displacement-incidence angle behaviours and no shear-wave splitting (Fig. 11a, b). Displacement (energy) increases progressively with angle of incidence for P-waves at both interfaces, although somewhat more rapidly for the wall-rock-to-shear-zone interface. The displacement-incidence angle relationships for the shear waves are similar, with both showing zero
0
2 G7 4
displacement at normal incidence and significant displacements at 30–408; there is considerably more energy associated with the wall-rock-to-shearzone interface, however. Displacements for both interfaces decrease to zero at c. 608 before increasing again significantly for wide-angle reflections. Displacement-incidence angle behaviours for the anisotropic model based on the petrofabricderived elastic properties are significantly different to the isotropic cases, with both interfaces exhibiting two shear waves and hence shear-wave splitting (Fig. 11c, d). In addition, the wall-rock-to-shearzone and shear-zone-to-wall-rock interfaces also exhibit different behaviours, with the latter characterized by greater displacements for all three mode conversations. This suggests that it may be possible to differentiate between interfaces via their mode conversion characteristics (e.g. Lloyd & Kendall 2005). The P–S1 and P–S2 conversions for both interfaces show the greatest variation between 0– 608 incidence angles, particularly for the former case. In contrast, the P–P waves show little variation with azimuth. Perhaps the best way to identify anisotropic differences is therefore to observe larger azimuthal variations at different angles of incidence (Guest et al. 1993; Lloyd & Kendall 2005). Furthermore, from the reflection coefficient plots, high amplitudes are expected for the converted waves. To assist interpretation further by providing information on the amplitudes of seismic waves,
0.6
0.6 P-P wall rock to shear zone interface
0.8
Travel time (secs)
B1
*
Depth km
(a)
Travel time (secs)
(b)
65
P-S1
1.0
P-S2
Azimuth = 0°
1.2 P-P
1.4 1.6
shear zone to wall rock interface
1.8
P-S1 P-S2
2.0 0
1
6
0.8
wall rock to shear zone interface P-S1
1.0
P-S2
1.2 1.4
Azimuth = 30°
P-P
1.6
shear zone to wall rock interface
1.8 2.0
P-S1 P-S2
0
1
3 2 4 Model x (km)
5
Travel time (secs)
2 4 Model x (km)
Travel time (secs)
0
wall rock to shear zone interface P-S1 P-S2
Azimuth = 60°
1.2 1.4
P-P
1.6
shear zone to wall rock interface
1.8
P-S1 P-S2
0
1
5
3 2 4 Model x (km)
0.6
0.6 P-P
B1
P-P
1.0
2.0
5
3 2 4 Model x (km)
0.8
0.8
P-P wall rock to shear zone interface P-S1
1.0
P-S2
Azimuth = 90°
1.2 1.4
P-P
1.6
shear zone to wall rock interface
1.8 2.0
P-S1 P-S2
0
1
3 2 4 Model x (km)
5
Fig. 10. Example of seismic modelling of reflection coefficients and mode conversions at the interfaces between a 2 km wide shear zone (sample Gn7 elastic constants) and its wall rock (sample B1 elastic constants). (a) Elastic model and ray paths. (b) Travel-time plots for anisotropic model interfaces for azimuths of 08, 308, 608 and 908.
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G. E. LLOYD ET AL.
P-waves (c)
displacement
S-waves
0.03
0 0.2
0.02 0.01 0 0
20
40
60
80
angle of incidence
0.50
0
S-waves
20
60
80
angle of incidence
P-S1
0.02
0.01
P-S2
0.02
0.01
0 0
20
40
60
80
angle of incidence
azimuths 0° 60° 30° 90°
0.06
0 0
40
0.50
0.03
P-S2
0
0 0.12
displacement
0 0.2
P-waves
P-P
0.03
P-S1
displacement
displacement
(b) 1.00
displacement
0
displacement
normal incidence
1.00
P-P
displacement
0.20 0.10
(d)
0.2
displacement
0.30
displacement
displacement
(a) 0.40
20
40
60
80
angle of incidence Fig. 11. Example of seismic modelling of reflection coefficients and mode conversions at the interfaces between a 2 km wide shear zone (sample Gn7 elastic constants) and its wall rock (sample B1 elastic constants) for azimuths of 08, 308, 608 and 908 (anisotropic cases only). (a) Isotropic reflection coefficients for P- and S-waves at the wall-rock-toshear-zone interface. (b) Isotropic reflection coefficients for P- and S-waves at the shear-zone-to-wall-rock interface. (c) Anisotropic reflection coefficients for P – P, P –S1 and P – S2 waves for different angles of incidence at the wall-rockto-shear-zone interface (note key below). (d) Anisotropic reflection coefficients for P– P, P–S1 and P–S2 waves for different angles of incidence at the shear-zone-to-wall-rock interface (note key below).
and in particular their mode conversions, the elastic model shown in Figure 11a was used to produce synthetic three-component (3C) seismograms for the two interfaces (Fig. 12). In both cases, the CPOderived elastic constants were rotated into the correct geographical orientation for 0–908 azimuths. The presence of the two interfaces is seen clearly in the 3C-seismograms, although P–P conversions are only recognized on the vertical component. In particular, they show considerable S-wave converted phase energy on the transverse component. This behaviour is indicative of anisotropy as there is no energy on the transverse
component in an isotropic case. However, the main conclusion to be drawn is that boundaries between tectono-lithological units are likely to produce significant azimuthal variations (as well as polarity reversals) in 3C-seismograms.
Deformation determination from reflection coefficients and mode conversions Three elastic models were designed to test whether it is possible to differentiate between different types of CPO-dependent deformations via seismic anisotropy. All three models were populated with the
FROM CRYSTAL TO CRUSTAL Radial
0
(b)
1.0
2.0
2.0
Transverse
0 1.0 2.0
Vertical
0
Radial
0
1.0
Travel time (secs)
Travel time (secs)
(a)
67
Transverse
0 1.0 2.0
Vertical
0 1.0
1.0
2.0
2.0 0
500
1000
1500
2000
2500
0
500
Offset (m)
(c)
Radial
(d)
2000
2500
Radial 1.0
2.0
2.0
Transverse
1.0 2.0
Vertical
0
1500
0
1.0
Travel time (secs)
Travel time (secs)
0
0
1000
Offset (m)
Transverse
0 1.0 2.0
Vertical 0
1.0
1.0
2.0
2.0 0
500
1000
1500
2000
2500
Offset (m)
0
500
1000
1500
2000
2500
Offset (m)
Fig. 12. Example of seismic modelling of reflection coefficients and mode conversions at the interfaces between a 2 km wide shear zone (sample Gn7 elastic constants) and its wall rock (sample B1 elastic constants) – travel-time graphs and three-component (3C) synthetic seismograms for wall-rock-to-shear-zone-to-wall-rock (i.e. B1 –Gn7–B1) interfaces using geographic elastic constants at azimuths of (see also Fig. 10b): (a) 08; (b) 308; (c) 608; and (d) 908. Note: (1) the presence of the two interfaces; (2) P –P conversions on the vertical component only; (3) considerable S-wave converted phase energy on the transverse component; and (4) significant azimuthal variations and polarity reversals at the boundaries between the tectono-lithological units.
elastic constants of sample B2c to represent deformation due to brittle faulting in the uppermost crust. In the first model, a second lower layer was populated with the elastic constants of sample B1 rotated such that the foliation was horizontal to represent a simple-shear deformation (Fig. 13a). In contrast, the lower layer of the second model was populated with the elastic constants of sample B1 rotated so the foliation was vertical to represent a pure-shear deformation (Fig. 14a). The third model was a hybrid of the other two models and represented a transition from coaxial pure-shear deformation at depth into partitioned and localized zones of ductile simple shear and brittle faulting deformations at progressively shallower levels (Fig. 15a). The thickness of each layer was
arbitrarily chosen at 2 km, although this is not significant because it was shown previously that reflections and mode conversions are most sensitive to the properties of the interface. A seismic source was placed at the surface in the centre of each model, from which travel times (Figs 13b, 14b & 15b), geometrical spreading reflection coefficients and mode-converted reflections for P– P, P –S1 and P –S2 were calculated (Figs 13c, 14c & 15c) and used to construct 3C-synthetic seismograms (Figs 13d, 14d & 15d). Although travel times do not differ between Models 1 and 2 and there is little or no shear-wave splitting (Figs 13b & 14b), the 3C-seismograms show much greater variation. For Model 1 (Fig. 13d), P-wave and mode-converted S-waves are
68
G. E. LLOYD ET AL.
(b) 0.6
B1-horiz
2
Depth km
B2-cat
0
Travel-time (s)
(a)
(c) P-P
0.7
0
0.8
0.3
0.9
P-S1 P-S2
1.0 1.10
0.2 5
1 2 3 4 Model x (km)
4
Incidence angle 40 20 60
80
P-P reflection coefficients 0 degs 30 degs 60 degs 90 degs
0.1
(d) 0 0.3 Radial
0 0.5 1.0 1.5 2.0 2.5 0 0.5 1.0 1.5 2.0 2.5
Displacement
Travel-time (s)
0.5 1.0 1.5 2.0 2.5
P-S1 reflection coefficients
0.2 0.1
Transverse 0.3
P-S2 reflection coefficients
0.2 0.1
Vertical 0
0.5
1.0
1.5
2.0
2.5
Offset (km)
3.0
0
Fig. 13. Determination of CPO-dependent deformation styles using seismic reflection coefficients and mode conversions. Model 1: brittle cataclastic fault overlying horizontal ductile (simple-shear) foliation. (a) Model geometry and ray tracing. (b) Travel times for P –P, P– S1 and P – S2 wave conversions. (c) Three-component (3C) synthetic seismograms. (d) Reflection coefficients and mode-converted reflections for P– P, P–S1 and P–S2 waves (key is the same for all plots).
present on the radial component, with the latter having the greatest amplitude. Only modeconverted waves are seen on the transverse component, while the P-wave is prominent on the vertical component with a small amount of modeconverted energy. There is significant variation in the reflection coefficients, especially on the mode conversions but less so on the P reflection. In contrast, Model 2 (Fig. 14d) has much less energy on the vertical component of the 3C-seismogram. The P-wave has the highest amplitude which decreases with offset, while the S-wave increases with offset. There is a large amount of mode-converted energy on the transverse component. The radial component has both P- and S-waves, although the former is very small and the latter are far more prominent. Unsurprisingly, Model 3 shows the same results as Model 1 for the reflections from the upper interface between the cataclasite and horizontal foliation (compare Figs 13d & 15d). However, the lower interface between the horizontal and
vertical foliations has the most energy on the transverse component, although less than in Model 2 (Fig. 14d) which corresponds to the mode-converted waves. Mode conversions are seen more clearly on the radial component than the P-waves, which only have a small amount of energy that disappears by 600 m offset. The vertical component has only P-wave energy and this is stronger than that observed in Model 2. The transition from horizontal to vertical foliation therefore appears to enhance the P-wave on the vertical component and reduce the S-wave on the transverse component of the seismogram. It appears from the three models (Figs 13–15) that differences in amplitude of P-wave and modeconverted phases may help to determine deformation style. However, it is important to recognize that the models represent simplified geometries and consider only three types of interface. For example, as the inclination of an interface increases, there will be some energy on the transverse component. The seismic models described here
FROM CRYSTAL TO CRUSTAL
)
4
Depth km
B2-cat B1-vert
2
0.6
0
0.8
0.3
0.9
P-S1 P-S2
1.0 1.1 0
1 2 3 4 Model x (km)
Travel-time (s)
(d) 0 0.5 1.0 1.5 2.0 2.5
0.2 5 0.1
Radial
0 0.5 1.0 1.5 2.0 2.5 0 0.5 1.0 1.5 2.0 2.5
(c)
P-P
0.7
Displacement
(b)
0
Travel-time (s)
(a)
69
0.3
Incidence angle 40 20 60
80
P-P reflection coefficients 0 degs 30 degs 60 degs 90 degs
P-S1 reflection coefficients
0.2 0.1
Transverse 0.3
P-S2 reflection coefficients
0.2 0.1
Vertical 0
0.5
1.0 1.5 Offset (km)
2.0
2.5
3.0
Fig. 14. Determination of CPO-dependent deformation styles using seismic reflection coefficients and mode conversions. Model 2: brittle cataclastic fault overlying vertical ductile (pure-shear) foliation. (a) Model geometry and ray tracing. (b) Travel times for P –P, P– S1 and P– S2 wave conversions. (c) Three-component (3C) synthetic seismograms. (d) Reflection coefficients and mode converted reflections for P–P, P–S1 and P– S2 waves (key is the same for all plots).
therefore show only the potential of this approach rather than precise solutions. Nevertheless, the results are encouraging and illustrate how the use of arrays of 3C-seismometers is crucial in enhancing understanding of crustal anisotropy. In the next section the experience gained from the CPO-derived seismological modelling is used in an interpretation of the seismology and tectonics of the NPM.
Discussion This section assesses whether it is possible to use CPO-derived seismic properties and models to discriminate between the different models to explain the kinematics and geodynamics of the NPM. It begins with a consideration of the known seismology of the NPM before considering the known tectonic configuration prior to a test of the pureversus simple-shear models based on the lessons and results gained earlier in this contribution.
NPM seismology There have been several studies of the seismic characteristics of the NPM based on either direct measurements of natural seismicity (e.g. Meltzer et al. 2001; Weeraratne et al. 2004) or ultrasonic laboratory measurements of samples collected from the region (e.g. Meltzer & Christensen 2001). Natural seismicity indicates Vp and Vs values of 5.5–6.5 and 3.0–3.7 km/s respectively, similar to the ultrasonically measured values after all (expansion) cracks have closed (Fig. 16a). These values are in excellent agreement with the CPO-derived measurements obtained in this study (see also Fig. 4). However, although there is good agreement between ultrasonic- and CPO-derived Vp anisotropy estimates, the former indicate considerably greater shear-wave splitting (Fig. 16b). It has been shown previously that variation in mica content is the main control on the amount of
70
G. E. LLOYD ET AL.
(b) 0
2
4
0.6 P-P
0.8
Travel-time (s)
B1-vert B1-horiz B2-cat
*
Depth km
(a)
P-S1 P-S2
1.0
1.6
0
1
lower interface
0.2
P-S1, P-S2
0.1
0 degs 30 degs 60 degs 90 degs
Radial
0 0.5 1.0 1.5 2.0 2.5
80
5
3 2 4 Model x (km)
Displacement
Travel-time (s)
0.3
0.3
(d) 0 0.5 1.0 1.5 2.0 2.5
Incidence angle 40 20 60 P-P reflection coefficients
P-P
1.4
2.0
0
(c)
1.2
1.8 6
upper interface
P-S1 reflection coefficients
0.2 0.1
Transverse 0.3 P-S2 reflection coefficients 0.2
0 0.5 1.0 1.5 2.0 2.5
Vertical
0
0.5
1.0 1.5 Offset (km)
2.0
2.5
0.1
3.0
0
Fig. 15. Determination of CPO-dependent deformation styles using seismic reflection coefficients and mode conversions. Model 3: brittle cataclastic fault overlying horizontal ductile (simple-shear) foliation overlying vertical (pure-shear) foliation. (a) Model geometry and ray tracing. (b) Travel times for P–P, P– S1 and P–S2 wave conversions. (c) Three-component (3C) synthetic seismograms. (d) Reflection coefficients and mode converted reflections for P– P, P–S1 and P –S2 waves (key is the same for all plots).
anisotropy exhibited, with foliation orientation and development also making contributions (Figs 5–7; see also Lloyd et al. 2009, 2011). As the mica content is up to 10% greater in the samples measured via ultrasonics (Meltzer & Christensen 2001), the discrepancy between the two estimates is considered to be mainly due to differences in mineral composition between the two sample sets. Meltzer & Christensen (2001) estimated c. 1.5 s of shear-wave splitting for a 40 km thick crust with vertical foliation, via ultrasonic measurements. This value is only slightly greater than the c. 1.3 s estimated from the CPO-derived model (Fig. 6). Again, the small discrepancy can be explained by differences in mica content and/or foliation development/orientation (Figs 5–7). As splitting delay times increase with distance travelled through anisotropic material, they can provide a means of
mapping rock fabric at depth (e.g. Kern & Wenk 1990). However, the range of delay times can also be influenced by compositional heterogeneity, lateral variation in anisotropy, changes in regional foliation orientation and velocity variance due to non-axial propagation through a wide range of event-station azimuths, as shown in this contribution. Meltzer et al. (2004) however argue that because the composition of the NPM is basically homogeneous and its structure is well constrained, while ray-paths are restricted to the crust and source-receiver geometries sample a range of azimuths with respect to structure, seismic data is ideal for studying and quantifying the affect of nonaxial propagation through the regional foliation. They argue further that as current tomography codes do not generally account for anisotropic effects, and may potentially under- or overestimate
FROM CRYSTAL TO CRUSTAL
velocity structure in the crust, this type of analysis has important implications for understanding crustal dynamics. In particular, Vp, Vs and Vp/Vs ratios are typically used to infer both lithology and rheology of subsurface materials providing constraints for thermo-mechanical models of deformation (e.g. Christensen & Mooney 1995). The potential for 1.3–1.5 s of shear-wave splitting due to CPO-induced seismic anisotropy should be easy to distinguish via natural seismicity, such as local events or teleseismic P–S conversions at the Moho. Indeed, Weeraratne et al. (2004) observed such strong anisotropy in teleseismic and regional shear phases from a seismic array deployment across the NPM. Stations outside the NPM recorded 1.5–2.3 s delay times, with WNW –ESE fast directions in SKS and related core phases; regional S-phases from the Hindu Kush with source depths of 200 –300 km had similar c. east– west fast directions but delay times of ≤0.5 s. As the depth range sampled by the latter lies mainly in the high-velocity lithosphere (which extends to .200 km beneath the NPM), it appears that the lithospheric contribution to the total shear-wave splitting observed for the teleseismic phases is only c. 0.5 s with c. 1.0–1.5 s originating in the sublithospheric mantle (Weeraratne et al. 2004). Furthermore, while SKS paths from a wide range of back-azimuths produce null measurements within the interior of the NPM, laboratory studies of gneiss samples suggest that as much as 21% shearwave anisotropy with north–south fast axis may exist in the crust (Meltzer & Christensen 2001). In addition, mantle lithosphere deformation consistent with east –west compression of the NPM may also contribute to shallow north–south anisotropy (Weeraratne et al. 2004). The null observations in the NPM interior may therefore be due to the mutual cancellation of north –south and east –west shear-wave splitting effects (see below).
NPM tectonics Seismic velocities and velocity ratios are typically used to infer both subsurface lithology and rheology, thereby providing constraints on thermomechanical models of deformation, tectonics and geodynamics. A prominent low-velocity zone (compared to surrounding regions) has been recognized beneath the NPM and extends through the crust into the upper mantle (e.g. Meltzer et al. 2001, 2004). One explanation for this behaviour, especially as composition is considered to be effectively homogeneous, is the presence of hot rocks at depth. The variable seismic waveforms and slightly lower Vp/Vs ratios observed therefore suggest the existence of super-critical fluids in small regions of limited extent, which are more consistent with
71
partial melts and/or aqueous fluids rather than the presence of large magma bodies. This is supported by a shallow (c. 2–5 km bsl) brittle–ductile transition that bows towards the surface under the NPM, which is consistent with rapid advection from depth of hot crust into the massif along shallow detachments. Furthermore, magnetotellurics indicate that the lower crust is atypically resistive (Park & Mackie 1997, 2000). Each of these anomalies exhibits a ‘bulls-eye’ pattern centred on the NPM. It has therefore been suggested that the magnitude and extent of the low-velocity zone within the NPM constrains crustal flow paths, thereby focusing exhumation and concentrating crustal strain and potential zones of partial melting in the crust. This led Zeitler et al. (2001) to describe the geodynamics of the NPM as a ‘tectonic aneurysm’. In contrast to the ‘tectonic aneurysm’ model involving hot rocks, high thermal gradients and/or pore pressures in typical plutonic and/or metamorphic rocks, Meltzer & Christensen (2001) proposed an alternative explanation for the seismic structure and characteristics of the NPM. They suggest that as the mid-lower crust is typically layered with well-defined foliations and fabrics on multiple scales, in situ velocities from refracted or turning rays that spend substantial portions of their travel paths either parallel or normal to the foliation plane may systematically either over- or underestimate seismic velocities. Whereas observed velocities of 6.0– 6.5 km/s that are interpreted normally as indicating rocks of intermediate composition could also be indicative of waves propagating (sub-) parallel to foliation in felsic rocks, observed velocities of 5.6–6.0 km/s could indicate propagation at high angles to foliation in similar lithologies (e.g. Figs 6 & 7; see also Lloyd et al. 2011). Such observed velocity variations only require changes in foliation properties (i.e. deformation) with depth rather than composition. Crucial to the validation of this alternative model are the seismic anisotropy characteristics of the NPM. Although Weeraratne et al. (2004) observed up to 1.5 s of shear-wave splitting within the NPM, they also recognized null observations within the interior of the massif, from which they derived a two-layer anisotropic model with north– south anisotropy in the crust and lithosphere due to east–west compression of the Nanga Parbat orogen cancelling splitting from c. east –west sublithospheric anisotropy (see above). The results of the CPO-derived seismological models described in this contribution can be used to test (although perhaps as yet only qualitatively) the various tectonic models for the NPM. The differences in amplitude of P-wave and mode-converted phases on the transverse component help to distinguish deformation style, while mode conversions
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and transverse component energy are diagnostic of the degree and orientation of foliation (which have dramatic effects on seismic waves). Steeply inclined foliation produces large mode-converted amplitudes on the transverse component of 3C-seismograms as well as significant shear-wave splitting, compared to shallowly inclined foliation. Such differences should be easy to distinguish in local events or teleseismic P –S conversions at the Moho. Although high-amplitude P-to-S-wave converted reflections have been observed in wide-angle reflection data in the Himalayas, they have been interpreted as indicating the presence of partial melt accumulation because they are believed to be uncharacteristic of crustal reflections (e.g. Makovsky et al. 1996). However, the results of the elastic models presented here suggest that they can indeed be associated with interfaces between different lithologies and/ or different deformation styles. These results not only indicate the importance of considering seismic anisotropy in crustal seismic surveys, but also that it should be possible to discriminate between different deformation processes active at depth using seismic measurements. As a first approximation, the results of the CPOderived seismological models described in this contribution appear to be more consistent with the Meltzer & Christensen (2001) model for NPM rather than other models such as the ‘tectonic aneurysm’ model of Zeitler et al. (2001).
NPM: pure- versus simple-shear tectonics This contribution has referred throughout to two simple alternatives (perhaps ‘end-member’ models) for the tectonics of the NPM, namely the penetrative pure-shear model (e.g. Butler et al. 2002) and the localized simple-shear model (e.g. Burg 1999). If pure shear is persistent at depth, it would require a steep to subvertical fabric to persist throughout the NPM due to horizontal shortening and vertical extension. In contrast, the simple-shear model would induce a crystal alignment in the direction of shearing at much shallower angles, perhaps with much narrower length scales measured in the (sub-) vertical sense. The question therefore arises as to whether it is possible to discriminate between the two models and hence to make inferences about the kinematics using real seismic data
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based on the experienced gained via the CPOderived seismic modelling described above. To begin to answer this question it is necessary to consider the effect of foliation orientation on the magnitude of shear-wave splitting, as steep foliations result in considerably greater shear-wave splitting than shallow foliations (i.e. Figs 6 & 7). Butler et al. (2002) envisaged an increasing strain gradient from the core of the NPM towards the northwest, within which subvertical pure-shear stretching flanks and passes into an inclined simpleshear zone (e.g. Fig. 17a). The impact of the changes in foliation orientation due to this strain gradient on shear-wave splitting can be assessed from the results of the seismic modelling described previously. From Figure 6d, most of the NPM (including its core) is dominated by steep vertical foliations and therefore should exhibit .1.0 s shear-wave splitting (e.g. Fig. 17b). However, a zone of lower shearwave splitting values (i.e. ,1.0 s) should also be observed, particularly near to the surface towards the northeast (e.g. Figs 1 & 2). In practice, the regional variation between pure-shear and simple-shear tectonics shown in Figure 17a can be recognized on all scales. For example, anastomosing simple-shear zones occur on many scales and separate domains of relatively low deformation that may be described as pure shear. Furthermore, many tectonites exhibit S–C foliations in which planar C-surfaces of concentrated deformation (i.e. ‘shear zones’) separate broader regions or ‘lithons’ containing S-surfaces inclined to the C-planes (e.g. Lister & Snoke 1984). Such scaleinvariant pure-shear–simple-shear deformation geometries are illustrated in Figure 17a, superimposed upon the regional tectonic traverse across the NPM. Their impact on the shear-wave splitting characteristics can be inferred from the seismic model described in Figure 7. This model considered the effect of varying the proportion of sample B2m elastic properties Elastic properties, which can be considered as a proxy for a simple-shear zone, to sample B1 elastic properties, which can be considered as a proxy for a pure-shear fabric. Together, these proxies could also represent various combinations of S–C foliation intensities (see also Lloyd et al. 2009). As the proportion of shear zones/S–C foliation increases, the amount of shear-wave splitting decreases (Fig. 7e). Thus, variations in shear-wave
Fig. 16. Comparison of seismic properties for the NPM. (a) Compressional (Vp) and shear (s1, s2) wave velocities: rainbow bars, CPO-derived values for samples B1, B2m and B2c (see Fig. 4); rectangular boxes, natural velocities measured in situ at NPM (Meltzer et al. 2001); solid and broken curves, ranges of experimental ultrasonic values for NPM samples (Meltzer & Christensen 2001). Also shown are the Vp/Vs ratio ranges per sample and the typical values of Vp expected in the middle crust according to Rudnick & Fountain (1995). (b) Compressional (AVp) and shear (AVs) wave anisotropies: open circles, absolute AVp values; black bars, range of AVs values; solid and broken double arrows, experimentally measured AVp and AVs values (Meltzer & Christensen 2001) (higher AVs values reflect higher mica contents and/or stronger mica CPO).
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splitting characteristics must also reflect the deformation state in terms of the development of shear zones and/or S–C foliations on the scale of the seismic wavelength and/or the volume of rock penetrated (Fig. 17b). On the basis of these data only, it appears that the tectonics and kinematics of the NPM can be explained by the model of Butler et al. (2002). Thus, pervasive pure-shear vertical stretching dominates the tectonics and is responsible for most of the exhumation and topography of the NPM (e.g. Fig. 17a). Superimposed onto this fabric is a localized (simple) shear and brittle faulting deformation, particularly near to the surface. However, confirmation of the viability of this model requires more field data, both geological and seismological. In particular, 3C-seismic array studies of reflected waves and mode conversions could be used to recognize and distinguish the two styles of deformation, and hence strain gradients, due to deformation changes and associated variations in petrofabric-derived elastic constants.
(4)
(5)
(6)
Conclusions (1)
(2)
(3)
Polymineralic high-strain zones responsible for large-scale orogenic displacement have different CPO and seismic properties depending on whether they are mylonitic or cataclastic. ‘Mylonitic’ shear zones exhibit specific nonrandom CPO and seismic characteristics. Girdle distributions develop typically parallel to mylonitic foliation, while maxima and/or minima in CPO develop parallel to mylonitic lineation. Such CPO characteristics result in anisotropic elastic properties and hence relatively large seismic anisotropy, with AVs girdles and AVs-maxima developing parallel to mylonitic foliation and lineation, respectively. The Vs1-min is typically normal to the mylonitic foliation. Where ‘mylonitic’ shear zones have evolved from a previously deformed (gneissic) protolith, they may exhibit uncharacteristically low seismic anisotropy compared to their wall rocks. It is therefore not sufficient to use high anisotropy as indicative of localized zones of concentrated ductile deformation.
(7)
(8)
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Cataclastic fault zones exhibit random CPO fabrics and hence very weak seismic anisotropy with isotropic symmetry, although a fracture-related and/or fault cement CPO can develop. The maximum seismic velocities probably lie in the fault plane and Vs2-max may develop parallel to the fault movement direction. Seismic modelling and ray-tracing has shown that it is possible to recognize differences in deformation style. Foliation orientation and intensity have dramatic effects on seismic wave propagation. A rock with 10% AVs and a vertically aligned foliation persistent throughout a 40 km thick crust induces c. 1.2 s of shear-wave splitting compared to a horizontal alignment that induces only 0.2 s splitting. This degree and/or difference in shear-wave splitting caused by CPOinduced anisotropy should be easy to distinguish in local events or teleseismic P–S conversions at the Moho. The model results also highlight the importance of considering anisotropy in crustal seismic surveys. Inclining foliation to match geological observations at the surface indicates that relative differences in shear-wave splitting may permit a strain gradient to be measured and mapped using seismic measurements. The results of the petrofabric-derived seismological models are consistent with a pervasive pure-shear vertical stretching model for the tectonics of the NPM (e.g. Butler et al. 2002), rather than the conventional shear and fault zones localization model (e.g. Burg 1999). In addition, they challenge the interpretation of low velocities beneath the NPM in terms of partial melts and hence cast doubt on the concept of ‘tectonic aneurysm’. Petrofabric-derived seismological modelling represents a combination of micro– meso– macro scale observations that can provide a quantification of the petrophysical properties involved in large-scale geodynamic processes. Small-scale petrofabric observations potentially can be effectively used as a proxy to investigate regional-scale geodynamics in the crust. However, this contribution has concentrated only on the input
Fig. 17. Relationship between deformation-induced shear-wave splitting and tectonics of the NPM according to the pure-shear vertical stretching model of Butler et al. (2002). (a) Symmetrical regional subvertical stretching passing into a restricted zone dominated by simple shear. Note also variation in pure-shear (P) and simple-shear (S) S –C foliations (Pc, Ps, Sc and Ss respectively) and/or shear zones that can occur on all scales. (b) Effect of foliation orientation on magnitude of shear-wave splitting in terms of relative proportions of pure- (P) and simple- (S) shears and S–C foliations. Shaded regions indicate range and relative probabilities of splitting for each type of deformation.
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from CPO. Other (micro-) structural elements are also known to contribute to the seismic properties of rocks (e.g. grain shape, fractures, lithological layering, grain boundaries, etc.) and must be included for a complete analysis. Fieldwork to Nanga Parbat was funded by a Royal Society research grant (RWHB). JH thanks the UK NERC for MRes funding. Part of the SEM/EBSD facilities was supported by the UK NERC Small Grant GR9/3223 (GEL, MC). We are grateful to K. Michibayashi, an anonymous reviewer and the Special Editor, D. Prior, for their reviews and comments that have helped to improve the original version of this manuscript.
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stretching and crustal thickening at Nanga Parbat, Pakistan Himalaya: a model for distributed continental deformation during mountain building. Tectonics, 21, doi: 10.1029.2001TC901022. Chlupacova, M., Skacelova, Z. & Nehybka, V. 2003. P-wave anisotropy of rocks from the seismic area in Western Bohemia. Journal of Geodynamics, 35, 45–57. Christensen, N. I. 1971. Fabric, seismic anisotropy and tectonic history of the Twin Sisters dunite, Washington. Geological Society of America Bulletin, 82, 1681– 1694. Christensen, N. I. 1984. The magnitude, symmetry and origin of upper mantle anisotropy based on fabric analyses of ultramafic tectonites. Geophysical Journal Royal Astronomical Society, 76, 89– 111. Christensen, N. I. & Mooney, W. D. 1995. Seismic velocity structure and composition of the continental crust: a global review. Journal of Geophysical Research, 100, 9761– 9788. Coward, M. P. 1994. Continental collision. In: Hancock, P. L. (ed.) Continental Deformation. Pergamon, New York, 264–288. Crampin, S. 1981. A review of wave motion in anisotropic and cracked elastic media. Wave Motion, 3, 242– 391. Crosson, R. S. & Lin, J. W. 1971. Voigt and Reuss prediction of anisotropic elasticity of dunite. Journal of Geophysical Research, 76, 570–578. Davis, P., England, P. & Houseman, G. 1997. Comparison of shear wave splitting and finite strain from the India– Asia collision zone. Journal of Geophysical Research-Solid Earth, 102, 27 511–27 522. George, M. T., Harris, N. B. W. & Butler, R. W. H. 1993. The tectonic implications of contrasting granite magmatism between the Kohistan arc and the Nanga Parbat-Haramosh Massif, Pakistan Himalaya. In: Treloar, P. J. & Searle, M. P. (eds) Himalayan Tectonics. Geological Society, London, Special Publications, 74, 173– 191. Guest, W. S. & Kendall, J.-M. 1993. Modelling seismic waveforms in anisotropic inhomogeneous media using ray and Maslov asymptotic theory: applications to exploration seismology. Canadian Journal of Exploration Geophysics, 29, 78–92. Guest, W. S., Thompson, C. J. & Spencer, C. P. 1993. Anisotropic reflection and transmission calculations with application to a crustal seismic survey from the East Greenland Shelf. Journal of Geophysical Research, 98, 14 161– 14 184. Helbig, K. 1993. Simultaneous observation of seismic waves of different polarization indicates subsurface anisotropy and might help to unravel its cause. Journal Applied Geophysics, 30, 1– 24. Helbig, K. 1994. Next moves in anisotropy. Oilfield Review, 6, 1 –1. Hess, H. H. 1964. Seismic anisotropy of the uppermost mantle under oceans. Nature, 203, 629. Hill, R. 1952. The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society, London, A, 65, 351– 354. Kendall, J.-M. 2000. Earth’s Deep Interior: Mineral Physics and Tomograpy from the Atomic to the Global Scale. Geophysical Monograph Series, 117, American Geophysical Union, Washington.
FROM CRYSTAL TO CRUSTAL Kern, H. 1982. P- and S-wave velocities in crustal and mantle rocks under the simultaneous action of high confining pressure and high temperature and the effect of microstructure. In: Schreyer, W. (ed.) High Pressure Researches in Geoscience. Verlagsbuchhandlung, Stuttgart, 15–45. Kern, H. & Wenk, H. R. 1990. Fabric-related velocity anisotropy and shear wave splitting in rocks from the Santa Rosa mylonite zone, California. Journal of Geophysical Research (Solid Earth), 95, 11 213– 11 223. Khananehdari, J., Rutter, E. H., Casey, M. & Burlini, L. 1998. The role of crystallographic fabric in the generation of seismic anisotropy and reflectivity of high strain zones in calcite rocks. Journal of Structural Geology, 20, 293–300. Kocks, U. F., Tome, C. N. & Wenk, H.-R. 1998. Texture and Anisotropy, Preferred Orientation in Polycrystals and the Effect on Material Properties. Cambridge University Press, Cambridge. Lister, G. S. & Snoke, A. W. 1984. S-C mylonites. Journal of Structural Geology, 6, 617–638. Lloyd, G. E. & Kendall, J.-M. 2005. Petrofabric derived seismic properties of a mylonitic quartz simple shear zone: implications for seismic reflection profiling. In: Harvey, P. K., Brewer, T., Pezard, P. A. & Petrov, V. A. (eds) Petrophysical Properties of Crystalline Rocks. Geological Society, London, Special Publications, 240, 75–94. Lloyd, G. E., Butler, R. W. H., Casey, M. & Mainprice, D. 2009. Deformation fabrics, mica and seismic properties of the continental crust. Earth and Planetary Sciences Letters, 288, 320 –328. Lloyd, G. E., Butler, R. W. H., Casey, M., Tatham, D. J. & Mainprice, D. 2011. Constraints on the seismic properties of the middle and lower continental crust. In: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anistropy. Geological Society, London Special Publications, 360, 7–32. Mahan, K. 2006. Retrograde mica in deep crustal granulites: implications for crustal seismic anisotropy. Geophysics Research Letters, 33: Art. No. L24301 Dec 16. Mainprice, D. 1990. An efficient Fortran program to calculate seismic anisotropy from the lattice preferred orientation of minerals. Computers and Geosciences, 16, 385–393. Mainprice, D. 2003. http://www.gm.univ-montp2.fr/ PERSO/mainprice Mainprice, D. & Nicolas, A. 1989. Development of shape and lattice preferred orientations: application to the seismic anisotropy of the lower crust. Journal of Structural Geology, 11, 175–189. Mainprice, D. & Silver, P. G. 1993. Interpretation of SKS waves using samples from the subcontinental lithosphere. Physics of Earth and Planetary Interiors, 78, 257–280. Mainprice, D. & Humbert, M. 1994. Methods of calculating petrophysical properties from lattice preferred orientation data. Survey Geophysics, 15, 575–592. Makovsky, Y., Klemperer, S. L., Ratschbacher, L., Brown, L. D., Li, M., Zhao, W. J. & Meng, F. L.
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1996. INDEPTH wide-angle reflection observation of P-wave-to-S-wave conversion from crustal bright spots in Tibet. Science, 274, 1690– 1691. Meltzer, A. & Christensen, N. 2001. Nanga Parbat crustal anisotropy: implications for interpretation of crustal velocity structure and shear-wave splitting. Geophysical Research Letters, 28, 2129– 2132. Meltzer, A., Sarker, G., Beaudoin, B., Seeber, L. & Armbruster, J. 2001. Seismic characterization of an active metamorphic massif, Nanga Parbat, Pakistan Himalaya. Geology, 29, 651–654. Meltzer, A., Christensen, N. & Okaya, D. 2004. Crustal seismic anisotropy: implications for understanding crustal dynamics. In: Proceedings of American Geophysical Union, Fall Meeting 2003, abstract #S11C-0302. http://adsabs.harvard.edu//abs//2003 AGUFM.SIICO302W. Michibayashi, K. & Mainprice, D. 2004. The role of preexisting mechanical anisotropy on shear zone development within oceanic mantle lithosphere: an example from the Oman ophiolite. Journal of Petrology, 45, 405– 414. Michibayashi, K., Ina, T. & Kanagawa, K. 2006. The effect of dynamic recrystallization on olivine fabric and seismic anisotropy: insight from a ductile shear zone, Oman ophiolite. Earth and Planetary Sciences Letters, 244, 695–708. Nicolas, A. & Poirier, J.-P. 1976. Crystalline Plasticity and Solid-State Flow in Metamorphic Rocks. Wiley, New York. Park, S. K. & Mackie, R. J. 1997. Crustal structure at Nanga Parbat, northern Pakistan, from magnetotelluric soundings. Geophysics Research Letters, 24, 2415– 2418. Park, S. K. & Mackie, R. J. 2000. Resistive (dry?) lower crust in an active orogen, Nanga Parbat, northern Pakistan. Tectonophysics, 316, 359–380. Passchier, C. W. & Trouw, R. A. J. 1996. Microtectonics. Springer-Verlag, Berlin. Poage, M. A., Chamberlaine, C. P. & Craw, D. 2000. Massif-wide metamorphism and fluid evolution at Nanga Parbat, northwestern Pakistan. American Journal of Science, 300, 463–482. Prior, D. J., Boyle, A. P. et al. 1999. The application of electron backscatter diffraction and orientation contrast imaging in the SEM to textural problems in rocks. American Mineralogist, 84, 1741–1759. Ramsay, J. G. 1980. Shear zone geometry. A review. Journal of Structural Geology, 2, 83– 89. Reuss, A. 1929. Berechnung der Fließgreze von Mischkristallen auf Grund der Plastizitatsbedinggungen fur Einkristalle. Zeitschrift fur Angewandte Mathematik unf Mechanik, 9, 49–58. Rudnick, R. L. & Fountain, D. M. 1995. Nature and composition of the continental crust – a lower crustal perspective. Reviews in Geophysics, 33, 267– 309. Shapiro, N. M., Ritzwoller, M. H., Molnar, P. & Levin, V. 2004. Thinning and flow of Tibetan crust constrained by seismic anisotropy. Science, 305, 233–236. Sibson, R. H. 1977. Fault rocks and fault mechanisms. Journal Geological Society of London, 133, 191– 213. Siegesmund, S., Takeshita, T. & Kern, H. 1989. Anisotropy of Vp and Vs in an amphibolite of the deeper
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Deformation mechanisms of plagioclase and seismic anisotropy of the Acebuches metabasites (SW Iberian massif) ´ NDEZ3 MANUEL DI´AZ-AZPIROZ1*, GEOFFREY E. LLOYD2 & CARLOS FERNA 1
Departamento de Sistemas Fı´sicos, Quı´micos y Naturales. Universidad Pablo de Olavide, Crtra. Utrera, 41013 Seville, Spain 2
School of Earth & Environment, University of Leeds, Leeds, LS2 9JT UK
3
Departamento de Geodina´mica y Paleontologı´a, Facultad de Ciencias Experimentales, Universidad de Huelva, Campus de El Carmen, 21071 Huelva *Corresponding author (e-mail:
[email protected]) Abstract: Samples of the Acebuches metabasites (SW Spain), deformed under low-pressure/ medium-to-high temperature metamorphic conditions, have been analysed via electron backscattered diffraction (EBSD) to obtain their plagioclase crystal lattice preferred orientations (LPO). Plagioclases from the highest temperature amphibolites show moderate LPO and a good correlation between 1808 misorientation angles and both the crystal and the kinematic coordinate systems, which is attributed to dislocation glide accommodated by mechanical albite þ pericline twinning. Plagioclases from medium-temperature amphibolites exhibit well-developed LPO, suggesting that dislocation creep was active during plagioclase deformation. Plagioclases from the more intensively deformed mafic schists exhibit weak LPO, indicating the activity of LPOdestroying deformation mechanisms. Evidence points to grain-boundary sliding accompanied by limited fracturing. The observed LPO are characterized by the alignment of [100] parallel to the kinematic X-direction. This association suggests that [100] was the preferential slip direction during dislocation creep of plagioclase, with (010) and/or (001) appearing to have acted as the dominant slip planes. The observed plagioclase LPO is combined with hornblende LPO to define the seismic fabric of the Acebuches metabasites. In samples with strong plagioclase LPO, the resulting seismic fabrics are highly influenced by this phase.
Plagioclase is a major component of the crust and it is therefore a common constituent of many shear zones, particularly in the lower crust. While lower crustal rheology remains debated, most models assume that in felsic lithologies it is mainly controlled by plagioclase (e.g. Ord & Hobbs 1989). Understanding of the deformation behaviour of plagioclase feldspar is therefore of crucial significance. With the advent of misorientation analysis using the scanning electron microscope (SEM) electron backscattered diffraction technique (EBSD), low symmetry minerals such as plagioclase can be investigated in much more detail than possible previously. Consequently, naturally deformed plagioclase has received more attention recently (e.g. Prior & Wheeler 1999; Jiang et al. 2000; EgydioSilva et al. 2002; Baratoux et al. 2005; Kanagawa et al. 2008) although comparatively little is known as to how this mineral accommodates deformation, as stated by different authors (e.g. Prior & Wheeler 1999; Stu¨nitz et al. 2003; Kanagawa et al. 2008). Furthermore, there has been no attempt to construct models that reproduce plagioclase lattice preferred orientations (LPO) development.
Deformation mechanisms or processes that have been recognized in experimentally and naturally deformed plagioclase under a wide range of temperatures include: brittle cataclastic flow, dislocation glide, dislocation creep accommodated by subgrain rotation and grain-boundary migration dynamic recrystallization, chemically induced recrystallization and grain-boundary sliding assisted by diffusion creep (e.g. Tullis & Yund 1985, 1987; Olesen 1987; Ji & Mainprice 1990; Heidelbach et al. 2000; Shigematsu & Tanaka 2000; Kruse et al. 2001; Egydio-Silva et al. 2002; Rosenberg & Stu¨nitz 2003; Baratoux et al. 2005; Kanagawa et al. 2008). Factors that determine the mechanism(s) or process(es) active include temperature and kinematics of deformation, plagioclase composition, rock mineralogy, phase strength contrast between layers, grain size and water content (e.g. Kruhl 1987; Ji & Mainprice 1990; Heidelbach et al. 2000; Rosenberg & Stu¨nitz 2003; Rybacki & Dresen 2004). In this study, LPO and misorientation analysis of plagioclases from the Acebuches metabasites (SW Spain) obtained via SEM/EBSD are presented.
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 79– 95. DOI: 10.1144/SP360.5 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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The Acebuches metabasites comprise a wide range of medium- to high-grade metamorphic rocks (amphibolites and mafic schists) of basic (hornblende and plagioclase) composition. The two principal minerals present are arranged in an interconnected weak layer structure with a high-strength contrast. Strain is partitioned between the phases, being concentrated in the plagioclase grains which represent the weaker phase, whereas hornblende grains (the strong phase) rotate in a rigid manner (Dı´az-Azpiroz et al. 2007). The principal objective of the present work is to characterize the deformation mechanisms that affected the plagioclase in the Acebuches metabasites. These rocks define a major contact in the southwest Iberian massif, interpreted as a Variscan suture (e.g. Crespo-Blanc & Orozco 1988; Castro et al. 1996; Simancas et al. 2003; Dı´az-Azpiroz & Ferna´ndez 2005) imaged in the IBERSEIS seismic profile (Simancas et al. 2003). The second objective is therefore to determine how the plagioclase and hornblende LPO (see also Dı´az-Azpiroz et al. 2007) combine to define the seismic fabric of these rocks.
Geological setting and sample details The Acebuches metabasites define a long (.200 km) and narrow (c. 1 km), WNW –ESE oriented and NNE-dipping, belt located at the contact between two of the main units of the Iberian Massif (SW European Variscan Chain), namely: the Ossa-Morena zone (OMZ) to the NNE and the South-Portuguese zone (SPZ) to the SSW (Fig. 1). Structural, petrological and geochemical features suggest that the protolith of the Acebuches metabasites was MORB-like oceanic crust (Bard & Moine 1979; Quesada et al. 1994; Castro et al. 1996), which was affected by two consecutive Variscan tectono-metamorphic events (Castro et al. 1996; Dı´az-Azpiroz & Ferna´ndez 2005). The first stage affected the entire rock sequence and involved an amphibolite to granulite facies metamorphism (M1) and a ductile shear deformation (D1). Subsequently, the structural base of the Acebuches metabasites was affected by the Southern Iberian Shear Zone (SISZ, Crespo-Blanc & Orozco 1988; Dı´az-Azpiroz & Ferna´ndez 2005), which produced a retrograde greenschist facies metamorphism (M2) and a transpressional ductile shear deformation (D2). The characteristics of the structures generated during D2 in the SISZ suggest that finite strain was higher toward the structural base of the Acebuches metabasites (Dı´az-Azpiroz & Ferna´ndez 2003, 2005). As a consequence of this tectono-metamorphic evolution, four different rock types can be distinguished in the Acebuches metabasites. These are, from top to bottom: (1) clinopyroxene-bearing
(Pl þ Hb þ Di + Qtz), coarse-grained banded amphibolites; (2) medium-grained banded amphibolites (Pl þ Hb + Qtz); (3) mylonitic amphibolites (Pl þ Hb + Qtz) and (4) mylonitic mafic schists (Pl þ Hb þ Act þ Ep þ Chl + Qtz). The structural features of (1) and (2) are due to D1 whereas the latter two rock types present structures formed by the superposition of D2 over D1. Also, the superposition of M1 and M2 gave place to a metamorphic field gradient across the Acebuches metabasites series ranging from the greenschist-lower amphibolite facies transition in the mafic schists, through the amphibolite facies, to the upper amphibolite-lower granulite facies transition in the clinopyroxenebearing amphibolites (Castro et al. 1996; Dı´azAzpiroz et al. 2006, 2007). In the four rock types, plagioclase shows a composition (as determined by EDS on a JEOL-JSM 5410 SEM, see Dı´azAzpiroz et al. 2007 for details) that ranges from An23 to An47, although most analyses show a narrower composition array (An32239) corresponding to andesine. Lesser amounts of oligoclase and rare labradorite have also been found. Ten samples from three different cross-sections of the Acebuches metabasites have been analysed in this study. From east to west and from north to south, the cross-sections and samples are (Fig. 1a): V1, V3 and V4 (Almonaster cross-section), A0 to A4 (Veredas cross-section) and PV3 and PV7 (Puerto de Veredas cross-section). All samples have been projected onto the Veredas cross-section (Fig. 1b), although distances between them are approximate. Grain size and shape, as well as mineral chemistry and thermometry of syndeformational assemblages of the Veredas cross-section are well documented in the literature (Dı´az-Azpiroz & Ferna´ndez 2003; Dı´az-Azpiroz et al. 2006, 2007) and are summarized in Figure 1c. According to the four rock types previously described, sample A0 is a clinopyroxene-bearing, coarse-grained banded amphibolite, samples A1 and A2 are medium-grained banded amphibolites, samples A3, V1, V3 and PV3 are mylonitic amphibolites and samples A4, V4 and PV7 are mylonitic mafic schists (Fig. 1, Table 1). The main foliation (S1) in the coarse- and medium-grained banded amphibolites (samples A0 to A2) is defined by a centimetre – decimetre-scale grain-size layering together with the shape preferred orientation of hornblende blasts, which also align to define a weak lineation (L1). S1 is better developed in the coarse-grained amphibolites than in the medium-grained amphibolites. In both rock types, the microstructure is mainly granoblastic, being mainly defined by plagioclase. Moreover, triple junctions defined by plagioclase and hornblende blasts (Fig. 2a) suggest chemical equilibrium between these two phases. Hornblende, plagioclase
DEFORMATION MECHANISMS OF PLAGIOCLASE 81
Fig. 1. (a) Geological map of the Acebuches metabasites (after Dı´az-Azpiroz & Ferna´ndez 2005. Copyright American Geophysical Union 2005) with lower hemisphere, equal-area projections of poles to S1 and S2, average foliation planes and L1 and L2. Location of the three studied cross-sections and samples is also shown. Inset is a general map of the European Variscan Chain with the location of the studied area. Geographical and UTM (zone S-29, data in kilometres) coordinates are shown for reference. (b) Veredas cross-section with projection of the studied samples. Please note that distances between samples are approximate. (c) Evolution with distance to the structural base of the Acebuches metabasite sequence, in the Veredas cross-section, of deformation temperature and finite strain related to the Southern Iberian shear zone (based on Dı´az-Azpiroz et al. 2006. Copyright American Geophysical Union 2006). Reproduced by permission of American Geophysical Union.
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Table 1. Main characteristics of the Acebuches metabasites samples used in this study, including rock type. (CGBA: coarse-grained banded amphibolites; MGBA: medium-grained banded amphibolites; MA: mylonitic amphibolites; and MS: mafic schists; deformation phase (D-phase) registered by the sample, approximate evolution of temperature and finite strain across the sequence, J-index of the ODF and slip systems determined by LPO. Temperatures of samples from the Veredas cross-section (samples A0 to A4) are from Dı´az-Azpiroz et al. (2007). Note that main slip planes and directions appear in bold, secondary or poorly defined slip planes and directions in normal text Sample A0 A1 A2 A3 V1 V3 PV3 A4 V4 PV7
Rock type
D-phase
CGBA MGBA MGBA MA MA MA MA MS MS MS
1 1 1 2 2 2 2 2 2 2
Temp (8C) 825 770 750 720
Strain
"? " #
655
and diopside rotated porphyroblasts are wrapped by S1. Plagioclase grain size shows a wide distribution with an average ranging from 240 to 365 mm in the coarse-grained amphibolites and from 140 to 160 mm in the medium-grained amphibolites (Dı´az-Azpiroz & Ferna´ndez 2003). Plagioclase porphyroblasts and matrix blasts can be either anhedral (Fig. 2b) or subhedral (Fig. 2a, c) and exhibit both growth and mechanical twinning, the latter indicated by tapering of the twin elements (Fig. 2a –c). Some mechanical twins appear to define conjugate orthogonal systems within individual grains (Fig. 2a). Other characteristic microstructures include micro-kinks (Fig. 2b, c), nucleated in some porphyroblasts about cataclastic fractures (Fig. 2c) and, within larger plagioclase grains, undulose extinction (Fig. 2b) and deformation bands (Fig. 2a). Several porphyroblasts show subgrains at the rims (Fig. 2a) and sometimes within them (Fig. 2b), and are surrounded by a plagioclase-rich mantle with grains of similar size (Fig. 2a– c). Bulging structures (Fig. 2a) suggest that grain-boundary migration (GBM) also occurred. Metabasites deformed by the SISZ (samples A3, A4, V1, V3, V4, PV3 and PV7) present a penetrative mylonitic foliation (S2) defined by the alignment of amphibole blasts (Fig. 2d, e). In the mafic schists, S2 is reinforced by the presence of plagioclase-rich, amphibole-rich and epidote þ chlorite-rich layers. On the foliation planes, a stretching lineation defined by plagioclase ribbons (L2) is commonly observed. Plagioclase blasts showing negligible internal deformation define a regular, fine-grained (40–75 mm in the mylonitic amphibolites and 10–21 mm in the mafic schists, Dı´az-Azpiroz & Ferna´ndez 2003) granoblastic matrix (Fig. 2d, f ), while foliation defined by elongated amphibole
J-index
Slip plane
1.4635 1.5486 1.4130 1.7212 2.0143 1.3374 1.6751 1.3239 1.3525 1.3740
(010) (001) (010) (001) (001) (001) (001) (010) (001) (010) (001) (010) (010) (001) (010)
Slip direction
[100] [100] [100] [100] [100] [100]
blasts wraps around hornblende and plagioclase porphyroclasts (Fig. 2e). Plagioclase porphyroclasts (Fig. 2e) exhibit numerous mechanical twins with tapered ends and sometimes undulose extinction and fracturing. In addition, they commonly show subgrains at the rims and are surrounded by grains of similar size as these subgrains. These core-andmantle structures give rise to s-type rotated porphyroclasts (Fig. 2e) that occasionally develop into completely recrystallized ribbons. Plagioclase matrix blasts also present fracturing (Fig. 2d, f ) and evidence for GBM (Fig. 2f ).
Methodology Plagioclase petrofabric XZ (perpendicular to the mylonitic foliation and parallel to the stretching lineation) sections from each sample were prepared via mechanical and chemical polishing, and then coated with a thin film of carbon deposited by vacuum evaporation (e.g. Lloyd 1987). The petrofabric of plagioclases was determined via EBSD in a CamScan SEM at the University of Leeds. SEM operating conditions were 20 keV accelerating voltage, c. 20 nA specimen current and a working distance of c. 40 mm, with specimens tilted at 758. Plagioclase EBSD patterns were automatically indexed using the Channel 5 software of HKL Technology incorporating specific plagioclase (andesine) composition crystal files obtained from the UK Engineering and Physical Science Research Council (EPSRC) Chemical Database Service (e.g. Fletcher et al. 1996) and the Cambridge Structural Database (e.g. Allen 2002). The analysed points were derived from a rectangular grid. A large step size was used to obtain LPO plots, thus preventing
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Fig. 2. Plagioclase microstructures from (a –d) the Acebuches banded amphibolites and (e, f ) the mylonitic metabasites. (a) Subhedral porphyroclast showing deformation bands and subgrains surrounded by a finer-grained granoblastic matrix. A grain with two orthogonal mechanical twin systems and bulging (suggesting GBM) can also be observed. (b) Porphyroclast showing mechanical twinning (tapering of the twin elements), undulose extinction and micro-kinking. Finer plagioclase subgrains and grains surround the porphyroclasts and appear also along specific planes within it. (c) Porphyroclast affected by mechanical twinning (tapering of the twin elements) and micro-kinking nucleated about a fracture. (d) Fine-grained granoblastic aggregate of slightly deformed plagioclase blasts, with some micro-cracks, and well-oriented amphibole prisms. (e) s-type rotated porphyroclast showing mechanical twinning with tapered elements, undulose extinction and fracturing. (f ) Plagioclase blasts from the matrix, showing granoblastic texture and several evidences for GBM, such as bulging and ‘pinning’ structures. All are cross-polarized light microscope images except (d), which is a SEM/BSE image.
oversampling of larger grains. In contrast, for misorientation analyses, the step size was approximately one-quarter of the average plagioclase grain
size for each sample. Automatically indexed EBSD patterns of plagioclase have often been dismissed because of their low reliability (e.g. Prior &
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Wheeler 1999), although this technique has improved significantly in the last years. Several analyses from different rock types were manually checked to preclude mis-indexing. Moreover, the plagioclase indexing rate (plagioclase analyses with MAD ,1 over total plagioclase indexed points) is over 70% for all studied samples, suggesting the results presented here are reliable. The EBSD data were processed into conventional LPO pole figures for selected crystallographic planes and directions using the program PFch5 (Mainprice 2010). Plagioclase is triclinic and, as suggested by Prior & Wheeler (1999), both lower and upper hemisphere projections are presented (Fig. 3). In addition, misorientation axis and misorientation angle data are presented (Fig. 4). Any two adjacent crystal lattices of a certain phase are related by what is known as misorientation (e.g. Randle 1993; Lloyd et al. 1997): a crystallographic direction (misorientation axis) about which different rotations can bring both lattices into parallelism. By convention, the lowest rotation angle is chosen as the misorientation angle. However, in triclinic symmetry phases, such as plagioclase, there is a unique misorientation axis/angle pair between two adjacent lattices (e.g. Kruse et al. 2001; Wheeler et al. 2001). In this study, neighbour-pair misorientation axis/angle pairs were derived using the Channel software on samples representative of the three studied rock types: A2 (banded amphibolites), V1 (mylonitic amphibolites) and A4 (mafic schists). Data are presented on angle frequency distributions, direct pole (sample coordinates) and inverse pole (crystal coordinates) figures.
Seismic fabrics The seismic properties of the studied metabasites have been calculated according to the assumption that they depend on the LPO single-crystal elastic constant and density of the constituent phases and their volume fraction within the rock (e.g. Mainprice & Humbert 1994). Plagioclase LPO has been obtained as described above and is shown in Figure 3. For comparative purposes, hornblende LPO (Dı´az-Azpiroz et al. 2007) of selected samples is presented in Figure 5. The LPO-derived seismic properties of each sample have been calculated using programs Anisch5 and VpG (Mainprice 1990; Mainprice & Humbert 1994). Single-crystal elastic parameters of plagioclase (Aleksandrov et al. 1974) and hornblende (Aleksandrov & Ryzhova 1961) have been used. The volume fraction of plagioclase (54%) and hornblende (46%) are, for all cases, the average modal fraction between these two phases in the studied samples. The obtained results of three selected samples representing the three main rock types (banded amphibolites,
mylonitic amphibolites and mafic schists) are displayed in pole figures showing the azimuthal distributions of the seismic properties (Fig. 6). The main velocity data from all analysed samples are summarized in Table 2. Reflectivity coefficients (Rc) between adjacent samples have been calculated using Equation 1 (Ji et al. 1993), considering density (r) as a constant: Rc ¼
r1 V1 r2 V2 V1 V2 ¼ r1 V1 þ r2 V2 V1 þ V2
(1)
where V1 and V2 are the Vp velocities in the vertical direction for any two adjacent samples.
Results Lattice preferred orientation The LPO of the banded amphibolite (samples A0 to A2, Fig. 3a–c) are of moderate intensity. In samples A0 and A1 [100] defines a single maximum, whereas [010] and [001] define several maxima that could be arranged in weak girdles only in the case of [001]. In sample A2 [100] aligns parallel to the XY-plane (i.e. parallel to the foliation) and defines a maximum subparallel to the Y-direction (i.e. normal to the lineation). The distributions of [010] and [001] show maxima parallel to the Z-direction and near the Y-direction, respectively. In samples A1 and A2, maxima of (010) and (001) close to the Z-direction are also observed. In turn, the distribution of planes in sample A0 is more ambiguous as maxima on the YZ-plane and between both directions could be deduced for (100) and (001). Pole figures from the Acebuches metabasites deformed by the SISZ (mylonitic amphibolites and mafic schists) are characterized by a monoclinic symmetry defined by a Y–Z mirror plane and a binary axis parallel to X. This introduces slight differences between lower and upper hemisphere projections of the crystallographic axes (Fig. 3e– i). The monoclinic appearance is mainly due to the common position of [010] maxima, which is slightly off-centre. A possible explanation for this could be related to the complex transpressional nature of the SISZ (e.g. Dı´az-Azpiroz & Ferna´ndez 2005), but further work should be carried out to better constrain this issue. The mylonitic amphibolites (samples A3, V1, V3 and PV3, Fig. 3d –g) exhibit well-developed LPO characterized by the alignment of [100] parallel to the tectonic X-direction (the stretching lineation related to the SISZ), although roughly describing a great circle around the Z-direction (i.e. parallel to the foliation plane). In contrast, the orientations of [010] and [001] are somewhat different. In sample V1, [010] is subparallel to the Y-direction (although it shows a secondary
DEFORMATION MECHANISMS OF PLAGIOCLASE
maximum near the Z-direction), whereas [001] shows different maxima on both sides of the Z-direction (poles to the mylonitic foliation of the SISZ) joined by a girdle. This is in agreement with the disposition of the (001) planes parallel to the foliation plane. In sample A3, [010] defines a girdle subparallel to XZ-plane and [001] concentrates in two maxima close to the YZ-plane. Finally, in samples V3 and PV3, [010] lies parallel to the Z-direction whereas [001] approximates the YZ-plane. The distribution of poles to (010) can be used as a sense of shear criterion (Ji et al. 1988). In the mylonitic amphibolites, the poles to (010) are either symmetric or slightly asymmetric, suggesting D2 involved both coaxial and noncoaxial strain. In asymmetric (010) fabrics, the deduced sense of shear is either synthetic with other shear sense criteria, that is sinistral (samples A2 and V3) or antithetic (i.e. dextral, sample PV3). Mafic schists (samples A4, V4, and PV7, Fig. 3h–j) show poorly developed LPO. Fabrics from samples V4 and A4 resemble those of the mylonitic amphibolites, with [100] aligning subparallel to the X-direction and [010] defining a girdle parallel to the YZ-plane and a maximum in an intermediate position between both axes. [001] pole figures are vague in both cases. Sample PV7 exhibits the weakest LPO among rocks affected by the SISZ. Its most remarkable feature is the clustering of the [010] direction onto the YZ-plane in an intermediate position between both directions. Poles to (010) and (001) from the three mafic schist samples plot within the YZ-plane. In this case, (001) plots closer to the Z-direction.
Misorientation analysis Frequency distribution diagrams of misorientation angles from three different rock types (banded amphibolites, mylonitic amphibolites and mafic schists) are presented in Figure 4, and are compared with the theoretical random distribution for plagioclase. The misorientation angle distributions of the three analysed samples (Fig. 4) are similar, being characterized by (1) a distribution close to the theoretical random distribution for most angles (10–1758), (2) slightly larger frequencies than the theoretical distribution for low angles (5 –108) and (3) significantly larger frequencies than the theoretical distribution for high angles (175 –1808). Low-angle misorientation axes show a random distribution when represented both in the kinematic and in the crystallographic reference frames (Fig. 4). In turn, high-angle (175 –1808) misorientation axes show a good correlation with the crystallographic directions (maxima parallel to [001], the pole to (010) and the direction orthogonal to [001] and parallel to (010), Fig. 4). Correlation of these
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misorientations with the kinematic reference frame (sample coordinates) is fairly good for sample A2 (Fig. 4a) where they adjust to a girdle parallel to the XZ-plane; weak for sample V1 (Fig. 4b) with maxima close to X-direction and a diffuse girdle around the XZ-plane; and non-existent for sample A4 (Fig. 4c).
Seismic fabrics The LPO-derived seismic properties of the studied samples (Fig. 6) show distinct azimuthal distributions depending on the samples. In samples A2 (banded amphibolites) and A4 (mafic schists), the maximum values of both Vp and AVs are subparallel to the fastest hornblende crystallographic direction [001]. This also holds true for the maximum value of AVs in sample V1. Therefore, as stated by other authors (e.g. Siegesmund et al. 1989; Kitamura 2006; Tatham et al. 2008), hornblende largely control the seismic properties of ductile deformed amphibolites such as the Acebuches metabasites. In contrast, the maximum Vp value of sample V1 shows a somewhat intermediate position between hornblende [001] and the fastest plagioclase crystallographic direction [010]. This suggests a higher influence of plagioclase LPO, which is more intense in sample V1 than in the other two (see also Table 1). Other minor deviations of the seismic fabrics from hornblende LPO (Fig. 5) can also be interpreted in terms of some influence exerted by plagioclase LPO (both strength and orientation). For example, the minimum Vp value of samples A2 and V1 presents an intermediate position between the slowest directions of hornblende and plagioclase ([100] in both minerals). Hornblende [100] is subparallel to Z in both samples (Fig. 5a, b), whereas plagioclase [100] is subparallel to Y in sample A2 (Fig. 3c) and to X in sample V1 (Fig. 3e). Vp in the vertical direction shows rather low values and a fairly restricted range from 6.33 (sample V4) to 6.53 (sample A0), being below 6.4 for most samples (important for seismic reflectivity). This is due to the structural attitude of the Acebuches metabasites, strongly dipping and with a low plunging stretching lineation. Moreover, slight differences in plagioclase –hornblende volume fractions (e.g. Castro et al. 1996; Dı´az-Azpiroz & Ferna´ndez 2005) cause very similar densities for all studied samples. For these reasons, the reflectivity coefficients between adjacent samples (Table 2) are rather low (Rc ¼ 0.000–0.004), except for that between samples A0 and A1 (Rc ¼ 0.011). This feature would account for the weak reflectors shown in the IBERSEIS seismic profile (Simancas et al. 2003). In the case of the Acebuches metabasites, the reflectors observed in this profile are probably related to the anisotropy produced by the grain-size
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(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Fig. 3. Lattice preferred orientation (LPO) plots for the studied plagioclases. Lower and upper hemisphere, polar data, equal-area projections of [100], [010] and [001] crystallographic directions, as well as lower hemisphere, equal-area projections of poles to (100), (010) and (001) planes are presented. Contour intervals counted as multiples of uniform
DEFORMATION MECHANISMS OF PLAGIOCLASE
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Fig. 3. (Continued) distribution (mud) ¼ 0.5; broken line, minimum (0.5 mud) contour interval; grey shading corresponds to a threshold mud ¼ 3.0; black squares and white triangles are maximum and minimum densities respectively; n, number of measurements. Samples are: (a) A0, (b) A1, (c) A2, (d) A3, (e) V1, (f ) V3 (g) PV3, (h) A4, (i) V4 and ( j) PV7.
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layering of the banded amphibolites and to limited fracturing occurring in the mafic schists (see below).
Discussion Deformation of plagioclase from the Acebuches metabasites took place mainly by crystal plasticity as deduced by the optically observed microstructures, such as: mechanical twinning, micro-kinking, deformation bands, subgrains and small grains
(a)
(b)
(c)
present at the mantles of porphyroclasts or polygonal regularly grained aggregates. Fracturing also appears as an important deformation mechanism, especially in lower temperature mafic schists. However, the relative importance of each deformation mechanism in the three main rock types studied (i.e. coarse and medium-grained banded amphibolites, mylonitic amphibolites and mafic schists) can be constrained better by LPO and misorientation analyses.
Slip-system transitions Plagioclase in the Acebuches metabasites accommodated deformation in part by crystal plasticity: specifically, their LPO (Fig. 3) indicate slip on the (010) and/or the (001) planes, with [100] as the main slip direction (Table 1). (010) [100] has also been recognized as a main slip system in the Acebuches metabasites by U-stage measurements (Duclo´s 2004). Secondary slip direction [010] could also have been active in sample A3 (Fig. 3d, Table 1). The latter coincide with slip directions reported by, for instance, Shigematsu & Tanaka (2000) and Kruse et al. (2001). Also, [201], [1 1 0] and [110] have been reported as secondary slip directions in similar samples from the Acebuches metabasites analysed via U-stage (Duclo´s 2004). Neither (010) [100] nor (001) [100] are common slip systems in plagioclase. However, both have been identified as secondary slip systems (Ji & Mainprice 1990; Kruse et al. 2001) and (010) [100] has also been described as a main slip system in naturally (Schulmann et al. 1996; EgydioSilva et al. 2002) and experimentally (Tullis & Yund 1977) deformed plagioclases. Moreover, Marshall & McLaren (1977) suggest (001) [100] as the main slip system in experimentally deformed
Fig. 4. Neighbour-pair misorientation analysis for samples (a) A2, (b) V1 and (c) A4. For each sample, misorientation angle frequency distribution diagram and theoretical random distribution curve for plagioclase are shown, together with pole figures of misorientation axis for angles ranging from 58 to 108 and from 1758 to 1808. Pole figures of misorientation axis for intermediate angles (55–608) of sample A4 are shown also. In each figure, the upper-hemisphere equal-area inverse-pole figure (crystal coordinates) appears at the top. In these, [001] is vertical and upwards, the positive pole to (010) points to the east and the direction orthogonal to [001] contained in (010) points to the south (Burri et al. 1967; Kruse et al. 2001). Contour intervals are multiples of uniform distribution (mud). Half width is 108 and cluster size is ,18. Minimum density is 0.00. Lowerhemisphere equal-area pole figures (sample coordinates) appear at the bottom in each figure. Contour intervals are multiples of uniform distribution (mud). Half width is 108 and cluster size is 38. Minimum density is 0.00.
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(a)
(b)
(c)
Fig. 5. Lower-hemisphere equal-area projections of (100) and (010) planes and [001] direction of amphiboles from samples (a) A2, (b) V1, and (c) A4 (from Dı´az-Azpiroz et al. 2007). Contour intervals: 0.5 multiples of uniform distribution (mud); broken line: minimum (0.5 mud) contour interval; black squares: maximum densities; white triangles: minimum densities; n: number of measurements.
andesine (the major plagioclase composition in the Acebuches metabasites) and (001) has been reported as the principal slip plane in plagioclases from gabbro mylonites (Kanagawa et al. 2008; Raimbourg et al. 2008). The simultaneous activity of multiple slip systems in plagioclase has been explained by Shigematsu & Tanaka (2000) by differences in the deformation conditions. Furthermore, Kruhl (1987) proposed a transition from slip on (001) to (010) associated with a temperature decrease. In the present case, the deduced change in the active slip plane could be related to (1) changes in the temperature conditions during deformation and/or (2) differences in deformation history. The latter could include (Dı´az-Azpiroz & Ferna´ndez 2003, 2005) the amount of finite strain, the occurrence of a single deformation phase (D1) versus a combination of two stages (D1 and D2) or the kinematics of deformation (e.g. Heidelbach et al. 2000 have determined different active slip planes during experimental deformation of albite by imposed axial compression and by simple shear). In the Acebuches metabasites affected by the SISZ, the data suggest that (001) is the dominant
slip plane for the medium-temperature amphibolites (samples A3 and V1, with marginal activity of (010) deduced for sample V1). As temperature decreases and finite strain increases within this shear zone (samples V3, PV3, A4, V4 and PV7), there appears to be a continuous transition in the main slip plane from (001), which still remains active, to (010). It could be hypothesized that if deformation conditions had continued changing in the same sense (i.e. temperature decrease and/or finite strain increase), (001) would have become completely inactive and (010) would have been the only slip plane. The combination of (010) as the main slip plane and (001) as the secondary slip plane can also be deduced from the banded amphibolites (samples A1 and A2). In this case, however, it seems improbable that the activity of (010) is related to a temperature decrease, as these rocks deformed under upper amphibolite facies conditions (Castro et al. 1996; Dı´az-Azpiroz et al. 2006, 2007). For that reason, it seems more likely the presence of (010) as the main slip plane in these samples could have been associated with differences in deformation history (geometry of the flow, intensity of strain, etc.).
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(a)
(b)
(c)
Fig. 6. Seismic fabrics calculated from samples (a) A2, (b) V1, and (c) A4. Lower hemisphere projections. Contour intervals counted as multiples of uniform distribution (mud). Maximum, minimum and vertical values are shown in the figure. Kinematic reference frame as in Figure 3.
It has been previously suggested that LPO orientation may influence to some extent the LPO-derived seismic fabrics of ductile deformed amphibolites (see also Dı´az-Azpiroz & Lloyd 2010). As the LPO orientation reflects deformation mechanisms and the slip systems that are active during deformation of a mineral, it follows that the seismic properties of amphibolites can be influenced by which slip systems are active during deformation of plagioclase. Numerous slip systems have been reported for plagioclase (e.g. Kruse et al. 2001) and it is common that some of them act
simultaneously or affect the same rock depending on deformation conditions (e.g. Kruhl 1987; Shigematsu & Tanaka 2000). Considering only the most common slip systems, (010) and (001) can act as the slip planes while [100] and [010] can act as the preferential slip direction. In consequence, the role that plagioclase LPO may play in the seismic properties of ductile amphibolites must be complex; it should be considered when dealing with the seismic interpretation of the lower continental crust, where plagioclase is a major phase.
Table 2. Main seismic properties of all the studied samples. Rc is the reflectivity coefficient (Ji et al. 1993), which has been calculated using the P-wave velocities in the vertical direction (Vpvert) and considering density as a constant (Equation 1) Sample A0 A1 A2 A3 V1 V3 PV3 A4 V4 PV7
Vpmax
Vpmin
Vpvert
Rc
AVp
AVsmax
AVsmin
AVsvert
6.61 6.46 6.52 6.49 6.59 6.51 6.54 6.58 6.66 6.58
6.47 6.34 6.30 6.32 6.28 6.31 6.30 6.24 6.21 6.30
6.53 6.39 6.34 6.37 6.38 6.38 6.36 6.37 6.33 6.37
0.011 0.004 0.002 0.001 0.000 0.002 0.001 0.003 0.003
2.1 2.0 3.5 2.8 4.8 3.2 3.7 5.4 7.0 4.4
2.39 2.66 3.55 6.01 2.77 4.07 5.01 6.81 6.13 3.74
0.06 0.03 0.09 0.03 0.09 0.09 0.00 0.12 0.00 0.00
0.6 1.4 0.8 2.5 1.9 1.0 1.8 2.1 1.8 1.0
DEFORMATION MECHANISMS OF PLAGIOCLASE
Mechanical twinning Optically, mechanical twinning in plagioclases from the Acebuches metabasites is suggested by tapering of the twin elements (Fig. 2). In samples A2, V1 and A4, 175 –1808 misorientation axes define maxima around distinct crystallographic directions (Fig. 4), which could be interpreted as due to the activity of different twinning laws. The maxima around [001] would correspond to Carlsbad twinning whereas the maxima at the normal to [001] included in (010) could be interpreted as due to complex twinning, probably including Ala and Manebach twin laws. These are growth twinning laws, which have also been determined by U-stage in similar samples from the Acebuches metabasites (Duclo´s 2004). The maximum parallel to [010] could tentatively be related to the activity of mechanical (albite and pericline) twinning. With respect to the kinematic reference frame, high-angle misorientation axes locate approximately parallel to the XZ-plane in samples A2 and V1 (Fig. 4a, b). They are randomly distributed in sample A4 (Fig. 4c), however, suggesting very low activity of mechanical twinning in this sample. LPO plots of sample V1 (Fig. 3f) show the [010] axis is concentrated around Y. The observed inconsistency between misorientation and LPO data could result from pseudosymmetry problems related to the possible artificial introduction of a symmetry centre during the analytical procedure and/or from some misindexing. In any case, it suggests low activity of mechanical twinning in sample V1. On the other hand, [010] axes of sample A2 show a maximum around Z (Fig. 3c), which is consistent with misorientation data (Fig. 4a). This suggests the activity of mechanical albite and pericline twinning, which produce 1808 rotations about the b* axis (the normal to (010) plane) and [010], respectively. Mechanical twinning in plagioclase has been attributed to uniform shear nucleated around cracks (Marshall & McLaren 1977; McLaren & Pryer 2001) and, alternatively, to dislocation glide (e.g. Starkey 1964; Smith & Brown 1988). In the latter sense, LPO in plagioclase has been ascribed to mechanical twinning (e.g. Olesen 1987). In sample A2, plagioclase twin axes and planes are intimately related to the active slip systems; it is therefore most likely that, in this case, mechanical twinning took place by dislocation glide. The glide plane deduced for sample A2 (Fig. 3c and Table 1), (010), is the glide plane related to the activity of albite twinning (Egydio-Silva & Mainprice 1999), suggesting the activity of Albite twin law was prominent in this sample. In contrast, twin laws determined by U-stage (Duclo´s 2004) in similar samples suggest pericline twinning is more common than albite twinning in the Acebuches
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metabasites. The higher activity of mechanical twinning related to dislocation glide in the highest temperature sample (A2) is in accordance with experimentally (Borg & Heard 1969; Seifert & Ver Ploeg 1977; Ross & Wilks 1995) and naturally (Suwa et al. 1974; Egydio-Silva et al. 2002) deformed plagioclases.
Dynamic recrystallization Several optical microstructures (deformation bands, subgrain structures, core-and-mantle structures, etc., Fig. 2) point to dynamic recrystallization as a major deformation mechanism in plagioclases from the Acebuches metabasites affected by the SISZ. Furthermore, dynamic recrystallization must have played a major role in grain-size reduction. Also, moderate-to-well developed LPO strongly suggest that dislocation glide (also evidenced by twinning and micro-kinking) and (probably) dislocation creep were active during deformation of these plagioclases. Despite optical features to the contrary (e.g. parallel subgrain boundaries), the misorientation analysis shows no evidence of subgrain rotation recrystallization, that is, scarce low-angle misorientation axes which exhibit no correlation with any preferred crystallographic direction (Fig. 4). It therefore seems likely this mechanism was marginal in the present case. Moreover, samples V1 and A4 show a slightly larger amount of small-to-intermediate misorientation angles (5–608) than the theoretical random distribution (Fig. 4b, c). These are related to misorientation axes that are neither crystallographically nor kinematically controlled. These features have been referred to as ‘no-host control’ (Olesen 1987; Ji & Mainprice 1990), which has been attributed to grain-boundary migration recrystallization starting from nucleation of new grains at sites of high stress concentration (e.g. Rosenberg & Stu¨nitz 2003) such as submicroscopic shear bands (Ave´ Lallemant 1985) or micro-fractures (Kruse et al. 2001). In the Acebuches metabasites, plagioclases show evidence pointing to grain-boundary migration recrystallization (Fig. 2) which likely nucleated at micro-kinks. High-defect densities would have favoured nucleation of new strain-free grains with no crystallographic relationship with the ‘parental’ kinked porphyroclast.
Grain-boundary sliding Field and microstructural features (sheath folds, fabric intensity, grain size, etc.) reported in previous works (Dı´az-Azpiroz & Ferna´ndez 2003, 2005) suggest the mafic schists from the Acebuches metabasites were more intensively deformed by the SISZ than the rest of the sequence (Fig. 1c). In contrast,
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these rocks show the weakest LPOs (Fig. 3h –j) although they resemble that of the mylonitic amphibolites (Fig. 3d– g) and form polygonal finegrained aggregates (Fig. 2d–f ). Moreover, low- and intermediate-angle misorientation axes of sample A4 show no correlation with the kinematic or the crystallographic frames (Fig. 4c). These features suggest diffusion-accommodated grain-boundary sliding as the most probable deformation mechanisms in these rocks (Stu¨nitz and Fitz Gerald 1993; Fliervoet et al. 1997; Jiang et al. 2000; Bestmann & Prior 2003; Raimbourg et al. 2008). The mafic schists of the Acebuches metabasites show finer grain sizes (10–21 mm, Dı´az-Azpiroz & Ferna´ndez 2003) than the rest of the rock sequence. An upper grain-size threshold of 10 mm has been suggested to activate grain-sensitive deformation mechanisms processes, such as grain-boundary sliding (Fliervoet & White 1995). The deformation mechanism map for wet anorthite (Rybacki & Dresen 2004) set this threshold, depending upon temperature, strain rate and shear stress, in slightly larger grain sizes. Indeed, grain-boundary sliding has been invoked as a deformation mechanism affecting plagioclase aggregates with grain size c. 10 mm (Raimbourg et al. 2008). The mafic schists deformed under lower temperature conditions than the rest of the Acebuches metabasites (e.g. Dı´az-Azpiroz et al. 2006, 2007, Fig. 1c). Both finer grains and lower temperatures favour grain-boundary sliding in opposition to dislocation creep in experimentally deformed wet anorthite (Rybacki & Dresen 2004). A weak LPO, but similar to that of higher-grade metabasites, and the presence of undulose extinction within some porphyroclasts (Fig. 2e) suggest limited dislocation creep. Dynamic recrystallization accommodating dislocation creep would therefore be responsible for development of new finer grains. Subsequently, grain-boundary sliding would have taken over producing (1) an increase in misorientation between adjacent grains, (2) randomizing of misorientation axes and (3) LPO weakening. A similar process has been previously described in plagioclase (Raimbourg et al. 2008) and calcite (Bestmann & Prior 2003). Considering the deformation mechanism map for wet anorthite of Rybacki & Dresen (2004) and assuming (1) the deformation temperature is c. 600 8C (Dı´az-Azpiroz et al. 2006) and (2) the grain-size threshold between the mylonitic amphibolites (deformed mainly by dislocation creep) and the mafic schists (deformed by grain-boundary sliding) is c. 20 mm, a minimum strain rate of 10214 s21 can be estimated for the SISZ. Switching deformation mechanism from dislocation creep in the mylonitic amphibolites to grainboundary sliding in the mafic schists, related to grain-size reduction, would have produced strain
weakening in the latter (see also Raimbourg et al. 2008). The mafic schists were therefore able to accommodate larger amounts of strain than the rest of the Acebuches metabasites and it is likely the deformation produced by the SISZ progressively localized at its lower limit. Lower temperature deformation in the mafic schists could also have favoured minor fracturing (e.g. Tullis & Yund 1987; Rosenberg & Stu¨nitz 2003), which is further evidenced by optical features (Fig. 2) and by phyllonitic foliation and striae found in the field (Dı´azAzpiroz & Ferna´ndez 2005).
Conclusions Plagioclases from the Acebuches metabasites accommodated the imposed strain by different combinations of deformation mechanisms, depending on the temperature of deformation and/or the kinematics of strain. Plagioclase from the banded amphibolites (e.g. sample A2), deformed by D1 at amphibolite to upper amphibolite facies conditions, accommodated deformation mainly by dislocation glide that gave rise to mechanical (albite and pericline) twinning, micro-kinking and a moderate LPO. The mylonitic amphibolites and the mafic schists were deformed by D1 þ D2 at the SISZ. Plagioclase from the mylonitic amphibolites (e.g. sample V1) was deformed by dislocation glide, evidenced by mechanical twinning and micro-kinking; moderate dislocation creep was accommodated by grain-boundary migration and, perhaps, limited subgrain rotation. Such a combination of deformation mechanisms would have resulted in grain-size reduction and development of a strong LPO. In this study, higher dislocation activity has been reported for the mylonitic amphibolites (samples A3, V1, V3 and PV3) than for the banded amphibolites (samples A0, A1 and A2), despite the fact that the latter are higher metamorphic grade rocks. The reason for this would not rely on temperature but on deformation history, which would be much more complex and intense in the mylonitic amphibolites. The most probable deformation mechanisms affecting plagioclase from mafic schists (e.g. sample A4) were limited dynamic recrystallization and subsequent grain-boundary sliding, accompanied by minor fracturing at the lowest grade rocks. This combination of processes would account for intense grain-size reduction, weakening of pre-existing LPOs and strain weakening. The latter produced progressive deformation localization at the lower structural boundary of the SISZ. Throughout the whole metabasite sequence, the crystallographic direction [100] acted as the predominant slip direction of the active slip systems. Few secondary directions were also active in
DEFORMATION MECHANISMS OF PLAGIOCLASE
some samples. The main slip plane was (010) in the higher-temperature banded amphibolites. It switched to (001) in the medium-temperature mylonitic amphibolites and again to (010) in the lower temperature mafic schists. It is very likely that strain, and not only temperature, exerted a strong influence on the preferred slip plane in each case. This study also focuses on the seismic properties of the Acebuches metabasites. The results presented here confirm hornblende largely control the seismic properties of ductile deformed amphibolites. However, some influence by plagioclase LPO strength and orientation, which depend on the deformation mechanisms and the active slip systems, needs to be considered. From the calculated seismic fabrics, it can be deduced that reflectors attributed to the Acebuches metabasites in the IBERSEIS seismic profile should be produced by grain-size layering in the banded amphibolites and fractures in the mafic schists, rather than by LPO of hornblende and plagioclase. Reviews by D. Prior, M. Pearce and two anonymous referees greatly improved the manuscript and are gratefully acknowledged. This study has received financial support from the Spanish Ministry of Education and Science (projects PB94-1085, BTE2003-05057-CO2-02, CGL2006-08638/ BTE, CGL 2009-11384 and TOPOIBERIA-CONSOLIDERINGENIO 2010-CSD2006-00041). The SEM/EBSD facility at the University of Leeds was part funded by UK NERC grant GR9/3223. We wish to acknowledge the use of the UK EPSRC Chemical Database Service at Daresbury and the Cambridge Structural Database System.
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Seismic velocity in antigorite-bearing serpentinite mylonites TOHRU WATANABE1*, YUHTO SHIRASUGI1, HIDEAKI YANO1 & KATSUYOSHI MICHIBAYASHI2 1
Department of Earth Sciences, University of Toyama, Japan 2
Institute of Geosciences, Shizuoka University, Japan
*Corresponding author (e-mail:
[email protected]) Abstract: The relationships between elastic wave velocities and petrofabrics were studied in two antigorite-bearing serpentinite mylonites. Rock samples with antigorite content of 37 and 80 vol% were collected from the Happo ultramafic complex, Central Japan. Compressional and shear-wave velocities were measured by the pulse transmission technique at room temperature and confining pressures of up to 180 MPa. Petrofabrics were examined by optical microscopy and scanning electron microscopy with electron backscattered diffraction (SEM-EBSD). Olivine a- and c-axes are weakly oriented perpendicular to the foliation and parallel to the lineation, respectively. Antigorite b- and c-axes are distinctly oriented parallel to the lineation and perpendicular to the foliation, respectively. Both samples show strong anisotropy of velocity. The compressional wave velocity is fastest in the direction parallel to the lineation, and slowest in the direction perpendicular to the foliation. The shear wave oscillating parallel to the foliation has higher velocity than that oscillating perpendicular to the foliation. As the antigorite content increases, the mean velocity decreases but both azimuthal and polarization anisotropies are enhanced. Measured velocities were compared with velocities calculated from petrofabric data by using Voigt, Reuss and Voight-Reuss-Hill (VRH) averaging schemes. All averaging schemes show velocity anisotropy qualitatively similar to measurements. There are large velocity differences between Voigt and Reuss averages (0.7–1.0 km/s), reflecting the strong elastic anisotropy of antigorite. Measured velocities are found between Reuss and VRH averages. We suggest that the relatively low velocity is due to the platy shape of antigorite grains, the well-developed shape fabric and their strong elastic anisotropy. The configuration of grains should be an important factor for calculating seismic velocities in an aggregate composed of strongly anisotropic materials, such as sheet silicates.
Serpentinites play key roles in subduction zone processes including transportation of water (Iwamori 1998; Schmidt & Poli 1998; Hacker et al. 2003a; Hattori & Guillot 2003; Hyndman & Peacock 2003), seismogenesis (Hyndman et al. 1997; Peacock & Hyndman 1999; Hacker et al. 2003b; Seno 2005), slab –mantle coupling (Peacock & Hyndman 1999; Wada et al. 2008; Hilairet & Reynard 2009) and exhumation of high-pressure rocks (Guillot et al. 2000; Hermann et al. 2000). Geophysical mapping of serpentinized regions in the mantle wedge leads to further understanding of these processes. The direct study of serpentinized peridotites is critical to the interpretation of indirect geophysical observations. There are three major forms of serpentine: lizardite, chrysotile and antigorite. Lizardite is the most abundant serpentine phase since it forms in oceanic crust in contact with sea water (e.g. O’Hanley et al. 1989). Chrysotile is a metastable variety (Evans 2004). Antigorite is the dominant serpentine at high pressures in subduction zones (Ulmer & Trommsdorff 1995; Hacker et al. 2003a). Christensen (1996) suggested that serpentinites be distinguished from other rocks by their anomalously high Poisson ratio (c. 0.35). Later,
Christensen (2004) and Watanabe et al. (2007) showed that antigorite-bearing serpentinized peridotites have distinctly higher mean velocity and lower Poisson ratio than lizardite-bearing peridotites with the same density. Only lizardite-bearing rocks show anomalously high Poisson ratios. Deformed antigorite-bearing rocks expected in the mantle wedge show strong seismic anisotropy (Kern et al. 1997; Watanabe et al. 2007). The sheet structure of antigorite can qualitatively explain the strong anisotropy. However, the lack of elastic constants and the difficulty in textural analysis (due to the small size of antigorite grains) has hindered the quantitative understanding of this anisotropy, required for the interpretation of geophysical observations. Quantitative understanding enables us to make a good prediction of seismic properties from petrofabrics. Progress has recently been made towards quantitative understanding. Antigorite single-crystal elastic constants have been revealed via the Brillouin scattering technique (Bezacier et al. 2010). Scattering electron microscopy (SEM) electron backscattered diffraction (EBSD) has enabled us to measure the orientation of fine antigorite grains
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 97– 112. DOI: 10.1144/SP360.6 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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(Katayama et al. 2009; Bezacier et al. 2010). Based on CPO data derived from EBSD measurements, Bezacier et al. (2010) calculated elastic properties of an antigorite-bearing serpentinite mylonite from Cuba. They presented elastic stiffness calculated as Voigt, Reuss and Voight-Reuss-Hill (VRH) averages and geometric mean. There are considerable differences between the elastic stiffnesses calculated with each of different schemes. Large uncertainty therefore exists in seismic properties predicted from petrofabrics. No information about the shape or position of neighbouring grains is used in Voigt and Reuss averaging (Mainprice 2007). The Voigt average is found by assuming a uniform strain field, while the Reuss average assumes a uniform stress field (Watt et al. 1976). The Voigt average gives an upper bound and the Reuss average a lower bound to elastic stiffness. Measured values should be found within these bounds. The VRH average is the arithmetic mean of Voigt and Reuss averages (Hill 1952). The geometric mean was first introduced by Hill (1952) as the geometric mean of Voigt and Reuss averages, and it has been generalized to take the orientation distribution function into account (Morawiec 1989; Matthies & Humbert 1993). The VRH average and geometric mean are very similar, and have no physical justification (Mainprice 2007). When component minerals have similar elastic properties and weak anisotropy, the Voigt, Reuss and VRH averages and geometric mean are nearly equal. These values can provide a good prediction of seismic properties. Examples of dunite and bronzitite were reported in Crosson & Lin (1971) and Babsˇka (1972). However, when component minerals have strong anisotropy, Voigt and Reuss values demonstrate large differences (Mainprice & Humbert 1994). The spatial distribution of mineral grains should be taken into account for further progress. It is therefore essential to measure the seismic velocities of deformed serpentinized peridotites to understand what microstructural information is taken into account. In this study, we made velocity measurements and petrofabric analyses in two serpentinite mylonite samples. Measured velocities are compared with velocities calculated from petrofabric data to constrain the microstructural information that is important for a good prediction.
understood (Nozaka 2005). The Happo ultramafic complex is composed of serpentinite me´langes, high-P/T schists, amphibolites, eclogites and weakly metamorphosed sedimentary rocks (e.g. Nakamizu et al. 1989). There are at least two types of ultramafic rocks – foliated and massive – with a transition type between them (Matsuhisa 1968). Most of the foliated rocks have mylonitic textures and contain more abundant serpentine than the massive rocks (Nozaka 2005). Based on petrological studies, Nozaka (2005) suggested that the mylonitization associated with high-temperature serpentinization (400 –600 8C) was superimposed on an early-stage peridotite mylonite at 700– 800 8C at a convergent plate boundary. Khedr & Arai (2010) concluded that the Happo ultramafic complex had been in the corner of the mantle wedge and suffered from low-T (,600 8C) retrogressive metamorphism during exhumation of the Renge metamorphic belt. Measurements were conducted on two serpentinite mylonites. The volume fraction of antigorite is 36.9 in Sample HPS-M and 80.2% in sample HKB-B. The direction of elongation is recognized on a polished foliation plane. Density and mineralogy of samples are summarized in Table 1. The density was calculated from the volume and mass measured at ambient conditions. The volume fraction of minerals was estimated by the point counting method: 2500– 3000 points were counted over an area of 126 mm2 for each rock. A cube (edge length c. 30 mm) was cut from each rock sample for velocity measurements. Two faces of a cube were parallel to the foliation plane, two faces were perpendicular to the elongation direction and the remaining two faces were perpendicular to the foliation plane and parallel to the elongation direction. The orthogonal axes were set for velocity measurements. The z-axis was perpendicular to the foliation plane and the x-axis was parallel to the
Samples Rock samples were collected from the Happo ultramafic complex, which is located in the Hida Marginal Tectonic Belt (Fig. 1). The Hida Marginal Tectonic Belt is characterized by medium- to low-P/T metamorphic rocks, although its development in relation to regional geology is still poorly
Fig. 1. Geographical map showing the Hida Marginal Tectonic Belt. The Happo ultramafic complex is located around Mount Happo.
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Table 1. Density and mineralogy of rock samples Sample
Density (g/cm3)
Minerals (vol%)
HPS-M HKB-B
3.00 + 0.02 2.80 + 0.02
Ol (57.5), Atg (36.9), Trm (4.5), Mgt (1.1) Ol (12.0), Atg (80.2), Mgt (7.8)
Ol, olivine; Atg, antigorite; Trm, tremolite; Mgt, magnetite
elongation direction. Velocity measurements on sample HPS-M were previously reported in Watanabe et al. (2007). In this study, measurements were made on a new cube cut from sample HPS-M.
Microstructural analysis The microstructure of rock samples was optically examined on thin sections cut perpendicular to the foliation and parallel to the lineation (zx-plane).
Crystal preferred orientation (CPO) of olivine and antigorite grains was analysed with a SEM-EBSD system (JEOL JSM6300 with HKL Channel 5) at the Centre for Instrumental Analysis, Shizuoka University. To analyse crystallographic orientation of antigorite, thin sections were ultrapolished with colloidal silica for no longer than a few hours. We measured 212 olivine and 213 antigorite crystal orientations in sample HPS-M and 121 olivine and 269 antigorite in sample HKB-B. The computerized
Fig. 2. Photomicrographs (crossed-polars) of rock samples. The width is 6 mm. (a) Sample HPS-M shows olivine porphyroclastic texture with olivine neoblasts and thin antigorite grains. Olivine porphyroclasts are elongated along the lineation. Antigorite grains are divided into two groups. The dominant group fills gaps between olivine porphyroclasts (triangle) and the other group intrudes into olivine porphyroclasts (arrow). (b) Olivine neoblasts are seen at the top and tail of porphyroclasts (sample HPS-M). (c) The fabric of sample HKB-B is dominated by antigorite grains. (d) Sample HKB-B with sensitive colour plate (DL ¼ 530 nm) showing the shape and crystal preferred orientation of antigorite. Some antigorite grains are oriented subperpendicular to the foliation.
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indexing of the diffraction pattern was checked visually for each orientation.
orientation of b-axes parallel to the lineation and c-axes normal to the foliation plane.
Sample HPS-M (Atg: 36.9 vol%)
Sample HKB-B (Atg: 80.2 vol%)
This sample shows olivine porphyroclastic texture with olivine neoblasts (c. 0.1 mm) and thin antigorite grains (Fig. 2a). Olivine porphyroclasts are elongated (2 –3 mm) along the lineation to have lenticular shapes. They show intense undulose extinction and well-developed subgrain boundaries. Olivine neoblasts are seen at the top and the tail of olivine porphyroclasts (Fig. 2b). Olivine CPO data (Fig. 3a) show a weak concentration of c-axes subparallel to the lineation, a-axes subperpendicular to the foliation and b-axes subperpendicular to the lineation and in the foliation plane. Antigorite grains have needle-like shapes (c. 0.1 mm in length and c. 0.01 mm in width) in the zx-plane, and are divided into two groups. The dominant group fills gaps between olivine porphyroclasts with their long axes mostly oriented along lenticular olivine grains (Fig. 2a). The other group intrudes into olivine porphyroclasts with their long axes subperpendicular to the lineation (Fig. 2b). Antigorite CPO data (Fig. 3a) show a distinct
Antigorite grains dominate the fabric of this sample (Fig. 2c). They have needle-like shapes (c. 0.1 mm in length and c. 0.01 mm in width) in the zx-plane, and most of the long axes are parallel or subparallel to the lineation. Olivine grains (12.0 vol%) are intensively elongated parallel to the lineation and have well-developed subgrain structures. Though the number is quite small, there are some antigorite grains with their long axes subperpendicular to the lineation. They can be seen in elongated olivine crystals and their relicts (Fig. 2d). Although the volume fraction is small (12.0 vol%), olivine shows a distinct orientation of c-axes parallel to the lineation, a-axes subperpendicular to the foliation plane and b-axes subperpendicular to the lineation and in the foliation plane (Fig. 3b). Antigorite has a strong orientation of b-axes parallel to the lineation and c-axes normal to the foliation plane. A c-axes girdle perpendicular to the lineation is also seen, which may reflect the weak folding of foliation planes.
Fig. 3. Pole figures of crystallographic orientation of olivine and antigorite. (a) Sample HPS-M and (b) sample HKB-B. Pole figures are equal-area lower-hemisphere projections with a Gaussian half-width angle of 8.58;
SEISMIC VELOCITY IN SERPENTINITES
Velocity measurements Method Compressional and shear-wave velocities were measured by the pulse transmission technique using Pb(Zr, Ti)O3 transducers with the resonant frequency of 2 MHz. A function generator (Hewlett Packard, 33120A) applied an electrical rectified pulse to one transducer to excite an elastic wave. The other transducer received the transmitted elastic wave and converted it to an electrical signal, which was digitized and averaged over 512 times by a digital oscilloscope (Agilent Technology, 54621A). The sampling interval was 5 ns. The digitized 12-bit signal was transferred to a computer for analysis. Hydrostatic pressure of up to 180 MPa was applied to cubic specimens. Cubic specimens with glued transducers were covered with silicone rubber and loaded into a pressure vessel (Riken, PV-2M-S6F). Silicone oil (Shin-Etsu Chemical, KF-96-100cs) was used as a pressure medium.
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All measurements were conducted at room temperature. One compressional wave velocity and two shear-wave velocities were measured in each of three orthogonal directions. Two shear waves propagating in one direction oscillate in mutually orthogonal directions. The number of electrical feedthroughs was 12; two loading cycles were needed for a specimen. The error in a measured velocity originates in errors in the travel time and path length as dv dt dl ≤ + v t l
(1)
where v, t and l are the velocity, travel time and path length, respectively. The errors in three values are denoted by dv, dt and dl, respectively. The error in a travel time depends on the quality of waveform data. The uncertainty in reading the first motion is 20 –30 ns for compressional waves. The first motion of the shear wave is likely to be obscured by preceding converted waves. The
Fig. 3. (Continued) contours represent the data points (multiples of uniform density, shading is linear). Maximum and minimum density values are shown. pfJ index represents the fabric strength after Mainprice & Silver (1993). N indicates the number of grains.
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uncertainty in the arrival time is 100 –150 ns for shear waves. Since the travel time is typically 2 and 4 ms for compressional and shear waves, the magnitude of the first term in equation (1) is 0.01 for the compressional wave and 0.03 for the shear wave. The error in the path length originates in the measurement error and the contraction under confining pressures. The length measurement has an error of the order of 0.05 mm. We estimate the dimensional change of a rock specimen by assuming isotropic elasticity. The isothermal compressibility is estimated to be 1 × 10211 Pa21 from elastic wave velocities and density. The isothermal compressibility at room temperature is estimated to be 0.8 × 10211 Pa21 for olivine (Kumazawa & Anderson 1969) and 1.6 × 10211 Pa21 for antigorite (Hilairet et al. 2006a). The value used is between the compressibilities of olivine and antigorite. The dimensional change is therefore of the order of 0.02 mm at the confining pressure of 180 MPa. The error in the path length is estimated to be less than 0.07 mm. Since the path length is around 30 mm, the magnitude of the second term in equation (1) is less than 0.003. The error in velocity is mainly due to the error in travel time.
Results Velocities of compressional and shear waves increase with increasing confining pressure. Velocity changes in sample HKB-B (Atg: 80.2 vol%) are shown in Figure 4. Most of the velocity increase was observed below 50 MPa, whereas there is only slight increase at higher pressures. The velocity increase below a few hundred MPa is due to the closure of pores (e.g. Birch 1960). If pores have an oblate spheroidal shape, the closure pressure depends on the aspect ratio of a pore (c/a; a: major axis, c: minor axis) (e.g. Walsh 1965). Pores with smaller aspect ratios close at lower pressure. The increase in velocity at lower pressures is larger for the compressional wave propagating normal to the foliation (Fig. 4b) and the shear wave oscillating normal to the foliation. This suggests that most of the thin pores are parallel or subparallel to the foliation. We think that the influence of pores on velocities is small at 180 MPa. Sample HKB-B shows strong anisotropy of elasticity. The fastest compressional wave velocity is observed in the direction parallel to the lineation (x-direction) and the slowest in the direction perpendicular to the foliation (z-direction). The magnitude
Fig. 4. Velocity with the confining pressure in sample HKB-B. Velocities were measured with increasing confining pressure. Velocities in x- and y-directions were measured during the first loading and those in z-direction during the second loading. (a) Compressional wave velocity. Propagating direction is indicated. (b) Compressional wave velocity normalized by the velocity at 180 MPa. Error bars are not indicated for clarity. (c–e) Shear-wave velocity for propagation in (c) x-direction, (d) y-direction and (e) z-direction. Oscillating and propagating directions are indicated by the first and second letters, respectively.
SEISMIC VELOCITY IN SERPENTINITES
of the azimuthal anisotropy is characterized by the anisotropy factor kp, defined:
of polarization anisotropy AVs is 0.08–0.09 for the propagation direction parallel to the foliation. The polarization anisotropy is not significant for the propagation direction normal to the foliation. The arithmetic mean of velocities is higher in sample HPS-M than HKB-B for both compressional and shear waves.
Vp (fastest) − Vp (slowest) . Vp (mean)
The factor kp is 0.28 at the confining pressure of 180 MPa. The shear wave propagating parallel to the foliation shows strong polarization anisotropy. The wave oscillating parallel to the foliation has higher velocity than that oscillating normal to the foliation. The polarization anisotropy is not significant for the shear wave propagating perpendicular to the foliation. The magnitude of the polarization anisotropy in one direction is characterized by the factor AVs, defined:
Crystal orientation of antigorite grains The crystal orientation of antigorite grains is an important factor to control seismic anisotropy in serpentinites. For a good interpretation of seismological observations, it is important to understand the antigorite fabric expected in the mantle wedge. In HPS-M and HKB-B, most of the antigorite c-axes are perpendicular to the foliation plane and b-axes parallel to the lineation (Fig. 3). Similar antigorite fabrics have been reported in an antigorite schist from the Western Alps, Italy (Vogler 1987) and in an antigorite-bearing serpentinite mylonite from the Sashu Fault, southwest Japan (Soda & Takagi 2010). Another type of CPO has also been reported. Antigorite c-axes are perpendicular to the foliation plane and a-axes parallel to the flow direction. Katayama et al. (2009) conducted deformation experiments on an antigorite serpentinite in which antigorite grains are randomly oriented, and showed that the deformation progressively formed this type of antigorite fabric. Bezacier et al. (2010) found a similar antigorite fabric in a serpentinite from Cuba. It is not known what causes this difference between two types of antigorite fabric. CPO develops during plastic deformation. Since
Vs (fastest) − Vs (slowest) . Vs (mean)
The factor AVs is 0.26, 0.23 and 0.016 for the x-, y-, and z-direction, respectively. Velocities of two samples measured at 180 MPa are shown in Figure 5. Sample HPS-M (Atg: 36.9 vol%) shows velocity anisotropy similar to sample HKB-B, although the magnitude is smaller. The factor of azimuthal anisotropy kp is 0.12 in sample HPS-M. In the fastest direction of the compressional wave (x-direction, parallel to the lineation), sample HPS-M has a lower velocity than sample HKB-B. In the slowest direction (z-direction, normal to the foliation), it has a higher velocity than sample HKB-B. The polarization anisotropy is also weaker in sample HPS-M than in sample HKB-B. The factor
HPS-M Vp = 6.72±0.08
AVs = 0.007 V
s
Vp = 5.83±0.02 AVs = 0.016
y Vs = 3.23±0.09
4.
1±
0.
1
AVs = 0.08 Vs
x (km/s)
=
Vs = 4.0±0.1
Vp = 7.6±0.1
Vs = 3.9±0.1
Vs = 4.1±0.1
HKB-B
z
=
4.
Vp = 7.70±0.02
V
s
=
3.
28
±0
.0
7
Vs = 3.3±0.2
AVs =
Discussion
Vs = 3.2±0.1
kp =
103
AVs = 0.26 V
s
2±
=
4.
1±
0.
1
Vs = 4.4±0.1
0.
2
Vp = 7.13±0.09 AVs = 0.09
Vs = 4.1±0.2
Vp = 6.82±0.07 AVs = 0.23
Vp(mean) = 7.1 (km/s), Vs(mean) = 4.1 (km/s)
Vp(mean) = 6.8 (km/s), Vs(mean) = 3.5 (km/s)
kp = 0.12
kp = 0.28
Fig. 5. Compressional and shear-wave velocities measured at a confining pressure of 180 MPa. Vp (mean) and Vs (mean) are the arithmetic means of three and six velocity values, respectively.
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the deformation of antigorite serpentinite is still poorly understood, we cannot yet answer this problem. We will thus consider the factors that affect CPO in antigorite serpentinites, and discuss the development of antigorite fabric. Antigorite fabric is formed through hydration (serpentinization) of peridotites and subsequent deformation. It should be affected by the initial fabric of olivine in peridotites, the topotactic relationships between olivine and antigorite and the deformation behaviour of antigorite serpentinite. Boudier et al. (2010) examined an antigorite schist from a diatreme in the Colorado Plateau, USA, and found that the structural relationships between olivine and antigorite come from the geometry of fluid pathways controlled by the initial olivine fabric and the topotactic relationships between olivine and antigorite. The primary topotactic relation is: [100]atg//[010]ol; [010]atg//[001]ol; and planes in contact as (001)atg//(100)ol. The secondary relation is: [100]atg//[100]ol; [010]atg//[001]ol; and planes in contact as (001)atg//(010)ol (Boudier et al. 2010). These relations control the orientation of antigorite grains during serpentinization. The antigorite fabric thus formed will be changed by subsequent deformation. Experimental studies showed that antigorite serpentinite deforms by semi-brittle processes at high pressure and temperature (Jung & Green 2004; Jung et al. 2009; Chernak & Hirth 2010). The deformation is accommodated by both frictional sliding and crystal plastic deformation within the antigorite stability field (Chernak & Hirth 2010). Jung & Green (2004) reported faulting of antigorite at 550 8C and 1.5 GPa. Jung et al. (2009) attributed acoustic emissions to faulting and subsequent frictional sliding up to 570 8C and 6 GPa, and reported the alignment of antigorite grains in shear zones. Similar antigorite fabric was observed by Chernak & Hirth (2010) in antigorite serpentinite deformed at 400 – 625 8C and 0.85 –1.5 GPa. They presented only photomicrographs with a gypsum plate to show the alignment of antigorite grains. Photomicrographs suggest that antigorite c-axes are aligned perpendicular to the shear plane, although the orientation of a- or b-axes cannot be inferred. The deformationDIA (D-DIA) experiments of Hilairet et al. (2007) concluded that antigorite deforms by macroscopically ductile processes at high pressure within its stability field. Chernak & Hirth (2010) suspected that the lack of strain localization in Hilairet et al. (2007) was due to the small strain. Experiments of Chernak & Hirth (2010) have demonstrated that an initial fabric can strongly affect the fabric development. They used natural foliated serpentinite samples and hot-pressed powdered samples. Core samples were made from a foliated serpentinite at different angles to the foliation. Antigorite
c-axes are thought to be oriented mostly perpendicular to the foliation. Core samples oriented with the compression axis (s1) at 458 to foliation and powered samples develop strong CPO along shear zones, while samples oriented with s1 perpendicular to foliation do not. If frictional sliding is completely suppressed, antigorite CPO develops uniformly in deformed samples. Since deformation is localized around a fault, antigorite grains with c-axes subperpendicular to it will develop strong CPO along the
Table 2. Calculated elastic stiffness tensors (in GPa) Sample HPS-M Voigt average 205.52 66.36 66.36 195.79 62.29 62.53 20.10 20.37 21.48 0.20 3.86 1.54
62.29 62.53 193.30 21.78 20.24 0.83
20.10 20.37 21.78 66.17 1.47 20.57
21.48 0.20 20.24 1.47 64.70 20.84
3.86 1.54 0.83 20.57 20.84 67.83
Reuss average 165.59 57.92 57.92 150.20 53.23 57.73 20.57 0.94 22.40 0.12 4.66 20.18
53.23 57.73 145.98 20.43 0.65 20.45
20.57 0.94 20.43 44.99 0.17 20.74
22.40 0.12 0.65 0.17 43.00 20.46
4.66 20.18 20.45 20.74 20.46 49.90
VRH average 185.55 62.14 62.14 173.00 57.76 60.13 20.33 0.28 21.94 0.16 4.26 0.68
57.76 60.13 169.64 21.11 0.20 0.19
20.33 0.28 21.11 55.58 0.82 20.66
21.94 0.16 0.20 0.82 53.85 20.65
4.26 0.68 0.19 20.66 20.65 58.86
Voigt average 192.25 49.35 49.35 156.90 41.70 42.36 24.55 26.91 8.04 0.71 9.78 1.84
41.70 42.36 141.62 24.28 1.11 0.19
24.55 26.91 24.28 53.48 0.01 20.06
8.04 0.71 1.11 0.01 51.91 23.72
9.78 1.84 0.19 20.06 23.72 59.13
Reuss average 149.78 42.67 42.67 111.09 39.74 46.36 21.23 24.03 7.06 21.16 12.12 20.22
39.74 46.36 103.22 22.08 20.15 22.14
21.23 24.03 22.08 30.67 0.12 0.07
7.06 21.16 20.15 0.12 29.52 23.40
12.12 20.22 22.14 0.07 23.40 38.37
VRH average 171.01 46.01 46.01 133.99 40.72 43.46 22.89 25.47 7.55 20.23 10.95 0.81
40.72 43.46 122.42 23.81 0.48 20.97
22.89 25.47 23.18 42.08 0.06 0.00
7.55 20.23 0.48 0.06 40.71 23.56
10.95 0.81 20.97 0.00 23.56 48.75
Sample KHB-B
SEISMIC VELOCITY IN SERPENTINITES
shear zone. However, we do not know the mechanism of alignment (rotation or recrystallization). Combining the topotactic relations between olivine and antigorite (Boudier et al. 2010) and the development of antigorite CPO along shear zones (Jung et al. 2009; Chernak & Hirth 2010), antigorite fabric in deformed serpentinites will depend on the initial olivine fabric in peridotite. The olivine fabric varies with deformation conditions. Type-A dominates at low stress and low water content, in which olivine b-axes are aligned subperpendicular to the shear plane and a-axes subparallel to the shear direction. Type-B dominates at high water content and/ or high stress, in which olivine b-axes are subperpendicular to the shear plane and c-axes are subparallel to the shear direction. Type-C dominates at high water content and modest stress, in which olivine a-axes are subperpendicular to the shear plane and c-axes are subparallel to the shear direction (Jung & Karato 2001). In addition to these, type-D and type-E are also known (Katayama et al. 2004).
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When the initial olivine fabric is type-A, the topotactic relationships predict that the primary orientation of antigorite has a-axes normal to the shear plane and c-axes parallel to the flow direction and that the secondary orientation has c-axes normal to the shear plane and a-axes parallel to the flow direction. Antigorite grains with c-axes normal to the shear plane can accommodate bulk deformation. Later deformation will develop shear zones with the latter type of antigorite CPO. When the initial olivine fabric is type-B, the primary orientation of antigorite has a-axes normal to the shear plane and b-axes parallel to the flow direction, and the secondary orientation has c-axes normal to the shear plane and b-axes parallel to the flow direction. Later deformation will develop shear zones with the former type of antigorite CPO. When the initial olivine fabric is type-C, the primary orientation of antigorite has c-axes normal to the shear plane and b-axes parallel to the flow direction, and the secondary orientation has a-axes normal to the shear plane
Fig. 6. Compressional wave velocity calculated from CPO data by using Voigt, Reuss and VRH averaging schemes. The same colour scale is used for all figures. Maximum and minimum velocity values are shown. The magnitude of azimuthal anisotropy is also shown. It should be noted that the maximum and minimum velocities are not seen in x-, yor z-directions. The magnitude of azimuthal anisotropy for calculated velocity therefore cannot be directly compared with that for measured velocity.
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and b-axes parallel to the flow direction. Later deformation will develop shear zones with the former type of antigorite CPO. The antigorite fabric with a-axes parallel to the lineation is therefore inherited from type-A olivine fabric, and that with b-axes parallel to the lineation is inherited from type-B or C. The experiments of Katayama et al. (2009) are similar to the powdered sample experiments of Chernak & Hirth (2010). Antigorite grains were randomly oriented in starting materials. Since frictional sliding was suppressed, probably due to lower temperatures (300–400 8C) and experimental configuration (simple shear), antigorite CPO developed uniformly with c-axes normal to the shear plane. We still do not know why antigorite a-axes align parallel to the flow direction. The intracrystalline deformation mechanism of antigorite needs to be understood in future studies. It is essential to understand the development of antigorite CPO.
Comparison between measured and calculated velocities We calculated elastic wave velocities of serpentinite mylonites from modal composition (Table 1) and CPO data (Fig. 3) to compare them to measured velocities. The comparison will teach us what microstructural information should be taken into account for a good prediction. Voigt, Reuss and VRH averages were calculated using Careware software developed by Mainprice (1990). The orientation distribution function of olivine and antigorite is calculated from CPO data. The orientation of tremolite and magnetite has not been measured. Because magnetite is almost isotropic in elasticity (Hearmon 1956), we assumed that magnetite grains are randomly oriented. Tremolite grains in sample HPSM are also assumed to be randomly oriented, since optical examination found no significant alignment of grains. Elastic moduli of olivine and magnetite
Fig. 7. Shear-wave velocities (50 . Vs1 . Vs2) calculated from CPO data by using Voigt, Reuss and VRH averaging schemes. The magnitude of polarization anisotropy (AVs) and the polarization plane of the fast shear wave are also shown. (a) Sample HPS-M and (b) sample HKB-B.
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are reported by Abramson et al. (1997) and Hearmon (1956), respectively. Elastic moduli of antigorite were calculated for the a-, b-, and c*- axes orthogonal reference frame. The c*-axis is perpendicular to the a–b plane of antigorite (Bezacier et al. 2010). Because of the absence of elasticity data of tremolite, elastic moduli of hornblende (Aleksandrov & Ryzhova 1961) were used instead. Calculated elastic stiffness tensors are listed in Table 2. Calculated compressional wave velocity is shown in Figure 6. All averaging schemes give azimuthal anisotropy, which is qualitatively similar to the measurements. The compressional wave velocity is fastest in the direction parallel to the lineation and slowest perpendicular to the foliation. This azimuthal anisotropy should be controlled by antigorite. The a-axes of olivine, along which the compressional wave velocity is fastest
Fig. 7.
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(Kumazawa & Anderson 1969), are weakly aligned perpendicular to the foliation in both samples. This cannot explain the observed azimuthal anisotropy. The b-axes of antigorite, along which the compressional wave velocity is fastest (Bezacier et al. 2010), are distinctly aligned parallel to the lineation; the c-axes, along which the compressional wave velocity is slowest, are aligned perpendicular to the foliation. The crystallographic orientation of antigorite grains qualitatively explains the observed azimuthal anisotropy. Neither tremolite nor magnetite contributes to the azimuthal anisotropy, since they are assumed to be randomly oriented. The stronger anisotropy in sample HKB-B is attributed to the larger amount and the stronger alignment of antigorite grains. Calculated shear-wave velocity is shown in Figure 7. The magnitude of polarization anisotropy
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Fig. 8.
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(AVs), the fastest shear-wave velocity (Vs1) and the slowest shear-wave velocity (Vs2) are indicated by colours as a function of the propagation direction. All averaging schemes give polarization anisotropy qualitatively similar to the observation. When the shear waves propagate in the foliation plane, the shear wave oscillating parallel to the foliation has the fastest velocity and the shear wave oscillating normal to the foliation has the slowest velocity. The polarization anisotropy is weakest for the shear wave propagating perpendicular to the foliation, although this is not clear in sample HPS-M. Stronger polarization anisotropy occurs in sample HKB-B. This is also attributed to the larger amount and the stronger alignment of antigorite grains in sample HKB-B. It should be noted that the fast shear-wave velocity (Vs1) is maximized around +458 from y-axis to x-direction. This suggests that velocity measurements in directions between the orthogonal directions are important for a more accurate characterization of seismic anisotropy (Godfrey et al. 2000). A comparison of measured and calculated velocities is made in the x-, y- and z-directions (Fig. 8). Velocities were measured at the confining pressure of 180 MPa. There are large velocity differences between Voigt and Reuss averages (0.7–1.0 km/s) in both samples, reflecting the strong elastic anisotropy of antigorite. In sample HPS-M, measured Vp agrees with the VRH average in the x-direction and with the Reuss average in the y-direction. In the z-direction, measured velocity is slightly lower than the Reuss average. The measured Vs1 is between the Reuss and the VRH averages in the x- and z-directions, whereas it agrees with the VRH average in the y-direction. The measured Vs2 is between the Reuss and the VRH averages in the x- and y-directions, whereas it agrees with the VRH average in the z-direction. In sample HKB-B, the measured Vp agrees well with the VRH average in the x- and y-directions, while it is slightly lower than the Reuss average in the z-direction. The measured Vs1 agrees with the VRH average in the x- and y-directions, but with the Reuss average in the z-direction. The measured Vs2 agrees with the Reuss average in all directions. In both samples, the measured velocities are between the Reuss and VRH averages. The measured values are closer to the lower bound than to the upper bound. Let us consider the cause of relatively low velocity. Possible causes are the influence of low confining pressure, the uncertainty in modal
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composition and the effect of geometry of antigorite grain which is not taken into account in calculations. First, we consider the influence of low confining pressure. Velocity calculations assume a polycrystalline aggregate without pores. Although most pores in rock samples are closed under the confining pressure of 180 MPa, some pores might remain open. Using a piston-cylinder-type high-pressure apparatus, we have made velocity measurements of sample HKB-B at a pressure of 1 GPa (Watanabe et al. 2010). Compressional wave velocity at room temperature is 7.68 + 0.04, 6.89 + 0.03 and 6.4 + 0.1 km/s in the x-, y- and z-directions, respectively. Velocity values are similar to those measured at 180 MPa in the x- and y-directions. Although a large increase in velocity was observed in the z-direction (perpendicular to the foliation), the compressional wave velocity of 6.4 km/s is still lower than the VRH average (Fig. 8). The velocity increase should not be attributed only to the closure of pores. The pressure effect on antigorite elasticity itself cannot be neglected. Hydrostatic compression curves of antigorite have shown that the linear compressibility in the c-axis direction significantly decreases with increasing pressure from 0 to 1 GPa (Hilairet et al. 2006b; Nestola et al. 2010). Since antigorite c-axes are aligned perpendicular to the foliation, compressional wave velocity in the z-direction depends strongly on the elastic stiffness along the c-axis. If the sample has no pores at atmospheric pressure, the compressional wave velocity in the z-direction will be lower than 6.4 km/s. We think that the relatively low velocity cannot be explained by the low confining pressure only. The modal composition estimated from microstructural examinations has an amount of uncertainty. If the volume fraction of olivine is overestimated or that of magnetite underestimated the calculation might give a higher velocity, leading to the relatively low measured velocity. Although it is difficult to evaluate the uncertainty in modal composition obtained by point counting, we do not think that it exceeds 5 vol%. The corresponding change in calculated velocity will be less than 0.2 km/s. This may partly explain the relatively low velocity. Finally, we consider the effect of the microstructure of antigorite grain. Studies on the effective elastic stiffness of a two-component composite material have shown that disc-shaped soft inclusions can reduce the bulk elastic stiffness down to the Reuss bound (Watt et al. 1976). Antigorite
Fig. 8. Comparison between measured and calculated velocities for sample HPS-M (left) and sample HKB-B (right). Propagation directions are indicated at the bottom. (a) Compressional wave, (b) fast shear wave and (c) slow shear wave.
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grains have a thin platy shape and the elastic stiffness is weakest in the direction perpendicular to the plate (c-axis direction). Antigorite grains can therefore work as platy soft inclusions and effectively reduce the bulk elastic stiffness. Although mineral grains are assumed to be uniformly distributed in the calculation, there are zones (Fig. 2a) and layers (Fig. 2c) of aligned antigorite grains. The bulk stiffness will be reduced in the direction perpendicular to these structures in comparison to a homogeneous microstructure. This might explain observed low velocities in the z-direction, especially in sample HKB-B. The platy shape of antigorite grains and the well-developed shape fabric were not taken into account in the calculation, and might be possible causes of the relative low velocity. Comparison between measured and calculated velocities in deformed rocks with sheet silicates (e.g. micas) will give us further information. Theoretical studies are also required to take into account the microstructural information in velocity calculations. Nishizawa & Yoshino (2001) and Nishizawa & Kanagawa (2010) have developed a method for calculating seismic velocity in mica-rich rocks. They modelled micas as thin inclusions and studied the effect of crystal shape. However, the host material was assumed isotropic in their study; the calculation method should be extended to treat elastically anisotropic host materials.
The change of the fast direction of compressional wave velocity could provide evidence for a serpentinized layer on top of the subducting slab. We suggest that northeast Japan is an ideal area in which to search for this change in the fast direction. Kawakatsu & Watada (2007) obtained the reflectivity profile by the receiver function technique. Reflectivities are measures of local shear-wave velocity jumps. One velocity jump at a depth of 80 – 120 km is interpreted to be the interface between serpentinized layer and oceanic crust, and the other velocity jump at a depth of 100–140 km is interprested to be the oceanic Moho. The shearwave velocity structure in the same vertical profile (fig. 3b in Tsuji et al. 2008) shows a weak lowvelocity anomaly (Vs ¼ 4.3– 4.4 km/s) at the top of the oceanic crust. It should be noted that velocities obtained by Tsuji et al. (2008) are assumed to be isotropic. The velocity of 4.3–4.4 km/s is consistent with the interpretation as serpentinized layer. The shallower part of top of the subducting slab has a higher velocity (Vs . 4.8 km/s), suggesting that this part is not serpentinized. The fast direction is perpendicular to the slab in the shallower part and parallel to the slab in the deeper part. We hope that seismologists will rise to the challenge of detecting this change.
Conclusions Implications for seismological observations Serpentinite mylonites from the Happo ultramafic complex were formed at a convergent plate boundary (Nozaka 2005). Let us consider the implications of our measurements for seismic observations in the mantle wedge. CPO data of olivine show that olivine a- and c-axes are weakly concentrated subperpendicular to the foliation and subparallel to the lineation, respectively. This type-C fabric of olivine was experimentally produced under conditions of high water content (Jung & Karato 2001). Before serpentinization, the fabric of olivine governs seismic velocities of mantle peridotites. For simplicity, we assume simple-shear deformation is induced by the subducting slab. The compressional wave velocity is fastest in the direction normal to the slab, and slowest along the slab. Hydrous fluids are likely to intrude planar defects created during deformation and potential slip planes (010)ol. Fluids may react to form antigorite, reduce seismic velocities and modify the seismic signal due to the olivine CPO. As serpentinization and deformation progresses, most c-axes of antigorite are oriented perpendicular to the slab. The compressional wave velocity is fastest in the direction parallel to the slab, and slow in the direction normal to the slab.
Elastic wave velocities and petrofabrics were measured in two antigorite-bearing serpentinite mylonites. Antigorite b-axes are strongly oriented parallel to the lineation and c-axes perpendicular to the foliation. Serpentinite mylonite samples show strong anisotropy for the propagation of compressional and shear waves, reflecting the crystal preferred orientation of antigorite and its strong elastic anisotropy. The compressional wave velocity is fastest in the direction parallel to the lineation, and slowest in the direction perpendicular to the foliation. The shear wave oscillating parallel to the foliation has a higher velocity than that oscillating perpendicular to the foliation. The rock sample with higher antigorite content has a lower mean velocity but stronger anisotropy. Measured velocities were compared to velocities calculated from petrofabric data using the Voigt, Reuss and VRH averaging schemes. All averaging schemes show velocity anisotropy qualitatively similar to measurements. There are large velocity differences between Voigt and Reuss averages (0.7–1.0 km/s), reflecting the strong elastic anisotropy of antigorite. Measured velocities are found between the Reuss and VRH averages. We suggest that the relatively low velocity is due to the platy shape of antigorite grains, the well-developed
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shape fabric and their strong elastic anisotropy. The configuration of grains should be an important factor for calculating seismic velocities in an aggregate composed of strongly anisotropic materials, such as sheet silicates. We would like to thank S. Arai, S. Ohtoh, T. Satsukawa and M. Shimojo for their help in sampling and microstructural analyses. D. Prior, S. Llana-Funez and an anonymous reviewer are thanked for their constructive comments and suggestions. This work was financially supported by Grants-in-Aid for Scientific Research ‘Rheology and Metamorphic Processes in Mantle Wedge Peridotites’ (22244062) and Cooperative Research Program (2008A-13, 2009-A-04, 2010-A-1-1428) of the Earthquake Research Institute, University of Tokyo.
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Obliteration of olivine crystallographic preferred orientation patterns in subduction-related antigorite-bearing mantle peridotite: an example from the Higashi –Akaishi body, SW Japan S. R. WALLIS1,2*, H. KOBAYASHI1, A. NISHII1, T. MIZUKAMI3 & Y. SETO4 1
Department of Earth and Planetary Sciences, Graduate School of Environmental Studies, Nagoya University, Nagoya 464-8601, Japan
2
Department of Earth Sciences, Bristol University, Wills Memorial Building, Queen’s Road, Bristol BS8 1RG, UK 3
Department of Earth Science, Graduate School of Environmental Studies, Kanazawa University, Kanazawa 920-1192, Japan
4
Department of Earth and Planetary Sciences, Faculty of Science, Kobe University, Nada Kobe 657-8501, Japan *Corresponding author (e-mail:
[email protected]) Abstract: Large parts of the mantle wedge near subduction boundaries are likely to be hydrated and contain antigorite. This mineral is acoustically highly anisotropic and potentially has a strong influence on seismic properties of the wedge. The Higashi–Akaishi body of SW Japan is an exhumed sliver of partially serpentinized forearc mantle, ideal for studying the effects of antigorite on the development of tectonic fabrics in the mantle. Samples with less than 1% antigorite show strong B-type olivine crystallographic preferred orientation (CPO) patterns. In contrast, samples with .10% antigorite deformed during the same tectonic event show much weaker olivine CPO patterns lacking the flow-normal a-axis concentration. These microstructural data suggest that the development of antigorite during deformation weakens olivine CPO due to phase boundary slip and associated rigid-body rotation of olivine grains. Antigorite and similar sheet silicates are likely to be present to some extent in the mantle wedge of all convergent margins. Our results suggest that even if this amount is only a few percent, strong olivine CPO is unlikely to develop and any pre-existing CPO is likely to be destroyed. Under these conditions, olivine CPO is unlikely to contribute significantly to seismic anisotropy in the mantle wedge.
Olivine crystallographic preferred orientation (CPO) is the dominant cause of seismic anisotropy in upper mantle and is widely used to estimate patterns of mantle flow (e.g. Tanimoto & Anderson 1984; Zhang & Karato 1995; Savage 1999). In oceanic areas, the fast polarization direction is typically parallel to plate movement direction and this observation can be explained by the development of olivine CPO with an a-axis maximum parallel to the flow direction in the mantle. This interpretation is supported by the common development of olivine CPO patterns with a-axis maxima parallel to mineral lineation in both ophiolites and mantle xenoliths (Nicolas et al. 1971; Ben Ismail & Mainprice 1998). However, convergent margins and continental rifts commonly show a seismic fast polarization direction perpendicular to plate motion and an alternative explanation is required. In continental rifts, many workers suggest the seismic anisotropy
is due to the presence of aligned melt-filled fractures (e.g. Kendall et al. 2006). In convergent margins, a variety of explanations have been proposed including complex flow parallel to the margin (e.g. Smith et al. 2001) and the development of unusual B-type CPO of olivine with a concentration of a-axes perpendicular to the flow direction (Jung & Karato 2001; Mizukami et al. 2004). Forearc mantle seismic anisotropy has also been attributed to aligned serpentine-filled faults within the slab (Faccenda et al. 2008) and the presence of fluidfilled cracks in the mantle (Healy et al. 2009). Here we focus on the possible role of CPO patterns within the mantle wedge. B-type olivine CPO was first observed in deformation experiments (Jung & Karato 2001); their development under natural conditions has also been verified (Mizukami et al. 2004; Skemer et al. 2006; Tasaka et al. 2008). Experiments show the development of B-type olivine CPO requires
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 113– 127. DOI: 10.1144/SP360.7 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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relatively high shear stresses. Combining these results with thermo-mechanical modelling leads to the suggestion that B-type fabrics may develop throughout large parts of the wedge (Kneller et al. 2005). Katayama (2009) shows that a 10 –20 km thick layer of mantle above the subducting slab that has a typical B-type olivine can account for the observed anisotropy in NE Japan (Okada et al. 1995; Nakajima & Hasegawa 2004). A number of recent studies have pointed out that the presence of hydrous sheet silicate minerals such as serpentine minerals can also have a strong influence on the seismic properties of mantle (e.g. Katayama et al. 2009; Mainprice & Ildefonse 2009). Serpentine minerals are formed from mantle under relatively low-T conditions and in the presence of H2O. In subduction zones, the inflow of cold lithosphere causes cooling of the overlying mantle. In addition, release of H2O-rich pore fluids and breakdown of hydrous phases both from subducted crust and within the subducted slab (Pawley & Holloway 1993; Schmidt & Poli 1998; Peacock & Hyndman 1999; Hyndman & Peacock 2003) cause hydration of the overlying mantle wedge. This hydration causes serpentine minerals to form by the reaction of water with olivine and a more siliceous phase, such as pyroxene. The hightemperature stability limit of serpentine is defined by the breakdown of antigorite at temperatures greater than c. 650 8C. The antigorite breakdown reaction is relatively insensitive to pressure changes, but at P above 2 GPa there is a gradual shift towards lower temperatures (Ulmer & Trommsdorff 1995; Hacker et al. 2003). Forearc mantle is both relatively cool and has a ready supply of H2O, implying that formation of serpentine in convergent margins is likely to be common. Widespread development of serpentine minerals in the forearc mantle is supported by the presence of serpentine diapirs (Fryer et al. 1985) and a variety of geophysical evidence (Takahashi et al. 1998; Kamimura et al. 2002; Blakely et al. 2005; Kawakatsu & Watada 2007; Tibi et al. 2008). Chrysotile –lizardite types of serpentine minerals are stable at relatively low temperatures. However, thermal modelling suggests that in both cool and relatively warm subduction zones the dominant stable polymorph of serpentine in the mantle wedge will be antigorite (Fig. 1). The wide stability in the mantle lithosphere (Hyndman & Peacock 2003), relatively low plastic yield strength (Hilairet et al. 2007) and very strong acoustic anisotropy (Bezacier et al. 2010) mean that antigorite potentially has a strong influence on the physical properties of the mantle. However, with some recent exceptions (Katayama et al. 2009; Mainprice & Ildefonse 2009) its presence is normally not considered in modelling of convergent margins, and it is unclear
Fig. 1. Calculated thermal structure of cool continental subduction zone based on data for NE Japan (adapted from Peacock & Hyndman 1999). The shaded region shows the part of the mantle wedge where serpentine (antigorite and chrysotile) is stable.
how the development of olivine CPO patterns is affected by the presence of antigorite. In this contribution, we present a study of olivine CPO patterns and fabric development in antigorite-bearing mantle wedge material and discuss the implications for understanding seismic properties of the mantle wedge.
Geology and deformation of the Higashi – Akaishi peridotite The Higashi –Akaishi (HA) peridotite body is part of the Cretaceous Sanbagawa high P/T metamorphic belt of SW Japan (Terabayashi et al. 2005; Wallis et al. 2009) and is exposed in the Besshi region, central Shikoku (Figs 2 & 3). As far as we are aware, the HA body is the only documented example of a kilometre-scale peridotite body from an oceanic subduction-type orogen that was demonstrably recrystallized in the garnet– lherzolite facies (Enami et al. 2004; Mizukami & Wallis 2005). The significance of this body is that
Fig. 2. (a) Index map of the Sanbagawa metamorphic belt. (b) Geological map of the Besshi region, central Shikoku (after Aoya 2001) (MTL: median tectonic line; HA: Higashi–Akashi peridotite body).
OLIVINE CPO IN ANTIGORITE PERIDOTITE
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Fig. 3. Geological map of the Higashi– Akaishi body and sample localities.
it represents a sliver of the forearc mantle exhumed from depths in excess of 100 km. This body preserves important information about processes operating deep in the forearc mantle above a subducting slab (Mizukami et al. 2004; Hattori et al. 2010; Sumino et al. 2010). Most other well-documented peridotite bodies in similar forearc tectonic settings have undergone extensive recrystallization and serpentinization during exhumation, and there is no clear evidence that they originated from such great depths.
Tectonic significance of deformation of the HA body Mizukami & Wallis (2005) use a combination of structural and petrological analyses to define four distinct deformational phases in the HA body: D1–D4. Mizukami & Wallis (2005) also show that the pressure–temperature (P–T ) path of the body can be constrained and linked to the deformation history by using a combination of geothermobarometry, microstructural observations and the structural relationship with surrounding metamorphic rocks (Enami et al. 1994; Aoya & Wallis 1999; Aoya 2001). The conditions associated with D1 are not well constrained and no clear tectonic interpretation has been proposed. However, application of garnet – pyroxene thermometry and orthopyroxene–garnet barometry to microstructural domains developed
during D2 reveals a record of deformation at temperatures of 700–800 8C and a pressure increase from 2.3 GPa to above 2.8 GPa – equivalent to ultra-high-pressure (UHP) conditions (Fig. 4; Mizukami & Wallis 2005). This increase of pressure at moderate temperatures during D2 clearly shows this phase of deformation is related to subduction rather than exhumation. The D2 P –T path lies mainly on the high-temperature side of the antigorite stability field; however, the higher P section of the path is constrained to shift to lower temperatures with the associated formation of antigorite. This type of P–T path is best explained as the result of subduction soon after initiation of subduction and before thermal steady state is achieved. Subsequent deformation phases, D3 and D4, are both related to exhumation and emplacement of the HA body at shallow levels in the orogen and represent exhumation processes. Further details are given in Mizukami & Wallis (2005). It was during D2 that the HA body reached peak pressure and maximum depth. Here, we focus on this phase of deformation and its associated fabrics.
Characteristics of D2 deformation D2 is the dominant phase of deformation throughout large parts of the HA body and is characterized by the development of a clear mesoscopic foliation S2 and mineral lineation L2. Mizukami & Wallis (2005) show that S2 is defined both by olivine
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S. R. WALLIS ET AL.
D2b
3
D2
e coesit quartz
Tr
D1 ?
2O
n+H
l2O3= 1.3 w t%)
D3
Ol + E
2
Atg (Al2O3= 0.1 wt%) Atg (A
Pressure (GPa)
D2a s on iti nd ary o c d T un P– o e on b 6) t a st cti 99 y- du , 1 ad ub ock e st e s ac of th Pe ge ong ( n a l R a
gt lherzolite facies sp lherzolite facies
1 Ds
D4
P–T Paths Higashiakaishi Besshi Unit
0
200
400
600
800
Temperature (°C) Fig. 4. P– T– D path of the Higashi–Akaishi peridotite body (Mizukami & Wallis 2005). gt, garnet; sp, spinel. Reaction curves: transition from spinel to garnet lherzolite from Ohara et al. (1971); antigorite stability from Ulmer & Trommsdorff (1995); quartz –coesite transition from Mirwald & Massonne (1980). Steady state P –T paths along subduction boundary (Peacock 1996) are calculated for subduction of 50 Ma oceanic lithosphere with convergence rates of 10, 30 and 100 mm/a and with shear heating where t ¼ 0 –5% P for 0 , T , 500 8C. The model assumes induced convective flow in mantle wedge at a depth corresponding to 2 GPa.
and varying amounts of aligned antigorite. This antigorite-bearing foliation is folded by the exhumation-related D3 deformation phase (Fig. 5), and Mizukami & Wallis (2005) suggest the antigorite parallel to S2 formed during the later stages of D2. This later stage of D2 can be referred to as D2b. An earlier part of the D2 stage is antigoritefree and is referred to as D2a. The earlier D2a is associated with subduction at moderate temperatures for mantle rocks, but still above the stability of antigorite. The development of antigorite during the later D2b stage therefore implies an anticlockwise P–T path with cooling at depths equivalent to UHP conditions (Fig. 4), suggestive of the conditions formed immediately after the onset of subduction and before thermal steady state is attained. Mizukami & Wallis (2005) further suggest that the development of buoyant weak serpentine is likely to have been the trigger for exhumation. The P– T evolution is preserved to different degrees in different D2 samples, offering the possibility of examining progressive structural changes
in peridotite as antigorite develops. Critical to this study is the interpretation that antigorite was present during late D2. The development of antigorite parallel to S2 is most readily explained as the result of syn-deformation formation, but the possible role of post-deformational mimetic crystallization also needs to be examined. The presence of D2 antigorite fabrics clearly folded by D3 folds (Fig. 5) suggests that this is not simply a case of late-stage antigorite growth mimicking an old fabric. In the following sections we present details of the microstructures of D2 tectonites and emphasize the evidence that supports the syn-D2 development of antigorite.
D2 microstructure Petrography In this study, four dunite samples preserving the subduction-related D2 fabric were examined (Fig. 3). The modal amounts of olivine and
OLIVINE CPO IN ANTIGORITE PERIDOTITE
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Atg
S2
Ol
S2
Atg
5m S2
Ol
d F3 fol e ac r axial t
Atg
S3
S2
S3
Ol
Field sketch of dunite foliations
2 mm
Fig. 5. Field sketch of tectonic fabrics in dunite with associated microstructres showing the S2b antigorite-bearing foliation folded by D3 deformation.
antigorite in these samples take values of 55 –99% and 0–40%, respectively (Table 1). The amount of olivine decreases as the amount of antigorite Table 1. Modal composition of D2 dunite Sample no. HC40
Hss44
5090402
Vol% (based on 1600 –2000 counted points) P-olivine 20.5 1.1 9.3 N-olivine 45.8 30.5 34.1 I-olivine 32.3 57.2 32.8 Antigorite – 9.8 21.2 Chromite 1.3 1.3 1.3 Clinopyroxene – – 1.7
HC117 1.7 32.5 22.7 42.2 0.9 –
increases. Chromite is present as an accessory mineral in all samples with a mode of approximately 1%. One sample contains a minor amount of clinopyroxene. Other serpentine minerals (chrysotile/lizardite) occur as late-stage veins, and were not included in the measurement of modal composition. The identification of all serpentine minerals was verified using Raman spectroscopy (Auzende et al. 2004).
Olivine The D2 deformational fabric typically displays a porphyroclastic microstructure consisting of dusty porphyroclastic olivine (P-olivine, 0.5– 2.0 mm) and finer-grained clear neoblastic olivine (Nolivine, 0.1–0.2 mm), which formed by dynamic
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S. R. WALLIS ET AL.
recrystallization of P-olivine (Mizukami et al. 2004; Mizukami & Wallis 2005) (Fig. 6). The difference between P- and N-olivine is clear in most cases, but some grains show an intermediate
grain size and texture. These subhedral olivine grains (,0.4 mm) with characteristics intermediate between P-olivine and N-olivine are classified as intermediate olivine (I-olivine).
Fig. 6. (a) Hand-specimen of antigorite-bearing dunite with tectonic fabrics defined by olivine (orange), antigorite (dark grey) and chrome spinel (black). (b) Antigorite-free dunite with dusty P-olivine and clear fine-grained N-olivine. Elongated N-olivine forms a preferred orientation defining the subduction-related fabric. Secondary lizardite and chrysotile serpentine minerals occur as late-stage veins along grain boundaries and cutting olivine grains (XZ section). (c) Microboudinage of chrome spinel in antigorite-bearing dunite. The space between the boudins is filled by olivine grains. The direction of elongated chrome spinel is parallel to the mineral lineation defined by an alignment of N-olivine and antigorite (section parallel to mineral lineation). (d) and (e) Antigorite-bearing dunite fabric (YZ sections). (f ) Platy antigorite and fine-grained olivine define the tectonic fabric, which wraps around P-olivine and chrome spinel (XZ section). P-Ol: porphyroclastic olivine; N-Ol: neoblastic olivine; I-Ol: intermediate olivine; Atg: antigorite; Spl: chrome spinel; V-Ser: late-stage serpentine veins.
OLIVINE CPO IN ANTIGORITE PERIDOTITE
In D2 samples, parallel alignment of elongate olivine grains defines both planar and linear features. Rod-like subhedral grains of chrome spinel (,2 mm) are locally found forming microboudins with their necks filled by olivine. The elongation direction of these grains shows the D2 stretching direction (Fig. 6c). Clinopyroxene (cpx) that is elongate in the same direction as N-olivine is also locally observed. This cpx is larger than P-olivine and occurs as porphyroclasts with maximum dimensions of 2–3 mm. Secondary chrysotile and lizardite veins also develop along grain boundaries or cutting P- and N-olivine grains (Fig. 6b).
Antigorite In D2b dunite, platy antigorite about 1 mm long is aligned parallel to the neoblastic olivine elongation direction. Antigorite grains are clear and show welldefined extinction positions in crossed polars. The subsequent deformation D3 folds the D2 fabrics including the antigorite foliation (see fig. 5 in Mizukami & Wallis 2005). We are therefore confident that the antigorite formed before D3, but there is the possibility that post-D2 mimetic growth produced the alignment of antigorite. We report here two microstructural observations that strongly suggest that the development of antigorite was largely syn-D2. (1)
(2)
There is an increase in the strength of antigorite alignment as the modal amount of this mineral increases (Fig. 6e, f ). If the antigorite formed after D2 (and before D3) and its alignment was simply due to formation along pre-existing grain boundaries of olivine, a decrease in preferred orientation of antigorite with increasing mode would be expected because the proportion of olivine boundaries would decrease. The increase in strength of the antigorite fabric is shown clearly by the shape preferred orientation (SPO) data (see later). Antigorite locally shows a clear deflection around porphyroclastic P-olivine and chromite (Fig. 6f), lacking any significant proportion of grains that develop at a high angle to this fabric. Locally this deflection is asymmetric, showing D2 deformation was non-coaxial. The localized deflection of the antigorite foliation around porphyroclastic olivine implies that the antigorite was affected by the local D2 strain field around the clasts and is therefore syn-D2.
Olivine and antigorite tectonic fabrics Shape preferred orientation (SPO) patterns of Nolivine and antigorite were determined to make a
119
quantitative comparison between them. N-olivine SPOs were also determined to accurately define the tectonic foliation. The foliation orientation is important for our studies because it is the reference frame for plotting the CPO patterns of N-olivine.
Shape preferred orientation measurements of olivine and antigorite Photomicrographs were taken of groups of grains of N-olivine and antigorite in YZ sections. Relatively homogeneous domains largely lacking P- and Iolivine were selected. Grain boundaries were then traced and digitized to determine shapes and define the SPO. For each sample the numbers of measured olivine and antigorite grains are 500–600 and 500– 800, respectively. The results of SPO analyses are represented as histograms, showing the orientation distribution of the elongation directions for each mineral (Fig. 7) as reported in Table 2. The centre of each histogram coincides with the mean direction. The mean direction for olivine in the same sample is also indicated in the antigorite histograms. The degree of preferred dimensional orientation of grains can be quantified using a concentration parameter k (Masuda et al. 1999). The circular equivalent to a Gaussian or normal distribution is the von Mises distribution, which is defined: f (u) =
1 ek cos (u−u) 2pI0 (k)
(1)
where u is the orientation, u is the mean orientation of the data and I0(k) is the modified Bessel function of order 0. The parameters u and k are equivalent to the mean (m) and the inverse of variance (1/s 2) in the standard Gaussian distribution. A stronger SPO corresponds to a smaller variation in the orientations of grains about the mean, and hence a high value of k. The olivine and antigorite SPOs all show unimodal distributions (Fig. 7) and the antigorite concentration ratios (k) increase with greater amounts of antigorite. Greater modal amounts of antigorite are associated with more elongate olivine grain shapes (Table 2). The difference in the mean directions of the olivine and antigorite distributions is small (2–68), suggesting that olivine and antigorite SPOs can be considered as forming part of the same foliation. The very similar SPO patterns of the two minerals are consistent with the olivine and antigorite fabrics having formed during the same deformation. It is the replacement of olivine by antigorite that leads to the elongate shape of the olivine. No evidence was observed that this replacement was accompanied by crystal plastic deformation of the olivine.
120
Fig. 7.
S. R. WALLIS ET AL.
OLIVINE CPO IN ANTIGORITE PERIDOTITE
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Table 2. Shape preferred orientations of N-olivine and antigorite in YZ section of D2 dunite tectonites Sample no. HC40 Hss44 5090402 HC117
N-Ol CR (k )
Atg CR (k )
N-Olivine axial ratio
AD of MD between N-Ol and Atg
0.46 0.37 1 2.3
– 0.38 0.98 1.9
1.48 1.5 1.71 1.95
– 68 28 68
N-Ol, neoblastic olivine; Atg, antigorite; CR, concentration ratio (k); AD, angular difference; MD, mean direction.
Olivine crystallographic preferred orientation patterns CPO patterns of neoblastic olivine in samples of D2 dunite were measured in samples with varying amounts of antigorite both with a universal stage and a scanning electron microscope (SEM) electron backscatter diffraction (EBSD) system. The results of the different methods give consistent and very similar results and we combined the results of both techniques to determine the CPO patterns. Before combining the results, it is necessary to convert U-stage measurements to Euler angles. The software we wrote for this purpose can be downloaded from http://web.me.com/westerngate/pubs.html. By measuring olivine CPO patterns in samples with varying amounts of antigorite, we aim to examine how the development of antigorite in hydrated peridotite affects olivine CPO. P-olivine is volumetrically small and its associated crystallographic orientation was not measured in this study. Mizukami et al. (2004) have documented olivine CPO in D1 dunite with an a-axis maxima parallel to the mineral lineation. D2a dunite that lacks significant amounts of antigorite shows a clear olivine CPO pattern with a c-axis maximum subparallel to the lineation, a b-axis maximum subperpendicular to the foliation and an a-axis maximum within the foliation and subperpendicular to the lineation (Fig. 8). This result is very similar to the D2 olivine CPO pattern reported by Mizukami et al. (2004) and represents an example of B-type olivine CPO of Jung & Karato (2001). Three samples of D2b dunite that contain significant amounts of antigorite (Fig. 9) show much weaker olivine CPO patterns (Fig. 8). For our study it is important to quantify the strength of the CPO patterns. The method of statistical analysis followed here treats spherical data as
unit vectors, which can be combined as an orientation tensor defined: ⎛ 2 li /N ⎝ mi li /N ni li /N
i /N li m 2 m /N i ni mi /N
⎞ li ni /N ⎠ m2i ni /N ni /N
(2)
where li, mi, ni are the direction cosines for each of the individual orientation data and N is the total number of measurements (Anderson & Stephens 1972; Woodcock & Naylor 1983). This tensor is symmetric and the three eigenvectors li are therefore mutually perpendicular. The largest of these vectors l1 gives an estimate of the mean orientation and the smallest l3 lies perpendicular to any girdle present. The third intermediate axis of the distribution is given by l2. The eigenvalues Si associated with li define the shape of the distribution: a cluster is characterized by S1 . S2 S3 and girdles are characterized by S1 S2 . S3. Values of Si can be used to test whether the data distribution is significantly distinct from random when compared to that expected for either a cluster or girdle distributions (Anderson & Stephens 1972). Woodcock & Naylor (1983) develop this approach and proposed the use of a concentration ratio, C ¼ ln [S1/S3]. This robust test is not based on assumptions about the distribution of the data (Woodcock & Naylor 1983) and is appropriate for testing complex distributions such as CPO patterns for non-randomness. Applying the Woodcock & Naylor C-test to our olivine CPO patterns shows that, in most cases, they differ significantly from random at the 95% confidence level (Table 3). Some features are still preserved in the olivine CPO patterns that can be compared to the original B-type pattern such as in HC117 (the sample containing 42% antigorite)
Fig. 7. Shape preferred orientation data of N-olivine (light grey) and antigorite (dark grey). Orientation data are separated into 158 bins and the 08 orientation is aligned with the mean direction for each sample. Black arrows indicate the mean orientation directions for N-olivine. The sample name, number of measurements (n) and concentration ratios (k) are also shown for each diagram.
122 S. R. WALLIS ET AL. Fig. 8. (a) Olivine neoblast CPO patterns in D2a and D2b dunite. The crystal axes are plotted with respect to the stretching lineation (L) oriented east–west and the tectonic foliation (S) defined by the SPO oriented vertical. Modal amount of antigorite is shown on the left of each set of CPO patterns. Greyscale shows contour intervals and contours are in multiples of uniform distribution. Maximum density of the distributions is shown in the bottom-right corner of each CPO. N: number of measurements. Equal-area lower-hemisphere projection. (b) Anisotropic elastic properties of dunite under ambient conditions with CPO of the samples shown in (a). Contours in km/s for Vp (light) and in % anisotropy for AVs (middle). Vp anisotropy and maximum AVs are shown on the bottom-right corner. Right: the direction of polarization of the fast shear wave (Vs1). These figures were prepared using software provided by D. Mainprice (Mainprice 1990).
OLIVINE CPO IN ANTIGORITE PERIDOTITE
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Fig. 9. Photomicrographs of dunite (XZ sections) used for the CPO measurements with varying amounts of antigorite.
(Fig. 8). The olivine a-axes in antigorite-bearing samples do not show a strong concentration. The patterns they do show are complex, including concentrations subparallel to the lineation and a girdle-type distribution subnormal to the lineation. These features show a fundamental difference to the B-type CPO. Although our samples are interpreted to represent a sequence formed in association with progressive development of antigorite, there is also no obvious weakening of the fabrics as the amount of antigorite increases from c. 10% to 40%. The sample with the greatest amount of antigorite (42%) actually shows slightly stronger concentrations of three olivine
axes than those of the other two antigoritebearing samples.
Anisotropic seismic properties of olivine CPO Anisotropic elastic properties of the antigoritefree D2a olivine CPO (Fig. 8) show that the fast direction of Vp corresponds to the direction of the a-axis concentration, which lies within the foliation and subnormal to the lineation. The direction of fast S-wave polarization direction also lies
Table 3. Strength of olivine CPO patterns. The value C (¼ ln[S1/S3]) expresses the strength of the preferred orientation in the data sample (Woodcock & Naylor 1983). This parameter can be used to test whether a CPO pattern is significantly different from random at specific confidence levels. Numbers in bold are significantly different from random distributions at the 95% confidence level Sample no. HC40 Hss44 5090402 HC117
a-axis
b-axis
c-axis
U-stage
EBSD
U-stage
EBSD
U-stage
EBSD
1.08 0.36 0.32 0.29
1.1 0.36 0.48 0.26
1.08 0.61 0.71 0.81
1.11 0.43 0.58 0.77
1.08 0.6 0.56 0.65
1.12 0.59 0.47 0.61
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S. R. WALLIS ET AL. 05090402 (21% Atg) AVs (%)
Vp (km/s)
Vs1 polarization
8.43
5.87 5.0
8.30 8.20
4.0
8.10
3.0
8.00
2.0
7.90
1.0
7.78
0.13
8.0%
5.86%
HC117 (42% Atg) AVs (%)
Vp (km/s)
Vs1 polarization
8.31
17.08 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0
8.08 7.92 7.76 7.60 7.44 7.28 7.17
14.8%
0.34
17.08%
Fig. 10. Seismic anisotropy calculated for samples 05090402 (21% antigorite) and HC117 (42% antigorite) using software provided by D. Mainprice (see Fig. 8). These calculations include the effects of both antigorite and olivine. Plots are all lower-hemisphere equal-area and are oriented so the foliation is horizontal and the mineral elongation lineation is east–west. The maximum values of anisotropy are given beneath each diagram.
approximately perpendicular to the lineation. In addition, both the Vp anisotropy and maximum difference between the fast and slow S-wave velocities (AVs) for this sample have relatively high values of 7.8 and 4.83%, respectively. In contrast, anisotropic elastic properties of D2b olivine CPO show a Vp fast direction and fast S-wave polarization direction subparallel to the lineation and roughly perpendicular to those shown by the D2a sample. The values shown by the D2b samples for Vp anisotropy and maximum AVs are 1.8–3.3 and 1.43 –2.63%, respectively. These are small and significantly less than the values shown by the D2a sample. The above figures for the seismic anisotropy refer to the olivine CPO only. Antigorite has strong acoustic anisotropy and its presence will affect the anisotropy of the samples. Determining this anisotropy requires that the antigorite CPO is combined with information about its elastic constants. The results for olivine and antigorite in the same sample must then be combined, taking into account their modal proportions. A detailed
discussion of this anisotropy is beyond the scope of the present contribution, but two examples serve to illustrate the effects of adding antigorite in moderate proportions (Fig. 10). In our sample with about 21% antigorite, both AVs and Vp anisotropy are more than twice that expected for olivine alone. The sample with around 42% antigorite shows maximum AVs values over 17%. It is clear that even at these levels of antigorite concentration, it is the antigorite that controls the anisotropy.
Discussion Interpretation of olivine CPO How can we account for the much weaker development of olivine CPO in the antigorite-bearing samples? Antigorite has a much lower plastic yield strength than olivine (e.g. Hilairet et al. 2007) and when hydrated peridotite undergoes plastic deformation the deformation will be concentrated in the antigorite. The process by which the antigorite
OLIVINE CPO IN ANTIGORITE PERIDOTITE
becomes aligned is not clear, but is likely to include mechanical rotation. Once the platy minerals are aligned close to the X –Z plane of finite strain and the shear plane, they will have a high syndeformational resolved shear stress along their basal planes and the grain contacts with olivine are likely to undergo grain-boundary sliding. The olivine grains can then undergo rigid body rotation. In the absence of a clear relationship between grain shape and crystallographic orientation, rigid body rotation will tend to randomize the distribution of crystallographic axes. Some process of growth such as diffusion is required to fill the voids created by rigid body rotation; if this growth occurs preferentially in some particular crystal orientation, the result may show a weak CPO (e.g. Wheeler 2009). Our results suggest that rigid body rotation and weakening of the olivine CPO patterns can occur when the amount of antigorite in peridotite is of the order of a few percent (≤10%). It is intriguing to note that the olivine CPO patterns in antigorite-bearing peridotite do show a statistically significant difference from that expected for a population of randomly oriented grains. In particular, they show a weak concentration of a-axes subparallel to the lineation. The reason for this is unclear. It is noteworthy that the sample with the highest proportion of antigorite shows the strongest olivine fabric. This suggests that there is a process acting to cause a weak alignment (perhaps related to a weak relationship between grain shape and crystallographic orientation).
Implication for seismic anisotropy in forearc mantle It is generally accepted that plastic deformation of the upper mantle causes widespread development of strong olivine CPO patterns. This preferred alignment of olivine crystallographic orientations is thought to be the dominant causes of upper mantle seismic anisotropy (Nicolas & Christensen 1987; Mainprice 2007). Seismic anisotropy in the wedge mantle in convergent margins is also generally interpreted as the result of flow-related olivine CPO (e.g. Jung & Karato 2001; Mizukami et al. 2004). In this contribution, we have shown that even relatively small amounts of antigorite (and presumably other sheet silicates such as lizardite, talc and chlorite) in the mantle prevent development of strong olivine CPO patterns and effectively remove any that were already present. The wedge mantle of convergent margins is generally thought to contain significant amounts of antigorite (e.g. Hyndman & Peacock 2003). Antigorite and other hydrous phases are likely to be particularly strongly developed close to the subduction boundary between
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the downgoing slab and the overlying wedge mantle. Our study therefore implies that seismic anisotropy in these regions is unlikely to be due to the development of strong olivine CPO patterns close to the subduction boundary. Strong olivine CPO patterns will only develop in parts of the mantle that are free from antigorite, that is, at temperatures above 650 8C. These regions are distant from mechanical boundaries in all but the hottest subduction zones and are unlikely to be associated with the high differential stresses required for the formation of B-type olivine fabrics. If strong olivine CPO patterns – in particular B-type fabrics – do not develop in the forearc mantle, then some other cause is required to account for the seismic anisotropy in this region. In our samples olivine is the dominant mineral but antigorite is aligned and strongly anisotropic; it is the volumetrically less significant antigorite that largely determines the seismic anisotropy. The preferred orientations of this and other sheet silicates in the wedge mantle close to subduction boundaries are likely to be a more important source of seismic anisotropy than olivine.
Conclusions The HA peridotite body of SW Japan preserves deformation fabrics reflecting subduction processes occurring in the mantle close to the plate boundary. Dunite samples strongly deformed by the synsubduction D2 deformation contain varying amounts of antigorite aligned parallel to the foliation. Asymmetric microstructures of antigorite around olivine porphyroclasts and the stronger development of antigorite shape preferred orientation (SPO) with increasing modal proportions both indicate that antigorite developed during D2 deformation and therefore at high pressure. The importance of these observations is that a suite of D2 tectonites with varying amounts of serpentine can be used to study the progressive changes in olivine CPO patterns of antigorite-bearing hydrated forearc mantle. In D2 dunite that lacks significant amounts of antigorite, olivine CPO patterns have B-type characteristics with an a-axis concentration perpendicular to the mineral lineation. The olivine CPO of antigorite-bearing dunite is much weaker for all three crystallographic axes. These weak olivine fabrics will only have a minor influence on the seismic anisotropy of the forearc mantle. The weakening or randomization of olivine CPO in antigorite-bearing dunite can be explained by grainboundary sliding that occurs between olivine and antigorite and causes associated rigid body rotation of the olivine grains. Antigorite is predicted to be present throughout large parts of forearc mantle
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regions and our results suggest these regions are unlikely to be associated with the development of strong olivine CPO. This implies that olivine CPO patterns are unlikely to be a prime cause of seismic anisotropy in forearc regions. Even at relatively moderate modal proportions, antigorite will dominate the anisotropy of mantle material. We are grateful to S. Llana-Fu´nez and M. Drury for careful and constructive reviews of this manuscript. We also thank K. Michibayashi for his assistance with the EBSD measurements and D. Mainprice for comments on this research.
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Dissolution precipitation creep versus crystalline plasticity in high-pressure metamorphic serpentinites ¨ CKHERT & CLAUDIA A. TREPMANN SARA WASSMANN*, BERNHARD STO Collaborative Research Centre 526, Institut fu¨r Geologie, Mineralogie und Geophysik, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany *Corresponding author (e-mail:
[email protected]) Abstract: Serpentinite is widely assumed to constitute weak material in subduction zones and to play an essential role for the development of a subduction channel. Information on deformation mechanisms and appropriate rheological models to describe these large-scale flow processes can only be obtained from natural serpentinites exhumed from ancient subduction zones. We examine the microstructural record of HP-metamorphic (P c. 2 + 0.5 GPa, T c. 550 + 50 8C) serpentinites exposed in the Zermatt–Saas zone, Western Alps, using optical and scanning electron microscopy with electron backscatter diffraction (EBSD). The schistose and compositionally layered rocks show pervasive small-scale folding. There is no evidence for any significant deformation by dislocation creep. Instead, the microfabrics including strain shadows and crenulation cleavage indicate that high strain is accumulated by dissolution precipitation creep. In terms of rheology, this suggests Newtonian behaviour and a low viscosity for the long-term flow of serpentinites in deeper levels of subduction zones. This does not preclude dislocation creep and a power law rheology at higher stress levels, as realized at local sites of stress concentration and transient episodes of post-seismic creep.
High and ultrahigh pressure (HP and UHP) metamorphic rocks exhumed from ancient subduction zones are identified in many orogenic belts worldwide (e.g. Coleman & Wang 1995; Harley & Carswell 1995; Schreyer 1995; Carswell & Compagnoni 2003; Ernst & Liou 2008). Thermochronometric results indicate that subduction and exhumation proceed at rates of centimetres per year, similar to plate velocity (Rubatto & Hermann 2001). As such, material cycling through subduction zones must be a rapid process. A feasible mechanism allowing high rates of burial and exhumation is provided by the subduction channel model, extending concepts originally developed for accretionary prisms (Shreve & Cloos 1986; Cloos & Shreve 1988a, b) to greater depth. Numerical simulations support the principal feasibility of the concept, subject to the validity of the chosen input parameters (e.g. Guillot et al. 2001; Schwartz et al. 2001; Gerya et al. 2002; Gerya & Sto¨ckhert 2006; Raimbourg et al. 2007; Warren et al. 2008a; Beaumont et al. 2009). Such models yield pressure –temperature (P–T ) paths for high-pressure metamorphic rocks (e.g. Gerya et al. 2002; Warren et al. 2008b, c; Beaumont et al. 2009) similar to those derived by petrological thermobarometry for exhumed natural rocks (e.g. Barnicoat & Fry 1986; Reinecke 1991; Schertl et al. 1991; van der Klauw et al. 1997; Perchuk & Philippot 2000; Terry et al. 2000; Cartwright & Barnicoat 2002; Carswell et al. 2003;
Parrish et al. 2006; Epard & Steck 2008). Furthermore, the simulations yield a crustal structure characterized by juxtaposition of units with contrasting P–T history and with an overall pattern grossly comparable to that established for the European Alps (Sto¨ckhert & Gerya 2005) within realistic timescales. The subduction channel simulations rely on rheological models for the different materials involved. In the simulations by Gerya et al. (2002) and Raimbourg et al. (2007), the material in the subduction channel was assigned a Newtonian behaviour with a constant low viscosity. This is motivated by the fact that (1) many HP and UHP metamorphic rocks appear to be little deformed, or even essentially undeformed (undistorted pillows: e.g. Oberha¨nsli et al. 2002; undeformed granite: e.g. Biino & Compagnoni 1992; Renner et al. 2001; Lenze & Sto¨ckhert 2007; undeformed gabbro: e.g. Zhang & Liou 1997), or (2) show evidence of deformation by dissolution precipitation creep (Sto¨ckhert 2002). Hence, deformation must generally be localized into weak shear zones (Sto¨ckhert & Renner 1998; Sto¨ckhert 2002) and the level of differential stress in subduction zones must generally be too low to drive deformation by dislocation creep, even at high temperatures and all along the subduction zone trajectory of the exhumed rocks. Which materials and mechanisms provide this weakness under (U)HP conditions is a matter for debate.
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 129– 149. DOI: 10.1144/SP360.8 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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The preferred candidate to play a major role in the development of a subduction channel and exhumation of (U)HP metamorphic rocks is serpentinite, derived by hydration of mantle material in the hanging wall of the subducted slab (e.g. Hermann et al. 2000; Guillot et al. 2001; Schwartz et al. 2001; Auzende et al. 2006; Federico et al. 2007; Agard et al. 2009; Angiboust et al. 2009; Hilairet & Reynard 2009). Accordingly, in numerical simulations, serpentinites are generally assigned a low viscosity (Guillot et al. 2001; Schwartz et al. 2001; Gerya et al. 2002; Hilairet & Reynard 2009). Laboratory experiments exploring the mechanical behaviour of serpentinite address the deformation mechanisms active during the progress of dehydration and simultaneously increasing porosity (e.g. Rutter & Brodie 1988; Tenthorey & Cox 2003; Llana-Fu`nez et al. 2007; Arkwright et al. 2008; Rutter et al. 2009) or the frictional strength of serpentinite (Raleigh & Paterson 1965; Escartin et al. 1997, 2001; Reinen 2000; Moore & Lockner 2007). These experiments address the brittle field and/or high differential stress conditions, which are probably not representative of the deeper levels of subduction zones. In novel experiments at confining pressures of 1 and 4 GPa, temperatures of 200 and 500 8C and strain rates of 1024 to 1026 s21 using ground natural antigorite, a power law rheology has been obtained by Hilairet et al. (2007). Microscopic inspection of their samples supports the importance of dislocation glide. The pressures and temperatures of these experiments correspond to conditions expected for subduction zones, yet the differential stresses are of the order 0.1–1 GPa. Extrapolating to strain rates of 10210 –10214 s21, Hilairet et al. (2007) conclude that a viscosity of 1019 Pa s represents a realistic upper bound for serpentinite in a subduction channel between 0.3 and 10 km thick. Gerya et al. (2002) proposed a constant low Newtonian viscosity of the order of 1018 –1020 Pa s to describe the material in a subduction channel, assuming dissolution precipitation creep in the presence of an aqueous fluid and increasing temperatures being compensated by increasing grain size. These estimates are based on the following two arguments. (1) Viscosities of the order 1019 Pa s or below are estimated for polyphase metasedimentary material deforming by dissolution precipitation creep, using the apparent viscosity contrast with embedded folded quartz layers deformed by dislocation creep with a power law available (Sto¨ckhert et al. 1999; Trepmann & Sto¨ckhert 2009). (2) Taking exhumed undeformed (U)HP metamorphic granites as stress gauges, the experimentally derived flow laws for dislocation creep of quartz (e.g. Paterson & Luan 1990; Gleason & Tullis 1993) and coesite (Renner et al. 2001) pose an upper bound
to the long-term level of differential stress acting on these rocks dependent on temperature, if evidence for crystal plastic deformation is absent. For the maximum temperatures derived by thermobarometry this upper bound would be a few MPa (Sto¨ckhert 2002). For strain rates in a subduction channel (of the order 10211 –10213 s21), which can be estimated based on assumptions on subduction rate and on the degree of localization of deformation, the predicted viscosity would then again be of the order 1019 Pa s or below. In contrast, in their numerical simulations Hilairet & Reynard (2009) prefer a power law rheology for serpentinite in a subduction channel, using the experimental flow law derived by Hilairet et al. (2007). Although the general patterns of material flow do not appear to be severely affected by the chosen rheological model, this may not be true for important details. Insight into deformation mechanisms and predictions on the rheology of materials in subduction zones can be gained by examination of the microfabrics of exhumed (U)HP metamorphic rocks. In the present study, we examine the microfabrics of intensely deformed serpentinites from the HPmetamorphic Zermatt–Saas zone in the Western Alps, using optical and scanning electron microscopy (SEM) with electron backscatter diffraction (EBSD). The aim is to evaluate the contribution of dissolution precipitation creep as opposed to dislocation creep in the accumulation of strain. Based on the microstructural record, we argue which rheological model may be preferred in models and numerical simulations on the deeper portions of serpentinite-dominated subduction channels.
Geological setting and sample location The Zermatt –Saas zone is located within the internal Penninic zone of the Western Alps (e.g. Coward & Dietrich 1989; Schmid et al. 2004), intercalated between the overlying units derived from the southern continent in the Alpine collision zone (i.e. Sesia zone and Dent Blanche nappe system) and the underlying continental Penninic units attributed to the northern continent (i.e. Gran Paradiso and Monte Rosa nappes), as depicted on the geological map in Figure 1. The Zermatt –Saas zone is interpreted to be a remnant of the once intervening oceanic lithosphere, which underwent eclogite facies metamorphism during Alpine subduction (e.g. Dal Piaz & Ernst 1978; Barnicoat & Fry 1986; Compagnoni & Rolfo 2003). Groppo et al. (2009) estimated the metamorphic conditions to pressures of 1.5 and 2.5 GPa at temperatures of approximately 500–600 8C. The Zermatt–Saas zone comprises huge bodies of serpentinite, derived from oceanic mantle, and metamorphosed oceanic crust
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Fig. 1. Geological map showing the oceanic units of the Zermatt–Saas zone in the region of the upper Val d’Aosta and the regional distribution of serpentinites (modified after Bigi et al. 1983). The sample locations are indicated.
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including sedimentary cover (Fig. 1). Conditions of UHP metamorphism in the stability field of coesite were identified in a slice of oceanic crust at Lago di Cignana (Reinecke 1991, 1998; Van der Klauw et al. 1997). In view of the variable P–T conditions and the heterogeneous structure of a dismembered ophiolite, the Zermatt– Saas zone has been interpreted as a potential candidate to represent a subduction channel (e.g. Bousquet 2008). In contrast, Angiboust et al. (2009) conclude from the similar P–T paths obtained for eclogites from widely spaced locations that the Zermatt–Sass zone was rather subducted and exhumed as a coherent slice of oceanic lithosphere. This interpretation may be in conflict with the dismembered structure, the intense deformation of a large portion of the rocks and the lack of a systematic gradient in the recorded P–T conditions parallel to the direction of subduction. The serpentinites investigated in this study have been sampled from outcrops in Val d’Aosta and Valtournanche (Fig. 1). Samples SW 32, SW 33, SW 35 and SW 44 –48 (N 458 53′ 38′′ , E 0078 37′ 16′′ ) taken about 1.5 km north of Valtournanche comprise antigorite, diopside, tremolite, chlorite and opaque oxide phases. Samples SW 36 and SW 43 taken from a road cut about 1.8 km north of Valtournanche (N 458 53′ 43′′ , E 0078 37′ 18′′ ) are primarily composed of antigorite and forsterite. Sample St 1009 taken in Val d’Aosta near the village of Breil (N 458 44′ 47′′ , E 0078 35′ 45′′ ) is composed of antigorite, forsterite, diopside, chlorite, tremolite, opaque oxide phases and inhomogeneously distributed titanoclinohumite.
Methods The samples were cut in two orientations perpendicular to the foliation and parallel and perpendicular to the lineation, respectively. Polished thin sections with a thickness of 30 mm were prepared. The microfabrics were analysed by optical and scanning electron microscopy. For analysis by EBSD (Prior et al. 1996, 1999, 2009) a LEO 1530 SEM was used. The thin sections were chemically polished with a silica suspension (Sytonw) for 15 min to reduce surface damage and then coated with carbon to avoid charging effects. The thin section was tilted at an angle of 708, the accelerating voltage was set to 20 kV and the working distance to 25 mm. The EBSD data were analysed using the HKLw software CHANNEL 5.0.
Microfabrics of serpentinites The serpentinites are characterized by a pronounced schistosity and pervasive small-scale folding. The prominent fold axes are parallel or slightly
oblique to the stretching lineation (Fig. 2). The folds are of multilayer type (Fig. 3). In pure antigorite layers, they can approach the geometry of chevron or kink folds. A later generation of folds with fold axes at a high angle to the stretching lineation is locally observed (Fig. 2). Vogler (1987) described similar structures in much detail from the Gouffre serpentinite body at Valtournanche and derived a tectonic history related to subduction and exhumation. Here, we focus on the deformation mechanisms in a set of representative samples. The schistosity of the serpentinites is defined by the combined shape preferred orientation (SPO) and crystallographic preferred orientation (CPO) of antigorite, which is the predominant mineral in most of the samples (Figs 3 & 4). In the hinges of smallscale folds the antigorite flakes are bent. In many cases, new grains have replaced deformed grains in regions of high curvature (Fig. 4b). In the majority of the samples a compositional layering is developed, with layers or highly elongate lenses of diopside, forsterite or chlorite (Fig. 4a, c, d). These minerals have crystallized during metamorphism of the original peridotite. This is evident wherever forsterite and, in places, also titanoclinohumite have overgrown antigorite or diopside that show a preferred orientation corresponding to the external schistosity (Fig. 5). Strongly elongate monomineralic aggregates of chlorite (Fig. 4d) probably represent pseudomorphs after spinel (e.g. Bucher & Frey 2002; Ferraris & Compagnoni 2003; Marocchi et al. 2007). Their aspect ratio in sections parallel to the stretching lineation and approximately normal to the foliation has been measured to be as high as 16 + 5 (N ¼ 25). In our samples, the only unequivocal relics of the original mantle mineralogy (apart from opaque minerals) are porphyroclasts of clinopyroxene up to a few millimetres in diameter, which are generally fragmented. The fragments are drifted apart in the direction of the stretching lineation. The space in between the fragments, here referred to as prismatic strain shadow (Fig. 2), is filled by the same mineral assemblage that constitutes the surrounding matrix with a similar microstructure, orientation and grain size (Figs 6–8). In some cases, the strain shadows are filled with pure forsterite aggregates (Figs 6b, c & 7a, c), in other cases with diopside and antigorite showing a pronounced preferred orientation (Figs 6a & 8). Likewise, these minerals fill the wedge-shaped strain shadows (Fig. 2) flanking the porphyroclasts at both sides. Forsterite shows some preferred orientation at these sites, which appears more pronounced within the prismatic strain shadows (Fig. 7b, d; upper row) compared to the surroundings of the clinopyroxene porphyroclasts (Fig. 7b, d; lower row).
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Fig. 2. Synoptic diagram (not to scale) showing the mesoscopic structures of representative sample St 1009. Foliation and schistosity, defined by layers of antigorite with preferred orientation, is deformed into small-scale folds. The fold axis (F1) is approximately parallel to the stretching lineation. (a) The stretching lineation is defined by strain shadows and fragments of porphyroclasts drifted apart as well as (b) strongly elongate chlorite aggregates, probably representing pseudomorphs after spinel crystals. (c) Large aggregates of olivine and titanoclinohumite with a grain size up to a centimetre have crystallized after formation of the schistosity, as antigorite or diopside with preferred orientation are overgrown. These aggregates formed prior to a second generation of folds (F2); in contrast, forsterite with a foam structure has crystallized in dilatant sites during this folding event. (d) A crenulation cleavage is locally developed in antigorite layers with forsterite precipitated from the pore fluid in dilational sites. The axes of the later generation of small-scale multilayer folds are oriented at a high angle to the stretching lineation.
In diopside-rich layers, the individual crystals show a characteristic elongate shape with an aspect ratio of c. 5.0 + 1.8 (N ¼ 513) and a pronounced combined SPO and CPO with the long axis of the grains corresponding to the crystallographic c-axis [001] aligned parallel to the stretching lineation and (100) aligned in the foliation plane (Fig. 8c). Optically, the diopside grains generally appear devoid of low-angle grain boundaries, supported by the EBSD scans shown in Figure 9. In most cases diopside makes up between 70 and 90% of the layer, the rest being antigorite or tremolite in subordinate amounts. Diopside grains are preferentially bound by interphase boundaries, which are predominantly rational (low index planes) with respect to diopside (Fig. 10). Forsterite occurs with a variety of microstructures in sample St 1009, including aggregates with differing grain size and slightly sutured high-angle grain boundaries. Most aggregates of forsterite, however, are composed of equant grains with plane or slightly curved interfaces meeting at near
Fig. 3. Optical micrographs (crossed polarizers) showing multi-layer folds in antigorite- and diopside-rich layers with pronounced combined SPO and CPO. In places, undeformed hypidiomorphic tremolite overgrows antigorite (atg, antigorite; di, diopside; tr, tremolite).
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Fig. 4. Optical micrographs (crossed polarizers) showing microstructures of antigorite schists. Images (a) and (c) show elongate lenses of deformed original clinopyroxene with strain shadows of fine-grained diopside embedded in antigorite. (b) New antigorite flakes grow across fold hinges. (d) Folds in elongate chlorite lenses (probably pseudomorphs after original spinel) and forsterite layers (atg, antigorite; di, diopside; chl, chlorite; cpx, clinopyroxene; fo, forsterite; tr, tremolite).
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Fig. 5. Optical micrographs (crossed polarizers) (a) Large forsterite crystals overgrow antigorite grains that show preferred orientation. (b) Titanoclinohumite crystals overgrow antigorite grains that show preferred orientation. (c) Crenulation cleavage in antigorite schist; forsterite is concentrated in fold hinges while the fold limbs are nearly free of forsterite. (d) Close-up of area marked by the white rectangle in (c): the forsterite aggregate exhibits a foam structure (atg, antigorite; di, diopside; fo, forsterite; ti chu, titanoclinohumite).
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Fig. 6. Optical micrographs (crossed polarizers) showing strain shadows in antigorite schists. (a) Wedge-shaped strain shadow of antigorite, diopside and chlorite at the edge of a large clinopyroxene porphyroclast. (b) and (c) Forsterite crystallized in prismatic strain shadows between fragments of clinopyroxene porphyroclasts (atg, antigorite; cpx, clinopyroxene; chl, chlorite; di, diopside; fo, forsterite; tr, tremolite).
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Fig. 7. (a) and (c) EBSD maps of the fragmented clinopyroxene porphyroclasts shown in Figure 6(b, c). Colours correspond to different crystallographic orientations of forsterite precipitated in the prismatic strain shadows in between the fragments and in the surroundings. (b) and (d) Pole figures (equal-area lower-hemisphere projection) showing the crystallographic orientation of forsterite precipitated in the prismatic strain shadows between the fragments (upper row) and in the surroundings of the porphyroclast (lower row). Colours correspond to the EBSD maps shown in (a) and (c), respectively (cpx, clinopyroxene; fo, forsterite).
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1208 angles at the grain edges. Within the polycrystalline forsterite domains, the microstructure approaches a foam structure controlled by interfacial free energy (Figs 5d, 11 & 12a). Signs of inhomogeneous crystal plastic deformation, such as low-angle grain boundaries or undulatory extinction, are systematically absent. EBSD mapping of
the forsterite aggregates reveals that there is no CPO (Fig. 11b). Where single forsterite crystals are bound by interphase boundaries, approximately plane or stepped interfaces parallel to (001) of antigorite prevail. In this case, the elongate grain shape of forsterite is controlled by the preferred orientation of antigorite (Fig. 12b, c).
Fig. 8. (a) Optical micrograph (crossed polarizers) showing pyroxene porphyroclast with prismatic strain shadows between fragments. The curved clinopyroxene slices in the upper part of the image probably represent original exsolution lamellae within orthopyroxene, now completely serpentinized (Vogler 1987). (b) EBSD map of area shown in Figure 8(a). Colours correspond to different crystallographic orientations of diopside precipitated between porphyroclast fragments shown in (a). (c) The upper row of pole figures (equal-area lower-hemisphere projection) shows the crystallographic orientation of diopside in the prismatic strain shadows, with the three large greenish squares representing the orientation of the cpx porphyroclast fragments, obviously unrelated. The lower row of pole figures shows the crystallographic orientation of diopside in the foliated matrix for comparison, colours corresponding to the EBSD map shown in (b) (atg: antigorite; di: diopside; cpx: clinopyroxene).
DEFORMATION OF HP SERPENTINITES Fig. 9. Misorientation profiles across single grains of diopside demonstrate a uniform crystallographic orientation and the absence of low-angle grain boundaries. Offsets are related to fractures. EBSD maps (colours display the relative misorientation of up to 208 relative to a reference within the respective grain) and optical micrographs show locations of profiles. The optical micrographs and the SEM images show identical grains, although their shape in transmitted light may appear slightly different from the SEM image. 139
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Fig. 10. SEM image (backscatter electron signal) of diopside and antigorite layer with pronounced SPO and CPO. Diopside is preferentially bound by rational interphase boundaries. Cross-cutting hypidiomorphic tremolite crystals are restricted to the diopside layer poor in antigorite to the right (white, diopside; dark grey, antigorite; light grey, tremolite).
In folded antigorite layers, forsterite has preferentially crystallized in one limb of the multilayer folds (the other being nearly pure antigorite with a very pronounced SPO and CPO) (Figs 5c & 13a, b). The fold limb with forsterite can be the long limb (Fig. 5c) or the short limb (Fig. 13a), or both can be approximately equal in length (Fig. 13b). In all cases the folds are asymmetric with respect to the angle between limbs and axial plane, the geometry corresponding to an asymmetric crenulation cleavage. Again, forsterite grain shape is controlled by antigorite (001) planes following the fold outline.
As observed for forsterite in aggregates, the isolated forsterite grains in this microstructural setting reveal no CPO (Fig. 11b). Large porphyroblasts of forsterite and titanoclinohumite, measuring up to centimetres in diameter, are observed in sample St 1009. Both phases occur side by side, with a very similar microstructure. Both have partially overgrown antigorite and diopside layers, which show a preferred orientation (Fig. 5a, b). The large forsterite and titanoclinohumite grains are therefore a product of metamorphism postdating the development of schistosity. In
Fig. 11. (a) EBSD map of forsterite aggregate with foam structure. Colours correspond to crystallographic orientation of forsterite as shown in (b). (b) Pole figures (equal-area lower-hemisphere projection) show no CPO for the forsterite aggregates (upper row) and for the isolated grains in the matrix (lower row) (atg, antigorite; fo, forsterite).
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Fig. 12. Optical micrographs (crossed polarizers) showing microstructure of forsterite aggregates. (a) Equilibrium 1208 angles at grain edges in forsterite aggregate (foam structure). (b) and (c) Interphase boundaries between forsterite and antigorite predominantly parallel to (001) of antigorite; in this case the preferred orientation of antigorite controls the grain shape of forsterite (atg, antigorite; fo, forsterite).
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Fig. 13. (a) Optical micrograph (crossed polarizers) showing forsterite crystals grown in between antigorite in the hinges and short limbs of asymmetric folds, while the long limbs are nearly devoid of forsterite. (b) Optical micrograph (crossed polarizers) showing an asymmetric crenulation cleavage with an accumulation of forsterite in one limb, while the other is nearly devoid of forsterite (atg, antigorite; di, diopside; fo, forsterite; tr, tremolite).
turn, the porphyroblasts predate the later stages of folding as their presence is observed to affect the fold shape. In some samples, diopside grains are rimmed by tremolite in crystallographic continuity. Isolated idiomorphic tremolite crystals up to about 0.1 mm in diameter are also found in antigorite layers. They have grown across fold hinges in places, and therefore must have crystallized after formation of the folds (Fig. 14).
Discussion The samples of HP-metamorphic serpentinites from the Zermatt–Saas zone in the Western Alps investigated in this study show a pronounced compositional layering (foliation) and schistosity, which are deformed into small-scale folds. The high aspect ratio (16 + 5) of stretched polycrystalline chlorite pseudomorphs demonstrates that these serpentinite samples represent high-strain zones within subducted oceanic lithosphere. The spatial
transition from nearly undistorted peridotites with cumulate fabrics into schistose and folded serpentinites is described in detail by Vogler (1987). Since the structures of the serpentinite samples investigated here (Fig. 2) are similar to the highly strained samples depicted by Vogler (1987), the small-scale compositional layering of flaser type is interpreted to be derived from the original coarsegrained peridotite fabric. Vogler (1987) described fine-grained olivine layers with CPO and a porphyroclastic microstructure, which possibly originate from pre-subduction deformation of the oceanic lithosphere and escaped serpentinization. The same is suspected for some aggregates of forsterite in sample St 1009, where very few relatively large grains show low-angle grain boundaries and the high-angle grain boundaries are somewhat sutured. Layer-parallel shortening resulted in pervasive folding on the millimetre to centimetre scale, governed by the thickness of the folded layers of the order c. 0.1–1 mm. For thin layers, the folds display a multilayer geometry (Figs 2 & 3) and approach the similar type (Hobbs et al. 1976; Ramsay & Huber 1987) as in the case of crenulation cleavage (Fig. 13). For layers with a thickness of the order millimetres, fold shape suggests a lower viscosity for pure antigorite compared to diopside-rich compositions (Fig. 3). Fold shape is also affected by the anisotropy of antigorite layers with combined SPO and CPO, evident from the prominent kink bands (Fig. 3). While olivine of the original mantle peridotite appears not to be preserved (except for minor amounts in sample St 1009), probably being completely transformed into antigorite at an earlier stage of the rock’s history, metamorphic forsterite occurs in large porphyroblasts and granular aggregates with a typical grain size of the order 0.1 mm. These forsterite grains crystallized during metamorphism after significant deformation, as they have overgrown both antigorite and diopside which show a combined SPO and CPO (Fig. 5a). In sample St 1009, titanoclinohumite (Fig. 5b) shows essentially the same microstructural characteristics as forsterite. Where metamorphic forsterite forms pure aggregates, the grains are of near-isometric shape and the high-angle grain boundaries are plane or simply curved. This foam structure controlled by minimization of interfacial free energy (e.g. Evans et al. 2001) indicates low-stress annealing after recrystallization, or that differential stress was too low to drive deformation by dislocation creep any time after the formation of forsterite. The forsterite aggregates with foam structure do not show a CPO (Fig. 11b). This is taken to rule out an origin of the equigranular microstructure during a previous
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Fig. 14. Optical micrographs (crossed polarizers) showing new idiomorphic tremolite crystals grown in folded antigorite layers (atg: antigorite; tr: tremolite; di: diopside).
stage of deformation by dislocation creep and recrystallization, followed by annealing and grain growth. A CPO developed during deformation by dislocation creep has been found not to disappear during normal grain growth (e.g. Park et al. 2001; Heilbronner & Tullis 2002), and a strong CPO may only be destroyed by discontinuous grain growth (Sto¨ckhert & Duyster 1999). There is no indication supporting the latter process in the forsterite layers. Crystallization of forsterite in a low-stress environment, at a level of differential stress insufficient to drive crystal plastic deformation of forsterite, is therefore indicated. In folded antigorite layers, minor forsterite is systematically concentrated in one limb of asymmetric multilayer folds while the other limb is made up of essentially pure antigorite with very pronounced combined SPO and CPO (Figs 5c & 13a, b). Such microstructures correspond to a crenulation cleavage in siliciclastic meta-sedimentary rocks (e.g. Mancktelow 1994; Passchier & Trouw 2005; McWilliams et al. 2007; Naus-Thijssen et al. 2010), indicating that either (1) forsterite was present before folding and became redistributed by dissolution along contacts with the
(001) basal plane of antigorite in one limb and precipitation in the other limb, or that (2) forsterite a priori crystallized during folding in zones of dilation. In both cases, transport in and precipitation from the pore fluid are required. In the prismatic strain shadows between the fragments of the clinopyroxene porphyroclasts and in the external wedge-shaped strain shadows (Fig. 2), minerals are precipitated from the pore fluid during deformation. The strain shadows comprise either nearly equigranular aggregates of pure forsterite with plane or slightly curved grain boundaries (Figs 6b, c & 7a, c), or polyphase aggregates of diopside, antigorite and chlorite with a combined SPO and CPO (Figs 6a & 8). The observation that pure forsterite aggregates have grown in strain shadows (Figs 6b, c & 7a, c) indicates transport in and crystallization from a fluid phase. The CPO of such forsterite aggregates appears more pronounced in prismatic strain shadows compared to the surrounding matrix, with a more or less developed foam microstructure (Fig. 7b, d). External shape and microstructure suggest that the observed CPO is a result of oriented nucleation and growth and not of crystal plastic deformation.
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Diopside layers show a pronounced combined SPO and CPO, with the long axis of the diopside fibres [001] aligned parallel to the stretching lineation and (100) approximately parallel to the foliation (Fig. 8c). If the result of deformation by dislocation glide, the CPO could be interpreted to indicate activation of the (100)[001] glide system which is observed in experimental deformation (e.g. Raterron & Jaoul 1991). It is also inferred for naturally deformed rocks (Buatier et al. 1991; Godard & Van Roermund 1995; Ulrich & Mainprice 2005) though being not necessarily the predominant system. However, in the Zermatt –Saas serpentinites, the microstructure of the diopside layers does not support deformation by dislocation creep and, consequently, the CPO does not reflect an activated glide system. (1) There is no evidence of strain-induced grain-boundary migration. (2) Most grains are bound by interphase boundaries with antigorite, which constitutes a minor phase in the diopside-rich layers (Fig. 10). These interfaces tend to be low-index planes, predominantly {110}, with respect to diopside. (3) There is no evidence of inhomogeneous crystal plastic deformation of diopside; low-angle grain boundaries are not observed (Fig. 9). These microstructures are taken to rule out deformation of diopside by dislocation creep. Consequently, the combined SPO and CPO of diopside in these layers must be the result of oriented nucleation and growth, as proposed by, for example, Mainprice & Nicolas (1987), Godard & Van Roermund (1995), Mauler et al. (2000) and Sundberg & Cooper (2008), and not of reorientation during crystal plastic deformation with the (100)[001] glide system predominating. In this case, experimentally derived flow laws for clinopyroxene in the dislocation creep regime (e.g. Raterron & Jaoul 1991; Bystricky & Mackwell 2001; Dimanov & Dresen 2005; Moghadam et al. 2010) cannot be used to estimate effective viscosities for adjacent layers from fold shape (e.g. Trepmann & Sto¨ckhert 2009). Comparison reveals that the microstructure of the polyphase strain shadows corresponds to the microstructure of layers with the same mineral assemblage beyond the porphyroclasts. As shown in Figure 8c, the CPO of diopside is not much different. Based on this similarity, the strain shadows being definitely filled with minerals grown from the fluid phase, similar processes appear feasible for the rock matrix. The orientation relations depicted in Figure 8c preclude a control of the CPO in the diopside layers by mimetic growth on old clinopyroxene fragments which are – in addition – only locally preserved. In view of the above observations, discarding a deformation by dislocation creep in the forsterite aggregates and in the diopside-rich layers, the similarity between
microfabrics in strain shadows and in corresponding layers in the bulk rock suggests dissolution precipitation creep as the predominant deformation mechanism throughout. Microstructures of a fold hinge depicted in Figure 10 suggest that antigorite becomes dissolved along interphase boundaries with diopside and precipitated at sites of dilatation. In contrast, in the forsterite –antigorite system the opposite relations are observed. There, forsterite appears to be readily re-distributed while antigorite controls the fold shape. The inferred predominance of dissolution precipitation creep is not meant to exclude a contribution of basal glide in antigorite, as suggested by the widespread kink bands in pure antigorite layers with very pronounced combined SPO and CPO. Finally, the presence of a pore fluid and mineral dissolution and precipitation in an open system combined with reactions is well demonstrated by the microfabrics of tremolite. This mineral rims preexisting diopside grains in some samples, but also occurs as undeformed and euhedral crystals replacing antigorite. In this case, it is observed to grow across fold hinges which indicates its formation after folding and the replacement of antigorite volume by volume (Fig. 14). Summing up the observations discussed above, there is no evidence for significant deformation in the field of dislocation creep apart from a possible contribution of basal glide in antigorite. Instead, the observations indicate dissolution precipitation creep as the predominant deformation mechanism accumulating large strain and controlling the bulk rock rheology. Also, the presence of an aqueous solution and an open system on a local scale is implied. Such processes where metamorphic reactions are combined with dissolution and precipitation during deformation are referred to as incongruent pressure solution (Beach 1979; McCaig 1987; Wintsch & Yi 2002). From the rheological point of view, dissolution precipitation creep can be treated as analogous to Coble creep (Elliott 1973; McClay 1977; Evans & Kohlstedt 1995; Kohlstedt et al. 1995). Strain rate is predicted to be proportional to stress and inversely proportional to grain size to the power of three. The theory of dissolution precipitation creep is dealt with in a considerable number of studies (e.g. Elliott 1973; Rutter 1976, 1983; Robin 1978; Raj 1982; Cox & Etheridge 1983; Wheeler 1992; Shimizu 1995; Kruzhanov & Sto¨ckhert 1998; Renard et al. 1999; Spiers et al. 2004; Van Noort et al. 2008) and compared to the results of laboratory experiments on granular aggregates (e.g. Gratier & Guiguet 1986; Spiers et al. 1990) or pairs of minerals in an aqueous solution (e.g. Urai et al. 1986; Hickman & Evans 1991, 1995; Gratier et al. 1999, 2009; Martin et al. 1999; Lohka¨mper
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et al. 2003). In fact, natural microstructures suggest that interphase boundaries between unlike minerals act as effective sites of dissolution. Dissolution precipitation creep is therefore proposed to be particularly important in polyphase rocks (e.g. Schwarz & Sto¨ckhert 1996; Sto¨ckhert 2002; Trepmann & Sto¨ckhert 2009) and rarely described for nominally monomineralic material (e.g. Andreani et al. 2005). Also in the latter study, small particles of ironoxides are reported to be present in addition to serpentine. These observations on natural rocks are supported by experimental evidence (e.g. Hickman & Evans 1991, 1995; Bos & Spiers 2000) and considered in a model by Wheeler (1992). Although a polymineralic rock undergoing deformation by incongruent dissolution precipitation creep is a complex system, it is assumed that it nevertheless obeys the principles derived for simple monomineralic systems. If this is true, Newtonian behaviour can be predicted for serpentinite in the deeper level of subduction zones, as far as long-term viscous deformation is concerned. This does not preclude dislocation creep and the holding of a power law rheology, found in laboratory experiments on antigorite by Hilairet et al. (2007), for higher stress levels which may be realized at sites of stress concentration or during coseismic loading and subsequent short episodes of postseismic creep (Hilairet et al. 2007). For the long-term viscous deformation addressed in numerical simulation of subduction channels, the validity of the assumption of Newtonian behaviour and a viscosity of 1019 Pa s or below is however supported by the record of exhumed high-pressure metamorphic rocks. The long-term stress level along the plate interface in subduction zones appears to be insufficient for a significant contribution of dislocation creep to overall deformation.
Conclusions The microfabrics of high-pressure metamorphic serpentinites of the Zermatt– Saas zone (Western Alps) provide insight into the mechanisms of deformation at a depth of c. 50 –70 km in a subduction zone. The highly strained polyphase serpentinites are schistose, compositionally layered and folded, with sparse relics of the original mantle minerals. The microstructures invariably indicate deformation by dissolution precipitation creep in an open system, with an aqueous fluid involved. The strong combined SPO and CPO of fine-grained metamorphic diopside layers is related to nucleation and growth, not to dislocation glide. The same holds for antigorite, though here a contribution of basal glide is not excluded. Evidence for deformation by dislocation creep in metamorphic
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forsterite is absent; this mineral is re-distributed by dissolution and precipitation. For the long-term viscous deformation addressed in numerical simulations of a subduction channel, the assumption of Newtonian behaviour of serpentinite and a viscosity of 1019 Pa s or below is supported by the record of exhumed high-pressure metamorphic rocks. This is consistent with the upper bound to viscosity (1019 Pa s) predicted from deformation experiments on antigorite in the dislocation creep regime (Hilairet et al. 2007). Long-term stresses in subduction zones are probably too low to activate dislocation creep. Most of the high strain appears to be accumulated by dissolution precipitation creep at low differential stress. We thank E. Mariani and A.-M. Boullier for their detailed and constructive review. Financial support by the German Science Foundation within the scope of Collaborative Research Center 526 ‘Rheology of the Earth’ is gratefully acknowledged.
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Crystal fabric development and slip systems in a quartz mylonite: an approach via transmission electron microscopy and viscoplastic self-consistent modelling LUIZ F. G. MORALES1*, DAVID MAINPRICE1, GEOFFREY E. LLOYD2 & RICHARD D. LAW3 1
Ge´osciences Montpellier UMR CNRS 5243 & Universite´ Montpellier 2, Place Euge`ne Bataillon, Batıˆment 22, 34095, Montpellier cedex 05, France 2
Institute of Geophysics and Tectonics, School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK
3
Department of Geosciences, Virginia Tech., Blacksburg, VA, 24061, USA *Corresponding author (e-mail:
[email protected])
Abstract: We have applied transmission electron microscopy (TEM) analyses coupled with viscoplastic self-consistent (VPSC) numerical modelling to identify the active slip systems and to better understand the crystal preferred orientation (CPO) development of the Torridon quartz mylonite (NW Scotland). TEM analyses showed evidence of activation of 1/3kal{p′ }, 1/3kal{z} and possible kal(c) slip systems, as well as dislocation climb and dynamic recrystallization. All the CPOs generated by VPSC models share common characteristics with the Torridon quartz mylonite, but only Models 2 and 3 reproduce the [c]-axes maxima at low angle (,208) to the foliation pole along the YZ plane, as observed in the mylonite. In Model 2, this concentration only occurs at g ≥ 2.6, whereas in Model 3 this maxima occurs at lower shear strains. The models that start with a previous preferred orientation acquire very strong CPOs after small-imposed strains, followed by the rapid rotation of the fabric in relation to the new imposed finite strain axes. The combined activation of kal{p′ }, kal{z} and possibly kal(c) slip systems, as demonstrated by TEM analyses, suggests that the VPSC model that best predicts CPO development in the Torridon quartz mylonite is Model 2, where the critical resolved shear stress (CRSS) of kal{p/p′ } is assumed to be slightly stronger than kal(c).
Quartz mylonites are typical rocks of high-strain zones and result from localized deformation in quartzites and quartz veins and by the intense deformation of granitoids in the presence of fluids under upper to middle-crustal conditions (e.g. Dixon & Williams 1983; Law et al. 1986, 1990; Lloyd et al. 1992; Fitz Gerald & Stu¨nitz 1993; Goodwin & Wenk 1995; Hippertt 1998; Wibberley 1999; Lloyd 2004; Jefferies et al. 2006; Pennacchioni et al. 2010). Less common but no less important is the occurrence of quartz mylonites in the lower crust, suggesting high differential stresses for their generation and implying a contrasting geological behaviour for the lower crust under certain conditions (e.g. Fitz Gerald et al. 2006). In all of these cases, quartz mylonites play an important role in controlling lithospheric strength as they may allow deformation localization (e.g. Carter & Tsenn 1987). Independently of the processes in which the quartz mylonites are generated, a common feature observed in these rocks is the development of strong crystal preferred orientation (CPO) of quartz.
Quartz CPOs can be used to infer deformation temperatures, magnitude and symmetry of strain and kinematic framework and the mechanisms for deformation (e.g. Schmid & Casey 1986; Law 1990; Stipp et al. 2002; Law et al. 2004). The development of CPO usually implies the activation of dislocation creep. CPO patterns are usually interpreted in terms of the relative activation of different slip systems, essentially controlled by temperature and variation in the strain and kinematic conditions (e.g. Schmid & Casey 1986; Okudaira et al. 1995; Takeshita 1996; Kurz et al. 2002; Heilbronner & Tullis 2006). Nevertheless, quartz CPOs depart from these expected patterns in many cases given the deformation conditions of certain rocks, and their interpretation by purely intracrystalline plasticity mechanisms is not straightforward (e.g. Hippertt 1998). Mechanisms such as dynamic recrystallization, dissolution/precipitation, grainboundary sliding and diffusion creep have also been used to explain the variety of quartz fabrics observed in both nature and experiments (e.g.
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 151– 174. DOI: 10.1144/SP360.9 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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Wenk & Christie 1991; Gleason et al. 1993; Hippertt & Egydio-Silva 1996; van Daalen et al. 1999; Heilbronner & Tullis 2002, 2006; Stipp et al. 2002; Halfpenny et al. 2006; Vernooij et al. 2006). In addition, CPO patterns are typically a ‘post-mortem’ feature for which we usually do not know the deformation path that produced the preferred orientation. In general, CPO patterns are interpreted assuming initially randomly distributed crystal orientations in a volume of rock prior to deformation. This deformation, and the mechanisms responsible for its accommodation, is responsible for the development of a given CPO. Indeed, in many cases this is what happens (e.g. Pennacchioni et al. 2010). In other cases, for example, the assumption has to be made simply because of the lack of exposure from low- to high-strain conditions along a shear zone. The role of pre-existing CPO on fabric development in high-strain zones is only rarely considered (e.g. Lister & Williams 1979; Ralser et al. 1991; Lloyd et al. 1992; Toy et al. 2008; Pennacchioni et al. 2010). Such ‘initial’ CPOs may have a strong influence on the ‘final’ CPO patterns observed in shear-zone mylonites. The quartz mylonite studied here originated from intense simple-shear deformation of a quartzfeldspar vein infilling a joint in Lewisian gneiss of the Upper Loch Torridon area (NW Scotland). This vein is perhaps one of the most-studied quartzbearing mylonites in terms of microstructures and preferred orientations (Law et al. 1990; Lloyd et al. 1992; Trimby et al. 1998; Lloyd 2004; Lloyd & Kendall 2005). Nevertheless, specific points regarding the nature and evolution of the CPO of the mature mylonite (centre of the vein/shear zone) and its relation to the CPO of the original quartz vein (now only preserved at the margins of the quartz vein/shear-zone walls) remain poorly constrained. Specifically, it is not well understood whether the crystallographic orientation observed in the mature mylonite is the result of a single episode of deformation dominated by single or multiple slip systems (Law et al. 1990) or if the observed preferred orientation reflects an inherited initial fabric from the margin of the sheared quartz vein (Lloyd et al. 1992). The preferred orientation observed in the quartz mylonite from Torridon cannot be solely explained by intra-crystalline plasticity via mutual activation of basal, rhomb and prismatic slip in the kal direction, as commonly observed in quartz-rich tectonites deformed at relatively low temperatures (e.g. samples R-405, P-248 and C-156 of Schmid & Casey 1986); other mechanisms have to be active, such as Dauphine´ twinning and dynamic recrystallization (Law et al. 1990; Lloyd et al. 1992; Trimby et al. 1998; Lloyd 2004). The Torridon quartz mylonite is therefore a
good case study for testing, through systematic modelling of crystallographic fabric development via the viscoplastic self-consistent (VPSC) approach (Lebensohn & Tome´ 1993; Tome´ & Lebensohn 2004), whether CPO results from a single deformation episode (starting from a random orientation) or if the observed CPO is inherited from the margin of the sheared vein. To better constrain the CPO predictions by numerical modelling, transmission electron microscopy (TEM) images from the mature mylonite are also presented, providing new information regarding the microstructural evolution of this shear zone and complementing previous work carried out by Law et al. (1990), Lloyd et al. (1992), Trimby et al. (1998), Lloyd (2004) and Lloyd & Kendall (2005).
Sample description and quartz CPO The quartz feldspar vein from the head of Upper Loch Torridon (Fig. 1a, b) was sheared during relative motion of joint blocks of Lewisian gneiss on either side of the vein. Foliation (XY) associated with shearing increases in intensity traced in to the centre of the vein and simultaneously curves towards parallelism with the vein/shear-zone margins (sensu Ramsay & Graham 1970) but is still inclined at 98 to the shear-zone margins in the ‘mature’ high-strain centre of the vein (Law et al. 1990). Original vein textures are only locally preserved at the low-strain margins of the vein. The high-strain centre of the vein is characterized by a typical Type-2 S–C′ microstructure (Law et al. 1990). The vein microstructure is arranged in two planar domains aligned parallel to the macroscopic foliation. One domain consists of elongate dynamically recrystallized quartz grains (,100 mm) whose long axes are oblique to the main foliation, while the second domain is characterized by an equigranular texture of quartz and feldspar with grain size of less than 5 mm. The microstructure suggests that deformation occurred under constant-volume plane strain flow conditions very close to strict simple shear (Law et al. 1990; Wallis 1995, p. 1082). A band-contrast electron backscatter diffraction (EBSD) map (Fig. 1a) and cathodoluminescence (CL) image (Fig. 1b) illustrate development of the shear zone with a spatially rapid development of foliation, grain-size reduction and fracturing of feldspar crystals within the quartz matrix. The mature mylonite (MM) in the centre of the vein consists of dynamically recrystallized grains originating from a few larger grains preserved at the vein margins (Lloyd et al. 1992; Trimby et al. 1998). Close inspection of the boundary of the quartz vein material at the margins of the vein (starting
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Fig. 1. SEM/EBSD images of the Torridon Shear Zone (XZ plane) include (a) a digital band contrast map in which the greyscale reflects the quality of the electron back-scattered patterns (bright: good EBSPs; dark: weak EBSPs) and (b) a cathodoluminescence (CL) image (montage of multiple scans) of the whole sample. Both images reveal detailed features of the microstructures, including: grain-size reduction over a short distance; mylonitic foliation development; and foliation deflection and brittle behaviour of feldspars. (c) Pole figures from EBSD maps for the shear-zone wall rock (SZWR), shear-zone margin (SZM) and mature mylonite (MM). Pole figures are plotted in shear-zone margin reference frame, represented here by the horizontal east– west plane (horizontal line). Note in the microstructural images that the foliation is inclined c. 98 anticlockwise in relation to the reference east–west plane, as marked by the black line in the pole figures (see reference frame in inset). Lower hemisphere, equal-area projections, multiples of uniform distribution, MAD ¼ 1.38. Figure (a) modified from Lloyd (2004).
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are distributed conically around the maxima of [c]-axes with opening angles of 288 for the former and between 52 and 578 for the latter.
Transmission electron microscopy For TEM-based microstructural analyses of the mature mylonite, selected areas of a thin section similar to the sample T1 of Law et al. (1990), Lloyd et al. (1992) and Lloyd (2004) were thinned by standard argon ion bombardment procedure with an accelerating voltage of 5 kV. These areas were then examined on a JEOL 120 kV transmission electron microscope using conventional bright-field imaging techniques. Dislocation orientations in relation to the crystallographic directions of quartz were characterized by the trace analysis method, in which the orientation of linear and planar features is recorded for a number of goniometer tilt angles, allowing the determination of their directions. The orientation of the Burgers vector (b) was determined via the ‘invisibility criteria’ (e.g. McLaren 1991), in which pure edge and screw dislocation are invisible or in very weak contrast for diffraction vectors (g) normal to the Burgers vector (g . b 5 0 and g . b 3 u 5 0, where u is the line direction of the dislocation). Depending on the angles between the diffraction vectors, the Burgers vector and dislocation line, the contrast in images can vary from moderate to strong as the contrast is proportional to g . b 3 u 5 0 even when g . b 5 0 for mixed dislocations. In practice, g . b 5 0 images are very weak when the deviation from the exact Bragg condition (s ¼ 0) is large (s . 0), even when g . b 3 u 5 0. Thus, s . 0 was used to characterize the Burgers vectors. Using computer-simulated images of quartz, the symmetry of dislocation images about the dislocation line has been shown to be a useful indicator of the g . b 5 0 condition (Mainprice 1981). When g . b 5 0, the images are symmetric about the dislocation line when s ¼ 0 and very weak when s . 0. Dislocation analysis in Figure 2 demonstrates that the dislocations are out of contrast for a diffraction vector g2 ¼ (1101) (g2 . b ¼ 0), which means that g2 ⊥ b (Fig. 2b); however, with g1|g3 ¼ (01¯11) they show strong contrast (g1 . b 5 g3 . b . 0) (Fig. 2a, c). In the strong-contrast images, the dislocations are asymmetrical about the dislocation line (Fig. 2a, c) whereas they appear symmetrical in the weak-contrast image (Fig. 2b). The variation in dislocation image contrast in Figure 2b is due to crystal bending changing the Bragg conditions from s c. 0 in the top left to s . 0 in the bottom right, consistent with g . b ¼ 0. The dislocation lines are essentially parallel to each other and suggest that the structure of quartz controls /
material) and mature mylonite in the centre of the vein/shear zone revealed the presence of abundant Dauphine twinning in narrow intra-granular ‘bands’ (Lloyd 2004). A comparison between the pole figures of different parts of the shear zone suggest a rapid spatial transition of the quartz CPO between the original vein material at the margin of the vein/shear zone and the mature mylonite in the centre of the shear zone, in agreement with the observed microstructures. Note that all the pole figures in Figure 1 were plotted with the shear-zone margin as the reference plane, where the foliation of the mature mylonite is inclined c. 98 anticlockwise in relation to this reference plane (inset diagram in Fig. 1). In the shear-zone wall rock, the quartz c (0001)-axes are distributed in one dominant and two secondary single maxima (Fig. 1) at angles between 30 and 608 to each other; the dominant maxima lies at an angle of c. 458 from the pole to the shear plane. The poles for the first m {1010} and second a {2110} order prisms are distributed in three dominant point maxima forming a great circle girdle whose pole is the strongest concentration of [c]-axes. The poles to positive rhombs r {1011} are strongly concentrated near the centre of the stereonet (sample Y direction), while the poles to negative rhomb z {0111} are distributed in ten distinct maxima distributed over the equal-area net. The poles to acute rhombs p {1012} and p′ {0112} are distributed along small circle (conical) girdles around the [c]-axes maxima. The [c]-axes in the shear-zone margin occur in two maxima where the dominant maximum lies between X and Y and the secondary maximum lies on the primitive circle of the net, making an angle of c. 208 with the pole of the shear-zone margin. The poles to {m} and {a} prisms are distributed in five maxima, and the dominant maximum lies in the shear plane or at a low angle to it in the opposite direction to the [c]-maxima. The poles to {r}, {z}, {p} and {p′ } are roughly distributed as three maxima, where the rhombs {r} and acute rhomb {p′ } define the ‘single-crystal’ like character of the CPO in the shear-zone margin. In the mature mylonite (Fig. 1) the [c]-axes are distributed along a discontinuous symmetric single girdle, normal to the shear-zone margin (shear plane) pole and oblique to the mylonitic foliation, with a maximum concentration lying c. 188 from the shear plane pole towards the centre of the stereonet. The poles to prisms {m} and {a} are distributed along single girdles parallel to the shear plane, both with three distinct maxima reflecting the three symmetric kal and kml-directions dictated by crystallography where the kal-directions were equally efficient as the dominant slip direction. The poles of acute rhomb ({p}, {p′ }) and rhomb ({r}, {z})
/
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Fig. 2. (a– c) Bright-field TEM images of quartz grains at 120 kV in the mature mylonite and (d) stereoplot containing the geometrical analysis of dislocations, taking into account two-beam orientation (BM), three diffraction vectors (g1, g2, g3 – g1//g3) and the dislocation line direction (u). The resultant slip planes and Burgers vector is 102) plane in kal marked with (b). Dislocations in these pictures are predominantly of mixed-type gliding on p ′ (1 1/3[1120] direction.
their orientation. Given the diffraction vectors, the beam orientations (BM 1 ¼ [1011], BM2 ¼ [4156]) and dislocation line direction of u ¼ [1543] indicated by the tilting procedure, the dislocations of Figure 2 can be described as mixed dislocations with b ¼ 1/3[1120] (angle between b and u ¼ 45.738) gliding on the (1102) plane (Fig. 2d). Further dislocation analysis presented in Figure 3 using three different diffraction vectors (g1 ¼ (1100), g2 ¼ (0111) and g3 ¼ (1011)) was impaired by progressive radiation damage in the image and by the extreme variation in contrast of the bright-field images caused by crystal bending. Radiation damage (already visible in Fig. 3c) limited observation time to one zone axis with one beam direction, which does not allow the correct determination of the dislocation line direction u. However, most of the long dislocations nearly parallel to the specimen surface are out of contrast for the diffraction vector g1 (g1 . b ¼ 0), while the contrast is moderate for g2 (g2 . b = 0) and moderate to strong for g3 (g3 . b = 0). The symmetry of the dislocation in the images is not very clear due to crystal bending, but most of them seem to be asymmetric about the dislocation line (as observed in Fig. 3b, c) whereas they
are weak and symmetric in Figure 3a, further supporting that g1 . b ¼ 0. When the angles between the trace of the dislocation lines and the diffracting vectors (g1, g2 and g3) are taken into account and the Burgers vector b ¼ 1/3 [1120] is compatible with the g . b values, the two possible line directions are [5419] in the (1101) plane and [2423] in (1102) plane (Fig. 3d). The geometry of dislocations gliding in (1101) with line direction u ¼ [5419] indicates almost pure edge dislocations (angle b to u ¼ 87.738). Alternatively, the dislocations gliding in (1102) with line direction u ¼ [5419] may be mixed dislocations (angle b to u ¼ 64.028), similar to the dislocations observed in Figure 2. The microstructures observed with TEM include subgrain walls (Fig. 4a) and tilt walls, characterized by the regular alignment of parallel straight-edge dislocations. Subgrains are usually slightly elongated (2:1) and have low internal dislocation densities (Fig. 4b). The new grains are also elongate, often with straight boundaries and high densities of dislocations associated with crystal lattice bending (Fig. 4c, d). Dislocation lines in these crystals are either straight or curved, which suggests the lack of structural control on the orientation of
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Fig. 3. (a– c) Bright-field TEM images of quartz grains at 120 kV in the mature mylonite (d) and stereoplot containing the geometrical analysis of dislocations, taking into account one-beam orientation (BM), three diffraction vectors (g1, g2, g3) and the dislocation line direction (u). The resultant slip planes and Burgers vector is marked with (b). Dislocations in these pictures are predominantly of edge-type gliding on z(1 101) plane or mixed type in gliding in p ′ (1102) kal 1/3[1120] direction. Note the beam damage in (c).
dislocation lines at this scale. In some places, dislocations are piled up along crystal boundaries (Fig. 4c), suggesting their inability to propagate through grain boundaries or their absorption along these structures. In addition to the development of subgrain walls, bulging features are observed in some places (Fig. 4d). Hexagonal networks of dislocations resembling honeycombs may also be observed in some grains (Fig. 4e, f ). Some of these dislocations are out of contrast when g ¼ (0110), suggesting that these dislocations are of kal or [c]-type (Fig. 4e). However, when the diffraction vector is (2110) with the beam direction parallel to the [c]-axis of this particular crystal, all the dislocations become visible and indicate that the networks are formed by kal type dislocations in the basal plane of these crystals (Fig. 4f ). The presence of such structures implies that, after a certain amount of propagation in the basal plane, the dislocations in the three symmetrical kal directions ([a1], [a2] and [a3]) form nodes or triple junctions at 1208 to each other and are sessile, and hence can only contribute to deformation by climb (e.g. Tre´pied et al. 1980).
Modelling CPO development in aggregates In the last 30 years, different numerical approaches have been used to simulate the evolution of CPO in minerals (Etchecopar 1977; Lister et al. 1978; Lister & Paterson 1979; Lister & Hobbs 1980; Takeshita & Wenk 1988; Wenk et al. 1989; Canova et al. 1992; Casey & McGrew 1999; Takeshita et al. 1999). For the present paper we have made extensive use of the viscoplastic self-consistent approach (Lebensohn & Tome´ 1993; Tome´ & Lebensohn 2004) because of its strong physical basis. Although this approach only allows the simulation of CPO development via intra-crystalline slip, a careful comparison between the CPOs measured in natural rocks and those predicted via VPSC modelling can at least verify the likely contribution of dislocation glide. Different approaches that impose homogeneous stress (Sachs 1928) or strain (Taylor 1938) within the aggregate are often referred to as lower- and upper-bound constraints. In the VPSC approach, both the local stress and strain can be different from the macroscopic quantities. Compatibility in strain and stress equilibrium is only guaranteed at
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Fig. 4. (a– c) Bright-field TEM images of the microstructures of the mature Torridon mylonite include large grains with well-developed subgrains and small grains with high dislocation density. (a) Dislocation tilt walls are composed of one family of straight dislocations (marked by the arrows); (b) subgrains are either slightly elongated (white arrow) with average aspect ratios of 2:1 or more equant (black arrow); (c, d) note the low dislocation density within the subgrains and new elongate grains with high homogeneous dislocation density and variable image contrast due to crystal bending varying the diffraction conditions. Note the straight grain boundaries (black arrow) and the dislocations piled-up along certain boundaries (white arrow) on (c) and bulging feature highlighted by the white arrow on (d). (e, f ) Hexagonal networks of screw kal dislocations in the basal plane. The triple junctions (nodes) of dislocations are sessile and can only contribute to the deformation by climb. The asterisks are used to locate a common point between the two pictures.
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the aggregate scale. Deformation is homogeneous at the grain scale and it is only accommodated by dislocation glide. In addition, the viscoplastic approach used here is spatially disordered, and does not take into account the topology of the aggregate. The shear rate induced in a slip system s by a given local deviatoric stress is described by a viscoplastic law: s s ns rij sij t † † † g s = g 0 rs = g 0 . (1) t0 t0s †
In this equation, g 0 is the reference strain rate, n s is the stress exponent (material constant n ¼ 1 for Newtonian behaviour such as diffusion creep, n ¼ 3 to 5 for dislocation creep), trs is the critical resolved shear stress (CRSS) and t0s is the stress necessary to activate a given slip system s, whose orientation relative to the stress axes is expressed by its Schmid tensor r s. The sum of the shear rates over all slip†systems gives the grain microscopic strain rate 1. To determine the microscopic † state for each grain (s, 1) and the volume average = that †determines the response of the aggregate (S , E; equation (2)), the ‘one-site’ simplification = = (Molinari et al. 1987; Lebensohn & Tome´ 1993) is used in the VPSC formulation: ksl=S =
=
†
†
=
=
k1l = E
(2)
Through this method, the interaction between individual grains is not directly taken into account. The interaction between each grain and the surrounding crystals is progressively substituted by the interaction between an inclusion with a crystallographic orientation and an infinite equivalent homogeneous medium (HEM), whose behaviour is the weighted average of the behaviour of the aggregate. The interaction problem can be solved using an equation of Eshelby (1957), which assumes ellipsoidal grain shape and tangential behaviour of HEM: †
† ˜ ijkl (skl − Skl ) 1 − Eij = −aM
(3)
˜ is the interaction tensor that In this equation, M depends on the rheological properties of the aggregate and on the shapes of grains. The quantity a is a constant used to parameterize the interaction between the homogeneous medium and the grains and essentially imposes more or less kinematically rigid conditions on the aggregate. A zero value for a corresponds to the Taylor –Bishop–Hill approach (Taylor 1938; Lister et al. 1978). It assumes homogeneous strain within the aggregate and requires the activation of at least five independent slip systems in
each crystal to maintain homogeneity in deformation (Lister et al. 1978; Lister & Paterson 1979; Lister & Hobbs 1980). The crystals deform at the same rate and have the same shape at each imposed strain step, leading to stress incompatibilities at the grain boundaries which are assumed to be corrected by elastic strains (Wenk et al. 1989). On the other hand an a of infinity assumes that all crystals experience the same state of stress (Sachs 1928), which leads to microstructural incompatibility at the grain scale, and strain compatibility is only obtained at the averaged aggregate scale. The classical self-consistent tangent model is given by a ¼ 1. In VPSC models, the evolution of CPO, the activity of slip systems and the aggregate yield strength for a given macroscopic strain history can be calculated from a set of active slip systems, an initial set of orientations and the macroscopic rheological parameters t0s and n s. For the present paper, the aggregates are composed of 1000 individual orientations. We have modelled crystallographic fabric development starting from random orientations, but also consider a pre-existing preferred orientation as observed at the margins of the Torridon quartz vein. To create a pre-existing crystallographic orientation, we firstly carried out a simple-shear simulation using an initial random orientation dataset, assuming a very low CRSS for the kal{p′ } slip system with the other potential slip systems being virtually inactive, to a shear strain of g ¼ 1.73. We then rotated the resultant CPO by an angle of 458 anticlockwise around the centre of the net (finite strain Y-axis), to bring the maximum concentration of [c]-axes to the orientation observed in the margin of the sheared quartz vein. A simple-shear deformation was superimposed on this initial CPO up to a shear strain of g ¼ 1.73, resulting in a total equivalent strain of 2 (g ¼ 3.5). We emphasize that Lloyd et al. (1992) mentioned the possibility that the CPO of the mature mylonite could be developed from the intense deformation of a few large grains with an initial orientation. This is not exactly what we modelled here, but at least gives us an idea of how an initial CPO evolves during superimposed deformation. The deformation history of the aggregate is given by an imposed macroscopic velocity gradient tensor L. As the deformation responsible for the CPO development in the mature mylonite in the centre of the sheared quartz vein is thought to be simple shear (Law et al. 1990; Wallis 1995), the velocity gradient (Lss) is given by equation (4): ⎡
⎤ 0 1 0 Lss = ⎣ 0 0 0 ⎦ 0 0 0
(4)
TEM VPSC TORRIDON QUARTZ MYLONITE
The parameters that can be modified during VPSC modelling are the active slip systems, their critical resolved shear stresses, the stress exponents, the interaction between grains, the HEM (defined by a) and the shapes of grains. The stress exponent for quartz aggregates reported in the literature varies between 1.4 and 5.7 with a mean value of 2.75 (see Mainprice & Jaoul 2009, table 2). In all the VPSC models presented here we used a ¼ 1 (tangent model) and a stress exponent of n s ¼ 3. The VPSC model is not very sensitive to values of n s between 3 and 5, and almost all minerals except calcite have stress exponents in this range. Increasing the stress exponent results in the strength of CPO becoming more intense for a given finite strain. In our models, the equivalent strain is defined by 1eq = Deq (t) dt
(5)
where the Von Mises equivalent strain rate is Deq =
2 Dij Dij . 3
(6)
Active slip systems in quartz single-crystal and polycrystalline aggregates are constrained by experiments and TEM analyses carried out by a number of authors (e.g. Griggs & Blacic 1965; Bae¨ta & Ashbee 1969; Kronenberg & Tullis 1984; Linker et al. 1984; Mainprice & Paterson 1984, 2005; Rutter & Brodie 2004; Mainprice & Jaoul 2009). In our modelling, we have used the slip systems: a,21¯1¯0.c(0001); a,21¯1¯0.r{01¯11}; a,21¯1¯0. z{011¯1}; a,21¯1¯0. p{01¯12}; a,21¯1¯0. p′ {011¯2}; a,21¯1¯0.m{01¯10}; c[0001]m{101¯0} and c[0001] a{21¯1¯0}. Hard slip systems such as dipyramids with slip directions oblique to both [c] and kal (e.g. , 2113 . {1011}) are not considered in this paper because VPSC modelling does not require five independent slip systems, unlike the Taylor –Bishop– Hill model. In addition, we do not consider the effect of Dauphine´ twinning in our models, but we discuss its possible effects due to the widespread presence of these microstructures in the mature Torridon mylonite (Lloyd 2004).
Evolution of quartz CPO VPSC simulations from random orientations For comparison with VPSC models, we have run a lower-bound simulation (e.g. Chastel et al. 1993; Tommasi et al. 2000) in which simple-shear deformation was accommodated entirely by activation of the kal{p′ } slip system (Fig. 5). For this specific
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case of single slip, the lower-bound modelling is preferred due to the assumption of homogeneous stress in the aggregate with variable individual crystal strains and represents an ideal end-member case. In these models, the shear plane is vertical (horizontal line) in the pole figures (Figs 5– 10) and the shear direction (SD) is horizontal; both are oriented east –west. Foliation is marked by the tilted black line on these pole figures (Figs 5–10). Model 1 (Fig. 6) shows the predicted simple-shear quartz CPO for a postulated set of CRSS values for different slip systems for a deformation occurring under upper greenschist/lower amphibolite facies conditions. In this model, the predominant slip system is kal(c) with activities varying between 40 and 50% as function of strain followed by relatively high activities of rhomb slip {r/z} in the kal direction. We emphasize that, in natural or experimental deformation, additional mechanisms such as grain-boundary sliding and grain-boundary migration contribute to achieving strain compatibility. For this reason, the CPO calculated by VPSC modelling is generally stronger for a given strain than that obtained during natural or experimental deformation, because only dislocation glide is considered. Considering an aggregate of randomly oriented spherical grains of quartz, two possibilities were found for generating high concentrations of quartz c-axes in a similar position as the mature Torridon mylonite and the (p ′ ) poles near the foliation normal (Z). The first assumes a similar critical resolved shear stress value for kal(c) and kal{p ′ } and kal{p} (Model 2; Fig. 7). The second possibility arises when we assume a much lower CRSS for kal{p ′ } and kal{p} than for kal(c) slip (Model 3; Fig. 8). Dauphine´ twinning in trigonal quartz results in the reversal of positive and negative crystallographic forms due a rotation of 608 around the quartz c-axis (Frondel 1962). In this situation, the positive and negative rhombs ({r/z}) and acute rhombs ({p/p ′ } are in an equal position in relation to the applied stress and, for this reason, all models have the same critical resolved shear stresses for both rhombs and acute rhombs. The models shown in Figures 6 –8 have a similar CPO evolution; with increasing strain, the quartz [c]-axes are distributed along a symmetric single girdle crossing the reference foliation at Y (but asymmetric in relation to the shear plane). Although the maximum concentration of [c]-axes is located near the foliation normal (Z) in Models 2 and 3 (Figs 7 & 8), it is exactly parallel to Z in Model 1 (Fig. 6) for all strains. Under these conditions, the activity of basal slip increases progressively with increasing strain, and reaches values of more than 50% when g ¼ 3.5 (1eq ¼ 2) followed by a decrease on the activity of positive and negative rhomb slip
160 L. F. G. MORALES ET AL. Fig. 5. Lower bound (homogeneous stress) quartz CPO developed in simple shear at equivalent strains (1eq) of 0.5, 1.0 and 1.5 (shear strains given as g) assuming single kal{p ′ } slip system. Pole figures are lower hemisphere equal-area projections for 1000 orientations. Maximum and minimum finite strain axes are marked X and Z. The horizontal line is the shear plane, whereas the tilted black line marks the flattening plane. The shear sense is dextral, top to the right and SD is the shear direction. Contours of multiples of uniform distribution; inverse log shadings vary from white (minimum density) to black (maximum density). Slip-system activity for the individual slip systems of the kal{p ′ } family is also indicated. Note that at low strains, the activity of individual slip systems does not vary; with increasing strain however, there is always a dominance of one member over the others and a frequent interchange between dominant systems. The maximum concentration in the pole figures is indicated by the white square.
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Fig. 6. Quartz CPO developed in simple shear at equivalent strains (1eq) of 0.5, 1.0 and 1.5 (shear strains given by g). Anisotropic VPSC model (Model 1), a ¼ 1, ‘normal’ CRSS values for different slip systems of quartz as expected in low-to-intermediate deformation temperatures (see values in the box). Pole figures are plotted in lower hemisphere, equal-area projections for 1000 grains. East –west horizontal line marks the shear plane (SD is the shear direction), whereas the tilted black line marks the finite strain reference frame. Contours are multiples of uniform distribution; inverse log shadings vary from white (minimum density) to black (maximum density). Note the dominance of basal slip in kal direction and its increasing activity with progressive deformation, and how the maximum of [c]-axes remains subparallel to the foliation pole even at relatively high-strain magnitudes. The maximum concentration in the pole figures is indicated by the white square.
162 L. F. G. MORALES ET AL. Fig. 7. Quartz CPO developed in simple shear at equivalent strains (1eq) of 0.5, 1.0 and 1.5 (shear strains given by g). Anisotropic VPSC model (Model 2), a ¼ 1. The CRSS for basal, acute rhomb and rhomb slip systems in kal direction are approximately the same (the latter two are slightly stronger than the former; see values in the box). Pole figures plotted in lower hemisphere; equal-area projections for 1000 grains. East –west horizontal line marks the shear plane (SD is the shear direction), whereas the tilted black line marks the finite-strain reference frame. Contours are multiples of uniform distribution; inverse log shadings vary from white (minimum density) to black (maximum density). Note that the activity of kal(c) is just slightly higher than the activity of kal{p} and kal{p′ }. Also note the variation of [c]-axes maxima and how the maximum of [c]-axes rotates from Z about W towards Y with increasing strain. The maximum concentration in the pole figures is indicated by the white square.
TEM VPSC TORRIDON QUARTZ MYLONITE 163
Fig. 8. Quartz CPO developed in simple shear at equivalent strains (1eq) of 0.5, 1.0 and 1.5 (shear strains given by g). Anisotropic VPSC model (Model 3), a ¼ 1, CRSS for kal{p′ } slip system is much lower than for basal, positive acute and rhomb systems (see values in the box). Pole figures plotted in lower hemisphere; equal-area projections for 1000 grains. East– west horizontal line marks the shear plane (SD is the shear direction), whereas the tilted black line mark the finite strain reference frame. Contours are multiples of uniform distribution; inverse log shadings vary from white (minimum density) to black (maximum density). Note that the predominance activity of kal{p′ } over the other slip systems, following the same behaviour of basal slip activity in Figure 6. In this setup, the maximum concentration of [c]-axes is at a low angle with Z in 1eq ≤ 1. The maximum concentration in the pole figures is indicated by the white square.
164 L. F. G. MORALES ET AL. Fig. 9. Quartz CPO developed in simple shear at equivalent strains (1eq) of 0.5, 1.0 and (shear strains given by g), starting from a previous orientation. Anisotropic VPSC model (Model 4), a ¼ 1, CRSS for different slip systems are the same of Figure 7. Pole figures plotted in lower hemisphere; equal-area projections for 1000 grains. East –west horizontal line marks the shear plane (SD is the shear direction), whereas the tilted black line marks the finite strain reference frame. Contours are multiples of uniform distribution; inverse log shadings vary from white (minimum density) to black (maximum density). Note the abrupt changes in the activities of kal(c), kal{p′ } and kal{p}. In these simulations, the maximum of [c]-axes in the pre-existing orientation moves towards the foliation pole with increase strain and remains in that position even at high-strain magnitudes. The maximum concentration in the pole figures is indicated by the white square.
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Fig. 10. Quartz CPO developed in simple shear at equivalent strains (1eq) of 0.5, 1.0 and (shear strains given by g), starting from a previous orientation. Anisotropic VPSC model (Model 5), a ¼ 1. The CRSS for kal{p′ } is much lower than the values for others slip systems (see values in the box). Pole figures plotted in lower hemisphere; equal-area projections for 1000 grains. East – west horizontal line marks the shear plane (SD is the shear direction), whereas the tilted black line marks the finite strain reference frame. Contours are multiples of uniform distribution; inverse log shadings vary from white (minimum density) to black (maximum density). Although the activity of the negative acute rhomb slip system is dominant, the maxima of [c]-axes is not retained and, with increasing strain, it becomes subparallel to the foliation pole. The maximum concentration in the pole figures is indicated by the white square.
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systems (Fig. 6). In Model 2 a similar situation is observed when 1eq ¼ 1; with increasing shear strain, however, the maximum concentration of [c]-axes rotates by 188 about X from Z towards Y (Fig. 7). The activity of kal(c) predominates with an average of 30%; the level of activity of kal{p} and kal{p′ } are similar at around 25%. Again, the activity of kal(c) slip predominates and becomes more important with increasing strain. The maximum concentration of {p}/{p′ } poles are not exactly parallel to Z. They are distributed in double symmetrical maxima at low angles to Z, where one of these maxima is normal to the shear plane. The activity of basal slip is nearly zero in Model 3, and the predicted CPO is developed with the dominant activity of the kal{p} and kal{p′ } slip systems with activities around 47% for each of these systems and with low activities of the kal{m}, kal{r/z} and kal(c) slip systems. Note that the point maxima of [c]-axes is at a small angle to Z under relatively low shear strains (Fig. 8). The activity of the positive and negative acute rhomb slip systems does not vary substantially with increasing strain in Model 3. In this model, {p′ } and {p} poles are at low angles to the foliation pole (Z ) and oblique to the shear plane (Fig. 8). The distribution patterns of other important crystallographic directions of quartz are nearly the same in all simulations. The poles of first- {m} and second- {a} order prisms are distributed along a discontinuous girdle parallel to the foliation, with the maxima oriented either nearly parallel to the shear direction for {a} or parallel to the macroscopic lineation (X ) for {m}. The poles to {r} and {z} rhombs occur in four maxima at 558 to the Z-axis. The poles of acute rhombs {p} and {p′ } are distributed in conical girdles around Z, also developing four symmetrical point-maxima with similar pole concentrations. The maximum concentration in these pole figures also lies in the XZ plane, making an angle of 208 with Z. In contrast to Taylor– Bishop–Hill simulations for progressive bulk simple shear (Lister & Hobbs 1980), VPSC simulations indicate a symmetrical relationship between crystal fabrics and finite strain axes regardless of the imposed bulk shear strain (Figs 5–8).
VPSC simulations starting from a previous CPO Two models were tested. The first used the same set of CRSS values used in Model 2 (Model 4; Fig. 9). The second used a low CRSS of 1.0 for kal{p/p′ } slip systems with a CRSS of 100.0 for [c]{a} and [c]{m} slip systems and a CRSS of 10 for all other slip systems (Model 5; Fig. 10). The CPOs are nearly the same as those predicted from an initial random orientation, and the only qualitative
difference is the CPO strength which is much stronger than those modelled in Figures 6–8. In addition, only relatively small shear strains (g , 0.87) are necessary to bring the quartz crystallographic axes from their initial tilted position to a position nearly parallel to the finite strain axes used as the reference frame for the pole figures (Figs 9 & 10). Quartz [c]-axes are distributed in a single maximum subparallel to the foliation pole Z and oblique to the imposed shear plane. The initial c-axis maximum rotates in the direction of applied shear strain (dextral: top to the right) towards Z, while the a and m poles simultaneously rotate towards the XY plane. The poles to {a} and {m} are distributed in a continuous girdle nearly parallel to the XY plane but inclined in relation to the shear plane, with maximum concentrations within the foliation at about 208 from X and parallel to X. The poles to rhomb and acute rhombs are distributed as complete or incomplete small circle (conical) girdles centred around Z, and the predicted CPOs are no stronger than the CPOs presented for Models 1–3 in Figures 6 –8. In terms of activity, however, the models differ significantly and the relative activities of different slip systems in the models vary in magnitude. In Model 4 presented in Figure 9, the initial decrease of activity of kal(c) up to a shear stain of 0.5 and then its progressive increase with increasing imposed shear strain (reaching values of 45%) is much more accentuated than in Model 2 (Fig. 7). A reciprocal behaviour (increasing and decreasing activity with strain) is seen for all the other important slip systems (Fig. 9) and the activity of kal{p} and kal{p′ } is (on average) 25%. When kal{p/p′ } are the dominant slip systems (Fig. 10), there are no strong variations in their activities (around 50% for each) and a small decrease in the activity of one is compensated for by a small increase in the activity of the other.
Discussion TEM observations The microstructures observed by transmission electron microscopy indicate that dislocation glide was an important deformation mechanism in the mature Torridon mylonite, as documented by the high dislocation density in a number of grains (Fig. 4c, d) which confirmed the intense plastic deformation of this rock (Law et al. 1990; Lloyd et al. 1992). Some of the crystals exhibit an alignment of dislocations, suggesting that some of the dislocation lines have a strong crystallographic control. Other dislocation lines appear curved (Fig. 4d) however, which may result from dislocation pinning by crystal impurities or by the
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interference between dislocation lines in different planes (e.g. Poirier 1985). Geometrical analysis of dislocations using different diffraction vectors in the same crystal indicate that dislocation glide occurs in the {p′ } and {z} slip planes in the kal direction (Figs 2 & 3). The presence of hexagonal arrays of screw kal-type dislocations (Fig. 4e, f ) in the basal plane suggests that climb was active as these dislocation nodes can no longer glide. It is possible that the kal(c) slip system was active in accommodating deformation, but no direct evidence is provided by our TEM observations of the dislocation networks as the kal screw dislocations could have cross-slipped from other planes (Fig. 4e, f ). Taken together, the activation of these three slip systems confirms the importance of dislocation glide in the development of the strong quartz CPO in the mature mylonite shown in Figure 1 (Law et al. 1990; Lloyd et al. 1992; Lloyd 2004). In addition, these honeycomb-type dislocation networks and the presence of elongate subgrains also suggest the activation of significant dislocation climb, which is an efficient mechanism for allowing continuous dislocation propagation (Nicolas & Poirier 1976; Poirier 1985; McLaren 1991; Mainprice & Jaoul 2009). During glide-controlled deformation, dislocations in the basal plane may have three different coplanar Burgers’ vectors, [a1], [a2] and [a3], which lie at 1208 to each other due to the symmetry of quartz. These dislocations probably started to move independently but, once they intersected to form triple junctions, they form sessile junctions and dislocation glide stops. For continuous deformation and to avoid strain hardening, dislocations have to climb from their original planes to other crystallographic planes (a process driven by diffusion) where they interact with the dislocations present on these planes (e.g. Poirier 1985; Trepmann & Sto¨ckhert 2003). Despite the small direct impact of dislocation climb for the CPO development, it allows dislocation glide in a given plane to continue, which directly favours the development of strong preferred orientation related to a specific slip system. On the other hand, as suggested by the contrast variation observed in Figure 4c, d, crystal bending is insignificant both in terms of strain accommodation and CPO development but it does maintain local strain continuity in the sample and may have some importance during early stages of deformation. Gliderelated recovery and heterogeneous deformation are documented by the presence of a variety of subgrain boundaries with a single family of dislocations forming tilt walls (Fig. 4a) and more complex boundaries composed of several families of dislocations (Fig. 4e, f ). These subgrains are then subdivided into smaller subgrains, suggesting a process of creep polygonization (e.g. Poirier 1985). On the
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other hand, grain-boundary migration recrystallization is limited to a few bulging microstructures (Fig. 4d) and, in this case, does not contribute significantly to CPO development or recovery process. The optically detected grain sizes observed in the SA foliation (Law et al. 1990, fig. 2c) and those observed in the TEM images are essentially the same; this implies that, despite the strong preferred orientation, quartz crystals are easily individualized and their boundaries not masked by rotation of the polarizing stage of the optical microscope (in contrast to what was reported by Trepmann & Sto¨ckhert (2003) in other quartz-rich mylonites).
VPSC modelling The characteristic features of the mature Torridon mylonite (Fig. 1) CPO are: (1) the [c]-axes form a strong maxima near the pole of the shear-zone margin and oblique to foliation in the mature mylonite; (2) the {a} and {m} poles form girdles parallel to the shear-zone margin reference plane; (3) the {r} and {z} poles are positioned on conical surfaces of large opening angle centred about the pole of the shear-zone margin; and (4) the {p} and p ′ poles are oriented on conical surfaces of lower opening angle centred about Z. Most of the CPO features of the mature mylonite can be reproduced by homogeneous simple-shear VPSC modelling using two different approaches (Models 2 and 3; Figs 7 & 8), but cannot be reproduced in Model 1 where kal (c) dominates the slip activity and forms a CPO with a strong c-axis maxima parallel to Z at all strains. A similar situation is observed in Model 2 but, at equivalent strains .1, the maximum concentration of [c]-axes rotates from Z toward Y around X and becomes parallel to the [c]-axis maxima observed in the mature mylonite (Figs 1 & 7). The activity of basal slip dominates and becomes more important with increasing strain, but the difference between the activities of kal{p} and kal{p′ } is small. For Model 2 (Fig. 7), the critical resolved shear stress of the kal{p} and kal{p′ } slip systems are slightly stronger at 1.5 than the resolved shear stress for the kal(c) system at 1.0. In addition, the CRSS of positive and negative rhomb slip systems at 2.5 are higher than the value for the acute rhomb systems; otherwise, the maximum concentration of [c]-axes would become parallel to Z direction as in Model 1 or at higher angles toward Y (Fig. 6). In Model 3 on the other hand, the maximum concentration of [c]-axes near Z in the YZ plane develops at low strains (1eq ≤ 1). The acute rhomb slip systems kal{p/p′ } in this model have a low CRSS of 1.0 plus a CRSS of 3.0 for kal{m} and a CRSS value higher than 7.0 for all other slip systems (Fig. 8). Model 3 results in the slip activity being
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dominated by kal{p/p′ }, with small activity of kal{m}. The critical resolved shear stress for different slip systems of quartz depends on a number of physical parameters (e.g. temperature, stress/strain rate, presence of water) and, as shown by our models, qualitatively the CPO patterns are essentially the same (Figs 6– 8). Nevertheless, small variations of the CRSS may lead to slight but important modifications in positions of the maximum concentration of the quartz crystallographic axes. For example, a decrease in the CRSS for the kal{p} and kal{p′ } by half of its original value of 3.0 (considering the expected active slip systems at low-to-medium deformation temperatures for quartz; Fig. 6) induces a change in the [c]-axes maxima orientation from parallel to Z to an intermediary position at c. 188 from Z in the YZ plane under high-strain conditions (Fig. 7). On the other hand, a ten-fold increase of the kal(c) CRSS and a seven-fold increase in the kal{r} and kal{z} CRSS are necessary to produce a similar concentration under lower strain conditions, with no significant variations in the general CPO patterns (Fig. 8). In the simulated aggregates with initial crystallographic preferred orientation, the CPO becomes very strong at relatively low shear strains when compared to models with initial random orientations. The initial [c]-axis maximum is rotated to an intermediate position between the starting orientation and the pole to foliation Z. The activity of different slip systems has more accentuated increases and decreases than the simulations starting from a random orientation. One explanation for rapid rotation as a function of finite strain is that the starting orientations for the simulations with an inherited fabric have a relatively high number of orientations in geometrically favourable positions for slip compared to a random fabric. As an example, the activity of {p}, {p′ }, {r}, {z} and {m} in a common kal slip direction increases initially because the initial orientation, with the [c]-axis maxima aligned with the shear direction, favours the slip on these planes, even if their CRSSs are assumed to be higher than for kal(c) (Fig. 9). Slip on the (c)-planes is not initially favoured as it is oriented 908 from the slip direction. Note that slip does not occur in the [c]-direction, although this direction is at a low angle to the shear direction, because the resolved shear stress is assumed to be high (.20.0). When rotation proceeds and the general orientation of the basal planes becomes aligned closer to the finite strain flattening plane, basal slip is favoured again. Its activity increases substantially, inducing the progressive rotation from the original maxima direction towards the direction Z. In Model 5, where kal{p} and kal{p′ } are the only easy slip systems with
CRSS of 1.0 and even kal(c) is difficult, then the only slip planes that contribute significantly to slip activity are {p} and {p′ }. The sum of their activities reaches values around 100%, and the activities of the other slip systems are virtually negligible. The {r} planes are initially about 458 from the shear direction which explains why kal{r} is favourably oriented for slip at low strain, but once the rotation starts the {p′ } planes become more favourably oriented and dominate the slip activity. In summary, crystal preferred orientations predicted via VPSC modelling evolve continuously with strain, showing strong CPOs of quartz for shear strains ≥2, in a similar way to olivine fabrics using the same type of numerical modelling (Tommasi et al. 2000). Fabric strength indicated by the J-index (Bunge 1982) reaches values of around 14 and 25 for g ¼ 3.5 in simulations starting from random and non-random orientations, respectively. On the other hand the Torridon quartz mylonite, which has an anomalously strong preferred orientation, has a J value of 8.6 (Fig. 11). The shear strain estimations based on the inflection of foliation into the shear zone (Ramsay 1967) gives values of g . 5 (Lloyd et al. 1992) or g 8 at the shear-zone centre, when we consider the minimum angle of 98 between the mylonitic foliation (finite strain flattening plane) and the vein margins (shear plane). In our models, the calculated J-indices for the above variation in shear strain, based on the geometrical relationships between shear direction and the X finite strain axis, gives values from 17 –25 in the models with initial random orientation and 23 –30 in the models with an initial crystal orientation (Fig. 11). When plotted on the modelled curves, the J-index of the mature mylonite indicates a g of c. 2 which represents an angle of c. 368 between the shear direction and the X-axis. This indicates that the VPSC modelling underestimates the shear strain necessary to generate a CPO with the same J-index value as determined in the natural mylonite. On the other hand, the J-indexes calculated from our models are clearly overestimated when compared to the value of the natural sample. In both cases, this may be explained by the fact that VPSC models only take into account deformation by dislocation glide. Mechanisms such as dynamic recrystallization that in nature and experiments may cause crystal orientation dispersion (e.g. Karato 1988; Trimby et al. 1998; Fliervoet et al. 1999; Barnhoorn et al. 2004) are not considered in these models. Dislocation glide and the development of crystal preferred orientation, as simulated by viscoplastic self-consistent modelling, therefore has to be assisted by other material processes to accommodate deformation effectively in natural rocks. Due the pervasive presence of subgrains at different
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Fig. 11. Evolution of the CPO strength represented by the J-index (i.e. the integral of the square of orientation distribution function; Bunge 1982) as a function of the shear strain for the anisotropic VPSC models presented in Figures 6– 10. The black star on the Y-axis indicates the J-index of the mature Torridon mylonite. The respective curves for each model are marked directly on the figure. Note a drop in the J-index of Model 5 due to the lack of mathematical convergence during modelling.
scales in the Torridon mylonite, in addition to a misorientation angle peak at 158 (Lloyd 2004), this suggests that subgrain rotation recrystallization was the dominant recovery mechanism, aided by grain-boundary migration.
Implications for the Torridon Shear Zone In comparison with quartz mylonites developed in low- to medium-grade shear zones, the quartz preferred orientations observed in the mature Torridon mylonite present an anomalous maxima of [c]-axes at a low angle to the pole to the shear plane, suggesting that the dominant slip system was kal{p/p′ }. Nevertheless, a predominance of basal slip is to be expected under these conditions followed by rhomb slip and small amounts of prismatic slip, all of them sharing a common kal slip direction
(e.g. Fig. 6) where the acute rhomb slip systems are usually of secondary importance (Stipp et al. 2002). The differences between the CPO patterns expected in low- to intermediate-temperature quartz mylonites, and what is observed in the Torridon vein, lead us to two possible explanations for the strong concentrations of [c]-axes near the Z direction but slightly off the periphery of the pole figure. Analysis of potential slip-system orientations for different positions on the quartz [c]-axis fabric skeleton, in relation to the shear-zone geometry, indicate that the positive and negative acute rhomb slip systems are strongly aligned with the inferred simple-shear kinematic framework (Law et al. 1990). This strong alignment can be interpreted as evidence for the dominant activity of acute rhomb slip in the kal direction, although kal(c) and kal{z} slip systems are also operative (Law et al. 1990,
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p. 43). The CPO supports the bulk simple-shear kinematic framework indicated by the shear-zone geometry. On the other hand, as stated in Lloyd et al. (1992), the CPO patterns as a whole can be interpreted as resulting from the operation of multiple slip systems in a common kal slip direction. In addition, the mylonitic fabric of the mature Torridon mylonite was developed in a specific location within the sheared quartz vein/shear zone at a particular shear strain, and essentially depends upon the inherited CPO of the starting material (14 large quartz grains) at that specific location. Geometrical characterization of dislocations through TEM images demonstrates that kal{p′ } and kal{z} were the main slip systems active during flow in the mature mylonite (Figs 2 & 3), possibly together with kal(c) slip (Fig. 4e, f ). However, there is a lack of evidence for dislocations on prismatic planes {a} and {m}. The VPSC models in which kal{p} and kal{p′ } are dominant therefore seem to be physically unrealistic, given the strong activity of kal{z} and possibly kal(c) slip systems and given the thermal conditions of shear-zone development (Figs 8 & 10). On the other hand, the CPO predicted under higher shear strains (Fig. 7), where the CRSS for basal slip is slightly lower than for the acute rhomb and rhomb slip systems, is in agreement with the high relative shear strain determined by Lloyd et al. (1992) and the TEM observations described in the present paper. Although Dauphine´ twinning is not directly considered in our models and was not observed in TEM analyses, it plays an import role in deformation of the mature Torridon mylonite. In the early stages of deformation, it has an important influence both on grain-size reduction and on the development and maintenance of fine-grained mylonitic structure (Lloyd et al. 1992; Lloyd 2004). Due the rotation of 608 around the [c]-axis, Dauphine´ twinning causes an increase in the number of symmetrically equivalent kal{p/p′ } slip systems. If these slip systems are favourably orientated for slip, their activity can be stronger than the activity of basal slip in the kal direction (e.g. Menegon et al. 2010). If the number of symmetrically equivalent {p/p′ } planes in favourable orientations for slip due to Dauphine´ twinning increases, they may play an important role on the final [c]-axis pattern; this is characterized by a strong maximum close to the pole of the foliation but slightly off the periphery of the pole figures. This is consistent with the observations of Law et al. (1990), Lloyd et al. (1992) and Lloyd (2004) for the CPO of the mature Torridon mylonite, and is corroborated by our numerical modelling. In addition, as Dauphine´ twinning causes an interchange between positive and negative crystallographic forms, it results in the interchange between k+al and k2al slip directions and
consequently plays a role in determining whether the crystal is in a favourable position to slip by activation of crystal slip systems in the kal direction. The ubiquitous presence of subgrains in the mature mylonite as observed in the TEM images, and in the distinct peak of low misorientation angles (,158) presented by Lloyd (2004), suggests that dynamic recrystallization by subgrain rotation-assisted dislocation creep was dominant during formation of the mature mylonite. Subgrain rotation implies the subdivision of larger-strained crystals into new grains displaying misorientation angles ,158 by progressive incorporation of dislocations along subgrain boundaries (e.g. Poirier & Nicolas 1975; White 1979). This process is therefore capable of dispersing the preferred orientation and compensates the increasing CPO strength induced by dislocation glide, allowing the development of a stable orientation. Nevertheless, the scattering of the CPO by rotation recrystallization is relatively weak, and some degree of host control between parent grain and new grains is normally observed (e.g. Poirier 1985; Law et al. 1986; Lloyd et al. 1992; Lloyd & Freeman 1994). An inherited fabric, such as that observed in the original vein material preserved at the margins of the Torridon quartz vein/shear zone, may therefore be retained even in the case of a strong superimposed deformation as observed in the mature mylonite.
Conclusions We have applied transmission electron microscopy (TEM) and viscoplastic self-consistent (VPSC) modelling to better constrain the origin of CPO and microstructural development in the Torridon quartz mylonite previously studied by Law et al. (1990), Lloyd et al. (1992) and Lloyd (2004). TEM analyses showed evidence for kal{p′ } and kal{z} slip systems in quartz, and possible activity on kal(c), but no direct evidence for activation of kal{m}, [c]{m} or [c]{a} slip systems. There is significant TEM-based evidence for climb-induced recovery in this material, including the presence of hexagonal dislocation networks in the basal plane and subgrain boundaries. Several models for CPO development using variations in critical resolved shear stress for different slip systems of quartz were investigated with the VPSC modelling approach and, unlike previous models, we have not used the relatively ‘hard’ kc + al slip direction in our simulations. Three models using a standard initial random orientation are presented, and the CPOs generated by these models demonstrate many common characteristics. These include: (1) the high [c]-axis concentration near the finite shorting direction Z; (2){a} and
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{m} poles distributed in the XY flattening plane; (3) {r} and {z} poles distributed along conical girdles with large opening angles centred around Z; and (4) {p} and {p′ } poles also in conical girdles but with small opening angles centred about Z. Many of these features are shared with the mature Torridon mylonite sample. In detail, however, each model has slightly different choices of critical resolved shear stresses for different slip systems of quartz. The different assumed CRSS values lead to slight (but important) modifications in the orientation of the maximum concentration of quartz [c]-axes. Model 1 has high activities of kal(c), kal{r} and kal{z}, but only Models 2 and 3 reproduce the c-axis concentration at an angle of c. 188 with Z in the YZ plane and a high concentration of {p′ } poles near Z. Model 2 has high kal(c) and kal{p/p′ } slip activity. Model 3 has high a dominant kal{p/p′ } slip activity, but no basal slip in kal. The models that start with a previous crystal orientation develop a very strong CPO after relatively small strains, following a rapid rotation of the original fabric. This is explained by the fact that the initial non-random fabric has a number of orientations that are geometrically favoured for slip on certain slip systems, for example, favoured slip in kal{p/p′ } rather than kal(c) as similarly demonstrated by Pennacchioni et al. (2010). Nevertheless, when the rotation proceeds and the basal planes of quartz become aligned with the finite strain plane XY, slip on kal(c) is favoured again and explains why the maximum concentration of quartz [c]-axes at a low angle with Z is not retained in these simulations. Taking into account that recrystallization is not considered in our numerical models but is widely observed in nature (Lloyd et al. 1992; Lloyd 2004; Pennacchioni et al. 2010), we conclude that initial CPOs can be retained even in the case of strong superimposed deformation. The mutual activation of kal{p′ }, kal{z} and possibly kal(c) slip systems suggest that the more realistic VPSC model for the CPO development is the one where all of the above slip systems have assumed similar CRSS values (Model 2; Fig. 7). In this situation, the maximum of quartz [c]-axes at a low angle to the foliation pole (Z) develops under higher-strain conditions and becomes stable in this position. We would like to dedicate this paper to the memory of M. Casey, who sadly passed away on 10th June 2008. Martin was much more than a colleague; he was a great friend, a font of inspiration and a tutor, always ready to discuss geological and non-geological problems. We are not sure if he would agree with the conclusions of our paper, but he would undoubtedly have a number of ‘simple’ questions that would result in very fruitful discussions. The authors are grateful to L. Menegon and an anonymous reviewer for the thoughtful reviews carried out in
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less than a month, and to D. Prior for all the editorial work (including an unexpected ‘second version’ of the manuscript). LFGM is particularly grateful to L. Menegon for the copies of two ‘in press’ papers and all the discussion during the EURISPET in Zu¨rich. LFGM is also grateful to A. Tommasi for advice on VPSC modelling and her encouraging comments about our modelling, and to French Agence Nationale de la Recherche (ANR) for a post-doctoral fellowship (Project Crystaltex). We also thank R. Marshall (University of Leeds) for specimen preparation. The automated EBSD system was funded by UK NERC Grant GR9/ 3223 (GEL).
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Calculating anisotropic physical properties from texture data using the MTEX open-source package DAVID MAINPRICE1*, RALF HIELSCHER2 & HELMUT SCHAEBEN3 1
Geosciences Montpellier UMR CNRS 5243, Universite´ Montpellier 2, 34095 Montpellier Cedex 05, France
2
Fakulta¨t fu¨r Mathematik, Technische Universita¨t Chemnitz, 09126 Chemnitz, Germany
3
Mathematische Geologie und Geoinformatik, Institut fu¨r Geophysik und Geoinformatik, Technische Universita¨t Freiberg, 09596 Freiberg, Germany *Corresponding author (e-mail:
[email protected]) Abstract: This paper presents the background for the calculation of physical properties of an aggregate from constituent crystal properties and the texture of the aggregate in a coherent manner. Emphasis is placed on the important tensor properties of 2nd and 4th rank with applications in rock deformation, structural geology, geodynamics and geophysics. We cover texture information that comes from pole figure diffraction and single orientation measurements (electron backscattered diffraction or EBSD, electron channelling pattern, Laue pattern, optical microscope universal-stage). In particular, we provide explicit formulae for the calculation of the averaged tensor from individual orientations or from an orientation distribution function (ODF). For the latter we consider numerical integration and an approach based on the expansion into spherical harmonics. This paper also serves as a reference paper for the mathematical tensor capabilities of the texture analysis software MTEX, which is a comprehensive, freely available MatLab toolbox that covers a wide range of problems in quantitative texture analysis, for example, ODF modelling, pole figure to ODF inversion, EBSD data analysis and grain detection. MTEX offers a programming interface which allows the processing of involved research problems as well as highly customizable visualization capabilities; MTEX is therefore ideal for presentations, publications and teaching demonstrations.
The estimation of physical properties of crystalline aggregates from the properties of the component crystals has been subject of extensive literature since the classical work of Voigt (1928) and Reuss (1929). Such an approach is only feasible if the bulk properties of the crystals dominate the physical property of the aggregate and the effects of grainboundary interfaces can be ignored. For example, the methods discussed here cannot be applied to the electrical properties of water-saturated rock, where the role of interfacial conduction is likely to be important. Many properties of interest to earth and materials scientists can be evaluated from the knowledge of the single-crystal tensors and the orientation distribution function (ODF) of crystals in an aggregate, for example, thermal diffusivity, thermal expansion, diamagnetism and elastic wave velocities. The majority of rock-forming minerals have strongly anisotropic physical properties and many rocks also have strong crystal preferred orientations (CPOs, or textures as they are called in materials science; these terms are used interchangeably in
this paper as no possible confusion can result in present context) that can be described concisely in a quantitative manner by the orientation distribution function ODF. The combination of strong CPOs and anisotropic single-crystal properties results in a three-dimensional variation in rock properties. Petrophysical measurements are usually made under hydrostatic pressure and often at high temperatures to simulate conditions in the Earth, where presumably the micro-cracks present at ambient conditions are closed. The necessity to work at high pressure and temperature conditions limits the number of orientations that can be measured. Typically, three orthogonal directions are measured parallel to structural features, such as the lineation and foliation normal defined by grain shape. The evaluation of physical properties from CPO allows the determination of properties over the complete orientation sphere of the specimen reference frame. This paper is designed as a reference paper for earth and material scientists who want to use the texture analysis software MTEX to compute
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 175– 192. DOI: 10.1144/SP360.10 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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physical tensor properties of aggregates from constituent crystal properties and the texture of the aggregate. MTEX is a comprehensive, freely available MatLab toolbox that covers a wide range of problems in quantitative texture analysis, for example: ODF modelling, pole figure to ODF inversion, electron backscatter diffraction (EBSD) data analysis and grain detection. The MTEX toolbox can be downloaded from http://mtex.googlecode. com. Unlike many other texture analysis software, it offers a programming interface which allows for the efficient processing of complex research problems in the form of scripts (M-files). The MatLab environment provides a wide variety of high-quality graphics file formats to aid publication and display of the results. In addition the MTEX toolbox will work identically on Microsoft Windows, Apple Mac OSX and Linux platforms in 32 and 64 bit modes with a simple installation procedure. In MTEX texture analysis information such as ODFs, EBSD data and pole figures are represented by variables of different types. For example, in order to define a unimodal ODF with half-width 108, preferred orientation 108, 208, 308 Euler angles and cubic crystal symmetry, the command myODF = unimodalODF(orientation(’Euler’, 10*degree, 20*degree,30*degree), ... symmetry (’cubic’), ’halfwidth’, 10*degree)
is issued which generates a variable myODF of type ODF, displayed as myODF = ODF specimen symmetry: triclinic crystal symmetry : cubic Radially symmetric portion: kernel: de la Vallee Poussin, hw = 10 center: (10, 20, 30) weight: 1
We will use this style of displaying input and output to make the syntax of MTEX as clear as possible. Note that there is also an exhaustive interactive documentation included in MTEX, which explains the syntax of each command in detail. The outline of the paper is as follows. In the first section the basics of tensors mathematics and crystal geometry are briefly described and presented in terms of MTEX commands. In the second section these basics are discussed for some classical second-order tensors and the elasticity tensors. In particular, we give a comprehensive overview about elastic properties that can be computed directly from the elastic stiffness tensor. All calculations are accompanied by the corresponding MTEX commands. In the third section we are concerned with the calculation of average matter tensors from their single-crystal counterparts and
the texture of the aggregate. Here we consider textures given by individual orientation measurements, which lead to the well-known Voigt, Reuss and Hill averages, as well as textures given by ODFs, which lead to formulae involving integrals over the orientation space. We can compute these integrals in several ways: either we use known quadrature rule, or we compute the expansion of the rotated tensor into generalized spherical harmonics and apply Parseval’s theorem. Explicit formulae for the expansion of a tensor into generalized spherical harmonics and a proof that the order of the tensor defines the maximum order of this expansion is included in the Appendix.
Tensor mathematics and crystal geometry In what follows we give the necessary background to undertake physical property calculation for single crystals, without the full mathematical developments that can be found elsewhere (e.g. Nye 1985). We will restrict ourselves to linear physical properties, which are properties that can be described by a linear relationship between cause and effect such as stress and strain for linear elasticity.
Tensors Mathematically, a tensor T of rank r is a r-linear mapping which can be represented by an r-dimensional matrix Ti1 ,i2 ,...,ir . A rank zero tensor is simply a scalar value, a rank-one tensor Ti is a vector and a rank-two tensor Tij has the form of a matrix. Linearity means that the tensor applied to r vectors x1 , . . . , xr [ R3 , defines a mapping (x1 , . . . , xr ) −
3 3 i1 =1 i2 =1
···
3 ir =1
Ti1 ,..., ir x1i1 · · · xrir
which is linear in each of the arguments x1 , . . . , xr . Physically, tensors are used to describe linear interactions between physical properties. In the simplest case, scalar properties are modelled by rank zero tensor whereas vector fields (i.e. directiondependent properties) are modelled by rank-one tensors. An example for a second-rank tensor is the thermal conductivity tensor kij which describes the linear relationship between the negative temp∂T ∂T ∂T , , ), that is a erature gradient −gradT = −(∂x 1 ∂x2 ∂x3 first-order tensor, and the heat flux q = (q1 , q2 , q3 ) per unit area which is also a first-order tensor. The linear relationship is given by the equality qi = −
3 j=1
kij
∂T , ∂xj
i = 1, . . . , 3,
ANISOTROPIC PHYSICAL PROPERTIES
and can be seen as a matrix vector product of the thermal conductivity tensor kij interpreted as a matrix and the negative temperature gradient interpreted as a vector. In the present example the negative temperature gradient is called applied tensor and the heat flux is called induced tensor. In the general case, we define a rank r tensor Ti1 ,...,ir inductively as the linear relationship between two physical properties which are modelled by a rank s tensor A j1 , j2 , ..., js and a rank t tensor Bk1 ,k2 ,...,kt , such that the equation r = t + s is satisfied. The rank of a tensor is therefore given by the rank of the induced tensor plus the rank of the applied tensor. The linear dependency between the applied tensor A and the induced tensor B is given by the tensor product Bk1 ,...,kt =
3 3 j1 =1 j1 =1
···
3
Tk1 ,k2 ,..., kt , j1 ,...,
js
A j1 ,...,
js
js =1
= Tk1 ,k2 ,..., kt , j1 ,...,
js
A j1 ,..., js .
sigma = tensor (M, ’name’, ’stress’, ’unit’, ’MPa’); sigma = stress tensor (size: 3 3) rank: 2 unit: MPa 1.45 0.00 0.19 0.00 2.11 0.00 0.19 0.00 1.79
the
normal
n = vector3d (1,0,0) n = vector3d (size: 1 1) x y z 1 0 0
According to Cauchy’s stress principle, the stress vector T n associated with the plane normal n is then computed by Tjn = sij ni .
T = EinsteinSum (sigma, [-1 1], n, -1, ’unit’, ’MPa’) T = tensor (size: 3) unit: MPa rank: 1 1.45 0 0.19
Note that the 21 in the arguments of the command EinsteinSum indicates the dimension which has to be summed up and the 1 in the argument indicates that the second dimension of s becomes the first dimension of T. Using the stress vector T n , the scalar magnitudes of the normal stress sN and the shear stress sS are given as Tin Tin − s2N .
In MTEX the corresponding calculation reads as sigmaN = double (EinsteinSum (T, -1, n, -1)) sigmaS = sqrt (double(EinsteinSum (T, -1, T, -1)) 2 sigma N^2) sigmaN = 1.4500 sigmaS = 0.1900
The crystal reference frame
M = [[1.45 0.00 0.19];... [0.00 2.11 0.00];... [0.19 0.00 1.79]];
Furthermore, we defined n = (1, 0, 0) to plane by
In MTEX this equation may be written as
sN = Tin ni = sij ni nj and sS =
In the right-hand side of the last equation we used the Einstein summation convention and omitted the sum sign for every two equal indexes; this will be default in all further formulae. In MTEX a tensor is represented by a variable of type tensor. In order to create such a variable, the r-dimensional matrix has to be specified. As an example we consider the 2nd rank stress tensor sij , which can be defined by
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Tensors can be classified into two types: matter tensors describing physical properties such as electrical or thermal conductivity, magnetic permeability, etc. of a crystalline specimen, and field tensors describing applied forces such as stress, strain or a electric field to a specimen. Furthermore, it is important to distinguish between single-crystal tensors describing constituent crystal properties and tensors describing averaged macroscopic properties of a polycrystalline specimen. While the reference frame for the latter is the specimen coordinate system, the reference frame for single-crystal tensor properties is unambiguously connected to the crystal coordinate system. The reference frames and their conventions are explained below. We will restrict ourselves to tensors of single or polycrystals defined in a Cartesian reference frame comprising T , Y T , Z T . The use of an the three unit vectors X orthogonal reference frame for single crystals avoids the complications of the metric associated with the crystal unit cell axes. In any case, almost all modern measurements of physical property tensors are reported using Cartesian reference frames.
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Table 1. Alignment of the crystal reference frame for the tensors of physical properties of crystals. The corresponds to crystallographic notation a, b, c, m directions in the direct lattice space, whereas the notation a∗ , b∗ , c∗ denotes the corresponding directions in the reciprocal lattice space, which are parallel to the normal to the plane written as ⊥a for a∗ , etc. Note that there are at least two possible reference choices for all symmetries except orthorhombic, tetragonal and cubic T X
Crystal symmetries Orthorhombic, tetragonal, cubic Trigonal, hexagonal Monoclinic Triclinic
Y T
Z T
a
b
c
a m a∗ a a∗ a Y T × Z T Y T × Z T
m −a b b T Z T × X T Z T × X b∗ b
c c c c∗ c c∗ c c∗
We next discuss how the single-crystal tensor reference frame is defined using the crystal coordinate system. In the general case of triclinic crystal symmetry, the crystal coordinate system is specified by its axis lengths a, b, c and inter-axial angles a, b, g resulting in a non-Euclidean coordinate system a, b, c for the general case. In order to align the T , Y T , Z T in Euclidean tensor reference frame X the crystal coordinate system, several conventions are in use. The most common conventions are summarized in Table 1. In MTEX the alignment of the crystal reference frame is defined together with the symmetry group and the crystal coordinate system. All this information is stored in a variable of type symmetry. For example by cs_tensor = symmetry(’triclinic’ [5.29], 9.18, 9.42],... [90.4, 98.9, 90.1]* degree, ’X||a*’, ’Z||c’, ’mineral’, ’Talc’); cs_tensor = symmetry(size: 1) mineral : Talc symmetry : triclinic ( 2 1) a, b, c : 5.3, 9.2, 9.4 alpha, beta, gamma : 90.4, 98.9, 90.1 reference frame : X||a*, Z||c
we store in the variable cs_tensor the geometry of Talc which has triclinic crystal symmetry, axis lengths 5.29, 9.18, 9.42, inter-axial angles 90.48, 98.98, 90.18 and the convention for a Cartesian a∗, Z|| c; right-handed tensor reference frame X||
for the alignment of we therefore have Y = Z × X the crystal reference frame. In order to define a crystal constituent property tensor with respect to this crystal reference frame, we append the variable cs_tensor to its definition, that is, M = [[219.83 59.66 -4.82 -0.82 -33.87 -1.04];... [59.66 216.38 -3.67 1.79 -16.51 -0.62];... [-4.82 -3.67 48.89 4.12 -15.52 -3.59];... [-0.82 1.79 4.12 26.54 -3.60 -6.41];... [-33.87 -16.51 -15.52 -3.60 22.85 -1.67];... [-1.04 -0.62 -3.59 -6.41 -1.67 78.29]]; C = tensor(M, ’name’, ’elastic stiffness, ’unit’, ’GPa’, cs_tensor) C = elastic stiffness tensor(size : 3 3 3 3) unit: GPa rank: 4 mineral: Talc (triclinic, X||a*, Z||c) tensor in Voigt matrix representation 219.83 59.66 -4.82 -0.82 -33.87 -1.04 59.66 216.38 -3.67 1.79 -16.51 -0.62 -4.82 -3.67 48.89 4.12 -15.52 -3.59 -0.82 1.79 4.12 26.54 -3.60 -6.41 -33.87 -16.51 -15.52 -3.60 22.85 -1.67 -1.04 -0.62 -3.59 -6.41 -1.67 78.29
defines the elastic stiffness tensor in GPa of Talc. This example will be discussed in greater detail in the section ‘Elasticity tensors’.
Crystal orientations c , Y c , Z c be a Euclidean crystal coordiLet X nate system assigned to a specific crystal and let s , Y s , Z s be a specimen coordinate system. In X polycrystalline materials, the two coordinate systems generally do not coincide. Their relative alignment describes the orientation of the crystal within the specimen. More specifically, the orientation of a crystal is defined as the (active) rotation g that rotates the specimen coordinate system into coincidence with the crystal coordinate system. From another point of view, the rotation g can be described as the basis transformation from the crystal coordinate system to the specimen coordinate system. Let h = (h1 , h2 , h3 ) be the coordinates of a specific direction with respect to the crystal coordinate system. Then r = (r1 , r2 , r3 ) = gh are the coordinates of the same direction with respect to the specimen coordinate system. Crystal orientations are typically defined by Euler angles, either by specifying rotations with
ANISOTROPIC PHYSICAL PROPERTIES
angles f1 , F, f2 about the axes Z s , X s , Z s (Bunge convention) or with angles a, b, g about the axes Z s , Y s , Z s (Matthies convention). Both conventions, and also some others, are supported in MTEX. In order to define an orientation in MTEX we start by c , Y c , Z c used fixing the crystal reference frame X for the definition of the orientation, cs_orientation = symmetry(’triclinic’ [5.29], 9.18, 9.42],... [90.4, 98.9, 90.1]* degree, ’X||a*’, ’Z||c’, ’mineral’, ’Talc’); cs_orientation = crystal symmetry (size: 1) mineral : talc symmetry : triclinic (-1) a, b, c : 5.3, 9.2, 9.4 alpha, beta, gamma: 90.4, 98.9, 90.1 reference frame : X||a, Z||c*
Now an orientation can be defined as a variable of type orientation, it is common practice to use the letter g to denote an orientation, derived from the German word Gefu¨ge used by Sander (1911). g = orientation (’Euler’, 10*degree, 20*degree, 5*degree, ’Bunge’, cs_orientation) g = orientation (size : 1 1) mineral : talc crystal symmetry : triclinic, X||a, Z||c* specimen symmetry: triclinic Bunge Euler angles in degree phi1 Phi phi2 10 20 5
Note that for the definition of an orientation the crystal reference frame is crucial. The definition of the variable of type orientation therefore includes a variable of type symmetry, storing the relevant information. This applies in particular if the orientation data (i.e. Euler angles) are imported from third-party measurement systems such as EBSD and associated software with their own c , Y c , Z c , which should specific conventions for X be defined when using the MTEX import wizard. In order to demonstrate the coordinate transform between the crystal and the specimen coordinate system, we choose a crystal direction in the reciprocal lattice h = h a∗ +kb∗ + ℓc∗ (pole to a plane) by defining a variable of type Miller: h = Miller (1, 1, 0, cs_orientation, ’hkl’) h = Miller (size: 1 1) mineral : talc (triclinic, X||a, Z||c*) h 1 k 1 l 0
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and express it in terms of the specimen coordinate system for a specific orientation g ¼ (108, 208, 58) r=g*h r = vector3d (size: 1 1), x y z 0.714153 0.62047 0.324041
The resulting variable is of type vector3d reflecting that the new coordinate system is the specimen coordinate system. Note that in order that the coordinate transformation rule makes sense physically, the corresponding crystal reference frames used for the definition of the orientation and the crystal direction by Miller indices must coincide. Alternatively, a crystal direction u = ua + vb + wc in direct space can be specified: u = Miller (1, 1, 0, cs_orientation, ’uvw’) h = Miller (size: 1 1), uvw mineral: talc (triclinic, X||a, Z||c*) u 1 v 1 w 0
and expressed in terms of the specimen coordinate system r=g*u r = vector3d (size: 1 1), x y z 0.266258 0.912596 0.310283
This obviously gives a different direction, since direct and reciprocal space do not coincide for triclinic crystal symmetry.
The relationship between the single-crystal physical property and Euler angle reference frames Let us consider a rank r tensor Ti1 ,...,ir describing some physical property of a crystal with respect to a well-defined crystal reference frame T , Y T , Z T . We are often interested in expressing X the tensor with respect to another, different Y, Z, which might be Euclidean reference frame X, (1) (2) (3)
a crystallographically equivalent crystal reference frame, a different convention for aligning the Euclidean reference frame to the crystal coordinate system or a specimen coordinate system.
Let us first consider a vector h that has the representation h = hT X T + hT2 Y T + hT3 X T 1
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with respect to the tensor reference frame T , Y T , Z T , and the representation X h = h1 X + h2 Y + h3 X Y, Z. with respect to the other reference frame X, Then the coordinates hT1 , hT2 , hT3 and h1 , h2 , h3 satisfy the transformation rule ⎛
⎞ ⎛ ⎞⎛ T ⎞ · Y T X · Z T ·X T X h1 h1 X ⎝ h2 ⎠ = ⎝ Y · X T Y · Y T Y · Z T ⎠⎝ hT2 ⎠, T Z · Y T Z · Z T h3 hT Z · X
3 =:R
(1) that is, the matrix R performs the coordinate transformation from the tensor reference frame T , Y T , Z T to the other reference frame X, Y, Z. X The matrix R can also be interpreted as the rotation matrix that rotates the second reference frame into coincidence with the tensor reference frame. Considering hTj to be a rank-one tensor, the transformation rule becomes hi = hTj Rij . This formula generalizes to arbitrary tensors. Let TiT1 ,...,ir be the coefficients of a rank r tensor with respect to the crystal reference frame X T, Y T, Z T and let Ti1 ,...,ir be the coefficients with respect to another reference frame X, Y, Z. Then the linear orthogonal transformation law for Cartesian tensors states that Ti1 ,...,ir = T Tj1 ,..., jr Ri1 j1 · · · Rir jr .
(2)
Let us now examine the three cases for a new reference frame as mentioned at the beginning of this section. In the case of a crystallographically equivalent reference frame, the coordinate transform R is a symmetry element of the crystal and the tensor remains invariant with respect to this coordinate transformation, that is T˜ i1 ,...,ir = Ti1 ,...,ir . Y, Z In the case that the other reference frame X, follows a different convention in aligning to the crystal coordinate system, the transformed tensor T˜ i1 ,...,ir is generally different to the original tensor. In MTEX this change of reference frame is carried out by the command set. Let us consider the elastic stiffness tensor Cijkl of talc (as defined above) as: C = elastic stiffness tensor (size: 3 3 3 3) unit: GPa rank: 4
mineral: talc (triclinic, X||a*, Z||c) 219.83 59.66 -4.82 -0.82 -33.87 -1.04
59.66 216.38 -3.67 1.79 -16.51 -0.62
-4.82 -3.67 48.89 4.12 -15.52 -3.59
-0.82 1.79 4.12 26.54 -3.60 -6.41
-33.87 -16.51 -15.52 -3.60 22.85 -1.67
-1.04 -0.62 -3.59 -6.41 -1.67 78.29
and let us consider the reference frame cs_orientation as defined in the previous section cs_orientation = symmetry(Size : 1) mineral : talc symmetry : triclinic (-1) a, b, c : 5.29, 9.18, 9.42 alpha, beta, gamma: 90.4, 98.9, 90.1 reference frame : X||a, Z||c*
Then the elastic stiffness tensor C ijkl of talc with respect to the reference frame cs_orientation is computed by setting cs_orientation as the new reference frame, that is C_orientation = set(C, ’CS’, cs_orientation) C_orientation = elastic stiffness tensor (size: 3 3 3 3) unit: GPa rank: 4 mineral: talc (triclinic, X||a, Z||c*) tensor in Voigt matrix representation 231.82 63.19 -5.76 0.76 -4.31 -0.59 63.19 216.31 -7.23 2.85 -5.99 -0.86 -5.76 -7.23 38.92 2.23 -16.69 -4.3 0.76 2.85 2.23 25.8 -4.24 1.86 -4.31 -5.99 -16.69 -4.24 21.9 -0.14 -0.59 -0.86 -4.3 1.86 -0.14 79.02
Finally, we consider the case that the second reference frame is not aligned to the crystal coordinate system but to the specimen coordinate system. According to the previous section, the coordinate transform then defines the orientation g of the crystal and equation (2) tells us how the tensor has to be rotated according to the crystal orientation. In this case we will write Ti1 ,...,ir = T Tj1 ,..., jr (g) = T Tj1 ,..., jr Ri1 j1 (g) · · · Rir jr (g),
(3)
to express the dependency of the resulting tensor from the orientation g. Here Rir jr (g) is the rotation matrix defined by the orientation g. In order to apply equation (3), it is of major importance that the tensor reference frame and the crystal reference frame used for describing the orientation coincide. If they do not coincide, the tensor has to be transformed to the same crystal reference frame used for describing the orientation. When working
ANISOTROPIC PHYSICAL PROPERTIES
with tensors and orientation data it is therefore always necessary to know the tensor reference frame and the crystal reference frame used for describing the orientation. In practical applications this is not always a simple task, as this information is sometimes hidden by the commercial EBSD systems. If the corresponding reference frames are specified in the definition of the tensor as well as in the definition of the orientation, MTEX automatically checks for coincidence and performs the necessary coordinate transforms if they do not coincide. Eventually, the rotated tensor for an orientation g ¼ (108, 208, 58) is computed by the command rotate: C_rotated = rotate (C,g) C_rotated = elastic_stifness tensor (size: 3 3 3 3) unit: GPa rank: 4 tensor in Voigt matrix representation 228.79 56.05 1.92 19.99 -13.82 6.85 56.05 176.08 11.69 50.32 -7.62 4.42 1.92 11.69 43.27 5.28 -19.48 2.33 19.99 50.32 5.28 43.41 -1.74 0.24 -13.82 -7.62 -19.48 -1.74 29.35 17.34 6.85 4.42 2.33 0.24 17.34 73.4
Note, that the resulting tensor does not contain any information on the original mineral or reference frame; this is because the single-crystal tensor is now with respect to the specimen coordinate system and can be averaged with any other elastic stiffness tensor from any other crystal of any composition and orientation.
Single-crystal anisotropic properties We now present some classical properties of single crystals that can be described by tensors (cf. Nye 1985).
Second-rank tensors A typical second-rank tensor describes the relationship between an applied vector field and an induced vector field, such that the induced effect is equal to the tensor property multiplied by the applied vector. Examples of such a tensor are † the electrical conductivity tensor, where the applied electric field induces a field of current density, † the dielectric susceptibility tensor, where the applied electric field intensity induces electric polarization, † the magnetic susceptibility tensor, where the applied magnetic field induces the intensity of magnetization,
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† the magnetic permeability tensor, where the applied magnetic field induces magnetic induction, † the thermal conductivity tensor, where the applied negative temperature gradient induces heat flux. As a typical example for a second-rank tensor we consider the thermal conductivity tensor k, ⎛ ⎞ k11 k12 k13 k = ⎝ k21 k22 k23 ⎠ k31 k32 k33 ⎞ ⎛ ∂x1 ∂x2 ∂x2 q1 − q1 − q1 − ⎜ ∂T ∂T ∂T ⎟ ⎟ ⎜ ∂x2 ∂x2 ⎟ ⎜ ∂x1 = ⎜− q2 − q2 − q2 ⎟ ⎜ ∂T ∂T ∂T ⎟ ⎝ ∂x ∂x2 ∂x2 ⎠ 1 q3 − q3 − q3 − ∂T ∂T ∂T which relates the negative temperature gradient −gradT = −(∂T/∂x1 , ∂T/∂x2 , ∂T/∂x3 ) to the heat flux q = (q1 , q2 , q3 ) per unit area by qi = −
3 j=1
kij
∂T ∂T = −kij . ∂xj ∂xj
(4)
In the present example the applied vector is the negative temperature gradient and the induced vector is the heat flux. Furthermore, we see that the relating vector is built up as a matrix where the applied vector is the denominator of the rows and the induced vector is the numerator of columns. We see that the tensor entries kij describe the heat flux qi in direction Xi given a thermal gradient ∂T/∂xj in direction Xj. As an example we consider the thermal conductivity of monoclinic orthoclase (Hofer & Schilling 2002). We start by defining the tensor reference frame and the tensor coefficients in W m21 K21. cs-tensor = symmetry(’monoclinic’, [8.561, 12.996, 7.192] ,... [90, 116.01, 90]*degree, ’mineral’, ’orthoclase’,’Y||b’,’Z||c’); M = [[1.45 0.00 0.19];... [0.00 2.11 0.00];... [0.19 0.00 1.79]];
Now the thermal conductivity tensor k is defined by k = tensor (M, ’name’, ’thermal_ conductivity’, unit’, ’W_1/m_1/K’, cs_tensor) k = thermal conductivity tensor (size: 3 3) unit : W 1/m 1/K rank : 2 mineral: orthoclase (monoclinic, X||a*, Y||b, Z||c)
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X
2.1
2.1
2
2
1.9
1.9
1.8
1.8
1.7
1.7
1.6
1.6
1.5
1.5
Y Z
min: 1.37
max: 2.1
1.4
min: 1.37
max: 2.1
1.4
Fig. 1. The thermal conductivity k of orthoclase visualized by its directionally varying magnitude for left: the tensor in standard orientation and right: the rotated tensor.
1.45 0 0.19 0 2.11 0 0.19 0 1.79
Using the thermal conductivity tensor k we can compute the thermal flux q in W m22 for a temperature gradient in K m21: gradT = Miller(1,1,0, cs_tensor, ’uvw’) gradT = Miller (size: 1 1), uvw mineral: orthoclase (monoclinic, X||a*, Y||b, Z||c) u 1 v 1 w 0
by equation (4). In MTEX, this becomes q = EinsteinSum(k, [1 -1],gradT, -1, ’name’, ’thermal_flux’, ’unit’, ’W_1/ m^2’) q = thermal flux tensor (size: 3) unit : W 1/m^2 rank : 1 mineral: orthoclase (monoclinic, X||a*, Y||b, Z||c) 0.672 1.7606 -0.3392
Note that the 21 in the arguments of the command EinsteinSum indicates the dimension which has to be summed up and the 1 in the argument indicates that the first dimension of k becomes the first dimension of q; see equation (4). A second-order tensor kij can be visualized by plotting its magnitude R(x) in a given direction x, R(x) = kij xi xj .
In MTEX the magnitude in a given direction x can be computed via x = Miller (1,0,0, cs_tensor, ’uvw’); R = EinsteinSum (k, [-1 -2],x,- 1,x,- 2) R = tensor (size:) rank : 0 mineral: orthoclase (monoclinic, X||a*, Y||b, Z||c) 1.3656
Again, the negative arguments 21 and 22 indicate which dimensions have to be multiplied and summed up. Alternatively, we can use the command magnitude, R = directionalMagnitude (k, x)
Since in MTEX the directional magnitude is the default output of the plot command, the code plot(k) colorbar
plots the directional magnitude of k with respect to any direction x as shown in Figure 1. Note that, by default, the X axis is plotted in the north direction, the Y axis is plotted in the west direction and the Z axis is at the centre of the plot. This default alignment can be changed by the commands plotx2north, plotx2east, plotx2south, plotx2west. When the tensor k is rotated the directional magnitude rotates accordingly. This can be checked in MTEX by g = orientation(’Euler’,10*degree, 20*degree,30*degree,cs_tensor); k_rot = rotate(k,g); Plot(k_rot)
ANISOTROPIC PHYSICAL PROPERTIES
The resulting ploat is shown in Figure 1. Furthermore, from the directional magnitude we observe that the thermal conductivity tensor k is symmetric, that is, kij = k ji . This implies that the thermal conductivity is an axial or non-polar property, which means that the magnitude of heat flow is the same in positive or negative crystallographic directions. We want to emphasize that there are also second-rank tensors which do not describe the relationship between an applied vector field and an induced vector field but relate, for instance, a zero-rank tensor to a second-rank tensor. The thermal expansion tensor a, defined ∂1ij , aij = ∂T is an example of such a tensor which relates a small applied temperature change ∂T (a scalar or zero-rank tensor) to the induced strain tensor 1ij (a second-rank tensor). The corresponding coefficient of volume thermal expansion becomes 1 ∂V = aii . V ∂T
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tensor, both tensors are symmetric, that is, sij = s ji and 1ij = 1 ji . For the elastic stiffness tensor, this implies the symmetry Cijkl = Cijlk = C jikl = C jilk reducing the number of independent entries of the tensor from 34 ¼ 81 to 36. Since the elastic stiffness Cijkl is related to the internal energy U of a body by Cijkl =
∂ ∂U , ∂1kl ∂1ij
assuming constant entropy, we obtain by the Schwarz integrability condition that allows the interchanging of the order of partial derivatives of a function: Cijkl
2 2 ∂ ∂U ∂U ∂ U = = = ∂1kl ∂1ij ∂1ij ∂1kl ∂1kl ∂1ij = Cklij .
Hence,
This relationship holds true only for small changes in temperature. For larger changes in temperature, higher-order terms have to be considered (see Fei 1995 for data on minerals). This also applies to other tensors.
Elasticity tensors We will now present fourth-rank tensors, but restrict ourselves to the elastic tensors. Let sij be the second-rank stress tensor and let 1kl be the second-rank infinitesimal strain tensor. Then the fourth-rank elastic stiffness tensor Cijkl describes the stress sij induced by the strain 1kl , defined
sij = Cijkl 1kl ,
(5)
which is known as Hooke’s law for linear elasticity. Alternatively, the fourth-order elastic compliance tensor Sijkl describes the strain 1kl induced by the stress sij , defined 1kl = Sijkl sij .
Cijkl = Cklij which further reduces the number of independent entries from 36 to 21 (e.g. Mainprice 2007). These 21 independent entries may be efficiently represented in the form of a symmetric 6 × 6 matrix Cmn , m, n = 1, . . . , 6 as introduced by Voigt (1928). The entries Cmn of this matrix representation equal the tensor entries Cijkl whenever m and n correspond to ij and kl according to: m or n ij or kl
1 2 3 11 22 33
4 5 6 23, 32 13, 31 12, 21
The Voigt notation is used for published compilations of elastic tensors (e.g. Bass 1995; Isaak 2001). In a similar manner, a Voigt representation Smn is defined for the elastic compliance tensor Sijkl . However, there are additional factors when converting between the Voigt Smn matrix representation and the tensor representation Sijkl . More precisely, for ij, kl, m, n which correspond to each other according to the above table, we have the identities:
The above definitions may also be written as Cijkl =
∂sij ∂1kl
and
Sijkl =
∂1kl . ∂sij
In the case of static equilibrium for the stress tensor and infinitesimal deformation for the strain
Sijkl
⎧ ⎪ ⎨ P = 1, = P · Smn , P = 12, ⎪ ⎩ P = 14,
⎫ if both m, n = 1, 2, 3 ⎪ ⎬ if either m or n are 4, 5, 6 ⎪ ⎭ if both m, n = 4, 5, 6
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Using the Voigt matrix representation of the elastic stiffness tensor (equation (5)) may be written as ⎛
⎞ ⎛ C11 C12 s11 ⎜ s22 ⎟ ⎜ C21 C22 ⎜ ⎟ ⎜ ⎜ s33 ⎟ ⎜ C31 C32 ⎜ ⎟ ⎜ ⎜ s23 ⎟ = ⎜ C41 C42 ⎜ ⎟ ⎜ ⎝ s13 ⎠ ⎝ C51 C52 s12 C61 C62 ⎛ ⎞ 111 ⎜ 122 ⎟ ⎜ ⎟ ⎜ 133 ⎟ ⎜ ⎟. ×⎜ ⎟ ⎜ 2123 ⎟ ⎝ 2113 ⎠ 2112
C13 C23 C33 C43 C53 C63
C14 C24 C34 C44 C54 C64
C15 C25 C35 C45 C55 C65
⎞ C16 C26 ⎟ ⎟ C36 ⎟ ⎟ C46 ⎟ ⎟ C56 ⎠ C66
The elastic compliance S in GPa21 can then be computed by inverting the tensor C. S = inv(C) S = elastic compliance tensor (size: 3 3 3 3) unit : 1/GPa rank : 4 mineral: talc (triclinic, X||a*, Z||c) tensor in Voigt matrix (×1023) representation 6.91 -0.83 4.71 0.74 6.56 0.35 -0.83 5.14 1.41 -0.04 1.72 0.08 4.71 1.41 30.31 -0.13 14.35 1.03 0.74 -0.04 -0.13 9.94 2.12 0.86 6.56 1.72 14.35 2.12 21.71 1.02 0.35 0.08 1.03 0.86 1.02 3.31
Elastic properties The matrix representation of Hooke’s law allows for a straightforward interpretation of the tensor coefficients Cij . For example the tensor coefficient C11 describes the dependency between normal stress s11 in direction X and axial strain 111 in the same direction. The coefficient C14 describes the dependency between normal stress s11 in direction X and shear strain 2123 = 2132 in direction Y in the plane normal to Z. The dependency between normal stress s11 and axial strains 111 , 122 and 133 along X, Y and Z is described by C11 , C12 and C13 , whereas the dependencies between the normal stress s11 and shear strains 2123 , 2113 and 2112 are described by C14 , C15 and C16 . These effects are most important in low-symmetry crystals, such as triclinic and monoclinic crystals, where there are a large number of non-zero coefficients. In MTEX the elasticity tensors may be specified directly in Voigt notation as we have already seen in the ‘Tensor mathematics and crystal geometry’ section. Alternatively, tensors may also be imported from ASCII files using a graphical interface called import wizard in MTEX. Let C be the elastic stiffness tensor for talc in GPa as defined in ‘The crystal reference frame’ section. C = elastic_stiffness tensor (size: 3 3 3 3) unit : GPa rank : 4 mineral: talc (triclinic, X||a*, Z||c) tensor in Voigt matrix representation 219.83 59.66 -4.82 -0.82 -33.87 -1.04 59.66 -216.38 -3.67 1.79 -16.51 -0.62 -4.82 -3.67 48.89 4.12 -15.52 -3.59 -0.82 1.79 4.12 26.54 -3.6 -6.41 -33.87 -16.51 -15.52 -3.6 22.85 -1.67 -1.04 -0.62 -3.59 -6.41 -1.67 78.29
The fourth-order elastic stiffness tensor Cijkl and fourth-order elastic compliance tensor Sijkl are the starting point for the calculation of a number of elastic anisotropic physical properties, which include † Young’s modulus, † shear modulus, † Poisson’s ratio, † linear compressibility, † compressional and shear elastic wave velocities, † wavefront velocities, † mean sound velocities, † Debye temperature, and, of course, their isotropic equivalents. In the following we provide a short overview of these properties. Scalar volume compressibility. First we consider the scalar volume compressibility b. Using the fact that the change of volume is given in terms of the strain tensor 1ij by ∂V = 1ii , V we determine, for hydrostatic or isotropic pressure (which is given by the stress tensor skl = −Pdkl ), that the change of volume is given by ∂V = −PSiikk . V The volume compressibility is therefore
b=−
∂V 1 = Siikk . V P
Linear compressibility. The linear compressibility b(x) of a crystal is the strain, that is the relative change in length ∂l/l, for a specific crystallographic
ANISOTROPIC PHYSICAL PROPERTIES
direction x when the crystal is subjected to a unit change in hydrostatic pressure −Pdkl . From ∂l = 1ij xi xj = −PSiikk xi xj l we conclude
b(x) = −
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Wave velocities. The Christoffel equation, first published by Christoffel (1877), can be used to calculate elastic wave velocities and the polarizations in an anisotropic elastic medium from the elastic stiffness tensor Cijkl or, more straightforward, from the Christoffel tensor Tik which is, for a unit propagation direction n, defined by Tik (n) = Cijkl nj nl .
∂l 1 = Sijkk xi xj . lP
Since the elastic tensors are symmetric, we have Young’s modulus. Young’s modulus E is the ratio of the axial (longitudinal) stress to the lateral (transverse) strain in a tensile or compressive test. As we have seen earlier when discussing the elastic stiffness tensor, this type of uniaxial stress is accompanied by lateral and shear strains as well as the axial strain. Young’s modulus in direction x is given by E(x) = (Sijkl xi xj xk xl )−1 . Shear modulus. Unlike Young’s modulus, the shear modulus G in an anisotropic medium is defined using two directions: the shear plane h and the shear direction u. For example, if the shear stress s12 results in the shear strain 2112 then the corresponding shear modulus is G = s12 /2112 . From Hooke’s law we have 112 = S1212 s12 + S1221 s21 , and hence G = (4S1212 )−1 . The shear modulus for an arbitrary, but orthogonal, shear plane h and shear direction u is given by G(h, u) = (4Sijkl hi uj hk ul )−1 . Poisson ratio. The anisotropic Poisson ratio is defined by the elastic strain in two orthogonal directions: the longitudinal (or axial) direction x and the transverse (or lateral) direction y. The lateral strain is defined by −1ij yi yj along y and the longitudinal strain by 1ij xi xj along x. The anisotropic Poisson ratio n(x, y) is given as the ratio of lateral to longitudinal strain (Sirotin & Shakolskaya 1982) as n(x, y) = −
1ij yi yj Sijkl xi xj yk yl =− . 1kl xk xl Smnop xm xn xo xp
The anisotropic Poisson ratio has recently been reported for talc (Mainprice et al. 2008) and has been found to be negative for many directions at low pressure.
Tik (n) = Cijkl nj nl = C jikl nj nl = Cijlk nj nl = Cklij nj nl = Tki (n), and hence the Christoffel tensor T(n) is symmetric. The Christoffel tensor is also invariant upon the change of sign of the propagation direction, as the elastic tensor is not sensitive to the presence or absence of a centre of crystal symmetry (being a centro-symmetric physical property). Because the elastic strain energy 1/2Cijkl 1ij 1kl of a stable crystal is always positive and real (e.g. Nye 1985), the eigenvalues l1 , l1 and l3 of the Christoffel tensor Tik (n) are real and positive. They are related to the wave velocities Vp , Vs1 and Vs2 of the plane P-, S1- and S2-waves propagating in the direction n by the formulae l1 Vp = , r
l2 Vs1 = , r
l3 Vs2 = , r
where r denotes the material density. The three eigenvectors of the Christoffel tensor are the polarization directions, also called vibration, particle movement or displacement vectors, of the three waves. As the Christoffel tensor is symmetric, the three polarization directions are mutually perpendicular. In the most general case there are no particular angular relationships between polarization directions p and the propagation direction n. However, the P-wave polarization direction is typically nearly parallel and the two S-waves polarizations are nearly perpendicular to the propagation direction. They are termed quasi-P or quasi-S waves. The S-wave velocities may be identified unambiguously by their relative velocity Vs1 . Vs2. All the elastic properties mentioned in this section have direct expressions in MTEX: beta = volumeCompressibility (C) beta = linearCompressibility (C,x) E = YoungsModulus (C,x) G = shearModulus (C,h,u) nu = PoissonRatio (C,x,y) T = ChristoffelTensor (C,n)
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X
X 200
0.045 0.04
180
0.035
160
0.03
140
0.025
Y Z
120
Y Z
0.02
100
0.015
80
0.01
60
0.005
min: −0.0025
max: 0.05
0
40
min: 17
max: 213
20
Fig. 2. Left: the linear compressibility in GPa21 and right: Young’s modulus in GPa for talc.
Note that all these commands take the compliance tensor C as basis for the calculations. For the calculation of the wave velocities the command velocity [vp, vs1, vs2, pp, ps1, ps2] = velocity (C,x ,rho)
allows for the computation of the wave velocities and the corresponding polarization directions.
Visualization In order to visualize the above quantities, MTEX offers a simple, yet flexible, syntax. Let us demonstrate it using the Talc example of the previous section. In order to plot the linear compressibility b(x) or Young’s Modulus E(x) as a function of the direction x, we use the commands plot (C, ’PlotType’, ’linearCompressibility’) plot (C, ’PlotType’, ’YoungsModulus’)
The resulting plots are shown in Figure 2. Next we want to visualize the wave velocities and the polarization directions. Let us start with the P-wave velocity in km s21 which is plotted by rho = 2.78276; plot (C, ’PlotType’, ’velocity’, ’vp’, ’density’, rho)
Note that we had to pass the density rho in g cm23 to the plot command. We now want to plot the P-wave polarization directions on top, so use the commands
hold on and hold off to prevent MTEX from clearing the output window: hold on plot(C, ’PlotType’, ’velocity’, ’pp’, ’density’, rho); hold off
The result is shown in Figure 3. Instead of only specifying the variables to plot, we can also perform simple calculations. From the commands plot (C, ’PlotType’, ’velocity’, ’200*(vs1-vs2)./(vs1+vs2)’, ’density’, rho); hold on plot (C, ’PlotType’, ’velocity’, ’ps1’, ’density’, rho); hold off
the S-wave anisotropy in percent is plotted together with the polarization directions of the fastest S-wave ps1. Another example illustrating the flexibility of the system is the following plot of the velocity ratio Vp /Vs1 together with the direction of the S1-wave polarizations. plot (C, ’PlotType’, ’velocity’, ’vp./vs1’, ’density’, rho); hold on plot (C, ’PlotType’, ’velocity’,’ps1’, ’density’, rho); hold off
Anisotropic properties of polyphase aggregates In this section we are concerned with the problem of calculating average physical properties of
ANISOTROPIC PHYSICAL PROPERTIES
(a)
9
(b)
187
(c)
1.7
80
8.5 70
1.6
7.5
60
1.5
7
50
8
6.5
1.4
40
1.3
6 30
5.5
1.2 20
5
max: 9.1
P-wave velocity
4
1.1
10
4.5
min: 3.94
min: 0.2
max: 86
S-wave anisotropy
min: 1
max: 1.7
Vp/Vs1 ratio
Fig. 3. Wave velocities of a Talc crystal plotted on seismic colour maps: (a) P-wave velocity together with the P-wave polarization direction; (b) S-wave anisotropy in percent together with the S1-wave polarization direction; and (c) the ratio of Vp/Vs1 velocities together with the S1-wave polarization direction.
polyphase aggregates. To this end, two ingredients are required for each phase p: (1) (2)
the property tensor Tip1 ,...,ir describing the physical behaviour of a single crystal in the reference orientation, the orientation density function (ODF) f p (g) describing the volume portion DV/V of crystals having orientation g or a representative set of individual orientations gm , m = 1, . . . , M (e.g. measured by EBSD).
As an example we consider an aggregate composed of two minerals (glaucophane and epidote) using data from a blueschist from the Ile de Groix, France. The corresponding crystal reference frames are defined by cs_glaucophane = symmetry (’2/m’, [9.5334, 17.7347, 5.3008],[90.00, 103.597, 90.00] * degree, ’mineral’, ’glaucophane’); cs_epidote = symmetry (’2/m’, [8.8877, 5.6275, 10.1517],[90.00, 115.383, 90.00] * degree, ’mineral’, ’epidote’);
For glaucophane the elastic stiffness was measured by Bezacier et al. (2010) who provided the tensor C_glaucophane = tensor (size: 3 3 3 3) rank : 4 mineral: glaucophane (2/m, X||a*, Y||b, Z||c) tensor in Voigt matrix representation 122.28 45.69 37.24 0 2.35 0 45.69 231.5 74.91 0 -4.78 0 37.24 74.91 254.57 0 -23.74 0 0 0 0 79.67 0 8.89 2.35 -4.78 -23.74 0 52.82 0 0 0 0 8.89 0 51.24
For epidote, the elastic stiffness was measured by Aleksandrov et al. (1974): C_epidote = tensor (size: 3 3 3 3) rank : 4 mineral: epidote (2/m, X||a*, Y||b, Z||c) tensor in Voigt matrix representation 211.5 65.6 43.2 0 -6.5 65.6 239 43.6 0 -10.4 43.2 43.6 202.1 0 -20 0 0 0 39.1 0 -6.5 -10.4 -20 0 43.4 0 0 0 -2.3 0
0 0 0 -2.3 0 79.5
Computing the average tensor from individual orientations We start with the case that we have individual orientation data gm, m ¼ 1, . . ., M, that is, from EBSD or U-stage measurements, and volume fractions Vm, m ¼ 1, . . ., M. The best-known averaging techniques for obtaining estimates of the effective properties of aggregates are those developed for elastic constants by Voigt (1887, 1928) and Reuss (1929). The Voigt average is defined by assuming that the induced tensor (in broadest sense, including vectors) field is everywhere homogeneous or constant, that is, the induced tensor at every position is set equal to the macroscopic induced tensor of the specimen. In the classical example of elasticity, the strain field is considered constant. The Voigt average is sometimes called the ‘series’ average by analogy with Ohm’s law for electrical circuits. The Voigt average specimen effective tensor kTlVoigt is defined by the volume average of the individual tensors T(gcm ) with crystal orientations gcm and volume fractions Vm : M Vm T(gcm ). kTlVoigt = m=1
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Contrarily, the Reuss average is defined by assuming that the applied tensor field is everywhere constant, that is, the applied tensor at every position is set equal to the macroscopic applied tensor of the specimen. In the classical example of elasticity, the stress field is considered constant. The Reuss average is sometimes called the ‘parallel’ average. The specimen effective tensor kTlReuss is defined by the volume ensemble average of the inverses of the individual tensors T −1 (gcm ): kTlReuss =
M
−1 Vm T −1 (gcm )
.
m=1
The experimentally measured tensor of aggregates is generally between the Voigt and Reuss average bounds as the applied and induced tensor fields distributions are expected to be between uniform induced (Voigt bound) and uniform applied (Reuss bound) field limits. Hill (1952) observed that the arithmetic mean of the Voigt and Reuss bounds kTlHill = 12(kTlVoigt + kTlReuss ), sometimes called the Hill or Voigt –Reuss–Hill (VRH) average, is often close to experimental values for the elastic fourth-order tensor. Although the VRH average has no theoretical justification, it is widely used in earth and materials sciences. In the example outlined above of an aggregate consisting of glaucophane and epidote, we consider an EBSD dataset measured by Bezacier et al. (2010). In MTEX such individual orientation data are represented by a variable of type EBSD which is generated from an ASCII file containing the individual orientation measurements by the command: ebsd = loadEBSD (’FileName’, {cs_glaucophane, cs_epidote}) ebsd = EBSD (Groix_A50_5_stitched. ctf) properties: bands, bc, bs, error, mad phase orientations mineral symmetry crystal reference frame 1 055504 glaucophane 2/m X||a*, Y||b, Z||c 2 63694 epidote 2/m X||a*, Y||b, Z||c
It should be noted that for both minerals the crystal reference frames have to be specified in the command loadEBSD. The Voigt, Reuss and Hill average tensors can now be computed for each phase separately by the command calcTensor: [TVoigt, TReuss, THill] = calcTensor (ebsd, C_epidote, ’phase’, 2)
C_Voigt = tensor (size: 3 3 3 3) rank: 4 tensor in Voigt matrix representation 215 55.39 66.15 -0.42 3.02 -4.69 55.39 179.04 59.12 1.04 -1.06 0.06 66.15 59.12 202.05 0.94 1.16 -0.77 -0.42 1.04 0.94 60.67 -0.86 -0.55 3.02 -1.06 1.16 -0.86 71.77 -0.65 -4.69 0.06 -0.77 -0.55 -0.65 57.81 C_Reuss = tensor (size: 3 3 3 3) rank: 4 tensor in Voigt matrix representation 201.17 56.48 65.94 -0.28 3.21 -4.68 56.48 163.39 61.49 1.23 -1.58 -0.13 65.94 61.49 189.67 1.29 0.75 -0.64 -0.28 1.23 1.29 52.85 -0.99 -0.38 3.21 -1.58 0.75 -099 65.28 -0.6 -4.68 -0.13 -0.64 -0.38 -0.6 50.6 C_Hill = tensor (size: 3 3 3 3) rank: 4 tensor in Voigt matrix representation 208.09 55.93 66.05 -0.35 3.11 -4.69 55.93 171.22 60.31 1.13 -1.32 -0.04 66.05 60.31 195.86 1.11 0.96 -0.71 -0.35 1.13 1.11 56.76 -0.93 -0.46 3.11 -1.32 0.96 -0.93 68.52 -0.62 -4.69 -0.04 -0.71 -0.46 -0.62 54.21
If no phase is specified and all the tensors for all phases are specified, the command [TVoigt, TReuss, THill] = calcTensor (ebsd, C_glaucophane, C_epidote)
computes the average over all phases. These calculations have been validated using the Careware FORTRAN code (Mainprice 1990). We emphasize that MTEX automatically checks for the agreement of the EBSD and tensor reference frames for all phases. In case different conventions have been used, MTEX automatically transforms the EBSD data into the convention of the tensors.
Computing the average tensor from an ODF Next we consider the case that the texture is given by an ODF f. The ODF may originate from texture modelling (Bachmann et al. 2010), pole figure inversion (Hielscher & Schaeben 2008) or density estimation from EBSD data (Hielscher et al. 2010). All these diverse sources may be handled by MTEX. Given an ODF f, the Voigt average kTlVoigt of a tensor T is defined by the integral kTlVoigt =
T(g)f (g)dg SO(3)
(6)
ANISOTROPIC PHYSICAL PROPERTIES
whereas the Reuss average kTlReuss is defined as kTl
Reuss
−1
=
T
−1
(g)f (g)dg
.
(7)
SO(3)
Equations (6) and (7) can be computed in two different ways. First, we can use a quadrature rule: for a set of orientations gm and weights vm , the Voigt average is approximated by kTlVoigt ≈
M
T(gm )vm f (gm ).
m=1
Clearly, the accuracy of the approximation depends on the number of nodes gm and the smoothness of the ODF. An alternative approach to compute the average tensor, avoiding this dependency, uses the expansion of the rotated tensor into generalized spherical harmonics, Dℓkk′ (g). Let Ti1 ,...,ir be a tensor of rank r. It is well known (cf. Kneer 1965; Bunge 1968; Ganster & Geiss 1985; Humbert & Diz 1991; Mainprice & Humbert 1994; Morris 2006) that the rotated tensor Ti1 ,...,ir (g) has an expansion into generalized spherical harmonics up to order r, Ti1 ,...,ir (g) =
r ℓ ℓ=0 k,k′ =−ℓ
Tˆ i1 ,...,ir (l, k, k′ )Dℓkk′ (g). (8)
The explicit calculations of the coefficients Tˆ i1 ,...,ir (l, k, k′ ) are given in the Appendix. Assume that the ODF has an expansion into generalized spherical harmonics of the form f (g) =
r ℓ ℓ=0 k,k′ =−ℓ
fˆ (l, k, k′ )Dℓkk′ (g).
The average tensor with respect to this ODF can then be computed by the formula 1 Ti ,...,i (g)f (g) 8p2 SO(3) 1 r 1 Ti ,...,i (g) f (g)dg = 2 8p SO(3) 1 r =
ℓ 1 Tˆ i1 ,...,ir (l, k, k′ )fˆ (l, k, k′ ). 2ℓ + 1 ′ ℓ=0 k,k =−ℓ
r
By default MTEX uses the Fourier approach, which is much faster than using numerical integration (quadrature rule) which requires a discretization of the ODF. Numerical integration is applied only in the cases when MTEX cannot
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determine the Fourier coefficients of the ODF in an efficient manner. At the present time, only the Bingham distributed ODFs pose this problem. All the necessary calculations are done automatically, including the correction for different crystal reference frames. Let us consider once again the aggregate consisting of glaucophane and epidote and the corresponding EBSD dataset as mentioned in the first section. For any phase, we can then estimate an ODF by odf_epidote = calcODF (ebsd, ’phase’, 2) odf_epidote = ODF (ODF estimated from Groix_A50_5 _stitched. ctf) mineral : epidote crystal symmetry : 2/m, X||a*, Y||b, Z||c specimen symmetry: triclinic Portion specified by Fourier coefficients: degree: 28 weight: 1
Next, we can compute the average tensors directly from the ODF [TVoigt, TReuss, THill] = calcTensor (odf_epidote, C_epidote) C_Voigt = tensor (size: 3 3 3 3) rank: 4 tensor in Voigt matrix representation 212.64 56.81 65.68 -0.25 2.56 -4 56.81 179.21 59.64 0.93 -0.83 -0.27 65.68 59.64 201.3 0.87 1.12 -0.71 -0.25 0.93 0.87 61.33 -0.79 -0.35 2.56 -0.83 1.12 -0.79 71.1 -0.48 -4 -0.27 -0.71 -0.35 -0.48 59.29 C_Reuss = tensor (size: 3 3 3 3) rank: 4 tensor in Voigt matrix representation 197.91 57.92 65.4 -0.09 2.53 -4 57.92 163.68 61.84 1.13 -1.27 -0.4 65.4 61.84 188.53 1.21 0.7 -0.58 -0.09 1.13 1.21 53.39 -0.9 -0.26 2.53 -1.27 0.7 -0.9 64.34 -0.42 -4 -0.4 -0.58 -0.26 -0.42 51.7 C_Hill = tensor (size: 3 3 3 3) rank: 4 tensor in Voigt matrix representation 205.28 57.36 65.54 -0.17 2.54 -4 57.36 171.45 60.74 1.03 -1.05 -0.33 65.54 60.74 194.92 1.04 0.91 -0.64 -0.17 1.03 1.04 57.36 -0.84 -0.31 2.54 -1.05 0.91 -0.84 67.72 -0.45 -4 -0.33 -0.64 -0.31 -0.45 55.49
Note that there is a difference between the average tensors calculated directly from the EBSD data and the average tensors calculated from the
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estimated ODF. These differences result from the smoothing effect of the kernel density estimation (cf. van den Boogaart 2001). The magnitude of the difference depends on the actual choice of the kernel. It is smaller for sharper kernels, or more precisely for kernels with leading Fourier coefficients close to1. An example for a family of well-suited kernels can be found in Hielscher (2010).
Conclusions An extensive set of functions have been developed and validated for the calculation of anisotropic crystal physical properties using Cartesian tensors for the MTEX open-source MatLab toolbox. The functions can be applied to tensors of single or polycrystalline materials. The average tensors of polycrystalline and multi-phase aggregates using the Voigt, Reuss and Hill methods have been implemented using three methods: (a) the weighted summation for individual orientation data (e.g. EBSD); (b) the weighted integral of the ODF; and (c) using the Fourier coefficients of the ODF. Special attention has been paid to the crystallographic reference frame used for orientation data (e.g. Euler angles) and Cartesian tensors, as these reference frames are often different in lowsymmetry crystals and dependent on the provenance of the orientation and tensor data. The suite of MTEX functions can be used to construct projectspecific MatLab M-files and to process orientation data of any type in a coherent workflow from the texture analysis to the anisotropic physical properties. A wide range of graphical tools provides publication quality output in a number of formats. The construction of M-files for specific problems provides a problem-solving method for teaching elementary to advanced texture analysis and anisotropic physical properties. The open-source nature of this project (http://mtex.googlecode.com) allows researchers to access all the details of their calculations, check intermediate results and further the project by adding new functions on Linux, Mac OSX or Windows platforms. The authors gratefully acknowledge that this contribution results from scientific cooperation on the research project ‘Texture and Physical Properties of Rocks’ which was funded by the French-German program EGIDEPROCOPE. This bilateral program is sponsored by the German Academic Exchange Service (DAAD) with financial funds from the federal ministry of education and research (BMBF) and the French ministry of foreign affairs. We would like to dedicate this work to the late M. Casey (Leeds) who had written his own highly efficient FORTRAN code to perform pole figure inversion for lowsymmetry minerals using the spherical harmonics approach of Bunge (Casey 1981). The program source code was freely distributed by Martin to all interested
scientists since about 1979, well before today’s opensource movement. Martin requested in July 2007 that DM made his own FORTRAN code open-source; in response to that request we have extended MTEX to include physical properties in MatLab, a programming language more accessible to young scientists and current teaching practices. Finally, the authors would like to thank the editors D. J. Prior, E. H. Rutter and D. J. Tatham for the considerable work in compiling this special volume dedicated to Martin Casey, their kind consideration in accepting our late submission and their detailed comments, which improved the manuscript.
Appendix: Fourier coefficients of the rotated tensor In this section we are concerned with the Fourier coefficients of tensors, since they are required in equation (8). Previous work on this problem can be found in Jones (1985). Here we present explicit formulae for the Fourier coefficients Tˆ m1 ,...,mr (J, L, K) in terms of the tensor coefficients Tm1 ,...,mr (g). In particular, we show that the order of the Fourier expansion is bound by the rank of the tensor. Let us first consider the case of a rank-one tensor Tm . Given an orientation g [ SO(3), the rotated tensor may be expressed as Tm (g) = Tn Rmn (g) where Rij (g) is the rotation matrix corresponding to the orientation g. Since the entries of the rotation matrix R(g) are related to the generalized spherical harmonics D1ℓk (g) by Rmn (g) = D1ℓk Umℓ Unk , ⎛ √1 ⎞ i 0 − √12 i 2 ⎜ ⎟ U = ⎝ − √12 i 0 √12 i ⎠, 0
i
0
we obtain Tm (g) = Tn D1ℓk Umℓ Unk . The Fourier coefficients Tˆ m (1, ℓ, k) of Tm (g) are therefore given by Tˆ m (1, ℓ, k) = Tn Umℓ Unk . Next we switch to the case of a rank-two tensor Tm1 m2 (g). In this case we obtain Tm1 m2 (g) = Tn1 n2 Rm1 n2 (g)Rm2 n2 (g) = Tn1 n2 D1ℓ1 k1 (g)Um1 ℓ1 Un1 k1 D1ℓ2 k2 (g)Um2 ℓ2 Un2 k2 .
ANISOTROPIC PHYSICAL PROPERTIES
With the Clebsch Gordan coefficients k j1 m1 j2 m2 |JMl (cf. Varshalovich et al. 1988), we have Dℓj11 k1 (g)Dℓj22 k2 (g) =
j 1 +j2
k j1 ℓ1 j2 ℓ2 |JLlk j1 k1 j2 k2 |JKlDJLK (g)
(9)
J=0
Tm1 m2 (g) =
2
Tn1 n2 Um1 ℓ1 Un1 k1 Um2 ℓ2 Un2 k2
k1ℓ1 1ℓ2 |JLlk1k1 1k2 |JKlDJLK (g). Finally, for the Fourier coefficients of Tmm′ , we obtain Tˆ m1 m2 (J, L, K) =Tn1 n2 Um1 ℓ1 Un1 k1 Um2 ℓ2 Un2 k2 k1ℓ1 1ℓ2 |JLlk1k1 1k2 |JKl. For a third-rank tensor we have Tm1 m2 m3 (g) = Tn1 n2 n3 Rm1 n1 (g)Rm2 n2 (g)Rm3 n3 (g) = Tn1 n2 n3 D1ℓ1 k1 Um1 ℓ1 Un1 k1 D1ℓ2 k2 Um2 ℓ2 Un1 k2 D1ℓ3 k3 Um3 ℓ3 Un1 k3 . Using equation (9) we obtain Tm1 m2 m3 (g) Tn1 n2 n3 D1ℓ1 k1 Um1 ℓ1 Un1 k1 D1ℓ2 k2 Um2 ℓ2 Un1 k2 D1ℓ3 k3
Um3 ℓ3 Un1 k3 2
Tn1 n2 n3 Um1 ℓ1 Un1 k1 Um2 ℓ2 Un1 k2 Um3 ℓ3 Un1 k3
J1 =0
k1ℓ1 1ℓ2 |J1 L1 lk1k1 1k2 |J1 K1 lDJL11 K1 (g)D1ℓ3 k3 =
2 J 1 +1
Tn1 n2 n3 Um1 ℓ1 Un1 k1 Um2 ℓ2 Un1 k2 Um3 ℓ3 Un1 k3
J1 =0 J2 =0
k1ℓ1 1ℓ2 |J1 L1 lk1k1 1k2 |J1 K1 l kJ1 L1 1ℓ3 |J2 L2 lkJ1 K1 1k3 |J2 K2 lDJL22 K2 (g). The coefficients of Tm1 m2 m3 are therefore given by Tˆ m1 m2 m3 (J2 , L2 , K2 ) =
Tm1 ,m2 ,m3 ,m4 = Tn1 ,n2 ,n3 ,n4 D1m1 ,n1 D1m2 ,n2 D1m3 ,n3 D1m4 ,n4 = Tn1 ,n2 ,n3 ,n4
2 2
k1m1 1m2 |J1 M1 lk1n1 1n2 |J1 N1 l
DJM11 ,N1 k1m3 1m4 |J2 M2 lk1n3 1n4 |J2 N2 lDJM22 ,N2
J=0
=
Finally, we consider the case of a fourth-rank tensor Tm1 ,m2 ,m3 ,m4 . Here we have
J1 =0 J2 =0
and hence
=
191
2
Tn1 n2 n3 Um1 ℓ1 Un1 k1 Um2 ℓ2 Un1 k2 Um3 ℓ3 Un1 k3
J1 =J2 −1
k1ℓ1 1ℓ2 |J1 L1 lk1k1 1k2 |J1 K1 l kJ1 L1 1ℓ3 |J2 L2 lkJ1 K1 1k3 |J2 K2 l.
= Tn1 ,n2 ,n3 ,n4
4 2 2
k1m1 1m2 |J1 M1 l
J0 =0 J1 =0 J2 =0
k1n1 1n2 |J1 N1 lk1m3 1m4 |J2 M2 lk1n3 1n4 |J2 N2 l kJ1 M1 J2 M2 |J0 M0 lkJ1 N1 J2 N2 |J0 N0 lDJM00 ,N0 and hence Tˆ m1 ,m2 ,m3 ,m4 (J0 , M0 , N0 ) =
2 2
k1m1 1m2 |J1 M1 lk1n1 1n2 |J1 N1 l
J1 =0 J2 =0
k1m3 1m4 |J2 M2 lk1n3 1n4 |J2 N2 l kJ1 M1 J2 M2 |J0 M0 lkJ1 N1 J2 N2 |J0 N0 l.
References Aleksandrov, K. S., Alchikov, U. V., Belikov, B. P., Zaslavskii, B. I. & Krupnyi, A. I. 1974. Velocities of elastic waves in minerals at atmospheric pressure and increasing precision of elastic constants by means of EVM (in Russian). Izvestiya of the Academy of the Sciences of the USSR, Geologic Series, 10, 15–24. Bachmann, F., Hielscher, H., Jupp, P. E., Pantleon, W., Schaeben, H. & Wegert, E. 2010. Inferential statistics of electron backscatter diffraction data from within individual crystalline grains. Journal of Applied Crystallography, 43, 1338–1355. Bass, J. D. 1995. Elastic properties of minerals, melts, and glasses. In: Ahrens, T. J. (ed.) Handbook of Physical Constants. American Geophysical Union, Special Publication, 45–63. Bezacier, L., Reynard, B., Bass, J. D., Wang, J. & Mainprice, D. 2010. Elasticity of glaucophane and seismic properties of high-pressure low-temperature oceanic rocks in subduction zones. Tectonophysics, 494, 201–210. ¨ ber die elastischen konstanten Bunge, H.-J. 1968. U kubischer materialien mit beliebiger textur. Kristall und Technik, 3, 431– 438. Casey, M. 1981. Numerical analysis of x-ray texture data: an implementation in FORTRAN allowing triclinic or axial specimen symmetry and most crystal symmetries. Tectonophysics, 78, 51–64. ¨ ber die Fortpflanzung van Christoffel, E. B. 1877. U Sto¨ssen durch elastische feste Ko¨rper. Annali di Matematica pura ed applicata, Serie II, 8, 193– 243.
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Fei, Y. 1995. Thermal expansion. In: Ahrens, T. J. (ed.) Minerals Physics and Crystallography: A Handbook of Physical Constants. American Geophysical Union, Washington, DC, 29–44. Ganster, J. & Geiss, D. 1985. Polycrystalline simple average of mechanical properties in the general (triclinic) case. Physica Status Solidi (B), 132, 395– 407. Hielscher, R. 2010. Kernel density estimation on the rotation group, Preprint, Fakulta¨t fu¨r Mathematik, TU Chemnitz. http://www.tu-chemnitz.de/mathematik/ preprint/2010/PREPRINT_07.php. Hielscher, R. & Schaeben, H. 2008. A novel pole figure inversion method: specification of the MTEX algorithm. Journal of Applied Crystallography, 41, 1024–1037, doi: 10.1107/S0021889808030112. Hielscher, R., Schaeben, H. & Siemes, H. 2010. Orientation distribution within a single hematite crystal. Mathematical Geosciences, 42, 359–375. Hill, R. 1952. The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society A, 65, 349– 354. Hofer, M. & Schilling, F. R. 2002. Heat transfer in quartz, orthoclase, and sanidine. Physics and Chemistry of Minerals, 29, 571– 584. Humbert, M. & Diz, J. 1991. Some practical features for calculating the polycrystalline elastic properties from texture. Journal of Applied Crystallography, 24, 978– 981. Isaak, D. G. 2001. Elastic properties of minerals and planetary objects. In: Levy, M., Bass, H. & Stern, R. (eds) Handbook of Elastic Properties of Solids, Liquids, and Gases, Volume III: Elastic Properties of Solids: Biological and Organic Materials, Earth and Marine Sciences. Academic Press, New York, 325– 376. Jones, M. N. 1985. Spherical Harmonics and Tensors for Classical Field Theory. Research Studies Press, Letchworth, England. 230. ¨ ber die berechnung der elastizita¨tsmoKneer, G. 1965. U duln vielkristalliner aggregate mit texture. Physica Status Solidi, 9, 825 –838. Mainprice, D. 1990. An efficient FORTRAN program to calculate seismic anisotropy from the lattice preferred
orientation of minerals. Computers & Geosciences, 16, 385–393. Mainprice, D. 2007. Seismic anisotropy of the deep Earth from a mineral and rock physics perspective. In: Schubert, G. (ed.) Treatise in Geophysics. Elsevier, Oxford, 2, 437– 492. Mainprice, D. & Humbert, M. 1994. Methods of calculating petrophysical properties from lattice preferred orientation data. Surveys in Geophysics, 15, 575–592. Mainprice, D., Le Page, Y., Rodgers, J. & Jouanna, P. 2008. Ab initio elastic properties of talc from 0 to 12 GPa: interpretation of seismic velocities at mantle pressures and prediction of auxetic behaviour at low pressure. Earth and Planetary Science Letters, 274, 327–338, doi: 10.1016/j.epsl.2008.07.047. Morris, P. R. 2006. Polycrystalline elastic constants for triclinic crystal and physical symmetry. Journal of Applied Crystallography, 39, 502– 508. Nye, J. F. 1985. Physical Properties of Crystals: Their Representation by Tensors and Matrices. 2nd edn., Oxford University Press, England. Reuss, A. 1929. Berechnung der Fließrenze von Mischkristallen auf Grund der Plastizita¨tsbedingung fu¨r Einkristalle. Zeitschrift fu¨r angewandte Physik, 9, 49–58. ¨ ber Zusammenha¨nge zwischen Sander, B. 1911. U Teilbewegung und Gefu¨ge in Gesteinen. Tschermaks Mineralogische und Petrographische Mitteilungen, 30, 381– 384. Sirotin, Yu. I. & Shakolskaya, M. P. 1982. Fundamentals of Crystal Physics. Mir, Moscow, 654. van den Boogaart, K. G. 2001. Statistics for individual crystallographic orientation measurements. PhD Thesis, Shaker, Freiburg University of Mining & Technology. Varshalovich, D. A., Moskalev, A. N. & Khersonskii, V. K. 1988. Quantum Theory of Angular Momentum. World Scientific Publishing Co., Singapore. Voigt, W. 1887. Theoretische studien u¨ber die elastizita¨tsverha¨ltnisse. Abhandlungen der Akademie der Wissenschaften in Go¨ttingen, 34, 48–55. Voigt, W. 1928. Lehrbuch der Kristallphysik. TeubnerVerlag, Leipzig.
The microstructural and rheological evolution of shear zones NICHOLAS J. AUSTIN1,2 1
Department of Earth Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2
Present address: Imperial Oil Resources, 237 4th Ave SW, Calgary, Alberta, Canada (e-mail:
[email protected]) Abstract: Evidence of localized strain is ubiquitous in deformed lithospheric rocks. Recent advances in laboratory deformation techniques, including the use of torsion experiments, have enabled the coupling of microstructural and rheological evolution to be investigated in experiments run to strains approaching those reached in many natural shear zones. Further, the increased use of electron backscatter diffraction to quantify crystallographic preferred orientation (CPO) has significantly increased understanding of CPO formation and evolution. Combined, these laboratory and field observations support the assertion that a rock’s microstructure is strongly linked to its rheology. However, complete quantification of the coupling between microstructure and rheology is complicated by the fact that rocks have inherently complex microstructures. This paper reviews recent work focused on quantifying the rates of microstructural evolution and the attainment of steady state for two key microstructural parameters: grain size and crystallographic preferred orientation. Theoretical considerations, laboratory measurements and field observations suggest that a full description of all relevant microstructural parameters, and the appropriate evolution equations for these parameters, may be needed to link microstructural and rheological evolution and therefore to quantify the bulk rheology of the lithosphere.
The localization of strain is ubiquitous within the lithosphere. For well over a hundred years, structural geologists have recognized that significant displacements are accommodated in shear zones with widths of centimetres to metres (Heim 1878; Bertrand 1884; Peach & Horne 1884; Boullier & Quenardel 1981; Milton & Williams 1981; Harris et al. 1983; Siddans 1983; Groshong Jr et al. 1984). More recently, geodetic data have shown that, globally, tectonic strains are accommodated in relatively few discrete zones (Kreemer et al. 2000, 2003). Cumulatively, these observations suggest that the mechanics of shear zones and the mechanisms controlling the distribution of strain in the lithosphere are important elements of geodynamics. However, the mechanisms of strain localization are still being debated. It is possible (perhaps probable) that many shear zones, across a wide range of length scales, are geometrically constrained or are seeded in zones of pre-existing weakness (Mancktelow & Pennacchioni 2005; Kelemen & Hirth 2007; Newman & Drury 2010). However, theoretical considerations do suggest that, under most circumstances, system weakening is necessary for strain localization to occur (Poirier 1980; Hobbs et al. 1990; Burg 1999; Monte´si & Zuber 2002). Potential candidates for weakening mechanisms include: changes in the geometry of the shear zone; variations in loading path (e.g. constant stress versus constant displacement rate); local changes in the pressure or temperature conditions;
or material softening. During non-dilatant deformation, in the absence of significant geometric effects material (rheological) weakening is required (Hobbs et al. 1990). Broadly, the goal of experimental, theoretical and field geologists alike is to describe quantitatively all material properties, external thermodynamic conditions and microstructural characteristics, and to incorporate the temporal and spatial variation of these parameters into theoretically derived flow laws so that shear-zone rheology may be accurately predicted and modelled, and so that all aspects of shear localization may be investigated quantitatively (Evans 2005). The coupling between microstructure, the deformation conditions and rheology must be understood in order to interpret the mechanics of natural deformation. Significant progress has been made in recent years in the quantification of two key microstructural parameters: grain size and crystallographic preferred orientation (CPO), and in defining the appropriate evolution equations. In this article, I will attempt to assess the current state of knowledge regarding the kinetics of grain size and CPO evolution in monomineralic rocks, and the coupling between these and the rheology of shear zones.
State variable evolution and rheology The rheology of a given rock mass is dependent on the external thermodynamic conditions (e.g.
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 193– 209. DOI: 10.1144/SP360.11 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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pressure, temperature, fluid activity, etc.), the mechanical properties of the mineral constituents and the microstructural or textural characteristics of the mineral framework. Common flow laws incorporate both external thermodynamic conditions (e.g. stress, pressure, temperature, fluid activity) and material properties (e.g. elastic moduli, activation energy, Burgers vector, unit cell volume, diffusivity) (Poirier 1985; Evans & Kohlstedt 1995; Evans 2005). However, a detailed description of microstructural and/or textural characteristics is seldom taken into account. The one exception is grain size, which is incorporated in flow laws for diffusion creep (Poirier 1985; Evans & Kohlstedt 1995) and some dislocation creep mechanisms (Walker et al. 1990; Goldsby & Kohlstedt 1997, 2001; Renner et al. 2002; Hirth & Kohlstedt 2003). Grain size is usually incorporated into the flow law as the mean grain size at a specific interval in time (Poirier 1985; Evans & Kohlstedt 1995); however some authors have accounted for grain-size distributions (Freeman & Ferguson 1986; Ter Heege et al. 2002, 2004; Herwegh et al. 2005) and have incorporated grain-size evolution relationships into models of shear localization and shear-zone evolution (Kameyama et al. 1997; Braun et al. 1999; Monte´si & Zuber 2002; Hall & Parmentier 2003; Monte´si & Hirth 2003; Kelemen & Hirth 2007). Other authors have accounted for the evolution of additional microstructural and/or textural characteristics empirically, by incorporating a strain term into conventional flow laws (Luton & Sellars 1969; Rutter 1999; Barnhoorn et al. 2004). In order to quantify rheological weakening, a quantitative description of rock (micro)structure is required. These internal state variables need to be included in the appropriate flow laws, along with the relevant material properties and external thermodynamic conditions (Covey-Crump 1994, 1998; Stouffer & Dame 1996; Stone et al. 2004; Evans 2005). Microstructural characteristics (internal state variables) can be related to rheology and finally to finite strain through three sets of equations (Stouffer & Dame 1996; Evans 2005): (i) evolution equations which describe the evolution of internal state variables as a function of external thermodynamic conditions and the internal state of the material (equation 1); (ii) kinetic equations which relate all necessary material state variables and external thermodynamic parameters to the rate of deformation (equation 2); and (iii) a kinematic equation which relates finite strain to the deformation path and instantaneous strain rates (equation 3):
z˙ m = f (˙1, s, T, P, fp , zr , . . . )
(1)
1˙ = f (s, T, P, fp , zr , . . . ) 1T = 1EL + 1˙ dt
(2) (3)
(after Evans 2005; see Table 1 for notation definitions). Internal state variables may be described using either implicit or explicit state variables (CoveyCrump 1994, 1997a, 1998; Stouffer & Dame 1996; Stone et al. 2004; Evans 2005). Implicit internal state variables describe, empirically, some macroscopic aspect of a rock’s structure. For example, the hardness parameter (Hart 1976) has been used to describe the mechanical evolution of calcite (CoveyCrump 1994, 1997a, 1998) and halite (Stone et al. 2004). Stone et al. (2004) further argued that the hardness parameter depends on the mean subgrain size. Explicit state variables require some aspect of the microstructure to be quantified, and thus may be better suited to direct comparison between deformation in the laboratory and under natural conditions (Evans 2005). The most commonly investigated explicit state variables include: subgrain structure (e.g. Poirier & Nicolas 1975; White 1979; Karato et al. 1980; Stone et al. 2004); dislocation density/structure (e.g. White 1973; Goetze & Kohlstedt 1977; de Bresser 1996); CPO (e.g. Prior et al. 1999; Bystricky et al. 2000; Pieri et al. 2001b); and grain size (e.g. Karato et al. 1980; Schmid et al. 1980; Karato 1989; van der Wal et al. 1993; Rutter 1995; Stipp et al. 2001, 2010; Stipp & Tullis 2003; Rutter & Brodie 2004), although many other microstructural characteristics such as grain-boundary morphology (e.g. Nishikawa et al. 2004) may be considered. This paper will focus on the evolution equations for grain size and CPO and the impact of coupling these evolution relationships with the kinetic equations used to describe creep processes. The importance of evolution equations has been articulated well by Shimizu (2008) who noted: ‘Studies of the evolution of grain size from transient to steady states would be required to understand the gradual change in grain size that occurs from the margins of protomylonites to the centers of associated ultramylonite zones’. Evolution equations are also required to understand deformation that occurs under non-steady-state conditions. Examples will be drawn from both laboratory and field studies on two well-studied minerals: calcite and olivine.
Grain-size evolution relationships During creep deformation in single-phase materials, grain size evolves through the process of dynamic recrystallization (Urai et al. 1986; Drury et al.
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Table 1. List of symbols and subscripts Description Symbol z ds do d s J d* M F b A 1˙ m D p K Q R T Tm P fp 1c 1o R g l b c
State variable Stabilized grain size Initial grain size Present grain size Differential stress Constant relating to subgrain distribution and material properties Limiting subgrain size Grain-boundary mobility ¼ Moexp(2Qm/RT) Driving force for grain-boundary migration Burgers vector Constant Strain rate Shear modulus Diffusivity ¼ Doexp(2QD/RT) Grain growth exponent Grain growth pre-exponential Activation enthalpy Gas constant (8.314 J K21 mol21) Temperature (K) Melting temperature Pressure (MPa) Fugacity Critical strain for microstructural evolution Strain at peak stress Empirical strain weakening exponent Grain-boundary energy Proportion of the energy associated with dislocation creep stored in the microstructure Fraction of the total mechanical work rate accommodated by dislocation creep Geometric constant
Subscript gb v dis p dif tot
Grain-boundary diffusion Volume diffusion Dislocation creep Peierls creep Diffusion creep Total
1989; Drury & Urai 1990; Humphreys & Hatherly 1995) whereby new grains nucleate and grow, resulting in the rearrangement of grain boundaries and the modification of material microstructure. Many studies have observed that the microstructure resulting from dynamic recrystallization is directly linked to the external thermodynamic conditions during deformation (Twiss 1977; Guillope & Poirier 1979; Karato et al. 1980; Schmid et al. 1980; Tullis & Yund 1985; van der Wal et al. 1993; Rutter 1995; de Bresser et al. 1998; Stipp et al. 2001, 2006, 2010; Ter Heege et al. 2002, 2004, 2005; Stipp & Tullis 2003; Drury 2005; Austin & Evans 2009). In its simplest form, this may be expressed as an empirical inverse relationship between grain size and flow stress, a piezometer, that has been supported by laboratory studies on metals (e.g. Luton & Sellars 1969;
Bromley & Sellars 1973; Twiss 1977), calcite (Schmid et al. 1980; Rutter 1995; Barnhoorn et al. 2004), quartz (Stipp & Tullis 2003) and olivine (Karato et al. 1980, van der Wal et al. 1993) (Fig. 1): −k ds s = Ao b m
(4)
(see Table 1 for notation definition). Extensive work has been carried out to develop explanations for this empirical relationship (Twiss 1977; Derby & Ashby 1987; Derby 1990, 1991, 1992; de Bresser et al. 1998, 2001; Shimizu 1998, 2008; Hall & Parmentier 2003; Austin & Evans 2007, 2009); the resulting equations are listed in Table 2 and shown in Figure 1. What is striking about the recrystallized grain-size relationships
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(b) 10-4 s-1 10-6 s-1 10-8 s-1
Dislocation Creep
10
n Cre
ep
10-10 s-1
1
10-11 s-1
10
Paleowattmeter (AE07) Field Boundary (dB1998) Twiss (1977) Hall and Parmentier (2003) Schmid et al. (1980) Rutter (1995SG) Rutter (1995MIG) DA1987 (relative) Sh1998IC (relative) Sh1998GB (relative)
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10-2 s-1
Dislocation Creep
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1
Diffu sio
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Calcite: 1023K
10-4 s-1
ep
Paleowattmeter (AE07) Field Boundary (dB1998) Twiss (1977) Hall and Parmentier (2003) Schmid et al. (1980) Rutter (1995SG) Rutter (1995MIG) DA1987 (relative) Sh1998IC (relative) Sh1998GB (relative)
n Cre
10-2 s-1
Diffu sio
Calcite: 673K
Stress (MPa)
(a)
10-5 s-1 10-6 s-1
10-12 s-1
10
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0
10
1
10
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(c)
(d)
Olivine: 1173K 10-8 s-1
10-6 s-1
10-10 s-1 Dislocation Creep
ary und Bo g ain lidin S
10-2 s-1
Gr
10-4 s-1
Dislocation Creep
10
10-12 s-1
2
10-6 s-1
10
10
1
Paleowattmeter (AE07) Field Boundary (dB1998) Twiss (1977) Hall and Parmentier (2003) Van der Wal et al. (1993) DA1987 (relative) Sh1998IC (relative) Sh1998GB (relative)
0
10
0
10
1
10
2
10
3
Grain Size (µm)
p Cree sion
10-8 s-1
Diffu
10-14 s-1
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Diffu
sion
Stress (MPa)
Cree
p
10
3
10-5 s-1
10-11 s-1 2
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y
ar und Bo ng i ain Gr Slid
2
Grain Size (µm)
Grain Size (µm)
10
10
1
Paleowattmeter (AE07) Field Boundary (dB1998) Twiss (1977) Hall and Parmentier (2003) Van der Wal et al. (1993) DA1987 (relative) Sh1998IC (relative) Sh1998GB (relative)
0
10
0
10
1
10-10 s-1
10-12s-1 10
2
10
3
Grain Size (µm)
Fig. 1. Deformation mechanism maps for (a, b) Calcite (T ¼ 673 and 1023 K, respectively); (c, d) olivine (T ¼ 1173 and 1573 K respectively; modified after Austin & Evans (2009) showing the stabilized grain sizes predicted by the various grain-size evolution relationships listed in Table 2, as well as those predicted by empirical piezometers. Temperatures were chosen to reflect common temperatures of deformation in both natural shear zones and in the laboratory. For the palaeowattmeter, the same values for each parameter are used as by Austin & Evans (2007) (l ¼ 0.1, c ¼ p, and g ¼ 1 J m22). The field boundary is the stable grain size predicted by de Bresser et al. (1998, 2001) (dB1998) (Table 2). The Derby & Ashby (1987) (DA1987), Shimizu (1998) intracrystalline nucleation (Sh1998IC) and Shimizu (1998) grain-boundary nucleation (Sh1998GB) relationships (Table 2) are shown in terms of their apparent k values. The prediction from Twiss (1977) (equation (4)) along with empirically calibrated stress– grain-size relationships are also shown for each mineral: (a, b) calcite (Schmid et al. 1980; Rutter 1995), (c, d) olivine (Van der Wal et al. 1993). Flow laws used to construct the deformation mechanism maps and to calculate the strain rate for each stress and grain size are: (a, b) dislocation creep (Renner et al. 2002) and diffusion creep (Herwegh et al. 2003), (c, d) Hirth & Kohlstedt (2003) for dry dislocation creep, dry grain-boundary sliding and dry diffusion creep. All flow laws are from compression experiments run to equivalent strains ,0.5.
shown in Figure 1 is that, despite substantially different formulations, they result in similar apparent k values. However, these relationships predict different sensitivities of the stabilized grain size to temperature (different apparent activation energies) as a result of variations in the relative contributions from a range of thermally activated processes including dislocation creep, diffusion creep and grain growth/grainboundary migration. This is illustrated by the change in the relative positions of the stabilized
grain-size relationships with changing temperature (e.g. Fig. 1a versus b for calcite at 673 K versus 1073 K, respectively). Further, despite the constant apparent k values, the stresses calculated for a given grain size by the different relationships are disparate by more than an order of magnitude (Fig. 1). At this juncture, it is important to note that when analysing grain-size data from laboratory experiments and naturally deformed rocks, both measurement uncertainties and potential differences that
MICROSTRUCTURE AND RHEOLOGY
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Table 2. Relationships for stabilized grain size shown in Figure 1 Label (Fig. 1)
Grain-size scaling relationship −k ds s = Ao m b
Mechanistic relationships Jd∗ MF DA1987 ds2 = 1˙
SH1998IC/GB
−k 1/j Dgb ds s = A2 m b Dv
Comments
Reference
Empirical relationship between flow stress and stabilized recrystallized grain size – Piezometer
Twiss (1977)
Stabilized grain size for bulge nucleation
Derby (1990, 1991, 1992), Derby & Ashby (1987) Shimizu (1998)
Stabilized grain size for subgrain rotation nucleation
Scaling relationships dB1998
AE07
−(n − 1) Qdis − Qdif Stabilized grain size sits at the exp ds = X s m mRT boundary between dislocation and diffusion creep – Field Boundary Hypothesis (asynchronous) 1 Adif m X= Cf Adis 1˙ dif Cf = 1˙ dis −Qg −1 p 1c Kg exp RT Grain-size reduction is statistically dsp = 1˙ more likely for high strain rate and coarse grains; grain growth follows normal growth kinetics (synchronous) −Qg −1 Kg exp p cg RT ds1+p = Grain-size reduction driven by bls1˙ tot dissipation of mechanical work; grain growth follows normal growth kinetics – Palaeowattmeter (synchronous) 1˙ dis b= 1˙ tot
may arise due to measurement techniques must be considered. In particular, some studies apply a stereological correction to linear intercept measurements (e.g. Olgaard & Evans 1986); other studies report grain size from linear intercept measurements with no stereological correction (e.g. Rutter 1995; Austin & Evans 2009). Several studies (e.g. Rutter et al. 1994; Austin & Evans 2009) showed that grain sizes from linear intercept measurements, without any corrections, are equivalent to grain diameters measured from grain area. Equivalent circular diameters, obtained from measured crosssectional areas in 2D images, have also been reported as both a number-weighted average and an areaweighted average (e.g. Herwegh et al. 2005).
De Bresser et al. (1998), de Bresser et al. (2001b)
Hall & Parmentier (2003)
Austin & Evans (2007), Austin & Evans (2009)
Notwithstanding measurement uncertainty, some of the variability observed in Figure 1 stems from the two types of formulations: mechanistic relationships and scaling relationships. Mechanistic relationships (Derby & Ashby 1987; Derby 1990, 1991, 1992; Shimizu 1998, 2008) are derived for specific mechanisms of dynamic recrystallization. Two primary mechanisms have been proposed by which new grains may nucleate during dynamic recrystallization: grain-boundary migration nucleation (GBM: discontinuous dynamic recrystallization which is also termed bulge nucleation or BLG) and subgrain rotation nucleation (SGR: continuous dynamic recrystallization). These occur under different external thermodyamic conditions (Guillope
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& Poirier 1979; Drury & Urai 1990; Rutter 1995; Bestmann & Prior 2003; Shimizu 2008). Grainboundary migration (bulge) nucleation occurs when a portion of a migrating grain boundary, driven by strain energy associated with the presence of dislocations within the parent grain, is pinned, either by impurities, other phases or dislocations/ subgrain boundaries. The grain boundary becomes bowed and, through local rotation of the crystal lattice, may lead to the formation of a new strain-free grain which then grows due to driving forces associated with both surface energy and internal strain energy (Drury & Urai 1990). Derby & Ashby (1987; see also Derby 1990, 1991, 1992) presented a model to explain the stabilized grain size that results from dynamic recrystallization by bulge nucleation, based on the premise that a steady-state grain size occurs when the rate of nucleation is balanced by the rate of grain-boundary migration (Table 2). Bulge nucleation is usually observed under low-temperature and/or high-stress conditions (Guillope & Poirier 1979; Stipp et al. 2010). Subgrain rotation recrystallization occurs when progressive lattice misorientation between subgrains produces a high-angle grain boundary (.10–158; White 1977; Guillope & Poirier 1979). Unlike bulge nucleation, which is constrained to occur on existing grain boundaries, subgrain rotation nucleation may occur either on grain boundaries or in grain interiors. Shimizu (1998, 2008) has developed models to predict the stabilized grain size from subgrain rotation recrystallization for both grain-boundary nucleation and nucleation within grain interiors. As with the Derby and Ashby model (Derby & Ashby 1987; Derby 1990, 1991, 1992), the Shimizu model (Shimizu 1998, 2008) predicts the stabilized grain size to occur when the rate of nucleation is balanced by the rate of grain-boundary migration (Table 2). At high temperature, or with large enough driving forces for grain-boundary migration, grain boundaries transition from pinned to free (Guillope & Poirier 1979). Under these conditions, grain boundaries rapidly sweep entire grains and subgrain rotation ceases to be an effective recrystallization mechanism. High-temperature grain-boundary migration recrystallization has been observed experimentally in calcite (Rutter 1995) and halite (Guillope & Poirier 1979); in both cases it was suggested that a different relationship between stress and grain size is required for this mechanism compared to subgrain rotation recrystallization. Based on analysis of quartzites deformed in natural shear zones under a wide range of external thermodynamic conditions, Stipp et al. (2010) argued recently that bulge nucleation, subgrain rotation nucleation and grain-boundary migration dominated recyrstallization may all require different calibrations for equation (4).
On the other hand, scaling relationships (de Bresser et al. 1998, 2001; Hall & Parmentier 2003; Monte´si & Hirth 2003; Austin & Evans 2007, 2009) explain the stabilized grain size that results from dynamic recrystallization through competing grain growth and grain-size reduction processes. Unlike the models described above, which are based on a microphysical description of specific recyrstallization processes, these scaling relationships rely on statistical or thermodynamic relationships to describe the recrystallized grain size. These relationships have the benefit of simultaneously describing the kinetics of the evolution of mean grain size and the resulting stabilized grain size; however, they do not explicitly tie these kinetic equations to microphysical models. One class, which has been termed either asynchronous or discontinuous, assumes that the stabilized grain size is determined either by the piezometer (equation (4)) (Monte´si & Hirth 2003) or by the boundary between dislocation creep and diffusion creep (de Bresser et al. 1998, 2001). When the grain size is too heavily weighted towards finer grains, the mean grain size will grow back towards the stabilized grain size. Alternatively, when the grain size is too heavily weighted towards coarse grains, the mean grain size will be reduced. The second class of scaling relationships, termed synchronous or continuous relationships, assumes that grain-size evolution is governed by competing grain growth and grain-size reduction processes and that the stabilized grain size occurs when grain growth and grain-size reduction rates are equal (Table 2; Hall & Parmentier 2003; Austin & Evans 2007, 2009). The fundamental difference between the various synchronous relationships is the equation used to describe grain-size reduction kinetics. Hall & Parmentier (2003) follow Derby (1990) in suggesting that grain-size reduction is a statistical process that is more likely for coarser grain sizes and for higher strain rates. We therefore have: −˙1 d . d˙ = 1c
(5)
Other authors (Kameyama et al. 1997; Braun et al. 1999; Monte´si & Zuber 2002; Monte´si & Hirth 2003) have proposed that the grain size on the righthand side of equation (5) might refer to the difference between the current grain size and the stable grain size, so that −˙1(d − ds ) . d˙ = 1c
(6)
MICROSTRUCTURE AND RHEOLOGY
Austin & Evans (2007, 2009) proposed that the rate of grain-size reduction is related to the rate at which mechanical work is done and the ability of a rock to either store or dissipate this energy. Based on this, they derived the grain-size reduction rate as: −bls1˙ d d˙ red = cg
2
(7)
(see Austin & Evans 2007, 2009 for full derivation). Laboratory measurements (Ter Heege et al. 2002; Austin & Evans 2009) of the grain-size reduction rate during high-temperature deformation of calcite support equation (7). In all of these scaling relationships, the grain growth kinetics are assumed to be the same as under static conditions: −Qg −1 1−p p d . d˙ = Kg exp RT
(8)
Experimental studies indicate that grain growth kinetics during diffusion creep are indistinguishable from those under static conditions in calcite and olivine (Karato et al. 1980; Walker et al. 1990; Austin & Evans 2009). However, Kellermann Slotemaker (2006) and Hiraga et al. (2010) recently suggested that, in single-phase and polyphase olivine aggregates respectively, grain growth might be enhanced during diffusion creep and/or superplastic deformation due to the geometrical relationship between grains (Sato et al. 1990). An added driving force for grain-boundary migration may also come from the internal strain energy associated with the presence of dislocations. For example, Shimizu (2008) argued that during dislocation creep the grain growth rate might be described by: −Qg . d˙ / s j exp RT
(9)
If grain growth is driven by surface energy minimization (Evans et al. 2001), larger grains will grow at the expense of finer grains; if grain growth kinetics are driven by internal strain energy, smaller grains will usually grow at the expense of larger grains (Shimizu 2008). This has implications for both the evolution of mean grain size and for changes to the grain-size distribution. Other driving forces, including those related to chemical potential gradients (Hay & Evans 1987), might also alter the grain-size evolution rate during dislocation creep deformation. Thus, in a single-phase material, equation (8) probably
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provides a minimum bound on the rate of grain growth during deformation accommodated by dislocation creep. The two classes of scaling relationships (Table 2) provide a range of kinetic equations (e.g. Equations (5–9)) which may be used to describe the evolution of mean grain size during creep deformation. However, these scaling relationships do not explicitly describe the microphysical processes; rather, they are based on statistical or thermodynamic concepts (e.g. conservation of energy; Austin & Evans 2007, 2009). Additional work is required to couple microphysical models of dynamic recrystallization with kinetic equations within thermodynamic constraints, in order to quantify grain-size evolution and the stabilization of grain size in response to the external thermodynamic conditions. This will require further experimental work, particularly targeting the kinetics of evolution of both mean grain size and grain-size distributions in many minerals under a range of thermodynamic conditions. These laboratory observations must be coupled with careful field measurements to constrain our ability to extrapolate between the laboratory and natural time and length scales.
Crystallographic preferred orientation It is well documented that materials deformed by dislocation creep develop a CPO with progressive strain (Wenk et al. 1973; Rutter et al. 1994; Casey et al. 1998; Bystricky et al. 2000; Pieri et al. 2001b; Barnhoorn et al. 2004; Barber et al. 2007; Skemer et al. 2011). Numerous models have been developed to describe both the rate of change of CPO during deformation and the characteristics of the CPO that develops and evolves. End-member models postulate that, during simple shear, grains rotate such that the easy slip system will either rapidly align with the shear plane or rotate in parallel with the long axis of the finite strain ellipse (McKenzie 1979; Ribe 1992). However, Warren et al. (2008) suggested that laboratory and field observations of deformed olivine were not consistent with either of these end-member models. Further, these end-member models do not explicitly predict the characteristics of the CPO that develops during creep deformation. It is likely that a full description of the probability distribution function of crystallographic orientations is required to link CPO to rheology (Evans 2005). Viscoplastic self-consistent (VPSC) models have often been used to simulate the formation of CPO in calcite rocks (Lebensohn & Tome 1993; Lebensohn et al. 1998; Wenk 1999; Pieri et al. 2001b; Barber et al. 2007) and in olivine (Wenk & Tome 1999; Tommasi et al. 2000). The premise of
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(a)
100
1 0.8
1
1.2
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n ow 1 kd sea -11 Br :10 aw ter r L me we att Po W &
10
Calcite n wn ow 1 do -1 kd s eak 1 s ea -5 Br -1 :10 Br 10 aw r w er: r L ete n m we La et r ow 1 Po iezo tm P kd swe at & ea -5 Po W & Br r:10 aw ete r L om we ez Po Pi &
Grain Size (µm)
1000
1.6
1.8
2
2.2
2.4
1000/T (1/K)
(b) 1000
Olivine
&
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Pi Po ez we om r ete Law r:1 0 -5 s -1
w -5 s La 0 er :1 w ter Po tme at W
1
1 0.6
&
&
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11
1
P ez ow om er ete Law r:1 0-
Pi
100
w 11 s La 0er :1 w ter Po tme at W
&
0.7
0.8
0.9
1
1000/T (1/K)
Fig. 2. Plots showing the relationship between stabilized grain size and temperature for (a) calcite and (b) olivine. Curves are calculated for strain rates of 10211 and 1025 s21 for both minerals. (a) The power-law breakdown equation is from Renner et al. (2002), the wattmeter is after Austin & Evans (2007, 2009) and the piezometer is after Schmid et al. (1980). The wattmeter parameters are after Austin & Evans (2009), using grain growth parameters from (Covey-Crump 1997b). (b) The power-law creep equation for dry olivine is from Hirth & Kohlstedt (2003). The wattmeter parameters are after Austin & Evans (2009), using modified grain growth parameters from Karato et al. (1980) (see Austin & Evans 2007) and the piezometer is after Van der Wal et al. (1993).
such models is that intracrystalline deformation is accommodated by slip on a limited number of discrete planes, with defined Burgers vectors. Two end-member model types define rigid bounds in strength: Taylor (iso-strain) and Sachs (iso-stress) models. Taylor models (e.g. Taylor 1938) are upper bounds for strength, in which all grains undergo the same deformation and grain boundaries do not slip. For iso-volumetric deformation, slip on five independent systems is required (von Mises criterion). Under specific external thermodynamic conditions and in the absence of e-twinning, computations using the Taylor formulation are consistent with measured crystallographic orientations from calcite rocks deformed in the laboratory (Lister
1978; Rutter et al. 1994). The lower-bound (Sachs models) (e.g. Chastel et al. 1993) require fewer slip systems but do not require displacement continuity across the grain boundaries. Consequently, grains oriented for easy slip will be preferentially deformed. Intermediate models between Taylor and Sachs have also been investigated (Molinari et al. 1987; Wenk et al. 1991; Lebensohn & Tome 1993; Lebensohn et al. 1998; Warren et al. 2008). The results of these studies suggest that relaxation of the requirement for displacement continuity across grain boundaries improves the relationship between calculated and observed CPOs. Other authors have attempted to incorporate dynamic recrystallization into calculations of CPO (Lebensohn et al. 1998; Wenk & Tome 1999; Kaminski & Ribe 2001; Pieri et al. 2001b; Blackman & Kendall 2002; Blackman et al. 2002), in some cases improving the ability of models to predict the CPO characteristics and intensity observed in samples deformed in the laboratory. However, these models do not explicitly account for the (micro)physics of dynamic recrystallization; to date, the recrystallization and grain-size evolution relationships discussed above have not been directly coupled with models of CPO formation and evolution. All theoretical models designed to predict CPO formation and evolution are constrained by the input data, including relative slip-system activity and strength and the effects of recrystallization. For example, in calcite, the relative importance of rk1210l+ , ck1210l+ and rk1210l+ (Lister 1978; Pieri et al. 2001b; Barber et al. 2007; de Bresser et al. 2008) are still being studied. The relative strength of these slip systems and the onset of etwinning at low temperature and/or for coarse grain size (Casey et al. 1978) have been shown to significantly impact the nature of the CPO that is produced (Pieri et al. 2001b; Barber et al. 2007) and may also have implications for bulk rheology.
Coupled microstructural and rheological evolution Grain size Once equations describing the stabilized mean grain size and the kinetics of evolution of mean grain size during creep deformation have been derived and supported by laboratory experiments, they may be coupled with existing creep equations in order to quantify the stabilized grain size that would exist at different temperatures within the Earth and the impact of grain-size evolution on the bulk rheology of a given monomineralic rock mass (Fig. 2). Two steady-state examples are considered: a dislocation
MICROSTRUCTURE AND RHEOLOGY
creep flow law coupled with (i) the palaeowattmeter (Austin & Evans 2007, 2009) and (ii) a piezometer (equation (4)). In Figure 2 examples are shown for both calcite and olivine at strain rates of 1 × 1025 and 1 × 10211 s21, representative of laboratory and tectonic strain rates respectively. For calcite (Fig. 2a), the Peierls flow law from Renner et al. (2002) is coupled with either the palaeowattmeter (equation AE07 in Table 2) or the piezometer from Schmid et al. (1980). For olivine (Fig. 2b), the dry power-law creep equation from Hirth & Kohlstedt (2003) is coupled with either the palaeowattmeter or the piezometer from Van der Wal et al. (1993). These indicate that, at constant strain rate, there is a systematic relationship between recrystallized grain size and temperature (Fig. 2), which is consistent with observations from naturally deformed calcite (de Bresser et al. 2002) and quartz (Hirth et al. 2001). Similarly, rocks deforming at different strain rates should be distinguishable on plots of grain size versus temperature as they will have similar slopes but different intercepts, as is seen when curves for a strain rate of 1 × 1025 s21 are compared to those for a strain rate of 1 × 10211 s21. However, uncertainties in temperature estimation using geothermometry, in grain-size measurement in deformed rocks, in the interpretation of grain-size distributions and in the relative timing of deformation compared to peak metamorphism and thus the preservation of deformation microstructures, along with the likelihood of nonsteady state deformation, must all be considered. The previous discussion has focused on the relationship between temperature, grain size and rheology for rocks deforming with a steady-state microstructure at constant external thermodynamic conditions (P, T, 1˙ ) (e.g. Fig. 2). This does not describe the evolution of microstructure and rheology in response to changing external thermodynamic conditions (e.g. changes in 1˙ , s or T ). When rocks are initially loaded in the laboratory they generally harden up to some peak stress, after which they may exhibit some amount of weakening until a ‘steady-state’ flow stress is attained which may or may not be associated with the attainment of a steady-state microstructure (Bystricky et al. 2000; Pieri et al. 2001a, b; Barnhoorn et al. 2004). The rate of grain-size change has important implications for shear localization and for the interpretation of microstructures in exhumed shear zones. For example, Prior et al. (1990) determined that the microstructure of quartz mylonites could reset in between 540 and 54 000 years, which would be equivalent to a finite equivalent strain of ,2. During triaxial deformation experiments on Carrara marble at temperatures between 1100 and 1220 K, Ter Heege et al. (2002) observed mean
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grain-size reduction from c. 100 to ,50 mm in a strain of ,1. This microstructural evolution was associated with weakening up to 41% (Fig. 3). Austin & Evans (2009) showed that the grain-size evolution rates taken from the experiments of Ter Heege et al. (2002) are consistent with the grain-size reduction kinetics predicted by equation (7). During torsion experiments on Carrara marble, Barnhoorn et al. (2004) observed major to complete recrystallization in samples deformed to equivalent strains between 3 and 4 (shear strain between 5 and 7). Those authors observed similar weakening (up to c. 27%) as in the triaxial compression experiments of Ter Heege et al. (2002; Fig. 3). The extensive studies on Carrara marble (Pieri et al. 2001a; Ter Heege et al. 2002; Barnhoorn et al. 2004) along with the analysis by Austin & Evans (2009) have demonstrated that the kinetics of grain-size reduction during dislocation creep, and the finite strain required to attain microstructural steady state, depend on the external thermodynamic conditions of deformation (T, s, 1˙ ) as predicted by equations (5– 7). Monte´si & Hirth (2003) modelled the rheological evolution of an olivine shear zone using either a synchronous grain-size evolution relationship with equation (5) coupled with equation (8) or an asynchronous relationship with equation (6) used to describe grain-size reduction. In both cases, only the dislocation creep strain rate was considered to contribute to grain-size reduction kinetics. As with the experimental studies discussed above this work predicted that, regardless of the starting grain size, a new stabilized grain size was reached by a strain of c. 4. Put together, these laboratory, field and modelling results appear to suggest that a stabilized grain size can be attained at equivalent strains between 0.5 and 5. However, in order to extrapolate laboratory observations to natural shear zones, more constraints are required on the kinetic equations for grain-size evolution for a range of minerals. For example, the grain-size reduction rates predicted by Equations (5) and (7) differ in magnitude by lsd/cg. For realistic values (s ¼ 100 MPa, d ¼ 50 mm, l ¼ 0.1, g ¼ 1 J/m2, c ¼ p), this results in a difference of greater then 2 orders of magnitude. Further, the coupling between grain-size reduction and grain growth processes needs to be better constrained. The difference between synchronous and asynchronous scaling grain-size evolution relationships will be small when the mean grain size is far from the stabilized grain size. Near the stable grain size, however, the magnitude of grain growth and grainsize reduction rates will be closer. The nature of the coupling between these two processes will therefore impact the ability of a microstructure to become reset in response to small perturbations in the deformation conditions. The finite strain required for a
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Fig. 3. (a) The amount of weakening reported during laboratory deformation experiments on Carrara marble is shown compared to the final equivalent strain from the experiment. Data are from Barnhoorn et al. (2004) and Ter Heege et al. (2002). The experiments from Barnhoorn et al. (2004) were run in torsion at temperatures of 773, 873 and 1000 K. The shear strains reported √ by Barnhoorn et al. (2004) were converted to von Mises equivalent strain using the relationship 1eq = g/ 3 (Schmid et al. 1987; Paterson & Olgaard 2000). The experiments from Ter Heege et al. (2002) were run in a triaxial compression configuration at temperatures of 1103, 1173 and 1223 K. Weakening refers to the comparison of flow stress to peak stress. (b) Weakening plotted against post peak equivalent strain for the same data as (a), except that data from Rutter (1995, 1999) are included as are fits to equation (10) with the r values labelled. ‘Post peak equivalent strain’ is the reported strain minus the strain at peak stress. For Barnhoorn et al. (2004), where strain at peak stress was not explicitly reported, it was taken as 1.5 for samples deformed at 773 K and 1.0 for samples deformed at 873 and 1000 K, based on the reported stress– strain curves. (c) As for (b), except only post peak equivalent strains up to 2 are shown for data from compression (Ter Heege et al. 2002) and extension (Rutter 1995, 1999) experiments.
stabilized grain size to be attained in monophase rocks will depend, for example, on the external thermodynamic conditions (particularly T, P, fp, s, 1˙ ), bulk mineralogy, mineral chemistry (Hay & Evans 1987) and the starting microstructure.
Crystallographic preferred orientation During laboratory deformation experiments, the formation and evolution of CPO is frequently observed (Bystricky et al. 2000; Pieri et al. 2001b; Barnhoorn et al. 2004; Fig. 4). Observations of natural shear zones also suggest that highly strained rocks have strong CPOs, particularly where these rocks are interpreted to have deformed by dislocation creep processes (Erksine et al. 1993; Mehl et al. 2003; Warren & Hirth 2006; Ebert et al. 2007; Austin
et al. 2008; Mehl & Hirth 2008; Warren et al. 2008). Figure 4 shows the CPO (quantified using the J-index) as a function of strain for laboratory torsion experiments on both calcite and olivine. It is important to note that the J-index is highly sensitive to the number of grains sampled (Matthies & Wagner 1996; Wenk 2002; Skemer et al. 2005); values reported in the literature therefore may reflect both the fabric strength and the details of the sample population and the calculation. Considering these tests were performed on different rock types under very different external thermodynamic conditions, and considering the uncertainties associated with using the J-index to quantify fabric intensity, the broad similarities in the rates of CPO evolution from these experiments is striking. The differences in the rates of CPO formation are as
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(a)45 CPO Intensity (J-Index)
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Skemer et al. (2011) observed that a pre-existing CPO could be modified with progressive strain, although equivalent strains greater than c. 0.6 may be required for the pre-existing CPO to begin to reset to new deformation kinematics. Field observations by Webber et al. (2008, 2010) suggested that even higher strains might be required to reset a pre-existing CPO. Based on comparison to observations from VPSC calculations, Skemer et al. (2011) suggested that the absence of recrystallization in coarse grains might explain the preservation of pre-existing CPOs to moderate strain. On the other hand, brittle or semi-brittle processes may result in significant rotations of crystallographic axes at relatively low strains (e.g. van Daalen et al. 1999). These studies clearly highlight the complexities in CPO evolution that may arise due to changing deformation kinematics. If additional complexities are superimposed on the deformation history, such as changing temperature, fluid activity, loading path or strain rate, then relative slip-system strengths and recrystallization kinetics may also change, further influencing the kinetics of CPO evolution and the impact of CPO evolution on shear-zone rheology.
Shear Strain
Fig. 4. CPO intensity, quantified using the J-index, is shown as a function of shear strain for laboratory deformation experiments performed on olivine (Bystricky et al. 2000; Zhang et al. 2000) and calcite (Pieri et al. 2001b; Barnhoorn et al. 2004). The dashed box in (a) highlights the region that has been expanded in (b). All J-index values are as reported in the original publication.
great for tests run on the same mineral under different external thermodynamic conditions as between different minerals (Fig. 4). Recently, the rate at which a pre-existing CPO may be reset has been investigated in the laboratory on both calcite (Delle Piane & Burlini 2008) and olivine (Skemer et al. 2011). Delle Piane & Burlini (2008) observed continued intensification of CPO following reversal of the strain direction during torsion experiments on Carrara marble. However, the CPO, which is characterized by c-axis maxima normal to the shear plane with one a-axis parallel to the shear direction, was systematically weaker than observed in similar experiments run without reversing strain (Pieri et al. 2001b; Barnhoorn et al. 2004). Delle Piane & Burlini (2008) also noted that, upon reversal of the shear direction, the yield stress was systematically lower than on initial loading. This could have important implications for shear-zone initiation, if previously deformed zones remain relatively weak even if later phases of deformation have different kinematics.
Mechanical evolution Flow laws for crystal plastic deformation mechanisms generally assume steady-state deformation; however, laboratory experiments frequently demonstrate that transient non-steady-state behaviour may persist to relatively high strains (Barnhoorn et al. 2004). A framework for quantifying non-steadystate deformation in rocks undergoing microstructural evolution is provided by Equations (1–3) (Stouffer & Dame 1996; Evans 2005). However, the relative importance of grain size, dynamic recrystallization and CPO on mechanical evolution is still being debated, and is likely to be material dependent. Experiments on calcite rocks deformed to large strains (shear strains up to 50) (Pieri et al. 2001a, b; Barnhoorn et al. 2004) suggest that weakening is likely due to recrystallization and the formation and intensification of CPO. This is consistent with laboratory experiments on ice performed by Glen (1955), which indicated that the creep rate also increased as a result of recrystallization and the development of CPO. Austin & Evans (2009) argued that observed weakening during triaxial experiments on synthetic fine-grained calcite rocks correlated with progressive intensification of c-axis alignment and did not appear to be related to an increased contribution of grain-sizesensitive creep. Using novel cycles of deformation plus annealing, Delle Piane & Burlini (2008) presented data that indicated only c. 1/3 of the observed weakening in Carrara marble could be
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due to CPO formation, with the remaining weakening associated with recrystallization. On the other hand, Ter Heege et al. (2004) argued that weakening observed in triaxial deformation experiments (also on Carrara marble) could be explained by an increased contribution from grain-size-sensitive creep, modelled using a composite flow law of dislocation creep + diffusion creep. Skemer et al. (2011) postulated that the presence of fine grains in olivine aggregates might also act to alleviate strain incompatibilities. This would reduce the number of slip systems required to accommodate strain assuming deformation is isovolumetric, potentially via an increased contribution of grainboundary sliding in finer-grained bands (Drury 2005). However, Rutter et al. (1994) observed that CPO intensity was lower in fine-grained calcite rocks undergoing superplastic deformation. If more slip systems are required to accommodate strain, perhaps in the absence of grain-boundary sliding, grain rotations may be inhibited (Tommasi et al. 2000) and creep strength may increase as the strongest slip system will be rate limiting. On the other hand computational work suggests that, even for material with uniform grain size, most deformation is accommodated on ≤3 slip systems (Etchecopar 1977; Lister et al. 1978) without invoking grainboundary sliding. Based on laboratory experiments on Carrara marble, Rutter (1999) argued that strain weakening might be empirically explained using a relationship of the form: r 1 1/n s=A 1˙ 1o
(10)
where 1o is the strain at peak stress and r is a weakening exponent (after Hutchinson & Neale 1977; Hu & Daehn 1996). This relationship is consistent with compression, extension and torsion experiments on Carrara marble (Fig. 3b, c), although other empirical relationships might also work (e.g. Luton & Sellars 1969; Barnhoorn et al. 2004). Flow laws with a strain weakening term are a valuable tool for investigating the relationships between material and system weakening. However, it is important to emphasize that empirical strain-dependent flow laws do not explicitly account for microstructural evolution. To develop theoretically sound and experimentally verified flow laws for rocks undergoing microstructural evolution, a philosophy similar to that described by Equations (1–3) is required. To accomplish this, continued laboratory, field and theoretical studies are needed to constrain the mechanisms of rheological weakening and the impact of microstructural evolution on rheology and on the localization of strain.
Conclusions/future directions The coupling of microstructural and rheological evolution through evolution, kinetic and kinematic equations (e.g. Equations (1–3)) has significant implications for our understanding of the mechanisms of strain localization during crystal plastic deformation and, consequently, for modelling of the bulk rheology of the lithosphere. This review has focused on the significant progress that has been made in recent years on the evolution equations for two of the most-studied microstructural parameters: grain size and crystallographic preferred orientation. Examples have been drawn from two of the best-studied minerals: calcite and olivine. By focusing on only two microstructural parameters in only two minerals in monomineralic rocks, without addressing the influence of metamorphic reactions, it could be argued that the scope of this review is not all-encompassing. That would be a fair criticism; however, despite these limitations, several conclusions may be drawn which help to define the important questions that need to be answered in order to fully explain the coupling between microstructural evolution and rheology. † Both the stabilized grain size and the rate of grain-size evolution during creep deformation respond to the deformation conditions including stress, strain rate, temperature, pressure and fugacity of the vapour phase. Different classes of relationships predict similar apparent sensitivities of the recrystallized grain size to the flow stress. However, there are substantial differences in the absolute magnitude of the mean grain size predicted by different recrystallized grain-size relationships. There are also differences in the temperature sensitivity, in the inferences regarding recrystallization mechanism and in the coupling between grain growth and grainsize reduction processes. All of these have important implications for extrapolation of laboratory data to natural conditions and for modelling the kinetics of microstructural evolution. † Laboratory experiments and modelling studies suggest that CPO forms and intensifies relatively quickly (below an equivalent strain of c. 5). It is also interesting to note that calcite and olivine do not show significantly different rates of CPO formation and intensification (quantified using the J-index). However, systematic a priori predictions of the rates of CPO formation are still problematic, despite significant advances in both analytical and modelling techniques. This is partly due to uncertainties around the role of grain-boundary sliding processes and relative slip-system strengths under wide ranges of
MICROSTRUCTURE AND RHEOLOGY
external thermodynamic conditions. Quantitative predictions of the rates of CPO formation and evolution, the characteristics of the resulting CPO and the impact on rheology are necessary in order to fully predict rheological evolution for rocks with evolving CPO. † Many studies have addressed the relative importance of grain size, dynamic recrystallization and CPO evolution on the rheological weakening commonly observed in laboratory experiments; however, even in monomineralic rocks, decoupling these effects is non-trivial. In order to quantitatively relate microstructural and rheological evolution, the relative and absolute contribution of these effects needs to be quantified and included in the appropriate kinetic equation to describe rheology during creep deformation and thus to model shear-zone formation and shear localization. This manuscript has benefited from numerous discussions with B. Evans, M. Herwegh, G. Hirth, A. Ebert and L. Mehl, among many others. I would like to thank D. Prior and E. Rutter for providing constructive reviews, which greatly improved the clarity and completeness of this manuscript, and for their patience.
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Torsion experiments on coarse-grained dunite: implications for microstructural evolution when diffusion creep is suppressed PHILIP SKEMER1*, MARSHALL SUNDBERG2, GREG HIRTH3 & REID COOPER3 1
Washington University in Saint Louis, Saint Louis, MO 63130 2
University of Minnesota, Minneapolis, MN 55455, USA
3
Department of Geological Sciences, Brown University, Providence, RI 02920, USA *Corresponding author (e-mail:
[email protected]) Abstract: Large strain deformation experiments in torsion were conducted on a coarse-grained natural dunite with a pre-existing lattice preferred orientation (LPO). Experiments were conducted at conditions where deformation by diffusion creep is initially negligible. Microstructural evolution was studied as a function of strain. We observe that the pre-existing LPO persists to a shear strain of at least 0.5. At larger strains, this LPO is transformed. Relict deformed grains exhibit LPO with [100] crystallographic axes at high angles to the shear plane. Unlike previous experimental studies, these axes do not readily rotate into the shear plane with increasing strain. Partial dynamic recrystallization occurs in samples deformed to moderate strains (g . 0.5). Recrystallized material forms bands that mostly transect grain interiors. The negligible rate of diffusion creep along relict grain boundaries, as well as the limited nature of dynamic recrystallization, may account for the relatively large strains required to observe evolution of microstructures. Our data support hypotheses based on natural samples that microstructures may preserve evidence of complex deformation histories. Relationships between LPO, seismic anisotropy and deformation kinematics may not always be straightforward.
As the most abundant mineral in the upper mantle, olivine plays a key role its rheological behaviour (Karato & Wu 1993; Hirth & Kohlstedt 2003). Moreover, olivine is essential to our interpretation of mantle kinematics because of its strong influence on seismic anisotropy (Nicolas & Christensen 1987; Karato et al. 2008). The generation and evolution of olivine lattice preferred orientation (LPO) has been well documented in laboratory studies (Carter & Ave Lallemant 1970; Nicolas et al. 1973; Zhang & Karato 1995; Bystricky et al. 2000; Zhang et al. 2000; Jung & Karato 2001), naturally deformed peridotites (Nicolas et al. 1971; Mercier 1985; Ben Ismail & Mainprice 1998; Warren et al. 2008; Skemer et al. 2010) and numerical simulations (Wenk et al. 1991; Wenk & Tome´ 1999; Tommasi et al. 2000; Kaminski & Ribe 2001; Blackman & Kendall 2002; Castelnau et al. 2009). However, there are still substantial differences between the microstructures generated or observed in these studies. Notably, studies of peridotite mylonites suggest that somewhat larger strains may be required to rotate LPO into concordance with deformation kinematics, in comparison to experimental studies and numerical simulations that have an initially random texture (Warren et al. 2008). This difference in the influence of strain on LPO may be a consequence of factors such as the initial
microstructure and the conditions of deformation. Furthermore, different numerical models show distinct patterns and rates of LPO development which may differ from experimental and geological observations (Castelnau et al. 2009). Variations between models arise from differences in numerical techniques, but also the inclusion or exclusion of specific dynamic recrystallization mechanisms (Wenk et al. 1991; Wenk & Tome´ 1999). To expand on existing experimental studies and provide further bases for numerical investigations, we conducted a series of deformation experiments on coarse-grained natural dunite. The objective of these experiments is to improve our understanding of microstructural processes in realistic mantle materials and to help bridge the gap between the laboratory and nature. For technical reasons, many of the experimental studies of olivine deformation have been conducted on relatively fine-grained synthetic materials (e.g. Zhang & Karato 1995; Bystricky et al. 2000). As a consequence, deformation in these experiments typically includes contributions from several parallel mechanisms including dislocation creep, diffusion creep and grain-boundary sliding (Fig. 1). In contrast, many geologically observable mantle rocks are interpreted to have deformed at conditions where dislocation creep is strongly dominant. It is therefore important to understand from an
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 211– 223. DOI: 10.1144/SP360.12 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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Differential stress (MPa)
GBS Dislocation creep
102
10–3 10–4 10–5 10–6 10–7
101
10–8
Diffusion creep
T = 1500 K P = 300 MPa
100 100
101
102 103 Grain size (microns)
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Fig. 1. Deformation mechanism map of olivine at T ¼ 1500 K, P ¼ 300 MPa and at dry conditions, using flow laws from Hirth & Kohlstedt (2003). Thick lines demark boundaries between deformation mechanisms: dislocation creep, diffusion creep and grain-boundary sliding (GBS). The dashed line is the location of the olivine recrystallized grain-size piezometer. Thin lines are contours of constant strain rate. The dark shaded region shows the conditions of experiments for typical hot-pressed aggregates. The light shaded region shows the conditions of the experiments in this study. Curvature in the strain-rate contours shows regions in deformation space where multiple mechanisms play a significant role in deformation. The white star on the piezometer line shows the recrystallized grain size for sample PIP-21.
alteration (Fig. 2). The initial grain size of the starting material is 510 + 100 mm, as determined by the mean intercept technique with a stereological correction factor of 1.7. The grains are equant, with straight or gently curved grain boundaries and 1208 triple junctions. A few grains have subgrain boundaries, but most do not exhibit any undulose extinction that would indicate the presence of many dislocations. The material contains c. 1% spinel and no other secondary phases are visible. The water content in the starting material is below the detection limit of Fourier Transform Infrared Spectroscopy (FTIR) (,30 ppm H/Si). Two experimental challenges must be overcome to deform coarse-grained dunite in torsion. First, due to the strength of these materials, a relatively large torque must be applied to the sample to achieve flow at reasonable experimental strain rates. Second, to apply a large torque to the sample there must be sufficient frictional traction between the sample and the forcing pistons to mitigate sliding along these interfaces. To solve these two problems we used a ‘dog-bone’ shaped sample which is wide at the ends and tapered to a narrow cylindrical central region (Fig. 3). The wide ends of the samples provide sufficient contact area with the pistons to support moderately large torques. These torques are then concentrated into the central cylindrical region of the sample, exploiting the strong radial dependence between shear stress (t) and torque (M) in a cylindrical
experimental perspective how mantle materials deform when grain-size-sensitive creep is more completely suppressed. To this end, we conducted laboratory deformation experiments on samples cored from a block of Balsam Gap Dunite, which was chosen for two reasons. First, it has a moderately large grain size which allows us to suppress activity of the diffusion creep mechanism. Secondly, it has a moderately strong pre-existing LPO. This allows us to consider how LPO might evolve in a situation where kinematics are changing, for example in a corner flow regime such as a mantle wedge or mid-ocean ridge.
Methods Coarse-grained olivine aggregates were deformed in torsion in the Paterson gas medium apparatus at Brown University and at the University of Minnesota. Paterson apparatus were used because of their relatively large sample volume (c. 1 cm3) and their ability to deform materials to large strains. Starting materials were cored from a clean block of Balsam Gap Dunite, which is largely devoid of visible
Fig. 2. Photomicrograph of the Balsam Gap Dunite, the starting material for these experiments.
TORSION EXPERIMENTS ON DUNITE
Fig. 3. Sample assembly. Dog-bone-shaped olivine samples are sandwiched between a series of alumina pistons and spacers. The white dashed line marks the location of the tangent sections from which the microstructure is analysed. Kinks that form when the iron jacket is swaged around the sample are used to measure shear strain. The example shown here corresponds to a maximum shear strain of g ¼ 1.8.
geometry (Paterson & Olgaard 2000) where d is the sample diameter and n is the stress exponent for a power-law material (Equation 1): M¼
pd3 t : 4(3 þ 1=n)
(1)
Relatively large stresses can therefore be achieved with relatively small torques. This is similar to the solution adopted by Bystricky et al. (2000), who allowed sintered hot-presses to collapse down into a tapered shape during densification prior
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to deformation (M. Bystricky, pers. comm. 2008). By introducing the tapered shape in a controlled fashion, we are able to achieve more consistent and reproducible results. To make the dog-bone shape, cores were placed in a lathe and a central section was ground down using a high-speed rotary tool and diamond burr. The narrow region had a constant diameter of 6.4 –7.0 mm over a length of c. 3.0 mm, which transitioned to the full diameter of 9.8 mm over a length of c. 2.0 mm. The transition between the centre and the ends of the sample has continuous curvature with a radius of c. 1 mm. This curvature was introduced to prevent puncturing the jacket material during compression. The ends of the sample were faced and sandwiched between porous alumina pistons, which help grip the sample (Fig. 3). Sample assemblies were dried overnight in a vacuum oven at 400 K prior to deformation. The narrow span of the sample was wrapped with Ni foil and the whole assembly was jacketed in iron, which was collapsed around the sample by briefly increasing the confining pressure to 110 MPa. This cinching of the jacket introduced creases, which are used for calculating shear strain (Fig. 3). Score marks, drawn with a razor blade normal to the shear plane, are also used to calculate shear strain and to confirm that slipping on sample –piston interfaces did not occur. Experiments were conducted at constant twist rates of 25–50 micro-rad/s at fixed temperature and pressure (1500 K; 300–315 MPa), to a range of shear strains. Because of slight differences in the effective sample length, these twist rates produced a range of strain rates as recorded in Table 1. All experiments were run until ruptures in the iron jacket occurred, reducing the effective confining pressure to zero and effectively terminating the experiment. The mechanical data directly recorded by the Paterson apparatus are the twist rate (with units of radians/second) and the torque applied to the
Table 1. Sample table Sample*
Pressure (MPa)
Temperature (K)
Maximum shear strain
PIP-19 PIP-20 PIP-21 PIP-22†
310 315 315 315
1500 1500 1500 1500
0.2 1.2 1.8 3.9
PT-0474 PT-0484
300 300
1500 1500
0.5 3.5
Maximum shear strain rate (1/s) 2.2 4.8 3.5 2.8 3.1 1.9 1.9
1025 1025 1025 1025 to 1024 1024 1024
*Experiments numbered PIP and PT conducted at Brown University and University of Minnesota, respectively. † Some strain-rate stepping was done at the beginning of this run.
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sample. The maximum strain rate is calculated from the imposed twist rate and the dimensions of the narrow span of the dog bone:
g_ max ¼
du_ 2l
(2)
where g_ max is the maximum shear strain rate, d is the sample diameter, l is the length of the sample and u_ is the imposed twist rate (Paterson & Olgaard 2000). The maximum strain rate is also determined from the deflection of the passive strain marker and the duration of the experiment. Strain calculated from the deflection of the strain marker tends to be slightly higher than strain calculated from equation (2), as some twist is inhomogeneously accommodated by the sample where it transitions from narrow to wide. We therefore report only the strains calculated from the deflection of the strain marker. Shear stress is calculated from the torque record using equation (1) assuming all deformation is partitioned into the narrow span of the sample. Shear strain and shear strain rate are a linear function of radius, with the maximum values occurring at the outer edge of the sample. Like strain, the stress state within the specimen varies both radially and axially due to the cylindrical geometry and tapered dog-bone shape. Because of the non-linear rheology predicted from flow laws and inferred from these measurements, most of the torque is supported by the outermost annulus of the cylindrical sample (Paterson & Olgaard 2000). Within this outermost annulus, torque is proportional to stress. Assuming that the total measured torque is proportional to the shear stress in the outermost shell of the specimen, the relationship between torque and maximum shear strain rate is analogous to a standard constitutive equation:
g_ max / M n
(3)
where M is torque and n is equivalent to the stress exponent. After recovering the samples, thin sections were made through a tangent section as close as possible to the edge of the deformed portion of the sample (Fig. 3). Tangent sections were used for analysis because the kinematics of deformation are nearly identical across the section. LPO was measured using electron backscatter diffraction (EBSD) on a JEOL 845 or JEOL 7001FLV scanning electron microscope with 20 mm working distances and 20 kV accelerating voltage. EBSD measurements were made on both the undeformed and deformed portions of the samples, to determine both the initial and final LPO. Undeformed regions were mapped automatically with 50 micron steps, while deformed regions were mapped with 10 micron
steps. Datasets were processed to eliminate erroneous data spikes, and reduced to one point per grain. Only grains with at least 5 contiguous pixels of the same orientation are used when plotting the pole figures of the relict grains. In experiments conducted to larger strains, EBSD was used to investigate the microstructures produced by dynamic recrystallization. These recrystallized regions were mapped with 0.25 micron steps, and the datasets were also processed to remove wild spikes. A minimal amount of smoothing was applied to fill gaps in the data.
Results Rheological data Figure 4 shows mechanical data from these experiments. In Figure 4a, stress –strain curves are shown for experiments conducted at strain rates of 2.2–4.8 1025 s21 (in black) and 1.9 1024 s21 (in grey). The higher strain-rate experiments achieve peak stresses at shear strains of c. 0.25, followed by significant decreases in strength. The lower strain-rate experiments do not show significant strain weakening. Shear stresses calculated from the torque record using equation (1) are much higher than predicted from flow laws (Fig. 1). As we discuss later, this is likely an artefact of the sample shape and deformation geometry. To constrain the deformation regime in these experiments, strain-rate stepping experiments were conducted at the beginning of run PIP-22 (Fig. 4b, c). A plot of torque (stress) versus shear strain rate is nearly linear in log–log space, and is well fit with an exponent of n ¼ 3.4. This is within error of the expected value for olivine deforming by dislocation creep (n ¼ 3.5 + 0.3) (Chopra & Paterson 1984; Hirth & Kohlstedt 2003). There is some deviation at the highest strain rates, as evidenced by the curvature in Figure 4c, which is interpreted as a transition to exponential Peierls creep (e.g. Evans & Goetze 1979; Katayama & Karato 2008). The remainder of the experiments discussed here were conducted at conditions within the n ¼ 3.4 regime. Based on the value of the stress exponent we conclude that deformation is accomplished by dislocation creep. However, calculations of shear stresses from the torque data using equation (1) gives values higher than flow laws would predict. For example, sample PIP-21 had an apparent steady-state shear stress of c. 320 MPa while the flow law would predict shear stresses of ,200 MPa. There are several possible explanations for the discrepancy. First, some torque was supported by deformation of the curved annulus where the dog-bone transitioned from narrow to
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reinforcing the suggestion that this is likely a systematic problem with the geometry of the experimental design. As we discuss below, other measures of stress such as recrystallization piezometers provide numbers that are closer to flowlaw predictions.
Appearent shear stress (MPa)
(a) 500
PIP–21
300
PT–0474 PIP–19
PT–0484
200
General microstructural observations 100 0
0
(b)
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Maximum shear strain PIP–22 1.3 × 10–4
1.9 × 10–4
600
2.7 × 10–4 30
25
6.2 × 10–5 6.2 × 10–5
400
20 15
2.8 × 10–5
10
200
5
0
Torque (N × m)
Appearent shear stress (MPa)
PIP–20
400
0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Maximum shear strain
(c) 10–3
Shear strain-rate (s–1)
215
10–4 n = 3.4
10–5 1 10
Torque (N × m)
102
Fig. 4. Mechanical data. (a) Stress–strain curves for experiments conducted at strain rates of 2.2– 4.8 1025 s21 (in black) and 1.9 1024 s21 (in grey). (b, c) Results from strain-rate stepping during experiment PIP-22. Apparent shear stress is calculated from the torque recorded by the apparatus, using equation (1).
A series of experiments from shear strains of 0.2– 3.9 were conducted to investigate the evolution of various microstructures (Table 1). In tangent sections through the specimens, both deformed and undeformed regions are observed (Fig. 5). Microstructures in the undeformed portion of the specimens appear unaltered in comparison to the starting materials. Deformed portions of the specimens show increasingly modified microstructures with progressive strain. As described in detail below, progressive strain modifies the pre-existing LPO. Progressive strain is also observed to alter the grain size and grain shape. Below g ¼ 0.5, we observe no fine-grained material that we interpret to be the product of dynamic recrystallization. Between g ¼ 1.2 and g ¼ 3.5, the amount of finegrained material increases from less than 1% to approximately 30% by volume. This fine-grained material forms narrow bands that grow and interconnect at larger strains, and we interpret this finegrained material to be the product of dynamic recrystallization. Relict grains often contain subgrains, but do not show any evidence of grainboundary bulging or migration. Some fractures are seen in thin section, generally parallel to the orientation of the maximum compressive stress. These fractures are Mode I (showing no shear offset) and exhibit clear cross-cutting relationships with ductile microstructures. They are also continuous features that intersect low-strain and high-strain regions of the specimen. We therefore conclude that the fractures form during the brief interval of deformation (generally 5–30 s) that occurred after the iron jacket ruptured and before the actuator that drives deformation was manually stopped. No other fracturing that is obviously related to experimental deformation is noted.
LPO of relict grains wide; this is clearly evident from the deflection of the strain marker (Fig. 3). Because of the cubic dependence of the stress –torque relationship on sample diameter, this could easily introduce errors to the stress calculation. Similarly, the ‘fins’ in the iron jacket produced by swaging around the dog-bone shape likely introduced some error. Bystricky et al. (2000) also observe elevated shear stresses in comparison to flow-law predictions,
In all samples, sufficient relict (unrecrystallized) material remains to make LPO measurements. These data can then be compared to the undeformed portions of the samples to determine how LPO in the unrecrystallized material evolved with progressive deformation (Fig. 6). Because of the relatively large grain size in the samples, datasets are quite small (50–100 unique grains); we therefore only describe the LPO in qualitative terms. Sample
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Fig. 5. Photomicrographs of samples PIP-19, PIP-21 and PT-0484 (left to right). The upper row of images shows the entire experimental charge. The middle row of images contains higher-magnification views of the deformed parts of the sample. The lower row of images shows high-magnification photomicrographs of the white inset boxes. Sense of shear is top to the right. Thin sections are nearly tangent to the edge of the deformed part of the dog-bone. Undeformed regions at the ends of the samples show no microstructural alteration, while the deformed centres of the samples show increasing microstructural alteration with strain. In the high-magnification images, recrystallized material is seen to form narrow bands which become more abundant and interconnected at higher strain.
PIP-19, which was deformed to a maximum shear strain of 0.2, exhibits little modification of the preexisting LPO. Minor rotations of the [100] and [001] axes are detected. Sample PT-0474, deformed to g ¼0.5, also shows little alteration of the preexisting fabric, although the maxima become somewhat less distinct. By contrast, in sample PIP-20
(which was deformed to a shear strain of 1.2) the pre-existing fabric has been completely transformed. The deformed portion of the specimen has a preferred orientation with [100] and [001] axes exhibiting partial girdles, including clusters of data at high angles to the shear plane and clusters of data in the shear plane normal to the shear direction.
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Fig. 6. Pole figures showing deformation fabrics in relict (unrecrystallized) grains and the pre-existing fabric for each sample. Pre-existing fabrics are plotted in the initial reference frame of the deformed portion of the sample. Projections are upper hemispheres, with the shear direction east–west and the pole to the shear plane north–south. Contour intervals are 1 Multiples of Uniform Distribution (MUD), with shading that saturates at 8 MUD. Small squares on each pole figure mark locations of individual data points. The sense of shear in these pole figures is dextral. Dashed lines show the orientation of the long axis of the finite strain ellipsoid. Two different sections of PIP-21 are shown; these were taken from different azimuths (separated by 1358) around the deformed sample.
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The [010] axes cluster at a single point maximum, also at a high angle to the shear plane. For sample PIP-21, deformed to g ¼ 1.8, data were collected from two slices taken from different azimuths around the cylindrical sample (separated by 1358). Various azimuths around the cylindrical sample had different relationships between the orthorhombic starting LPO and the kinematics of deformation at that position in the sample. Analysing these data allows us to directly test the influence of the initial preferred orientation in the starting material. For these two datasets, the LPO produced by deformation is similar in the [100] and [010] directions, but deviates somewhat in the [001] direction. The [100] axes are again at a very high angle to the shear plane. Sample PT-0484, which was deformed to maximum shear strain of 3.5, displays the best developed fabric with [100] and [010] axes clustered at high angles to the shear plane and [001] axes clustered in the shear plane normal to the shear direction.
Dynamic recrystallization Small amounts of fine-grained material (c. 1 vol%) first appear in sample PIP-20, which was deformed to a maximum shear strain of g ¼1.2. This finegrained material, which we interpret to be the product of dynamic recrystallization, is organized into 10–25 mm thick bands oriented c. 208 to the shear plane (Fig. 7a). These bands are only a few grains wide. The preferred orientation of the material in these bands forms a great circle of [100] and [010] axes, rotated about the [001] axis. This character of deformation is also evident in inverse pole figures of misorientation axes, which highlight the strong preference for rotation about the [001] axis (Fig. 7c, e). In sample PIP-21, deformed to g ¼1.8, finegrained material is more abundant (c. 10 vol%) with bands of these grains up to c. 100 microns thick dissecting relict grains (Fig. 7b, d, f ). Again, the crystal orientations within the bands form great circles of [100] and [010] axes, rotated about the [001] axis. Within these bands there is a complex microstructure. Some grains have very large aspect ratios, and are dissected by low-angle boundaries. The large-aspect ratio grains have serrated margins with small (c. 4 mm) equant grains present on the serrations. These microstructures are interpreted to be evidence of dynamic recrystallization by both subgrain rotation and nucleation and growth processes. There are varying magnitudes of internal deformation in these grains as indicated by the distortion of the crystal lattice in the EBSD data. The preferred orientation of these recrystallized grains is related to the orientation of the unrecrystallized grain through which the bands
pass (noted by white stars on the pole figures in Fig. 7). At g ¼3.5 (sample PT-0484), these bands of recrystallized grains become interconnected and form a network that dissects the relict grains into long ribbon shapes. This process results in a strongly layered microstructure within the sample, at an oblique angle to the shear plane (Fig. 5). The recrystallized grain size provides an independent constraint on the flow stress during deformation (e.g. Twiss 1977). Although the morphology of the recrystallized grains is complicated, the mean grain size for sample PIP-21 (calculated using the mean intercept technique with a stereological correction factor of 1.75 from the image in Figure 7) is c. 9.2 + 3.0 mm. Applying the recrystallization piezometry of van der Wal et al. (1993), this corresponds to a differential stress of 259 þ88/249 MPa (Fig. 1). We note that this grain size is within error of what would be expected for olivine deforming at these conditions, based on published dislocation creep flow laws (Hirth & Kohlstedt 2003). Even the stress predicted for the smallest neoblasts (c. 4 mm corresponding to differential stresses of only c. 480 MPa; van der Wal et al. 1993) is significantly below the flow stresses estimated from the torque record.
Discussion Normally it is assumed that, with increasing strain, olivine [100] crystallographic axes will tend to rotate towards the shear plane. Indeed, as we will discuss here, many experimental, numerical and geological observations have found this to be the case. However, other numerical studies (particularly those that do not explicitly include dynamic recrystallization) show that oblique LPO are retained at moderate strain. At the largest strains in this study we observe that, while the pre-existing LPO is modified, there is very little subsequent rotation of the LPO of relict unrecrystallized grains; sample PT-0484, deformed to a shear strain of 3.5, still has its [100] and [010] axes at a c. 458 angle to the shear plane. We now discuss our results in relation to previous studies.
Comparison of results to previous experimental and numerical studies When fine-grained olivine aggregates are studied at typical laboratory conditions, a fraction of deformation may be accommodated by mechanisms other than dislocation creep (Fig. 1). The experiments by Zhang & Karato (1995), the first to investigate evolution of olivine LPO in general shear, were conducted on polycrystalline aggregates with a mean grain size of ,50 mm at 1473 and 1573 K.
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Fig. 7. EBSD data for recrystallized shear bands cross-cutting unrecrystallized grains from (a, c, e) samples PIP-20 and (b, d, f) PIP-21. (a) An EBSD map showing incipient recrystallization in sample PIP-20. The shear band at this stage is only a few grains wide. Colouration is with respect to the microscope reference frame and the band contrast of the Kikuchi patterns. (b) An EBSD map from the centre of a larger shear band in sample PIP-21. (c, d) Pole figures showing the orientations of all data within this EBSD map. The white stars show the orientation of the host parent grain. The colouration is the same as in the EBSD maps. (e, f) Inverse pole figures of misorientation axes between adjacent pixels in the EBSD maps for a range of misorientation angles. The strong clustering of data indicate that most crystallographic rotations in these regions are about the [001] axis.
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At 1473 K, dynamic recrystallization was limited and occurred mainly on grain boundaries. The relict grains exhibited an LPO (specifically maxima of [100]) that nearly tracked the maximum elongation of the finite strain ellipse. At higher temperatures (1573 K), more dynamic recrystallization was observed and preferred orientations rotated into concordance with the shear plane at shear strains of c. 1. Even at lower temperatures, rotation of olivine LPO towards the shear plane was achieved at lower strains than we observe in our study. Some of the samples from the Zhang & Karato (1995) study showed two clusters of crystallographic orientations, one with [100] in the shear plane and one with [100] oblique to the shear plane. These peaks were subsequently shown by Lee et al. (2002) to arise from different mechanisms of recrystallization. In the Lee et al. (2002) study, grains oblique to the shear plane were found to have low dislocation densities and interpreted to form by grain-boundary migration recrystallization. The LPO of this low dislocation density population of grains is similar to the LPO that we observed in the unrecrystallized relict grains from our study. However, in our study, the relict grains show extensive internal deformation and subgrain development. We therefore do not believe that the obliquity of [100] axes in our samples results from grain-boundary migration. A series of experiments by Bystricky et al. (2000) also provide constraints on the evolution of olivine LPO. These experiments were conducted on fine-grained aggregates at 1473 K, in a torsion geometry that allowed the authors to achieve larger strains. Their results show grain-boundarydominated recrystallization and rapid development of LPO with strong [100] maxima in the direction of shear, and girdles of [010] and [001] maxima orthogonal to the shear direction. The LPO in the fine-grained portions of our experiments do not reproduce these observations. We suggest that this discrepancy arises from differences in the distribution of the recrystallized grains. In the experiments by Bystricky et al. (2000), recrystallized grains formed contiguous and extensive networks along grain boundaries (even at low strain). The recrystallized material was therefore oriented and distributed in a way that facilitates accommodation of large strain in fine-grained material. In our experiments, the bands of recrystallized material form at higher angles to the shear plane and are often not interconnected; they therefore may not have accumulated much strain. This may explain why these bands of recrystallized grains exhibit LPO that is primarily related to the orientation of the host grain, rather than the macroscopic kinematics of deformation. We suggest that the differences in the distribution of recrystallized grains may be a
consequence of the difference in the initial grain size, and therefore in the density of grain-boundary nucleation sites. Numerical models that explicitly include subgrain rotation or nucleation dominated recrystallization predict rapid rotations of the LPO towards the shear plane (e.g. Wenk & Tome´ 1999; Kaminski & Ribe 2001). However, models that do not include recrystallization mechanisms show that olivine LPO may remain oblique to the shear plane at moderate strains (Wenk et al. 1991; Tommasi et al. 2000). This is partly due to the sluggish rotation of unfavourably oriented grains (Wenk et al. 1991). Our observations of oblique LPO in relict grains agree with numerical models in the absence of dynamic recrystallization. The absence of recrystallization in the relict grains may therefore explain the long-lived obliquity of the LPO in our samples. Another important factor influencing the rate of LPO development may be the suppression of grain-size-sensitive processes. The large grain size in our samples means that strain incompatibilities caused by differential rotation of adjacent grains are not readily relaxed by diffusion creep, grainboundary sliding and/or grain-boundary migration. Thus, more slip systems may be required to accommodate arbitrary strain and crystallographic rotations with respect to kinematics may be inhibited (Tommasi et al. 2000). We also note that in our experiments dynamic recrystallization occurs primarily in the grain interiors rather than along grain boundaries. This may have a number of consequences. First, the absence of fine-grained materials along relict grain boundaries may change the subsequent deformation of those relict grains by inhibiting diffusion creep or grain-boundary sliding: both of these diffusion-controlled processes may act to relax incompatibilities along relict grain boundaries. Also, the formation of shear bands through grains rather than along grain boundaries changes the degree to which interconnectivity of microstructures occurs. Interconnectivity of weak material is essential for macroscopic weakening of a two-part composite material (e.g. Handy 1994).
Comparison of results with geological observations In general, geological observations of the relationships among LPO, foliation and shear plane are limited, due to the difficulty of measuring finite strain and identifying the shear plane in natural settings. Many measurements of olivine LPO from naturally deformed samples show a moderate obliquity between the [100] maximum and the foliation plane (Nicolas et al. 1971; Mercier 1985; Ben Ismail
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& Mainprice 1998). Because finite strain (as well as the orientation of the shear plane) is often unknown, it is not however clear how much natural variation there is in the relationship between olivine LPO and the kinematics of deformation. The kinematically best-constrained examples of natural shear zones show that, with sufficient strain, olivine [100] crystallographic axes tend to rotate into the shear plane (Warren et al. 2008; Skemer et al. 2010). However, the amount of strain required is somewhat larger than inferred from laboratory experiments, which was interpreted to be a consequence of the pre-existing LPO (Warren et al. 2008). Pre-existing LPO is observed, both in some natural shear zones and in our experiments, to be overprinted at relatively modest strain (g 1). However, other studies have suggested that under certain conditions still larger strains may be required to obliterate pre-existing LPO (Webber et al. 2008, 2010), or that deformation and LPO development may be otherwise modified (Toy et al. 2008). The relationship between the olivine LPO and the foliation and shear planes also depends on the specific type of LPO. For example, field and experimental observations have demonstrated that the sense of obliquity between the LPO and the shear plane is different for olivine A-type and E-type LPO (Katayama et al. 2004; Skemer et al. 2010). Our study suggests that rates of LPO development may vary more significantly than previously supposed. This is not too surprising: LPO is a very complicated phenomenon, the formation of which involves several distinct and sometimes competing physical mechanisms. LPO development is strongly influenced by the relative strengths of various slip systems (Wenk et al. 1991; Tommasi et al. 2000; Kaminski 2002; Karato et al. 2008). Changes in the predominance of these slip systems are known to induce changes in preferred orientation (Jung & Karato 2001). Details of the recrystallization mechanism (Urai et al. 1986; Wenk & Tome´ 1999; Lee et al. 2002) or modification of LPO due to the presence of secondary phases (Warren & Hirth 2006; Sundberg & Cooper 2008; Skemer et al. 2010) may also alter LPO development. Indeed, in specific natural settings, LPO development is seen to vary on outcrop scales (Skemer et al. 2010). Further study is required to understand whether the observations in this study are solely a consequence of the starting material, the conditions of deformation or both. Many recent geological and experimental studies have demonstrated that deformation microstructures continue to evolve at very large strains (Bystricky et al. 2000; Pieri et al. 2001; Heidelbach et al. 2003; Rybacki et al. 2003; Barnhoorn et al. 2005; Bystricky et al. 2006; Heilbronner & Tullis 2006; Warren & Hirth 2006; Skemer & Karato 2008; Warren et al. 2008; Skemer et al. 2010;
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Webber et al. 2010). Indeed, because of the strong feedbacks between microstructure and rheology, these studies suggest that at some conditions steady-state rheology may not be achieved in materials deformed to modest strains (Mackwell & Paterson 2002). Steady-state microstructure and rheology is even more difficult to achieve in situations where deformation conditions also continue to evolve. For example, materials advected through corner-flow regimes at mid-ocean ridges and subduction zones experience changes in deformation kinematics as well as changes in temperature, pressure and chemical conditions. Evolving kinematics coupled with sluggish microstructural evolution will greatly complicate the interpretation of geophysical data (Kaminski & Ribe 2002; Castelnau et al. 2009). Our experiments demonstrate that, under certain conditions, we cannot always assume simple parallel relationships between LPO, seismic anisotropy and deformation kinematics.
Conclusions We have conducted large-strain deformation experiments in torsion on a natural dunite. This starting material has a larger initial grain size and stronger initial LPO than the synthetic aggregates investigated in previous studies. These experiments are therefore conducted at conditions where secondary deformation mechanisms such as grain-boundary sliding and diffusion creep are largely suppressed. We observe that the character and rate of microstructural evolution is different than in previous studies. Specifically, we observe that considerable strain (at least g ¼ 1) is required to modify the preexisting LPO. We also observe little rotation of the LPO of relict (unrecrystallized) material towards the shear plane. The LPO we observe in relict grains is similar to LPO formed numerically in the absence of dynamic recrystallization (e.g. Wenk et al. 1991). Additionally, we note that dynamic recrystallization occurs mainly in the grain interiors rather than along grain boundaries. We note that the rate of microstructural evolution cannot be decoupled from the initial microstructure of a sample. We conclude that both the starting microstructure of a material and the conditions of deformation play a key role in determining how subsequent microstructural evolution occurs. The interpretation of seismic anisotropy, particularly in complicated kinematic environments, must be carefully evaluated in this context. The authors thank D. Kohlstedt for use of Paterson torsion apparatus at the University of Minnesota and T. Tullis for providing the starting material for these experiments. D. Prior, V. Toy and an anonymous reviewer are thanked for their constructive comments. This work was supported in part by NSF EAR-0911289 and EAR-0609869.
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Kaminski, E. & Ribe, N. M. 2002. Timescales for the evolution of seismic anisotropy in mantle flow. Geochemistry, Geophysics, Geosystems, 3, 1051. Karato, S. & Wu, P. 1993. Rheology of the upper mantle: a synthesis. Science, 260, 771– 777. Karato, S.-I., Jung, H., Katayama, I. & Skemer, P. 2008. Geodynamic significance of seismic anisotropy of the upper mantle: new insights from laboratory studies. Annual Review of Earth and Planetary Sciences, 36, 59–95. Katayama, I. & Karato, S. I. 2008. Low-temperature, high-stress deformation of olivine under watersaturated conditions. Physics of the Earth and Planetary Interiors, 168, 125– 133. Katayama, I., Jung, H. & Karato, S. I. 2004. New type of olivine fabric from deformation experiments at modest water content and low stress. Geology (Boulder), 32, 1045– 1048. Lee, K. H., Jiang, Z. & Karato, S. I. 2002. A scanning electron microscope study of the effects of dynamic recrystallization on lattice preferred orientation in olivine. Tectonophysics, 351, 331 –341. Mackwell, S. J. & Paterson, M. S. 2002. New developments in deformation studies: high-strain deformation. In: Karato, S.-I. & Wenk, H.-R. (eds) Plastic Deformation of Minerals and Rocks. Mineralogical Society of America, Washington, D.C., 1–19. Mercier, J. (ed.) 1985. Olivine and Pyroxenes. In: Wenk, H.-R. (ed.) Preferred Orientation in Deformed Metals and Rocks: An Introduction to Modern Texture Analysis. Academic Press, Orlando, 407–430. Nicolas, A. & Christensen, N. I. 1987. Formation of anisotropy in upper mantle peridotites; a review. In: Fuchs, K. & Froidevaux, C. (eds) Composition, Structure and Dynamics of the LithosphereAsthenosphere System. American Geophysical Union, Washington DC, United States, Geodynamics Series, 111–123. Nicolas, A., Bouchez, J. L., Boudier, F. & Mercier, J. C. 1971. Textures, structures and fabrics due to solid state flow in some European lherzolites. Tectonophysics, 12, 55–86. Nicolas, A., Boudier, F. & Boullier, A. M. 1973. Mechanisms of flow in naturally and experimentally deformed peridotites. American Journal of Science, 273, 853 –876. Paterson, M. S. & Olgaard, D. L. 2000. Rock deformation tests to large shear strains in torsion. Journal of Structural Geology, 22, 1341– 1358. Pieri, M., Burlini, L., Kunze, K., Stretton, I. & Olgaard, D. L. 2001. Rheological and microstructural evolution of Carrara marble with high shear strain: results from high temperature torsion experiments. Journal of Structural Geology, 23, 1393– 1413. Rybacki, E., Paterson, M. S., Wirth, R. & Dresen, G. 2003. Rheology of calcite-quartz aggregates deformed to large strain in torsion. Journal of Geophysical Research, 108, 2089. Skemer, P. & Karato, S. 2008. Sheared lherzolite xenoliths revisited. Journal of Geophysical Research, 113, B07205.
TORSION EXPERIMENTS ON DUNITE Skemer, P., Warren, J. M., Kelemen, P. B. & Hirth, G. 2010. Microstructural and rheological evolution of a mantle shear zone. Journal of Petrology, 51, 43– 53. Sundberg, M. & Cooper, R. F. 2008. Crystallographic preferred orientation produced by diffusional creep of harzburgite: effects of chemical interactions among phases during plastic flow. Journal of Geophysical Research, 113, B12208. Tommasi, A., Mainprice, D., Canova, G. & Chastel, Y. 2000. Viscoplastic self-consistent and equilibriumbased modeling of olivine lattice preferred orientations; implications for the upper mantle seismic anisotropy. Journal of Geophysical Research, B, Solid Earth and Planets, 105, 7893– 7908. Toy, V. G., Prior, D. J. & Norris, R. J. 2008. Quartz fabrics in the Alpine Fault mylonites: influence of preexisting preferred orientations on fabric development during progressive uplift. Journal of Structural Geology, 30, 602– 621. Twiss, R. J. 1977. Theory and applicability of a recrystallized grain size paleopiezometer. Pure and Applied Geophysics, 115, 227– 244. Urai, J. L., Means, W. D. & Lister, G. S. 1986. Dynamic recrystallization of minerals. In: Hobbs, B. E. & Heard, H. C. (eds) Mineral and Rock Deformation; Laboratory Studies; the Paterson Volume. American Geophysical Union, Washington DC, United States, Geophysical Monograph, 161– 199. van der Wal, D., Chopra, P., Drury, M. & Fitz, G. J. 1993. Relationships between dynamically recrystallized grain size and deformation conditions in
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experimentally deformed olivine rocks. Geophysical Research Letters, 20, 1479–1482. Warren, J. M. & Hirth, G. 2006. Grain size sensitive deformation mechanisms in naturally deformed peridotites. Earth and Planetary Science Letters, 248, 438–450. Warren, J. M., Hirth, G. & Kelemen, P. 2008. Evolution of lattice-preferred orientation during simple shear in the mantle. Earth and Planetary Science Letters, 272, 501–512. Webber, C., Little, T., Newman, J. & Tikoff, B. 2008. Fabric superposition in upper mantle peridotite, Red Hills, New Zealand. Journal of Structural Geology, 30, 1412–1428. Webber, C., Newman, J., Holyoke, C. W. III, Little, T. & Tikoff, B. 2010. Fabric development in cm-scale shear zones in ultramafic rocks, Red Hills, New Zealand. Tectonophysics, 489, 55– 75. Wenk, H.-R. & Tome´, C. N. 1999. Modeling dynamic recrystallization of olivine aggregates deformed in simple shear. Journal of Geophysical Research, 104, 25 513– 25 527. Wenk, H.-R., Bennett, K., Canova, G. R. & Molinari, A. 1991. Modeling plastic-deformation of peridotite with the self-consistent theory. Journal of Geophysical Research-Solid Earth and Planets, 96, 8337–8349. Zhang, S. & Karato, S. I. 1995. Lattice preferred orientation of olivine aggregates deformed in simple shear. Nature (London), 375, 774– 777. Zhang, S., Karato, S. I., Fitz, G. J., Faul, U. H. & Zhou, Y. 2000. Simple shear deformation of olivine aggregates. Tectonophysics, 316, 133–152.
Slip-system and EBSD analysis on compressively deformed fine-grained polycrystalline olivine R. J. M. FARLA1, J. D. FITZ GERALD1*, H. KOKKONEN1, A. HALFPENNY1, U. H. FAUL2 & I. JACKSON1 1
Research School of Earth Sciences, Building 61, The Australian National University, Canberra ACT 0200, Australia
2
Department of Earth Sciences, Boston University, 675 Commonwealth Ave., Boston, MA 002215, USA *Corresponding author (e-mail:
[email protected]) Abstract: A slip-system analysis was performed on two synthetic compressively deformed olivine aggregates, derived from experimental solution– gelation (sol– gel) and natural San Carlos precursors to determine how dislocation density relates to Schmid factor for slip in olivine. Individual grain orientations were measured with electron backscatter diffraction. Using decorated dislocations, grain populations were separated into subsets of high versus low dislocation density. Analysis of preferred orientations and distributions of Schmid factors suggests that there is only weak correlation between Schmid factor and dislocation density, slip on (010)[100] in San Carlos grains but (001)[100] in sol –gel material, with multiple slip or stress heterogeneity in both.
Olivine, the main constituent of the Earth’s upper mantle, has been thoroughly studied in the past decades to understand how crystallographic preferred orientations (CPO) develop under upper mantle conditions due to prevailing plate tectonic forces (Ben Ismaı¨l & Mainprice 1998; Tommasi et al. 2000; Jung et al. 2006; Karato 2008), but less attention has been given to fine-grained polycrystalline olivine. CPO development commonly occurs in coarse-grained olivine during plastic deformation (i.e. high-temperature dislocation creep, .1000 8C) where individual grains rotate and/or recrystallize, promoting anisotropy in seismic measurements (Karato 1988; Lee et al. 2002; Drury & Pennock 2007). For a fine-grained aggregate, however, stabilization of grain size below that at which extensive subgrain development takes place decreases the probability of recrystallization (White 1979) and is observed in essentially dry, solution–gelation (sol –gel) derived olivine (Faul et al. 2011) or ultra-mylonites in shear zones. Several slip systems potentially operate in olivine during plastic deformation. According to the von Mises criterion, five independent slip systems are needed for maximum strain compatibility (Von Mises 1928). For each slip system (hkl)[uvw], a geometrical Schmid factor S provides a measure of resolved shear stress (0 value 0.5) from the crystal orientation relative to external stress (Hull & Bacon 2001). For uniaxial compression, the Schmid factor is defined S ¼ cos a cos b
where a is the angle between compression direction and slip plane normal and b is the angle between compression direction and slip direction. For a polycrystalline aggregate, homogeneous stress must be assumed to calculate S for each grain. Studies on the fabric and seismic anisotropy of olivine have shown that dominant slip systems relate to the critical resolved shear stress on each slip system and temperature. Other dependencies can include chemical potential of components such as oxygen and silica (in the case of olivine) and ‘water’. The type-A CPO typically develops in low-OH natural olivine for high-temperature plastic deformation (Green & Radcliffe 1972; Goetze 1978; Ben Ismaı¨l & Mainprice 1998; Jung & Karato 2001). This CPO is characterized in shear deformation by [100] grain directions subparallel to the shear direction and [010] subnormal to the shear plane. Models employing the three most active slip systems in olivine, (010)[100], (001)[100] and (010)[001] (Ribe & Yu 1991; Wenk et al. 1991), can replicate this CPO type. Additional slip systems may also be operating, for example, (0kl)[100] (Passchier & Trouw 1998). The range of observed CPO types is summarized in Jung et al. (2006) for a variety of water concentrations in olivine. Two end-member states describe the interactions between grains during deformation, the uniform strain (Taylor) and uniform stress (Sachs) models (e.g. Winther et al. 1997; Dawson & Wenk 2000). In the Taylor model, some grains require higher
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 225– 235. DOI: 10.1144/SP360.13 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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Table 1. Rheological and microstructural data of sol –gel and San Carlos samples deformed at 1250 8C Sample Sol– gel San Carlos
Strain, 1total (%)
Differential stress, smax (MPa)
Strain rate, 1max/Dt (s21)
Grain size, d (mm)
rlow (mm22)
rhigh (mm22)
rmean (mm22)
18.9 15.0
266 227
1.1 1024 1.0 1024
4.2 + 0.2 13.0 + 0.5
c. 1 c. 2
c. 15 c. 16
8.5 + 0.2 11.4 + 2.9
grain-boundary stresses than others to produce uniform strain. In the Sachs model, some grains in favourable orientations deform more than neighbouring grains; some unfavourably oriented grains do not deform at all. In general, a positive correlation between dislocation density (r) and Schmid factor is more likely if grain interactions obey the Sachs model rather than the Taylor model (Karato & Lee 1999). However, it is likely that the deformation involves heterogeneous stress or strain as implemented in the Taylor –Bishop– Hill model (Lister et al. 1978) and the self-consistent model for a limited number of slip systems (Wenk et al. 1991; Tommasi et al. 2000). More recent modelling requires activation of four essential slip systems in olivine plus another degree of freedom to describe the microscopic stress heterogeneities (Castelnau et al. 2008). This study builds upon previous studies by Karato & Lee (1999) and Lee et al. (2002) and investigates to what extent the Taylor and Sachs models are applicable in deformed polycrystalline olivine by comparing fine-grained sol –gel olivine with a coarser-grained San Carlos for high-stress (but low-strain) compressive deformation. A robust analysis of slip systems from large populations of grains with high- and low-dislocationdensities has been possible because grain rotation and/or recrystallization is minor and CPO is weak or absent from our aggregates. Our new analyses have advantages over those from previous studies in two ways: firstly, Karato & Lee (1999) analysed only 5 –85 grains in each of four aggregates; secondly the CPO/dislocation density analyses conducted by Lee et al. (2002) involved two aggregates which both possessed a strong overall CPO (resulting from high shear strains).
Method Sol–gel olivine material was prepared from dissolution of Fe and Mg nitrates in an ethanol solution (Jackson et al. 2002; Faul & Jackson 2007). A second type of olivine material was prepared from crushed San Carlos phenocrysts (Tan et al. 2001). Cold-pressed pellets of each material were fired at 1400 8C at a 50/50 CO/CO2 gas mix and hotpressed in Ni70 –Fe30 foil at 1300 8C and 300 MPa
Ar confining pressure in a Paterson gas-medium apparatus (Paterson 1990). The porosity was reduced to 1–2%. After successful hot-pressing, sol –gel (H6529) and San Carlos specimens (H6694) were obtained. Compressive deformation experiments were carried out on cylindrical specimens using the same Paterson apparatus at 300 MPa confining pressure and 1250 8C. The sol–gel sample was deformed through a series of ramps and dwells of increasing load (Faul & Jackson 2007; run D6532, fig. 8). Care was taken to make sure (i) that steady-state creep was achieved for every dwell (visible in displacement versus time plots) and (ii) that the confining pressure was not exceeded in order to avoid embrittlement (as described by the Goetze criterion, e.g. Karato 2008). The maximum load was equivalent to a differential stress of 266 MPa and the total strain was 18.9%. The San Carlos specimen was deformed to a maximum differential stress of 227 MPa and a total strain of 15% (run D6701). The experiments were terminated with a furnace quench to preserve the dislocation microstructure. Table 1 contains a summary of the rheological data of the deformed specimens described above. Both specimens were sectioned perpendicular to the compressive axis, polished using colloidal silica (0.05 mm) and subsequently oxidized in air at 900 8C for 45 min to decorate dislocations (Kohlstedt et al. 1976; Karato 1987). Subsequent polishing for 10 min using either 0.05 mm alumina slurry (sol –gel olivine) or colloidal silica (San Carlos olivine) removed the surface oxide layer (,1 mm thick) plus 2–6 mm of material to expose suitably decorated grains and dislocations. The determination of dislocation density is an extension of previous work: dislocations are decorated by oxidation (e.g. Karato et al. 1986) then imaged using backscattered electrons in a scanning electron microscope (Karato & Lee 1999). The dislocationline-length per unit volume, a 3D dislocation density, was determined using procedures demonstrated by Farla (2010): the projected length of dislocations with a ‘depth of information’ value was obtained with Monte Carlo simulations of electron–olivine interactions. A Zeiss Ultraplus field-emission scanning electron microscope (FESEM) equipped with electron
SLIP-SYSTEM ANALYSIS IN DEFORMED OLIVINE
backscatter diffraction (EBSD) capabilities, was used in this study. An accelerating voltage of 20 kV was used with a nominal probe current of 0.6 nA. The stage was tilted to 708, creating a working distance of around 25 mm. An objective aperture of 30 mm with high-current mode was used. Diffraction patterns were collected using an HKL Nordlys S Camera, equipped with a forward-scattered electron detector to provide orientation contrast (OC) images. Manual indexing of one point per grain for sol – gel and San Carlos specimens was performed, guided by OC images. Approximate 3D dislocation density (with large uncertainty) was determined in grains to define two subsets: grains with high dislocation density (r 15 mm22) or low dislocation density (c. 1 mm22). San Carlos olivine with larger grain sizes readily allowed separation of grains into high- and low-dislocation-density classes so that no EBSD measurements were made on mediumdensity grains for this specimen. The same procedure was relatively difficult with the smaller grain size in the sol– gel aggregate. The grain size for the San Carlos specimen was determined from an EBSD map showing ‘grain’ boundaries of misorientations .108, whereas the grain size of sol– gel olivine was determined from an image of decorated grain boundaries. The average grain size was calculated as the diameter of a circle of equivalent area with a sectioning bias correction of 4/p (Jackson et al. 2002).
Results Fourier transform infrared (FTIR) measurements confirmed the deformed sol –gel sample had a low H2O content (,10 wt ppm H2O, at the detection limit) and an average concentration of 26 wt ppm H2O was measured for San Carlos olivine (D6701). The measurements were carried out by taking the height of the broad band absorption around 3400 cm21. The absorbance was calculated using the Bouguer –Beer –Lambert law (Aubaud et al. 2009) using a molar absorption coefficient of 1 ¼ 80 l mol21 cm21 (Leschik et al. 2004), calibrated for water in silicate glass. No evidence of peaks was found in the absorbance spectra for structurally bound OH in olivine.
Microstructural observations Figure 1 shows typical fine-grained olivine microstructures imaged using forward- and backscattered electron detectors. A heterogeneous distribution of dislocation density is evident. A binary contrast filter was applied to highlight the dislocations, which were manually recoloured in red (Fig. 1b, e).
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The orientation contrast images were captured during manual indexing to facilitate separation of grains with high versus low dislocation densities. During each EBSD session, the orientation contrast images preserved the location of the analysed data points which allowed identification of the indexed grains. These are indicated by green square symbols (as shown in Fig. 1f ). These grains were rediscovered using the backscatter electron detector (Fig. 1e) for further image processing and dislocation density determination. The yellow arrows indicate regions of low dislocation density, possibly resulting from grain-boundary bulging into adjacent high-dislocation-density grains (Fig. 1c, f ).
Pole figure analyses Pole figures were constructed for sol–gel and San Carlos olivine before and after the data were separated into subsets of grains of high or low dislocation density. Figure 2a shows the CPO in the unseparated data for sol–gel and San Carlos olivine, one manual EBSD measurement per grain. Note that the overall patterns for preferred orientation are very weak, consistent with deformation to just 15–19% strains. Multiples of uniform distribution (MUD) for the CPO patterns only have maximum values up to 1.75 (Fig. 2a) and should be used as a fabric strength indicator when comparing the CPO patterns from this study to others. A strong pattern will generally have a maximum MUD of 5–6. [100] lies near normal to the compression direction in either weak girdles (sol –gel) or point maxima (San Carlos), [010] is moderately concentrated subparallel to the compression direction but no pattern is identifiable for [001] (Fig. 2a). For reference, CPO from an undeformed specimen of the sol–gel material is given in Faul et al. (2011, fig. 6) In contrast, the CPOs strengthen considerably by data separation into dislocation-density subsets (Fig. 2b, d). The CPO for the sol–gel aggregate is generally weaker than that for San Carlos olivine. Turning first to the high-dislocation-density grains (Fig. 2b), San Carlos olivine shows a concentration of [010] oriented subparallel to the compression direction, but this is not evident in sol–gel olivine. San Carlos olivine also shows an [001] girdle normal to the compression direction while those in sol– gel olivine are possibly clustered in two separate girdles near 458 to the compression direction. Comparative plots (Fig. 2c) show contours of the maximum possible Schmid factor, only for (010)[100] slip, across all grain orientations. While there is some similarity to the measured patterns in Figure 2b (e.g. San Carlos [001]), it is not strong.
228 R. J. M. FARLA ET AL. Fig. 1. (a, d) Backscattered electron images (at 5 kV) of sol –gel and San Carlos olivine showing grain boundaries and grain-to-grain variations of dislocation density. (b, e) Same areas but showing the processed images with dislocations highlighted in red. Images like these were used to determine the 3D dislocation density. (c) Inverted orientation contrast image (at 20 kV) for sol –gel olivine for another area, illustrating a similar heterogeneous distribution of dislocation densities among grains. The yellow arrows show grain-boundary bulging and the circles enclose quadruple junctions. (f) Inverted orientation contrast image (at 20 kV) of San Carlos olivine of the same area in panels (d) and (e) for a comparison of grains of low and high dislocation density. The green square symbols indicate grains selected for indexing. These are low-dislocation-density grains seen in panel (d). The yellow arrows indicate grain-boundary bulging into high-dislocation-density grains.
SLIP-SYSTEM ANALYSIS IN DEFORMED OLIVINE
For the low-dislocation-density grains (Fig. 2d), pole figures for [100] show concentrations perpendicular to the compression direction in both materials. CPOs for [010] and [001] are different for sol –gel and San Carlos olivine; the strongest feature is a point maximum of [001] orientations parallel to the compression direction for San Carlos, corresponding to low Schmid factor for (010)[100] slip (Fig. 2c). Note also that for San Carlos CPOs, the [010] and [001] patterns show distinct complementarity between high- versus low-dislocation-density grains.
Schmid factor distributions Additional information can be obtained from the data through a direct comparison of calculated Schmid factor for the three main slip systems in olivine and the two dislocation-density subsets (Fig. 3). For each indexed grain, the HKL Channel software records Bunge Euler angles u1, F and u2 (Bunge 1982). With these Euler angles, the Schmid factor for any slip system can be determined. Histograms of the Schmid factor distribution for chosen slip systems across all grains were plotted as normalized frequencies for the high- (in red) and low(blue) dislocation-density subsets for sol– gel and San Carlos olivine (Fig. 3). Histogram data must be compared to a (green) baseline calculated for a uniform distribution of compression directions. Three main slip systems in olivine were considered using the histogram approach, (010)[100], (001)[100] and (010)[001] (the three columns of Fig. 3). The most obvious skews exist for San Carlos olivine with (010)[100]: in Figure 3d, not only is high S clearly indicated for high-dislocationdensity grains, but the reverse is also seen for low dislocation density. For sol– gel olivine, signs of activity exist for (001)[100] (Fig. 3b) but not for (010)[100] (Fig. 3a). In contrast, Figure 3a, c, e generally show trends that are indistinguishable from the uniform-distribution baseline. Figure 3f is different again; the distribution is skewed towards high S for low-dislocation-density grains and conflicts with that of the corresponding high-density grains where the trend is indistinguishable from uniform. (010)[001] slip is therefore not indicated for either olivine aggregate by this type of analysis.
Comparison of Schmid factor and dislocation density in inverse pole figures Where pole figures show how crystallographic directions of grains are distributed in the reference frame of the sample, inverse pole figures (IPFs) show how a chosen direction in the sample reference frame is distributed in the reference frame of the crystal. In Figure 4 the compression direction on
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each grain is plotted as an IPF for both high and low-dislocation-density subsets. In addition, Figure 4c shows (red regions in the same IPF space) orientations for compression directions that would produce high Schmid factors (.0.4) on each of the three slip systems already used in the previous section. The compression directions that yield maximum Schmid factor of 0.5 on single slip systems are indicated as [hkl]c where subscript c indicates the direction is in cartesian (not crystallographic) space; [101]c bisects [100] and [001] directions. No strong similarity to the measured IPFs is obvious for any one of the three slip systems chosen. In IPF representation, San Carlos olivine shows a CPO somewhat stronger (note maximum MUD between 2 and 3) than for sol–gel olivine. For both, there is distinct complementarity between low- and high-dislocation-density grains. Few grains of high dislocation density have [001] axes parallel to the compression direction, orientations with very low Schmid-factors for all of the slip systems. In San Carlos olivine, compression directions for high-dislocation-density grains lie mainly from [010] to [110]c, which partially corresponds to high Schmid factor values on (010)[100], (noting though that grains with a compression direction near [010] had low Schmid factors for all three slip systems). For low-dislocation-density grains in San Carlos, compression directions cluster between [001] and [011]c, the direction for maximum Schmid factor on (010)[001]. Sol– gel olivine shows a different pattern. In high-dislocation-density grains, many compression axes are near [101]c (possible (001)[100] slip) while few compression axes have high Schmid-factor values for (010)[100] slip. For low-dislocation-density grains, the most dominant clustering occurs near high Schmid factors for either (010)[100] or (010)[001] slip systems.
Rotation on sub-boundaries Lattice rotation (misorientation) axes across subgrain boundaries (2– 108) were obtained from EBSD map data (many measurements per grain) of the San Carlos specimen (Fig. 5) only. The data is represented in a crystal frame similar to the inverse pole figure above, but Figure 5 instead shows misorientation-axis distribution. Misorientation axes dominantly lie near [001] with others near [010].
Discussion The effect of rheology on slip-system activation Compressive deformation to modest strains in this study led to the overall development of very weak
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Fig. 2.
R. J. M. FARLA ET AL.
SLIP-SYSTEM ANALYSIS IN DEFORMED OLIVINE
CPO in sol –gel and San Carlos olivine with maximum MUD values of only 1.75, stronger for San Carlos olivine (Fig. 2a). This result is likely influenced by many parameters such as grain size, impurity content, water content and possibly melt fraction, the latter three essentially absent from sol –gel olivine. The strain rate/stress data for sol –gel olivine are presented in Faul et al. (2011) and for San Carlos olivine in Farla (2010). During deformation in the diffusion creep regime, both specimens demonstrated similar strain rates (5 1026 s21 at 110 MPa) despite the difference in grain size (see Table 1). In the dislocation creep regime, San Carlos olivine was significantly weaker than sol –gel olivine.
Slip-system analysis By separating high- and low-dislocation-density grains, we attempt to correlate Schmid factor with dislocation density in both olivine specimens. The grain orientations of both groups of high- and low-dislocation-density grains show weak CPO patterns and, quite noticeably, some complementarity of the CPO between the two groups. Complementarity is most pronounced in San Carlos for the [001] direction (Fig. 2b, c). Such an observation can be used to infer that grains in favourable orientations deformed via ‘easy’ slip and developed high dislocation densities. On the other hand, grains in unfavourable orientations for slip presumably deformed little and should have low dislocation densities. If this inference is true, then a signature of (010)[100] slip (the easiest system in olivine) should be observable in the CPO for grains of high dislocation density. As the schematic pole figures in Figure 2c demonstrate, the CPO of grains with high (010)[100] Schmid factor should feature two girdles for [100] and [010] at 458 from the compression direction; such girdles possibly exist for sol –gel olivine but certainly not for San Carlos olivine (Fig. 2). In addition, a girdle normal to the compression direction should occur for [001]. A pronounced feature of this type is seen for San Carlos olivine, but not for sol –gel olivine
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(Fig. 2b). Complementary patterns should be observed at low dislocation density given that the overall CPO is essentially uniform (see Fig. 2a). Parallel to the compression direction at low dislocation density (Fig. 2d), San Carlos grains show distinct [001] concentration whereas sol–gel olivine shows a weaker [010] maximum. Evidence for activation of a slip system can possibly be deduced more readily by skews in the histograms of Figure 3 relative to the uniform distribution if reverse trends exist for high- versus low-dislocation-density grains. On this basis, there is evidence for activation of (010)[100] in San Carlos olivine (Fig. 3d) and (001)[100] in sol–gel olivine (Fig. 3b). Both low- and high-dislocationdensity subsets are indistinguishable from the green baseline in three cases (Fig 3a, c, e). San Carlos olivine even shows some trend against activation of (010)[001] (Fig. 3f ). The histograms show no single slip system to have been dominant in either specimen. In the IPFs of Figure 4, concentrations of the compression direction for high-dislocation-density grains are expected to lie near [110]c for (010)[100] slip, near [101]c for (001)[100] slip and near [011]c for (010)[001] slip (Fig. 4c). Neither specimen clearly shows a pattern suggestive of single slip. However sol–gel olivine at high dislocation density does show clustering in the vicinity of [101]c, suggesting that (001)[100] has been active. San Carlos, on the other hand, appears to have deformed on (010)[100], although the highest concentration of the compression direction in high-dislocation-density grains lies near to [010]. Typically, this is observed when the grains are able to rotate during plastic deformation (Karato 2008). However, given the low strain in our experiments, multiple slip systems and grain-to-grain stress heterogeneities are more likely explanations. Misorientation-axis data (2–108 rotations) plotted in Figure 5 can be compared with those expected for the three main slip systems and two types of dislocation substructures, tilt versus twist boundaries. For a single slip system (i.e. tilt boundaries of edge dislocations), [001] misorientations measured on internal boundaries strongly indicate
Fig. 2. Pole figures presenting grain orientation data for both sol–gel and San Carlos olivine. The compression direction is represented by pairs of black arrows. The half-scatter width is 208 with 58 clustering. An equal-area lower-hemisphere projection was used. (a) Pole figures for all measured grains. The patterns are weak with low maximum multiples of uniform distribution (MUD) values (see legends). Concerning the number of grains analysed, for sol–gel olivine it includes grains of medium dislocation density but, for San Carlos olivine, it excludes them. (b) and (d) show pole figures separated into grain subsets of high or low dislocation density; note that higher maximum MUD values here indicate stronger preferred orientation. The CPO patterns in the pole figures between high and low dislocation density partially complement each other for both sol–gel and San Carlos olivine. (c) Pole figures contoured for the range of maximum possible Schmid factors on slip system (010)[100] plotted for all grain orientations (also equal-area projections).
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Normalized frequency
Normalized frequency
(e)
(f) Normalized frequency
Normalized frequency
(d)
(c)
Fig. 3. Schmid-factor distributions for sol–gel and San Carlos olivine for the three main slip systems (010)[100], (001)[100] and (010)[001]. The frequency of grains ( y-axis) has been normalized for comparison. The subsets of high- and low-dislocation-density grains are shown by the red and blue lines, respectively, presented as histograms at 0.05 Schmid factor intervals. As a reference, the green curve shows a baseline calculation of normalized frequency for Schmid factors of a uniform distribution of grain orientations. Significant deviations from the green curve could indicate activation of a slip system.
R. J. M. FARLA ET AL.
Normalized frequency
(b)
Normalized frequency
(a)
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Fig. 4. (a, b) Equal-area inverse pole figures in which the compression direction is plotted in the olivine crystal frame for sol–gel and San Carlos olivine of both dislocation density subsets. Projections are equal-area with a half-scatter width of 208 and 58 clustering. Also, n is the number of grains analysed. (c) An additional inverse pole figure shows orientations of compression axes for the highest Schmid factors (.0.4) on the three main slip systems in olivine.
slip in San Carlos olivine grains on (010)[100], but also minor contribution from (001)[100] required to explain the [010] rotations. Alternatively, a twistboundary geometry involving two families of screw dislocations with Burgers vectors either [100] or
[001] would explain this [010] rotation axis. These deductions are broadly consistent with results in Figures 3 and 4. Discrepancies between Schmid factor and dislocation density in Figures 2b, d, 3 and 4 suggest that a model for uniform stress (Sachs) cannot be valid. Put a different way, slip in high-Schmid-factor grains could possibly have been suppressed by stress transfer to surrounding ‘hard’ grains. In addition, stress heterogeneity is supported by considerable grain-to-grain variations in dislocation density and by weak patterns from the measured pole figures and IPFs. Note that conclusions about stress heterogeneity were similarly reached by Karato & Lee (1999) and Lee et al. (2002). Furthermore, there is evidence for operation of multiple slip systems in our deformed aggregates. For San Carlos olivine, we conclude that (010)[100] operated during deformation (Figs 3d & 4b and especially Fig. 5), but all the analyses suggest some contributions from other slip systems. For sol– gel olivine, (001)[100] is apparently active (Figs 3b & 4a) but again accompanied by other slip systems.
Fig. 5. Plot of misorientation-axis data for low angle (2– 108) internal sub-grain boundaries shown in the crystallographic reference frame; data computed from the San Carlos EBSD map. The half-scatter width is 208 and 38 clustering. Here, n is the number of points (not grains) analysed.
Comparison with previous studies The dislocation-density distribution and substructures that evolve during deformation of our fine-grained olivine appear to be different than in olivine aggregates undergoing dynamic
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recrystallization under high-strain simple shear (Karato & Lee 1999; Lee et al. 2002). Commonly, a heterogeneous distribution of dislocation density is observed in low-OH deformed olivine polycrystals. This may result from varying intra-granular stresses between grains, where grains which are favourably oriented are more likely to deform. However, from high-strain deformation experiments, such grains are less likely to dominate the final preferred orientations caused by grainboundary migration (Zhang et al. 2000). In this study, some evidence for grain-boundary bulging and sliding between grains is observed, most clearly for fine-grained sol –gel olivine (see Fig. 1) but not at the expense of a significant increase in grain-boundary area or an increase in low-dislocation-density grains. Grain shapes are not typical of high-temperature stress-induced grain-boundary migration and vary between equigranular polygonal or mild interlobate structures (Passchier & Trouw 1998). It is not well known to what degree the microstructural state of stress and strain plays a role between grains (Taylor 1938; Karato & Lee 1999). However, a second-order-estimate model has recently (Castelnau et al. 2008) simulated stress heterogeneities in polycrystalline olivine. This model essentially falls between the Taylor (uniform strain) and Sachs (uniform stress) end-member models. The second-order modelling highlights the likelihood of localization of stress and strain in a deforming olivine polycrystal with local stresses and strains significantly higher or lower than the imposed (bulk) macroscopic equivalents. All the grains in the model of Castelnau et al. (2008) were allowed five degrees of freedom via the three known slip systems in olivine plus two ‘arbitrarily’ assigned systems. While the details of the chosen parameters are debatable, the effects of strain localization, fluctuations and stress gradients in and between grains have important consequences for microstructural evolution in olivine aggregates.
density, of Schmid-factor distribution and of misorientation-axis data arising from an EBSD map have been analysed for evidence of correlation between Schmid factor (on three olivine slip systems) and dislocation density. The patterns show many complexities that defy explanation by activation of any one slip system, although the misorientation-axis analysis (San Carlos olivine only) does appear to allow a straightforward interpretation. For coarser-grained San Carlos olivine, there is strong evidence from all analyses that grains of high dislocation density are favourably oriented to deform via the slip system (010)[100], but probably with important contributions from other slip systems. For sol–gel olivine, equivalent analyses suggest (001)[100] slip but the evidence is weaker than for San Carlos olivine and also suggestive of contributions from multiple slip systems. Perhaps differences in grain size, OH and impurity content could have played a role in the slip-system contrast between these two types of olivine aggregate, although more experimental evidence is needed to confirm the patterns. Heterogeneous stress provides an explanation for deformation of grains in unfavourable orientations relative to the external compressive stress (i.e. with low Schmid factor). Variation in dislocation density between grains indicates that strain is heterogeneous. Furthermore, the CPO data described above re-affirm that neither the Sachs model of uniform stress nor the Taylor model of uniform strain are adequate to describe plastic deformation and textural evolution of olivine. Plastic deformation involves multiple slip systems and is more comparable to advanced simulations such as the second-order micromechanical model of Castelnau et al. (2008). We thank F. Brink for his assistance with FESEMs and EBSD at the ANU Centre for Advanced Microscopy in the Australian Microscopy & Microanalysis Research Facility. R. Farla acknowledges the award of an Endeavour International Postgraduate Scholarship by the Australian Government.
Conclusion This study for weakly strained (c. 15 –19%) synthetic olivine aggregates shows a near-uniform CPO, together with low maximum-MUD values. An overall near-uniform CPO offered a good possibility for identification of operating slip systems in olivine, partly through searching for complementarity expected between grain populations of high and low dislocation density without significant impacts from grain rotation or recrystallization during low-strain deformation. Patterns of preferred orientation from pointindexed grains of high and low dislocation
References Aubaud, C., Bureau, H., Raepsaet, C., Khodja, H., Withers, A. C., Hirschmann, M. M. & Bell, D. R. 2009. Calibration of the infrared molar absorption coefficients for H in olivine, clinopyroxene and rhyolitic glass by elastic recoil detection analysis. Chemical Geology, 262, 78–86. Ben Ismaı¨l, W. & Mainprice, D. 1998. An olivine fabric database: an overview of upper mantle fabrics and seismic anisotropy. Tectonophysics, 296, 145–157. Bunge, H. J. 1982. Texture Analysis in Materials Science: Mathematical Models. Butterworths, London.
SLIP-SYSTEM ANALYSIS IN DEFORMED OLIVINE Castelnau, O., Blackman, D. K., Lebensohn, R. A. & Ponte Castaneda, P. 2008. Micromechanical modeling of the viscoplastic behaviour of olivine. Journal of Geophysical Research, 113, B09202. Dawson, P. R. & Wenk, H. R. 2000. Texturing of the upper mantle during convection. Philosophical Magazine A, 80, 573–598. Drury, M. R. & Pennock, G. M. 2007. Subgrain rotation recrystallization in minerals. In: Prangnell, P. B. & Bate, P. S. (eds) Materials Science Forum, Fundamentals of Deformation and Annealing. Trans Tech Publications, Switzerland, 550, 95–104. Farla, R. J. M. 2010. An exploratory study of dislocation relaxation in polycrystalline olivine. PhD thesis, The Australian National University. Faul, U. H. & Jackson, I. 2007. Diffusion creep of dry, melt-free olivine. Journal of Geophysical Research, 112, 1 –14. Faul, U. H., Fitz Gerald, J. D., Farla, R. J. M., Ahlefeldt, R. & Jackson, I. 2011. Dislocation creep of fine-grained olivine. Journal of Geophysical Research, 116. http://dx.doi.org/10.1029/2009JB007174. Goetze, C. 1978. Mechanisms of creep in olivine. Philosophical Transactions of the Royal Society London, 288, 99–119. Green, H. W. & Radcliffe, S. V. 1972. Dislocation mechanisms in olivine and flow in upper mantle. Earth and Planetary Science Letters, 15, 239. Hull, D. & Bacon, D. J. 2001. Introduction to Dislocations. 4th edn. Butterworth-Heinemann, Oxford, UK. Jackson, I., Fitz Gerald, J. D., Faul, U. H. & Tan, B. H. 2002. Grain-size sensitive seismic wave attenuation in polycrystalline olivine. Journal of Geophysical Research B, 107, 2360. Jung, H. & Karato, S. 2001. Water-induced fabric transitions in olivine. Science, 293, 1460– 1463. Jung, H., Katayama, I., Jiang, Z., Hiraga, T. & Karato, S. 2006. Effect of water and stress on the latticepreferred orientation of olivine. Tectonophysics, 421, 1–22. Karato, S. 1987. Scanning electron microscope observation of dislocations in olivine. Physics and Chemistry of Minerals, 14, 245 –248. Karato, S. 1988. The role of recrystallization in the preferred orientation of olivine. Physics of the Earth and Planetary Interiors, 51, 107– 122. Karato, S. 2008. Deformation of Earth Materials: An Introduction to the Rheology of Solid Earth. Cambridge University Press, Cambridge. Karato, S. & Lee, K. H. 1999. Stress-strain distribution in deformed olivine aggregates: inference from microstructural observations and implications for texture development. In: Proceedings of the Twelfth International Conference on Textures of Materials, Montre´al, Canada, 1546–1555. Karato, S. I., Paterson, M. S. & Fitz Gerald, J. D. 1986. Rheology of synthetic olivine aggregates-
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influence of grain-size and water. Journal of Geophysical Research B, 91, 8151–8176. Kohlstedt, D. L., Goetze, C., Durham, W. B. & Vandersande, J. 1976. New technique for decorating dislocations in olivine. Science, 191, 1045– 1046. Lee, K., Jiang, Z. & Karato, S. 2002. A scanning electron microscope study of the effects of dynamic recrystallization on lattice preferred orientation in olivine. Tectonophysics, 351, 331– 341. Leschik, M., Heide, G. et al. 2004. Determination of HO and DO contents in rhyolitic glasses. Physics and Chemistry of Glasses, 45, 238 –251. Lister, G. S., Paterson, M. S. & Hobbs, B. E. 1978. The simulation of fabric development in plastic deformation and its application to quartzite: the model. Tectonophysics, 45, 107– 158. Passchier, C. W. & Trouw, R. A. J. 1998. Microtectonics. 2nd edn. Springer, Berlin Heidelberg, Germany. Paterson, M. S. 1990. Rock deformation experimentation. In: Duba, A. G. (ed.) The Brittle-Ductile Transition in Rocks. American Geophysical Union, Washington, Geophysical Monograph Series, 56, 187– 194. Ribe, N. M. & Yu, Y. 1991. A theory for plastic deformation and textural evolution of olivine polycrystals. Journal of Geophysical Research B, 96, 8325– 8335. Tan, B. H., Jackson, I. & Fitz Gerald, J. D. 2001. Hightemperature viscoelasticity of fine-grained polycrystalline olivine. Physics and Chemistry of Minerals, 28, 641– 664. Taylor, G. I. 1938. Plastic strain in metals. Journal of the Institute of Metals, 62, 307–324. Tommasi, A., Mainprice, D., Canova, G. & Chastel, Y. 2000. Viscoplastic self-consistent and equilibriumbased modeling of olivine lattice preferred orientations: implications for the upper mantle seismic anisotropy. Journal of Geophysical Research, 105, 7893– 7908. Von Mises, W. 1928. Mechanik der plastischen formanderung von kristallen. Applied Mathematics and Mechanics, 8, 161. Wenk, H. R., Bennett, K., Canova, G. R. & Molinari, A. 1991. Modelling plastic deformation of peridotite with the self-consistent theory. Journal of Geophysical Research B, 96, 8337–8349. White, S. H. 1979. Grain and sub-grain size variations across a mylonite zone. Contributions to Mineralogy and Petrology, 70, 193– 202. Winther, G., Jensen, D. J. & Hansen, N. 1997. Modelling flow stress anisotropy caused by deformation induced dislocation boundaries. Acta Materialia, 45, 2455– 2465. Zhang, S. Q., Karato, S., FitzGerald, J. D., Faul, U. H. & Zhou, Y. 2000. Simple shear deformation of olivine aggregates. Tectonophysics, 316, 133–152.
Characterization of microstructures and interpretation of flow mechanisms in naturally deformed, fine-grained anhydrite by means of EBSD analysis REBECCA C. HILDYARD*, DAVID J. PRIOR, ELISABETTA MARIANI & DANIEL R. FAULKNER Department of Earth and Ocean Sciences, University of Liverpool, 4 Brownlow Street, Liverpool L69 3GP, UK *Corresponding author (e-mail:
[email protected]) Abstract: Anhydrite-rich layers within foreland fold and thrust belts are frequently observed to be the weakest horizon of the sequence. Characterizing the microstructure of anhydrite is therefore important for interpreting the larger-scale deformation history of these rocks. Two microstructures from naturally deformed, fine-grained (,15 mm mean grain size) anhydrite samples from the Triassic Evaporites of the Umbria– Marche Apennines, Italy were analysed using electron backscatter diffraction (EBSD). Microstructural observations, misorientation analysis and crystallographic preferred orientation (CPO) determination were carried out on these samples. Both samples have a CPO characterized by alignment of k001l and distribution of k100l and k010l on a great circle normal to this. This anhydrite k001l ‘fibre texture’ has not been described before. Microstructure A is characterized by a moderate to weak CPO and a weak shape preferred orientation at 558 to 708 from the trace of the k001l maximum. Low-angle boundaries are revealed by misorientation analysis. A change in grain size from c. 10 to c. 7 mm corresponds to reduction in strength of CPO and reduction in the number of low-angle grain boundaries. Microstructure B is characterized by a very strong CPO. The orientation of the CPO changes between different microstructural domains. The k001l maximum is always perpendicular to the trace of a strong grain elongation and high-angle grain boundaries have misorientations close to k001l, suggesting that the CPO is geometrically controlled: anhydrite grains are platy with k001l short axes. The origin of the CPO is therefore unclear but it need not relate to dislocation creep deformation. Whether or not CPO relates to dislocation creep, both samples have a high number of lower-angle grain boundaries and internal grain distortions with k010l and k001l misorientation axes. These are indicative of dislocation activity and the data are best explained by slip on either (100)[010] (dominant) and (001)[100] or a combination of these. Neither of these slip systems has been recognized before. Both microstructures are interpreted to have undergone dynamic recrystallization, and the weakening of the CPO with decreasing grain size in microstructure A is suggested to be indicative of a grain-boundary sliding mechanism becoming active. Comparison with experimental data shows that creep mechanisms involving dislocations at the observed grain sizes require the differential stress magnitudes driving deformation to be greater than c. 100 MPa.
Evaporitic rocks have been recognized to act as detachment horizons in many fold and thrust belts globally, including the Northern Apennines in Italy (see e.g. Bally et al. 1986; Barchi et al. 1998), the Jura Mountains of Switzerland (Laubscher 1975; Jordan & Nuesch 1989) and the Antalya thrust system in southern Turkey (Marcoux et al. 1987). The deformation in these zones is thought largely to be plastic (e.g. Jordan 1992) (i.e. accommodated by viscous processes such as dislocation and diffusion creep) but brittle deformation also occurs. For example, it was recently observed that the main shocks of the 1997 Colfiorito earthquake sequence in the Umbria Marche Apennines nucleated on normal faults within the Triassic Evaporites (Barchi et al. 1998; Miller et al. 2004).
The Triassic Evaporites (Fig. 1) are a c. 2 km thick sequence of metre- to decimetre-scale interbeds of gypsum-anhydrite, dolostones and variable amounts of halite. During the Jurassic, original gypsum was deposited in a shallow marine environment and later replaced by anhydrite after c. 1 km of burial (Murray 1964; De Paola et al. 2007; Collettini et al. 2008). A complex post-depositional history of diagenesis and deformation followed as the Triassic Evaporites acted as the main detachment horizon of the Tuscan Nappe during the formation of the Northern Apennine chain (Bally et al. 1986; Boccaletti et al. 1987; Carmignani & Kligfield 1990). Regional contraction followed by extension, uplift and erosion, which heavily modified all primary sedimentary structures and textures (Lugli
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 237– 255. DOI: 10.1144/SP360.14 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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subsurface samples are therefore needed. Here, borehole-derived samples of anhydrite provide an opportunity to study the microstructures of the Triassic Evaporites. Two microstructures from very-fine-grained anhydrite samples (hereafter referred to as MS-A and MS-B) with some common characteristics are studied to constrain deformation mechanisms and rheological implications. The crystallographic preferred orientations (CPO) displayed in these two samples are used together with misorientation data and analysis of intracrystalline distortions to attempt to identify the deformation mechanisms (including dislocation slip systems) that were operative.
Anhydrite crystallography and rheology
Fig. 1. Generalized geological and location map of samples used in this study.
2001). The dehydrated evaporites were deformed at mean pressures in the region of c. 200 – 250 MPa and temperatures of c. 230 –315 8C (Barchi et al. 1998; Lugli 2001; De Paola et al. 2008). The anhydrite-rich rocks of the Triassic Evaporites are characterized by flow structures of layer-parallel millimetre- to centimetre-scale pseudo-lamination, which in very pure anhydrite layers consists of very tight and transposed isoclinal folds (Lugli 2001). Surface exposures of the Triassic Evaporites tend to be composed predominantly of gypsum as subsequent hydration reactions have re-replaced the anhydrite; to study anhydrite deformation,
Anhydrite has orthorhombic crystal symmetry and, as outlined by Hildyard et al. (2009a), has in previous textural studies been represented by the crystallographic reference frame of space group Bbmm (Mugge 1883; Ramez 1976; Mainprice et al. 1993; Dell’Angelo & Olgaard 1995; Heidelbach et al. 2001). Hildyard et al. (2009a) described the need to use a different crystal setting due to the electron backscatter diffraction (EBSD) software only containing standard crystal settings. Hildyard et al. (2009a) transformed the atomic positions of anhydrite in the space group Amma (Cheng & Zussman 1963; Hawthorne & Ferguson 1975) to a standard space group of Cmcm (table 3 in Hildyard et al. 2009a gives pre- and post-transformation crystal files). This was done via a transformation matrix in a routine made available by the Bilbao Crystallographic Server (Aroyo et al. 2006a, b). Due to this rotation all previously published crystallographic planes, directions and slip systems discussed below have been rotated into this standard (Cmcm) reference frame for comparison purposes and all crystallographic data in this paper are referred to in the standard (Cmcm) reference frame (Tables 1 & 2 give a summary). Bulk CPO analyses identified the following general intracrystalline deformation mechanisms
Table 1. Summary of slip systems from previous work, the rotated equivalents of previous work and the slip system from this work Slip systems in original reference frame (Bbmm) for example Muller et al. (1981) (001)[010] (012)[-1-21] (012)[1-21]
Slip systems converted into standard reference frame (Cmcm) Hildyard et al. (2009a)
Slip systems identified in Hildyard et al. (2009a)
(100)[001] (201)[1-1-2] (201)[11-2]
(100)[001]
EBSD ANALYSIS OF FINE-GRAINED ANHYDRITE
Table 2. Slip systems confirmed by Hildyard et al. 2009a and possible slip systems identified in this work Slip systems identified in Hildyard et al. 2009a (100)[001]
Possible new slip systems identified in this work (100)[010] R ¼ 001 (001)[100] R ¼ 010
and slip systems for polycrystalline anhydrite (Mugge 1883; Ramez 1976): (1) (2) (3)
translation glide on (100)[001]; translation glide on (201)[1-1-2] and [11-2]; twinning on {110} (a rotation of the [100] axis by 83.58 around [001]; Klassen-Neklyudova 1964).
Investigations into naturally deformed anhydrite rocks have recognized CPOs which reflect a preferential alignment of the {100} planes subparallel to the foliation and k001l parallel to the lineation (Schwerdtner 1970; Jordan 1992; Mainprice et al. 1993; Hildyard et al. 2009a), providing evidence for glide on the (100)[001] slip system in naturally deformed rocks. Numerous experimental investigations of the rheological behaviour of anhydrite have been conducted (e.g. Handin & Hager 1958; Ramez 1976; Muller & Briegel 1978; Muller et al. 1981; Dell’Angelo & Olgaard 1995; Heidelbach et al. 2001). A brief outline of some of the key experiments relevant to this work is provided below. In axial compression experiments, Muller et al. (1981) found that the strength of anhydrite increases with decreasing grain size up to 350 8C and decreases with decreasing grain size above 400 8C. No evidence for diffusion creep was found but the microstructures they observed (in the experiments conducted at 300 –450 8C) indicated dynamic recrystallization by grain-boundary migration, which was used to explain the apparent resistance to work hardening of the finegrained aggregates. Muller et al. (1981) measured strong preferred orientation of (010) poles parallel to shortening and argued that this was related to a twinning mechanism rather than to dislocation slip systems. Subsequent to this, in a seminal paper on the topic, Dell’Angelo & Olgaard (1995) used experimental results to characterize two high-temperature flow regimes and show evidence for both dislocation creep and diffusion creep within both natural and synthetic anhydrite. The first regime (regime 1) was characterized as a twinning and dislocation creep regime that showed strain hardening at stresses above 150 MPa and steady-state flow at
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low stresses. Regime 1 is split into two parts based on the microstructure such that regime 1a is identified by twinning, grain flattening, undulose extinction, high dislocation density, sutured grain and twin boundaries and a strong CPO with (010) poles aligned parallel to shortening. Regime 1b is identified by microstructures indicative of increased grain-boundary mobility and dislocation recovery, grain-size reduction, low dislocation density, unstrained recrystallized grains and strong CPO with (010) poles aligned parallel to shortening and (110) more weakly. Regime 2 was defined by diffusion creep accompanied by grain-boundary sliding where they observed no evidence of deformation on the grain scale despite shortening of 45%, no undulose extinction, straight grain boundaries, a very low dislocation density, no subgrains and almost no CPO. They also observed in the transition from regime 1 to regime 2, and therefore the transition from dislocation creep to diffusion creep, polygonal and subequant grains, some undulose extinction, straight to serrate grain boundaries and a weak CPO. They suggested that in this transition zone, recrystallization occurred by subgrain rotation and that, in general, the microstructures were indicative of both diffusion creep and dislocation creep mechanisms, hinting at a possible mechanism of grain-boundary sliding accompanying dislocation creep. It is not clear in their discussion whether Dell’Angelo & Olgaard (1995) relate the alignment of (010) poles to shortening as related to twinning (after Muller et al. 1981) or to dislocation glide, although Heidelbach et al. (2001) related this CPO in the Dell’Angelo and Olgaard experiments to twinning. It is worth noting that the CPO observed by Dell’Angelo and Olgaard differs from the CPO observed in studies of naturally deformed anhydrite (Schwerdtner 1970; Jordan 1992; Mainprice et al. 1993) and in later torsion experiments (Heidelbach et al. 2001). Heidelbach et al. (2001) deformed anhydrite in torsion. They observed a strong alignment of (100) subparallel to the shear plane and [001] subparallel to the shear direction, and ascribed this to the (100)[001] slip system. In detail, both (100) and [001] were not aligned exactly with foliation and lineation; they are rotated counter to the shear induced rotation. The sample developed a strong grain-shape fabric with grain long axes oriented c. 258 from the shear plane. The CPO does not start to develop until shear strains of 1.5 –2 and is rotated c. 408 from kinematic alignment. The CPO reaches maximum strength after shear strains of 3 –4 and is rotated c. 208 from the kinematic framework at this stage. This remained more or less unchanged up to a shear strain of c. 8, with a small reduction in back-rotation angle to 108. In
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this experiment, strain softening occurred between shear strains of 1 and 2. This was accompanied by a reduction in grain size and a change in stress exponent (measured by strain-rate stepping) from .3 to 1, interpreted as a change from dislocation creep to diffusion creep. It has been recognized that the transition zone that separates dislocation creep from diffusion creep may be a deformation mechanism field in its own right, in which grain-boundary sliding accompanies dislocation creep. This field has been recognized in experiments on ice (Goldsby & Kohlstedt 1997, 2001; Durham et al. 2001), olivine (Hirth & Kohlstedt 1995; Drury and Fitz Gerald 1998), calcite (Schmid et al. 1977, 1980; Walker et al. 1990; Rutter et al. 1994), perovskite (Mecklenburgh et al. 2010) in natural olivine peridotites (Warren & Hirth 2006) and naturally deformed plagioclase and pyroxene gabbros (Mehl & Hirth 2008). This field has been named DisGBS (Warren & Hirth 2006). In dislocation creep, the strain rate is independent of grain size (grain size insensitive) and the strength is controlled by the strongest slip system of several slip systems operating together. In DisGBS deformation is accommodated by glide on only one slip system (the weakest) plus grain-boundary sliding. Whichever of these two mechanisms is stronger will control the rheology (Goldsby & Kohlstedt 2001; Hirth & Kohlstedt 2003). DisGBS is grainsize sensitive.
Analytical techniques Sample preparation The samples used were cut from borehole cores. The sample containing MS-A was taken from a core from the Burano 1 borehole, from a depth of 2236 m, and the sample containing MS-B was taken from the Fossobrone borehole from 2064 m depth. Due to the method of core drilling and recovery, the specimens are unorientated and way up is not known. However, the cores were drilled vertically and the orientation of the core axis (X ) with respect to the thin sections is known (Fig. 2). Thin sections were orientated such that the core axis was parallel to the short axis of the thin section. The rock chips cut from the borehole core were made into 30 mm thick polished thin sections. The use of water was excluded during the preparation of the thin sections to avoid any hydration of the anhydrite. Only in the final stages of chemomechanical polishing was water-based SYTON (Fynn & Powell 1979) used for very short time intervals (15 min for each sample). BSE imaging of the samples was used to confirm that no hydration of the anhydrite had occurred during preparation. Finally, the samples were coated with a thin film
of carbon to prevent charging (Lloyd 1987; Prior et al. 1996).
EBSD data acquisition Full crystallographic orientation data were collected using a Philips XL30 SEM with a tungsten filament. In order to perform EBSD, samples are rotated to 708 from the horizontal while the electron beam is vertical. The EBSD patterns were collected using an accelerating voltage of 20 kV and beam current of 3 nA. The working distance between sample and pole piece was 24 mm. Samples were ‘mapped’ using automated beam scanning, where the specimen is stationary and the beam scans a user-defined area. The stage then drives to a new area and the beam scanning is resumed. Due to the fine grain size of both samples, step sizes of 1 and 2.5 mm were used for MS-A and MS-B, respectively. The EBSD patterns were indexed with the program Channel5 from Oxford Instruments. EBSD patterns were indexed using a minimum of 7 and a maximum of 8 bands and 74 reflectors.
Data processing All EBSD raw data were post-processed using the Channel5 software package to replace non-indexed points (where the software is unable to find a solution for the diffraction pattern) or remove misindexed (where the software applies the wrong solution, for example one point of dolomite in a large grain of anhydrite) data points. The process used here was first to replace wild spikes (i.e. an isolated mis-indexed pixel) with a zero solution (i.e. a non-indexed point). Non-indexed points with a minimum of 5 indexed neighbouring pixels with orientations within 108 of each other were then replaced with the average orientation of the neighbouring pixels. To avoid introducing artefacts by growing grains past their grain boundaries, this procedure was limited to a subset of data with high band contrast/pattern quality. Grain boundaries have low pattern quality and are excluded from the subset (Prior et al. 2009). This process was iterated until all non-indexed points with 5 or more indexed neighbours had been replaced, or until the edge of the subset (i.e. edge of the grain as defined by the band contrast/pattern quality threshold) was reached. Finally, to fill any remaining areas without any data, the full dataset was noise-reduced (again, replacing non-indexed points with 5 or more neighbours with a common orientation following the approach adopted by Bestmann & Prior 2003). Before noise reduction and processing, the datasets used here had 80 –85% indexing with very few mis-indexed points. Indexing increased to .95% after processing.
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Fig. 2. Transmitted light micrographs of the samples used in this study. Microstructure A: (a) plane-polarized light micrograph of the full thin section, laminations are highlighted by black lines, core axis is shown, the box labelled (b) shows the location of the micrograph in part (b); (b) plane-polarized and cross-polarized micrograph of the area studied with EBSD, the white line indicates the zone where a change in grain size occurs; (c) plane-polarized and cross-polarized micrograph of a close-up of the boxed area in (b) showing the change in grain size (white dashed line). Microstructure B: (d) cross-polarized light micrograph of the full thin section, arrows and ‘S’ highlight the micro-shear features; (e) crossed-polarized light micrograph of a close-up of one of the micro-shears, indicated by ‘S’ and arrow; (f ) plane-polarized and cross-polarized micrograph of a close-up of the micro-shear shown in (e). The reference frame for these samples is shown by the schematic with X parallel to the core axis.
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Fig. 3. A range of EBSD data from Microstructure A. (a) All Euler and band-contrast maps of the subsets (S1, S2, S3, S4) used in this analysis. Lower-hemisphere stereographic pole figures show the CPO of each subset. There is a marked increase in the strength of the CPO with increasing grain size (highlighted by the M-index). Data are plotted as one point per grain, contoured using a half-width of 208 and a cluster size of 58. (b) Histograms of the grain-size distribution in each subset. (c) Band-contrast map with colour-coded grain boundaries superimposed (see key on right for boundary intervals; note the increase in high-angle .158 boundaries with decreasing grain size). (d) Phase map
EBSD ANALYSIS OF FINE-GRAINED ANHYDRITE
Grain-size and grain-shape analysis Grain-size distributions were measured by means of high-resolution (1 mm) EBSD data. Grains were automatically determined to be grains rather than subgrains when they are completely surrounded by boundaries with misorientation angles .108. This figure is arbitrarily chosen as the misorientation angle that best reflects the transition between grain boundaries and subgrain boundaries (Shigematsu et al. 2006). Two-dimensional grain size is calculated as the diameter of a circle of equivalent area to the measured grain size. Grains with a diameter less than 4 times the step size were removed from analysis (e.g. if the step size used was 1 mm, grains with a diameter of 4 mm were not included in the analysis). From this information, determining average grain size in a statistically representative way is nontrivial. Grain size is often expressed as the arithmetic mean. However, most recrystallized grain distributions have a non-Gaussian distribution and have often been found to show a log-normal distribution; the geometric mean (not the same as the median) therefore gives a better representation of the grain microstructure (e.g. Ranalli 1984; Newman 1994; De Bresser et al. 2001). The geometric standard deviation is also given as a better approximation of the distribution of grain size. To assist in identifying the general grain shape preferred orientation in MS-A, a variation on Fry plots was used (Fry 1979). Enhanced Normalized Fry (ENFry) plots (Erslev & Ge 1990) modify the original approach to improve the definition of the central vacant field by normalizing centre-to-centre distances (Erslev 1988) and apply an object-pair selection factor (Erslev & Ge 1990) to determine which points are used in the calculation. The result of this is to reduce the amount of data generated by Fry analysis, to improve accuracy and remove subjectivity. The computer programs developed by Waldron & Wallace (2007) and grain data taken from the EBSD maps are used here to produce ENFry plots.
Results Both of the anhydrite microstructures described are from samples derived from boreholes drilled within the Triassic Evaporites (see Fig. 1); the thin
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sections and EBSD data are viewed relative to the core axis (X ).
Transmitted light microscopy Both samples are of a fine grain size (c. 10–50 mm) and consist of c. 95% anhydrite, plus minor amounts of dolomite. MS-A (Fig. 2a–c) is characterized by a change in grain size and MS-B (Fig. 2d –f ) is characterized by elongate grains aligned along features termed micro-shears. The sample which contains MS-A has fine-scale (1–2 mm) fairly regular lines interpreted as the traces of planar laminations. These are seen in one area of the thin section (Fig. 2a), which are truncated by a zone of irregular, discontinuous, chaotic structures. The greyscale variation seen in plane polarized light (Fig. 2a) relates to changes in grain size; Figure 2b, c highlights this change. Very little twinning or undulose extinction is seen and the overall population of grains is inequigranular with interlobate grain boundaries. The sample containing MS-B is more homogeneous in areas not containing the microshears and consists of equigranular grains and no obvious shape preferred orientation (Fig. 2d). The micro-shears (Fig. 2e, f ) that characterize MS-B have a strong shape preferred orientation with the long axes of elongate anhydrite crystals aligned parallel with the trace of the shear plane. It appears that the crystals with the highest aspect ratio (i.e. the longest) are generally concentrated towards the centre of the micro-shear. The micro-shears are approximately 1–2 mm wide.
EBSD analysis Microstructure A. EBSD data collected across a change in grain size are displayed as Euler maps (Fig. 3a). The change in grain size is highlighted by the graphs in Figure 3b; the grain size has a lognormal distribution (Fig. 3b). From the geometric mean and standard deviation values, depicted in Figure 3b, a measurable increase in grain size from S1 to S4 can be seen. At the finest grain sizes (S1), small numbers of larger grains are present in a matrix of small grains and, as the grain size becomes overall coarser (towards S4), large grains become more common. The grains in each subset exhibit a shape preferred orientation as shown by the average grain
Fig. 3. (Continued) highlighting the distribution of dolomite though the sample (yellow is dolomite, blue anhydrite). Basal plane and mirror plane orientations for all of the dolomite in areas S1 to S2. Point plots show all of the data. Contoured plots are based on one point for each of the 221 grains to avoid bias towards the large grains. Data mostly from S1 (170 points). (e) Average grain shape ellipses produced from Fry plots; the inset in each figure shows the average orientation of the long axis of the grain ellipse.
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shape ellipses produced from Fry plots. The long axis of the grains in all subsets is subparallel to the X axis of the sample (Fig. 3e) and close to parallel with the laminations’ observed in thin section (Fig. 2a). Lower hemisphere pole figures (Fig. 3a) of the CPO data in subsets S1 –S4 show: (i) a clustering of k001l axes between 208 and 358 to Z in the X –Z plane and a girdle of data points perpendicular to this in k100l and k010l and (ii) a decrease in the strength of the CPO from the coarse-grained (S4) to fine-grained (S1) areas, with a reduction in the angle of the k001l to Z from 30–358 to 208. The M-index, or misorientation index (Skemer et al. 2005), is calculated to show the strength of the CPO in each the subset. The CPO is moderate to weak, shown by the low M-index values (where 0 represents a random fabric and 1 a singlecrystal fabric). There is a higher proportion of low-angle (,108) grain boundaries (Fig. 3c) in the coarser-grained areas (S3 and 4) compared to the finer-grained areas (S1 and 2). Misorientation angle distribution profiles are plotted for each data subset and the two end-members (i.e. finest and coarsest) are shown in Figure 4a. The random-pair (pairs of noncontacting grains chosen at random) distribution is close approximates the theoretical curve for a random distribution of orthorhombic crystals (Wheeler et al. 2001), reflecting the weak CPO. The fit deviates from the theoretical more markedly in S4 (coarser grained) than in S1 (finer grained), mirroring the change in the strength of the CPO. The misorientation angle distribution for neighbourpairs (pairs of grains in physical contact) shows a significant deviation from the theoretical curve and from the random-pair distribution at low misorientation angles. This deviation becomes less pronounced when moving from S4 (coarse grained) to S1 (fine grained). To check whether this difference (between neighbour-pair and random-pair distributions) is statistically significant, the Kolmogorov– Smirnov (K– S) test (as advocated by Wheeler et al. 2001) was applied to the data from each subset. The results give d values of between c. 16 (S1) and c. 26 (S4) which means, in both cases, the probability of the neighbour- and random-pair distributions being samples of the same distribution is less than 0.01%. Misorientation axis/angle pairs are plotted on inverse pole figures (Fig. 4a) for subsets S1 and S4. In S1 at low misorientation angles of between 2 and 58, data are clustered at [001] and [010]. This changes to a strong maximum at [010] in S4. At misorientation angles of between 5 and 108 in S1, the same maxima at [001] and [010] are observed; in S4 we see a maximum at [010] only. There is no consistent shift of low-angle
misorientation axes between [001] and [001] when the data are limited to given grain-size ranges. At higher angles (between 10 and 408 misorientation) in S1 there is a more random distribution of data. Towards coarser grain sizes in S4, the maximum at [010] becomes significantly weaker after misorientation angles of 5–108 and a weak clustering at [001] and [100] in the 20 –408 pole figure. In both (S1) and (S4), there is a maximum at [001] between 40 and 508 misorientation. A similar maximum of [001] misorientations at 80– 908 is stronger in S4 (coarse) and is accompanied by a weak maximum at [010]. Examination of both point data and contoured data in misorientation axis figures (Fig. 4) is important. The appearance of peaks in contoured data that are not apparent in point data are a good indication that the misorientations represent ‘special’ misorientations (exact rotations around specific crystal directions), as the points overlie each other exactly in the point plot. Such ‘special’ misorientations most commonly arise due to systematic misindexing (Prior et al. 2009) or twinning. The apparent clustering of data at 40–508 around k001l is only apparent in the contoured data and is likely to be a mis-indexing problem. It must be emphasized that these plots are extremely sensitive to systematic misindexing and the level of mis-indexing that will give rise to the maximum in the contoured plot will relate to ,0.05% of the data. This mis-indexing is probably responsible for the slight rise in misorientations in the neighbour-pair misorientation angle plots at 40–508, but will not impinge on orientation (generates ,0.02 MUD (multiples of uniform distribution) error) or grain size (generates ,0.1 mm error) data. The known twin (83.58 rotation around k001l) could contribute to the k001l misorientation maximum at 80– 908. This is supported in S4 by the neighbour-pair misorientation angle plots having a peak at c. 83.58, even though no macroscopic twins are observed. However, the fact that a broad maximum around k001l is seen in both point plots and contoured plots suggests that a significant component of this signal is not exactly related to twin or mis-indexing. There is an inhomogeneous distribution of dolomite throughout this sample. There is a significantly higher concentration of dolomite in the fine-grained area of the sample, namely S1 and, to a lesser extent, S2 (Fig. 3d). The presence of higher percentages of dolomite (Fig. 3d) correlates with reduced grain size and a weaker CPO. The dolomite in these subsets has a weak to random CPO (Fig. 3d) with no obvious correlation to the anhydrite CPO. Dolomite is completely absent in subsets S3 and S4. Microstructure B. EBSD data were collected across the micro-shear and are shown as an Euler
EBSD ANALYSIS OF FINE-GRAINED ANHYDRITE
angle map (Fig. 5a). The geometric mean grain sizes of three subsets were calculated in the same way as for MS-A. The grain-size distributions show that this sample is coarser grained than MS-A (SH ¼ 12.1 mm; SK ¼ 13.06 mm and SL ¼ 9.37 mm). All sheared sub-areas and areas undisturbed by shearing in MS-B have a very strong clustering of k001l axes with corresponding girdles in k100l and k010l. The three subsets SH, SK and SL have differently oriented k001l maxima. In SL, the k001l maximum is in a similar orientation to the k001l maximum in the bulk background CPO seen in undisturbed areas of the samples (Fig. 5b). Figure 5d is colour coded to highlight which grains contribute to the texture seen in the pole figures in Figure 5a. Individual anhydrite crystals are clearly elongate and parallel to each other, outlining a grain shape preferred orientation. There is some crystallographic control on the orientation of the elongation; k001l specifically corresponds to grain short axes. A large number of low angle (2–158) grain boundaries are seen (Fig. 5c) across the micro-shear. Misorientation angle distribution graphs (Fig. 4b) confirm this observation and show that there is a significant deviation away from the theoretical random curve for both neighbour and non-neighbour pairs. Neighbour- and random-pair distributions for SH and SL are similar. The Kolmogorov–Smirnov (K –S) test was applied to the data from each subset. The results give d values of between c. 15 (SH) and c. 34 (SL) which, in all cases, means that the probability of the neighbour- and random-pair distributions being samples of the same distribution is less than 0.01%. The distribution of misorientation axes and angles (Fig. 4b) is very similar in SH and SL with k001l and k010l axes clusters at low misorientation angles (2–108) and a strong k001l maximum at higher angles (.208). Note that this maximum appears equally in both point plots and contoured plots, and is therefore not an artefact of misindexing (see earlier discussion of MS-A). There is no consistent shift of low-angle misorientation axes between [001] and [001] when the data are limited to given grain-size ranges. Although misorientation data show that lowangle distortions must be present, only a small number of grains were found to have intracrystalline distortion that could be analysed in more detail (Fig. 6). Pole figures of narrow bands of data through each grain show variable results. Small circle dispersions of data are seen around either a k001l or k010l rotation axis (grains 1 and 2, respectively), or all three principal crystallographic directions show two cross-cutting small circle dispersions at c. 908 (grain 3).
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Discussion Two fine-grained anhydrite microstructures both display microstructural characteristics different to other published descriptions. This may relate to difficulty in correctly imaging the microstructures of fine-grained anhydrite by methods other than quantitative EBSD. Initial collective interpretations of the microstructures resolved using EBSD indicate that deformation mechanisms and slip systems different to those previously recognized may have been operative within these samples. Both samples (MS-A and MS-B) have a distinct crystallographic preferred orientation and a grain shape preferred orientation (Figs 3 & 5). This is a k001l ‘fibre texture’ (a term used in material science) with a single maximum of k001l and all other directions evenly distributed by rotation around this maximum. The CPO in MS-A shows a maximum of data in k001l at c. 20 –358 to Z with corresponding girdles in k100l and k010l. The CPO in MS-B has a similar form to MS-A, with a CPO characterized by strong alignment of k001l. There are however two key differences: the CPOs for MS-B are much stronger and different sub-areas in MS-B have the k001l maxima aligned differently. In MS-A there is a weak grain-shape fabric with the average grain long-axis subparallel to X. In MS-B the grains are clearly elongated with grain long axes subparallel to X. MS-B has a very strong relationship between CPO and grain shape with a k001l maximum orientated at c. 908 to the long axis of grain elongation. There is a geometrical control on the CPO; the crystallographic k001l axis is the grain short axis. MS-A has a similar relationship between CPO and grain shape, although in this case the k001l maximum is at a high angle (but not 908) relative to the grain long axis. A first-order comparison of misorientation data presented in Figure 4a, b shows two different misorientation signatures for MS-A and MS-B. The misorientation analysis of MS-A shows a clustering of misorientation axes at low (,208) misorientation angles around k001l and k010l and some high-angle clustering around k001l at 80– 908 misorientation (clustering at 40–508 is eliminated as misindexing). The low-angle misorientation signature in MS-A is indicative of low-angle grain boundaries controlled by dislocations. MS-B shows a similar clustering of data at low misorientation angles; in contrast to MS-A, however, above 208 misorientation axes are strongly and consistently clustered around k001l. These cannot be explained by dislocation models alone (Sutton & Balluffi 1995). The high-angle misorientation signature in MS-B is related to the shape of the grains as discussed earlier. We suggest that the sample comprises aligned platy crystals of anhydrite (with (001)
246 R. C. HILDYARD ET AL. Fig. 4. Misorientation data for microstructures A and B. (a) MS-A misorientation axis angle pairs for subset S1 and S4 plotted in a crystal reference frame (inverse pole figures), equal-area upper-hemisphere plots. Data are separated into misorientation angle intervals that match some of the grain boundaries highlighted on the grain-boundary map. Contoured pole figures (half-width of 158) given as multiples of uniform density (MUD). Misorientation profiles displaying random-pairs, neighbour-pairs and theoretical random distribution for a population of orthorhombic crystals. (b) As for MS-A but misorientation data is for subsets SH and SL.
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Fig. 4. (Continued)
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Fig. 5. EBSD data of microstructure B. (a) Euler angle map, colour-coded with respect to Euler angles showing a micro-shear-type feature. The CPO of three distinct subsets across the micro-shear are shown (equal-area lower-hemisphere contoured plots) and grain-size distributions in SH, SK, SL. (b) Bulk CPO of an area (outside the field of view) unaffected by the micro-shear. (c) Typical subgrain and grain boundaries found across this setting (location given in (a), colour-coded relative to the key). (d) Texture component map (overlain on a greyscale pattern quality map) colour-coded relative to the orientation of a k001l fibre texture which is responsible for the CPO across the micro-shear zone. Sample and pole figure reference frame is shown.
faces against (001) faces). This geometry constrains the crystals’ short axes k001l to be strongly aligned and gives rise to the dominance of clear k001l highangle misorientation axes.
An explanation as to why the grains in MS-B have crystallographically controlled shapes is speculative at this stage. It has been suggested that a grain shape preferred orientation forms as a
EBSD ANALYSIS OF FINE-GRAINED ANHYDRITE
Fig. 6. Intra-grain distortions. (a) Band-contrast and texture component maps for distorted grains (1– 3); the cross in each map indicates the starting reference point to which all other measurements are taken relative to. (b) Upper-hemisphere equal-angle pole figures of narrow bands of data taken from grains 1– 3 showing small circle dispersions. Grain 1 shows a k001l rotation axis, grain 2 shows a k010l rotation axis and grain 3 shows dual smallcircle dispersion indicating more than one rotation axis. (c) Interpretations of the dispersions based on a tilt-boundary model.
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secondary feature in recrystallized grains (e.g. Lister & Snoke 1984) and a CPO-domain shape preferred orientation (as the microstructure in MS-B might be described) has been said to form by dynamic recrystallization of larger grains (Knipe & Law 1987; Herwegh & Handy 1998). The misorientation data for MS-B could be seen as evidence for the grains having inherited some pre-existing microstructure by a process such as dynamic recrystallization of a larger parent grain (see e.g. Jiang et al. 2000). The authors do not know of any instances where dynamically recrystallized grains have a very strong relationship between shape and crystal orientation. In contrast, such relationships are well documented for crystals grown in a melt (e.g. olivine) or hydrothermal fluids (e.g. quartz) for some metamorphic minerals (amphiboles), for highly anisotropic minerals (phyllosilicates) and for minerals developed from phase transformations. Given the likely complexities of the sample history, the origin of the evaporite as gypsum (Lugli 2001) and the potential for further transformations between the anhydrous (anhydrite) and hydrated phases (bassanite, gypsum), we think there is potential that this fabric may be inherited from another calcium sulphate phase. For example in MS-B, the reduction in volume created as gypsum dehydrates to anhydrite would result in a temporary reduction in vertical stress possibly allowing new anhydrite grains to grow with the grain long axes parallel to this vertical minimum stress direction (e.g. parallel to the core axis (X ) in these samples). Hildyard et al. (2009b) showed that gypsum can have a very strong CPO with (010) aligned with the foliation plane and a cluster of (100) in that plane, and that this CPO corresponds to a shape fabric. Furthermore, Hildyard et al. (2011) show that bassanite can have a strong CPO that relates to grain shapes and that there can be inheritance of orientations (and development of CPO) related to transformations between bassanite and gypsum. This topotactic or mimetic relationship has also been reported for other mineral phases such as magnetite and hematite (Barbosa & Lagoeiro 2010), but to test this hypothesis for the current sample set would require analysis of a sample that contains both the original gypsum and the resultant anhydrite. However, with the samples used in the analysis presented here, the only phase observed is anhydrite. There is evidence of dislocation activity in MS-B as shown by internal grain distortions in Figure 6. These distortions can be used to place some constraints on the likely slip systems active within these grains. The 2D trends in misorientation in the texture component maps in Figure 6a are taken to be the boundary trace of the low-angle boundaries present within the grain. A rotation axis of k001l and
k010l can be seen in grains 1 and 2, respectively, and no clear rotation axis is present within grain 3 (where there all three directions show cross cutting small circle dispersions at 908). The data were found to best fit a tilt boundary model (Lloyd et al. 1997; Prior et al. 2002) in the case of grain 1 and grain 2 (Fig. 6c), where the boundary plane contained the mapped boundary trace and the misorientation axis. In a tilt boundary model, the misorientation axis must be contained within the slip plane and at 908 to the slip vector, both of which are then at a high angle to the boundary plane. Adhering to the geometrical constraints of the tilt boundary model, the simplest solution to misorientation data seen in grain 1 would be (100)[010] slip (rotation axis k001l) and in grain 2 (001)[100] slip (rotation axis k010l). The known anhydrite slip systems are shown in Table 1. (100)[010] and (001)[100] are not known slips system (Table 2). For grain 3, there is an apparent two-way distortion which could indicate that both of these slip systems are active in the same grain. Given that the observed neighbour-pair low-angle misorientations are dominated by k010l misorientations with minor k001l misorientations, we could extrapolate the limited data in Figure 6 to suggest that distortion related to (001)[100] and another slip system (perhaps (100)[010]) are prevalent in both MS-A and MS-B. We do need to be wary here as k010l rotations can also be explained by known slip systems. The (100)[001] slip system is well known and is interpreted from imaged distortions in another Apennine evaporite by Hildyard et al. (2009a), where k010l low angle misorientations dominate. They pointed out that the distortions can also be explained by coupled activity on the (201)[1-1-2] and (201)[11-2]. It is worth noting that the unknown slip system (001)[100] could better explain the CPO (but see previous discussion) and the (001) planes are subparallel to foliation. The distortion of grains is not necessarily related to the process that aligns the grains and generates the CPO, but does indicate that deformation involving intragranular dislocation activity has occurred. A more detailed inspection of the orientation and misorientation data for MS-A links the CPO to grain size: there is a distinct reduction in the strength of the CPO with a reduction in grain size. There is a level of uncertainty here; we are making the assumption that the fine-grained microstructure of MS-A is generated from the coarse-grained microstructure in that sample. Furthermore, on the basis of similar CPO forms and reduced grain size of MS-A, we speculate that the MS-A samples might be equivalents of MS-B after further deformation. Since this is speculative, the implications will be
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explored only briefly. Grain size is known to be a controlling factor in determining deformation mechanisms. The mechanisms that involve grainboundary sliding (diffusion creep and DisGBS) are sensitive to grain size, favouring small grains (e.g. Stu¨nitz & Gerald 1993; Fliervoet & White 1995; Fliervoet et al. 1997; Kruse & Stu¨nitz 1999; Bestmann & Prior 2003; Warren & Hirth 2006; Mehl & Hirth 2008). A number of previous workers have recognized that a reduction in grain size coupled with a reduction in the strength of a crystallographic texture is attributed to an increased dispersion of misorientations between grains and the weakening of a CPO due to a grain-boundary sliding process (Fliervoet et al. 1999; Jiang et al. 2000; Bestmann & Prior 2003; Storey & Prior 2005; Skemer & Karato 2008). The weakening of the CPO signature in MS-A towards progressively finer grain sizes could be interpreted as being indicative of a shift in deformation mechanism from that which is grain-size insensitive (dislocation creep) towards that which is grain-size sensitive (grain-boundary sliding) and which has weakened, but not totally destroyed, the CPO. The presence of a higher percentage of dolomite in the finest grain sizes (Fig. 3d) may have inhibited grain growth by grainboundary pinning, thus enhancing the possibility of a shift towards grain-size-sensitive deformation mechanisms (De Bresser et al. 1998, 2001; Kruse & Stu¨nitz 1999; Newman et al. 1999; Herwegh & Berger 2004; Warren & Hirth 2006; Mehl & Hirth 2008). Two high-temperature flow regimes defined by Dell’Angelo & Olgaard (1995) showed evidence for dislocation creep and diffusion creep. In the transition between regimes 1 and 2 (i.e. between dislocation and diffusion-dominated processes), they suggested that the microstructures they observe were indicative of grain-boundary sliding accompanied by dislocation activity (DisGBS). The microstructures from MS-A observed in the present study match those that Dell’Angelo & Olgaard (1995) described in the transition from regime 1 to 2 including subequant, polygonal grains with slightly serrate boundaries, low dislocation density, unstrained recrystallized grains and a weakening of the CPO (microstructures indicative of increased grainboundary mobility and dislocation recovery during grain-size reduction). It is worth noting that the role of GBS in CPO evolution remains unclear. Although Dell’Angelo & Olgaard (1995) interpreted CPO weakening to diffusion creep, an experiment that reaches higher strains showed no CPO weakening in diffusion creep (Heidelbach et al. 2001), perhaps related to co-operative slip on aligned boundaries (Wheeler 2009). We speculate that the MS-A sample represents deformation by a DisGBS mechanism. Whether this is realistic
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in anhydrite requires further work on betterconstrained natural and experimental samples. The microstructures observed on these samples can be used to place some constraints on deformation conditions. A deformation mechanism map showing stress –grain-size relationships has been drawn up from experimental data (Fig. 7). Dell’Angelo & Olgaard’s (1995) regime 1 and regime 2 flow laws are used to generate constant strain-rate lines and regime boundaries for conditions that enable comparison with other creep data (Fig. 7a). The Muller et al. (1981) data plot entirely within regime 1 (dislocation creep and twinning) and match the projection of the Dell’Angelo & Olgaard (1995) flow law well, given realistic experimental uncertainties. Two points have been extracted from figure 1 of Heidelbach et al. (2001): a point at 80 MPa and 12 mm for dislocation creep and at 40 MPa and 6 mm for diffusion creep. Constant strain-rate lines put through these using grain-size exponents from Dell’Angelo & Olgaard (1995) would predict a regime boundary at lower stresses than calculations from Dell’Angelo & Olgaard (1995) for equivalent conditions (700 8C, 1023 s21). In Figure 7b the Dell’Angelo & Olgaard (1995) flow laws are used to calculate constant strain-rate lines at 230 8C and regime boundaries at 230 and 315 8C, corresponding to the range of estimated deformation temperatures (Barchi et al. 1998; Lugli 2001; De Paola et al. 2008). A further regime boundary (labelled MIN?, Fig. 7b) is drawn at lower stresses, corresponding to moving the 315 8C boundary by the estimated difference of boundaries at 700 and H700 8C in Figure 7a. The deformation of both MS-A and MS-B has involved dislocation activity, which means that the stresses of deformation must be above (for dislocation creep) or near (for DisGBS) the stresses predicted by the regime boundary. The grain sizes for MS-A (7–10 mm) and MS-B (13 mm) are plotted on Figure 7b and show that the stresses must be high in these grain sizes for deformation mechanism involving dislocation activity. The estimated minimum stress regime boundary line predicts differential stress magnitudes well in excess of 100 MPa at 315 8C. Lower temperatures will lead to higher stresses. Further constraint (Fig. 7b) is provided by room-temperature experiments (De Paola et al. 2009) that show that a fine-grained sample from 2064 m in the Fossombrone borehole (MS-B comes from the same sample block) does not fail with differential stresses over 200 MPa, even at low effective confining pressures. Pore fluid pressures can be very high (up to 85% of lithostatic) in these evaporitic rocks (Miller et al. 2004). At the estimated confining pressures (200 –250 MPa: Barchi et al. 1998; Lugli
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Fig. 7. Log stress (s) versus log grain size (D) deformation mechanism maps for anhydrite. (a) Regime boundaries (inclined top left to bottom right) and constant strain-rate lines calculated from flow laws of Dell’Angelo & Olgaard (1995) and drawn at conditions comparable to the experimental data of Muller et al. (1981) and Heidelbach et al. (2001). Form of line indicates data source (initials in legend correspond to the references above) and strain rate and labels show temperature in Celsius. For clarity, only 1026 s21 data of Muller are shown (the fit of 1025 s21 data to Dell’Angelo and Olgaard in equally good). Two points are plotted from Heidelbach et al. (2001). The constant strain-rate lines through these use the grain-size exponents from Dell’Angelo & Olgaard (1995) and the regime boundary (labelled H700 8C) is drawn through the intersection of these, parallel to the other regime boundaries. (b) Regime and constant strain-rate lines calculated from flow laws of Dell’Angelo & Olgaard (1995) and drawn at conditions relevant to the samples in this study. Constant strain-rate lines (labelled with the log of the strain rate in s21) are drawn for 230 8C, regime boundaries are drawn for 230 and 315 8C. A third regime boundary (labelled min?) represents an estimate of the minimum-stress regime boundary from the experimental constraints (by subtracting the stress difference between the 700 and H700 8C boundaries in part (a) of the figure from the calculated boundary for 315 8C). Lines show the mean grain sizes for areas S1 and S4 of MS-A and the largest mean grain size measured in MS-B. Another line (500 mm) gives the approximate grain size from samples that deformed in dislocation creep (Hildyard et al. 2009a). Peak stresses for deformation at ambient temperature give estimates of brittle strength at two different effective pressures (Pe) (De Paola et al. 2009).
2001; De Paola et al. 2008) present at the locations and depths from where these samples were derived, effective pressures were likely to have been high enough to allow high differential stresses without failure. (3)
Conclusions Interpretation of orientation and misorientation data obtained from two microstructures within very fine-grained anhydrite rocks from the Triassic Evaporites of the central Italian Apennines using EBSD analysis allows the following conclusions to be drawn. (1) We have identified a new CPO in anhydrite: a k001l fibre texture, with a single maximum of k001l and all other directions evenly distributed by rotation around this maximum. (2) In the sample with the strongest CPOs (MS-B), there is a very strong grain shape fabric and the k001l maximum is perpendicular to the trace of grain long axes. The k001l fibre texture and high-angle misorientation axes near k001l are best explained by alignment of
(4)
(5)
individuals grains that are platy with k001l short axes. The CPO does not need to relate to dislocation creep and may owe its origins to earlier phase transformations from the original gypsum evaporate. Both microstructures contain low-angle boundaries and distortions that attest to dislocation activity during deformation. The distortion can be explained by activity on the (100)[010] and/or (001)[100] slip systems. These are different to the recognized easy slip system for anhydrite (100)[001]. We speculate that in the sample MS-A, weakening of the CPO, reduction in grain-shape fabric, modification of the relationship of the CPO and the shape fabric and loss of crystallographically controlled high-angle misorientation axes relates to deformation by a mechanism involving grain-boundary sliding, accompanied by dislocation activity (DisGBS). Comparison of our data with extrapolated results from laboratory creep experiments requires creep to have occurred at relatively high differential stresses (.100 MPa).
EBSD ANALYSIS OF FINE-GRAINED ANHYDRITE The NERC studentship NER/S/A/2005/13332 and two NERC grants NER/A/S/2001/01181 and NE/ C002938/1 funded this work. The authors would like to thank L. Morales and E. Rutter for thorough reviews that improved the original manuscript.
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Grain growth and the lifetime of diffusion creep deformation MARK A. PEARCE1* & JOHN WHEELER2 1
Geospatial Research Ltd., Department of Earth Science, Durham University, Durham DH1 3LE, UK
2
Department of Earth & Ocean Sciences, University of Liverpool, Liverpool L69 3GP, UK *Corresponding author (e-mail:
[email protected]) Abstract: Extreme grain-size reduction due to cataclasis, neocrystallization or phase change results in a switch to diffusion creep and dramatic weakening in deforming rocks. Grain growth increases strength until dislocation creep becomes a significant deformation mechanism. We quantify the ‘lifetime’ of diffusion creep by substituting the normal grain growth law into the diffusion creep flow law to calculate the time taken for dislocation creep to become significant. Stress-temperature and strainrate-temperature space is outlined where diffusion creep may accommodate significant strain: these regions have an upper temperature limit beyond which grain growth is fast enough to move the rock quickly into the dislocation creep field. For plagioclase the limit lies in the amphibolite facies. Rocks in a mantle upwelling experience grain-size reduction during phase changes. Pressure-dependent grain growth limits the deformation that can be accommodated by diffusion creep. This time limit and associated strain limit is independent of starting grain size with a small dependence on upwelling rate and plume width. In both these tectonic environments, second phases are likely to play a role in the maximum achievable grain size due to grain-boundary pinning. Hence we predict the minimum lifetimes of diffusion-creep-dominated deformation following extreme grain-size reduction.
High-temperature rock microstructures evolve through a combination of creep, grain growth and grain-size reduction. Extreme grain-size reduction, by cataclasis (Hadizadeh & Tullis 1992; Nyman et al. 1992; Babaie & La Tour 1994; Imon et al. 2004; Keulen et al. 2004) or neocrystallization (Wayte et al. 1989; Berger & Stu¨nitz 1996; Stu¨nitz 1998; Stu¨nitz & Tullis 2001), has been posited as a cause of crustal weakening in both compressional (Brodie & Rutter 1985; Rubie 1990) and extensional (Handy & Stunitz 2002) settings and is due to fine-grained rocks deforming by grain-sizesensitive diffusion creep. In the mantle, similar extreme grain-size reduction may occur following phase change, e.g. spinel –olivine in an upwelling or serpentine –olivine and olivine –spinel in a downgoing slab (Riedel & Karato 1997). Weakening occurs because stresses are lower in diffusion creep than if the rocks were to deform by dislocation creep at the same strain rate (Rutter & Brodie 1988). However, boundary-curvature-driven grain growth should be fast at extremely small grain sizes leading to a grain-size increase and, due to grainsize sensitivity of diffusion creep, an increase in strength (Rubie 1983; Karato et al. 1986). This means that weakening by grain-size reduction is transient (Brodie & Rutter 1987) with the timescale determined by the grain growth rate, which is a function of initial grain size and temperature. Here, we quantify the transient time or ‘lifetime’ over which diffusion creep dominates in some
simple situations, and characterize the amount of strain it can therefore accommodate. Our aim is to illustrate the sensitivity of these lifetimes and strains to the various controlling parameters, and thus characterize the circumstances under which diffusion creep can accommodate ‘significant’ strain (significant values of strain vary with geological setting but in the field it is difficult to differentiate strains .10 so this is chosen as a significant value for crustal rocks). Our conceptual model is as follows. We begin with a fine-grained rock formed by metamorphism or cataclasis. The initial mean grain size (before any boundary-energy-driven grain growth has taken place) is highly dependent on the mechanism of grain-size reduction (Rubie 1990), although if nucleation during metamorphism is easy then extremely small (nanometre-scale) grain sizes may result (Rubie 1983; Brodie & Rutter 1985; Wayte et al. 1989; Kenkmann & Dresen 2002). Because of the many factors involved, rather than modelling the initial grain size in detail we will simply illustrate the sensitivity of subsequent evolution to the initial grain size. We explore situations in which the fine-grained aggregate deforms initially by diffusion creep. Grains grow in order to reduce grain-boundary energy. The rate of grain growth is temperature activated, but also has a dependence on pressure in some cases. Because of the grain-size sensitivity of diffusion creep, grain growth increases the rock strength and eventually moves it into, or at
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 257– 272. DOI: 10.1144/SP360.15 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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least to the border of, the dislocation creep field. We calculate the lifetime of evolution in the diffusion creep field, and the strains that can be built up by that process. Once rocks are deforming by dislocation creep, grains may not continue to grow because grain-boundary bulging or subgrain rotation recrystallization (Urai et al. 1986; Hirth & Tullis 1992) come into play to reduce grain size. These processes are likely to produce a grain size which is reasonably close to the diffusion creep/dislocation creep transition (De Bresser et al. 1998). We do not address this latter evolution; there are no indications that such grain-size reduction will move the rock back far into the diffusion creep field. Two different applications are given here which have different boundary conditions; both apply the central theme of incorporating grain growth into flow laws to create time-dependent rheologies. Firstly, neocrystallization of plagioclase is considered. This situation arises in mid-lower crustal rocks where calcic plagioclase in gabbros and granites reacts to form more albitic plagioclase which is subsequently deformed (Jiang et al. 2000). Secondly, a mantle upwelling going through the spinel –olivine phase change is considered. This is assumed to be adiabatic, but a time-dependent pressure dependence is incorporated as the material rises through the mantle.
Model Rocks undergoing creep can be constrained between two bounds: isostress and isostrain rate. The latter is usually used when considering rocks in shear zones undergoing weakening by grain-size reduction (Rutter & Brodie 1988; De Bresser et al. 2001). However, there is no a priori reason to choose either and both will be considered here for lower crustal rocks. In order to calculate the total strain achievable before a significant proportion of the strain rate is accommodated by dislocation creep, the strain rate needs to be defined as a function of time using the dependence of grain size on time. This is then integrated over the total time during which diffusion creep is the dominant deformation mechanism.
Isostress bound Creep flow laws, which describe the relationship between shear stress t (MPa) and shear strain rate (s21), g˙ are of the general form
g˙ = At n d −m e−Q/RT
(1)
where A is the pre-exponent (MPa2n mmm s21), n is the stress exponent (generally 1 for diffusion creep and ≥3 for dislocation creep), d is the mean grain
size (mm), m is the grain-size exponent (0 for grain-size-insensitive dislocation creep and c. 3 for grain-size-sensitive diffusion creep), Q is the activation energy (kJ mol21), R is the gas constant (kJ mol21 K21) and T is the absolute temperature. The evolution of the mean grain size by surface-energy-driven grain growth in a monophase rock can be described using the normal grain growth law (Brook 1976): d p − d0p = k0 e−Qgg /RT t
(2)
where d is the current grain size (m), d0 is the starting grain size (m), p is the grain growth exponent, k0 is the grain growth pre-exponent (mp s21), Qgg is the activation energy for grain growth (kJ mol21), and t is the time (s). Assuming that the strain rate and grain growth rate are independent of each other (Austin & Evans 2007), the grain growth law may be rearranged and substituted into the strain-rate equation to give a time-dependent strain rate:
g˙ = At n (d0p + k0 e−Qgg /RT t)−m/p e−Q/RT .
(3)
Integrating this with respect to time will give the total strain achievable before the grain size reaches some critical value. This value and therefore the upper limit of the integration is the time it takes for the grain size to reach a value where the dislocation creep rate is some proportion a of the diffusion creep rate, given by
g˙ dis = ag˙ diff Adis t ndis d−mdis e−Qdis /RT = aAdiff t ndiff d−mdiff e−Qdiff /RT (4) where subscripts diff and dis relate to grainboundary diffusion and dislocation creep, respectively. Since mdis ¼ 0, this can be rearranged to give the critical grain size dcrit where this condition is satisfied: Adiff ndiff −ndis (Qdis −Qdiff )/RT 1/m t e . dcrit = a Adis
(5)
This defines a line on a log10s – log10d deformation mechanism map parallel with the field boundary between dislocation and diffusion creep. The critical grain size will be reached at critical time tcrit obtained by rearranging equation (2) for d ¼ dcrit: tcrit =
p dcrit − d0p . k0 e−Qgg /RT
(6)
DIFFUSION CREEP LIFETIME
Assuming the contribution to the finite strain from dislocation creep is negligible because the strain rate is dominated by diffusion creep, the maximum strain achievable in the diffusion creep field is therefore:
gdiff tcrit = g˙ dt
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This gives the stress where equation (4) is satisfied as a function of the bulk strain rate. In order to find the grain size where this equation is satisfied, tcrit is substituted into equation (9) and written in terms of the diffusion creep strain rate:
g˙ bulk = (1 + a)g˙ diff
(11)
ndiff −mdiff −Qdiff /RT g˙ bulk = (1 + a)Adiff tcrit dcrit e .
0
=
tcrit
Adiff t ndiff (d0p + k0 e−Qgg /RT t′ )−m/p e−Qdiff /RT dt′
0
This is then rearranged to obtain the critical grain size:
⎡
⎢Adiff t ndiff e(Qgg −Qdiff )/RT =⎢ ⎣ mdiff k0 1− p
dcrit
× (d0p + tk0 e−Qgg /RT )
1−
mdiff p
ndiff −Qdiff /RT (1 + a)Adiff tcrit e = g˙ bulk
⎥ ⎥ ⎦ 0
Adiff t e mdiff k0 1− p
(d0p
+ tcrit k0 e
gbulk = )
1−
mdiff p
( p − mdiff )
− d0
(7) This is an isostress constraint because the stress is held constant throughout the integration. We also hold the temperature and the pressure constant (because grain growth rates are pressure dependent).
Isostrain-rate bound As well as the constraint from equation (4), the strain rates for the two mechanisms have to sum to the predefined strain rate g˙ bulk :
g˙ bulk = g˙ dis + g˙ diff .
(8)
These can be combined and rearranged to give an expression for the critical stress where equation (4) is satisfied, i.e.
⎛
(12)
tcrit
g˙ bulk dt = g˙ bulk tcrit .
(13)
0
−Qgg /RT
g˙ bulk
.
Finally, this critical grain size can be substituted into the grain growth law (equation (6)) to obtain the time that it takes for the grain size to reach this critical value tcrit. The total strain accommodated is then the time integration of the bulk strain rate which is constant for this bound:
⎤tcrit
ndiff (Qgg −Qdiff )/RT
=
1/mdiff
1 = 1+ g˙ a dis
(9)
⎞1/pdis
⎜ ⎟ g˙ ⎟ bulk tcrit = ⎜ ⎝ ⎠ 1 (−Q /RT) dis 1+ Adis e a
.
(10)
As above, we assume temperature and pressure are fixed.
Deformation at constant pressure and temperature Since the finite strain accomplishable before the condition in equation (4) is satisfied is a function of d0, the effect of the starting grain size will be investigated. As discussed above, the grain-size refinement that initiates deformation by diffusion creep may be accomplished by a number of processes. While the recrystallized grain size is predictable by the use of a piezometer, more extreme grain-size reduction occurs through the process of nucleation driven by chemical disequilibrium (contrast grain sizes predicted by Kenkman & Dresen 2002; Post & Tullis 1999). A range of starting grain sizes from a few microns to a few tens of microns is therefore likely to occur depending on the refinement mechanism. Examples of finegrained reaction products include jadeite and quartz formed by the breakdown of albite at the transition to eclogite facies (Rubie 1983), and pure albite formed from the reaction of calcic plagioclase (from deep level plutonic rocks) to pure albite at greenschist facies conditions, for example in metagabbro (Jiang et al. 2000). In order to apply the model derived above it is necessary to have constitutive equations which describe both deformation
260
M. A. PEARCE & J. WHEELER
and grain growth. These only exist for a restricted subset of minerals including plagioclase, calcite, quartz and olivine. Plagioclase will be used to illustrate the effects of temperature, stress and strain rate on diffusion creep ‘lifetime’ as it is likely to undergo extreme grain-size refinement in the situations outlined above.
Time-dependent pressure change We can examine the behaviour of olivine during deformation at constant P and T but find it instructive to consider its behaviour in a mantle upwelling undergoing a phase change from spinel to olivine. The olivine forms at some pressure P0 which then diminishes during further upwelling. Pressure dependence needs to be built into the constitutive equations for deformation and grain growth in order to model this effect. Assuming a vertical columnar upwelling of fixed width w undergoing ‘pipe-flow’ (Vogt 1976; Sleep 1992), a parabolic velocity field is defined by v = vmax
2 2x 1− w
(14)
where v is the velocity (m s ) at point x (m) for x ¼ 0 at the centre of the upwelling, where the upwelling rate is vmax. The shear strain rate is then given by the derivative of the velocity field with respect to x: dv 4xvmax =− . w2 dx
(16)
where rmantle is the density of the mantle (kg m23) and g is the acceleration due to gravity (m s22). The activation energy of the flow and grain growth laws can be rewritten to include pressure dependence: Q = E + PV
(17)
where E is the activation enthalpy (kJ mol21), P is the pressure (MPa) and V is the activation volume (1023 m3 mol21). Including the time-dependent pressure dependence yields Q = E + (P0 − vtrmantle g)V.
(19)
If Q is a function of t, equation (2) can be rewritten as Q(t′ )gg dt′ . = k0 exp − RT t
d − p
d0p
(20)
0
If k0, T and Qgg are constants throughout the deformation, the integral reduces to equation (2). Substituting equation (18) into equation (20) gives the grain size as a function of time
dcrit = ⎣d0p +
tcrit
⎤1/p E + (P0 − vt′ rmantle g)V ′⎦ dt k0 exp − RT
0
= d0p + k0
E + P0 V exp − vrmantle gV RT 1/p vtcrit rmantle gV × exp . −1 RT RT
(21)
(15)
The pressure change between the reference pressure and the pressure at time t is a function of the upwelling rate: P = P0 − vtrmantle g
⎤1/ndis ⎡ g˙ bulk tcrit = ⎢ . ⎥ ⎥ ⎢ 1+1 × ⎥ ⎢ a ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ Edis + (P0 − vtrmantle g)Vdis ⎦ ⎣ Adis exp − RT
⎡
21
g˙ =
The critical stress scrit at which equation (4) is satisfied is given by combining equations (10) and (18) as
(18)
Substituting equations (19) and (21) into equation (11) gives the diffusion creep strain rate as a function of time, including a time-dependent pressure. Solving iteratively for t gives the length of time before the condition in equation (4) is fulfilled, and therefore the length of time that the shear zone is dominated by diffusion creep. For any given point, the strain rate is time-independent (given by equation (15)) and therefore the strain accumulated can be calculated by using equation (13).
Results Results are presented for plagioclase-rich lower crustal rocks using the anorthite flow laws of Rybacki & Dresen (2000) modified for shear deformation (Rybacki & Dresen 2004). Grain growth is modelled using the experimental data of Dresen et al. (1996) with the parameters presented in Table 1. The deformation is constrained to last less than 50 Ma, as this was considered to be more than adequate for the length of most orogenic
DIFFUSION CREEP LIFETIME
261
Table 1. Grain growth and deformation parameters used in this study A
n
m
Q
E
V
k0
p
Law ‘Wet’ plagioclase diffusion creep (Rybacki & Dresen 2000) ‘Wet’ plagioclase dislocation creep (Rybacki & Dresen 2000) Plagioclase grain growth (Dresen et al. 1996) Olivine diffusion creep (Hirth & Kohlstedt 2003) Olivine dislocation creep (Hirth & Kohlstedt 2003) Olivine grain growth (Faul & Jackson 2007)
8.65
1
3
170
–
–
–
–
206.2
3
0
356
–
–
–
–
–
365
–
–
2.59 × 1024
2.6
23
–
– 1.5 × 10
– 9
1.1 × 106 –
1
3
–
375
6 × 10
–
3.5
0
–
530
18 × 1023
–
398
6 × 10
1 × 10
–
–
–
23
– 29
3.3
A is the deformation pre-exponent (MP a2n mmm s21); n is the stress exponent; m is the grain size exponent; Q is the activation energy (kJ mol21; for olivine the activation energy is given as in equation (17)); E is the activation enthalpy (kJ mol21); V is the activation volume (1023 m3 mol21); k0 is the grain growth pre-exponent (m2 s21); and p is the grain growth exponent.
Isostress deformation The critical grain size as a function of shear stress and temperatures is contoured in Figure 1. Critical grain sizes range from 1 to 10 000 mm depending on the temperature and stress of the deforming system. The critical grain size is inversely proportional to both stress and temperature, with the largest grain size occurring under the lowest stress and lowest temperature conditions. This occurs because the critical grain size is governed by the balance between diffusion and dislocation creep. While both creep rates are slower at low temperature, the lower activation energy of diffusion creep therefore means that larger grain sizes can be obtained before dislocation creep becomes significant. Where dcrit , d0 no deformation will occur in the diffusion-creep-dominated regime and therefore tcrit will be 0. Critical times are presented in Figure 2a–c for conditions where 0 , tcrit , 50 Ma. Since tcrit is a function of starting grain size d0 (equation (6)), results are presented for three different grain sizes of 5, 50 and 100 mm. The upper temperature–stress bound is defined by d0 ¼ dcrit. The lower temperature bound is given by the temperature where tcrit ¼ 50 Ma. The total shear strain accumulated before dislocation creep becomes significant is shown in Figure 2d –f. The upper temperature –stress limit is the same as the upper limit in the time plots; if there is no diffusion
creep deformation, no strain can be accumulated. Contours of strain form loops (explained further in ‘Causes of contour morphology’ section) with the maximum strain attainable inversely proportional to the starting grain size. For a starting grain size of 5 mm the maximum strain achievable within the constraints placed on the model is .106, dropping to 102 for a starting grain size of 100 mm. The range of strains attainable with no significant contribution from dislocation creep is approximately 5 orders of magnitude for any given starting grain size, thereby highlighting the importance of conditions of deformation and grain growth.
103
0.5 1.5
Shear Stress, MPa
events. The cut-off point where dislocation creep becomes significant was chosen to be where the dislocation creep strain rate is 10% of the diffusion creep strain rate, and is discussed below.
102 1
2
101 2.5
3.5
3
100 400
500
600 700 Temperature, °C
800
900
Fig. 1. Log10 contour plot of critical grain size (mm) where a ¼ 0.1 for plagioclase deforming under isostress boundary conditions. The critical grain size is independent of starting grain size for constant temperature and pressure deformation; these sizes therefore apply to all isostress plots.
262
M. A. PEARCE & J. WHEELER
Critical Time (tcrit)
(a)
Total Strain Achievable
(d)
3
3
10
10
d0 = 5μm
6
d0 = 5μm
Shear Stress, MPa
Shear Stress, MPa
Te 2
10
-6
-4
1
10
5
2
10
Td 4 1
10
3 Ta
-2
Tb
3
Tc
2
1
0 0
10 400
(b)
0
500
600 700 Temperature, °C
800
10 400
900
(e)
3
10
500
600 700 Temperature, °C
800
10
2
1
10
d0 = 50μm Shear Stress, MPa
Shear Stress, MPa
d0 = 50μm
10
1
2
10
3 2 1 1
10
1
0 -1 -2 -3
0
10 400
(c)
500
600 700 Temperature, °C
800
0
10 400
900
(f)
3
0
0
-4
10
500
600 700 Temperature, °C
800
900
3
10
d0 = 100μm
d0 = 100μm Shear Stress, MPa
Shear Stress, MPa
900
3
2
10
1
10
2
10
0
1
10
1
-1
1
2
0 -1
-2
0
10 400
500
600
700
800
-3 900
Temperature, °C
0
0
10 400
500
600 700 Temperature, °C
-1 800
900
Fig. 2. (a–c) Log10 contour plots of diffusion creep lifetime in Ma and (d– f) total strain achievable where a ¼ 0.1 for plagioclase deforming under the isostress boundary condition for starting grain sizes of (a, d) 5, (b, e) 50 and (c, f) 100 mm. Shading shows strains .10 and zigzag lines mark the boundary between the gdif ¼ agdis. For explanation of Ta – e see text.
Isostrain-rate deformation Critical grain sizes for the isostrain-rate bound are shown in Figure 3 as a function of bulk shear strain rate and temperature. In this model, the stress is allowed to be whatever is necessary to
accomplish the imposed shear strain rate. There is a variation over 3 orders of magnitude between 10 and 10 000 mm. The smallest critical grain sizes are found at the lowest temperatures and fastest strain rates. As the strain rate decreases or the temperature increases, dcrit increases. Contour plots of
DIFFUSION CREEP LIFETIME –10
10
1.5 2
Bulk Shear Strain Rate, s–1
–12
10
–14
10
2.5 –16
10
3
3.5
–18
10
400
500
600 700 Temperature, °C
800
900
Fig. 3. Log10 contour plot of critical grain size (mm) where a ¼ 0.1 for plagioclase deforming under isostrain-rate boundary conditions. The critical grain size is independent of starting grain size for constant temperature and pressure deformation; these sizes therefore apply to all isostrain-rate plots.
tcrit (Fig. 4a –c) again show a region of varying tcrit bounded by two extrema curves. The upper bound on time is again imposed as 50 Ma and the lower bound is constrained to be where the d0 ¼ dcrit. The length of the deformation episode with no significant component of dislocation creep increases with decreasing temperature and bulk strain rate until no grain growth occurs and the rocks are permanently in the diffusion creep field. Strains achievable are shown contoured in Figure 4d–f. Increased starting grain sizes leads to a reduction in the maximum strain achievable from .105 where d0 ¼ 5 mm to .102 where d0 ¼ 100 mm. In the region where diffusion creep lasts for 50 Ma, the contours for isostrain rate (horizontal) and strain achievable decrease with decreasing strain rate. This is because in this region, the strain rate is the same for the same amount of time and therefore the strain achieved is the same regardless of temperature. Only where dcrit is exceeded at some point during the 50 Ma allowed does the total strain vary as a function of temperature.
Deformation with time-dependent pressure For the example of mantle upwelling, the diffusion and dislocation creep flow laws of Hirth & Kohlstedt (2003) are used along with the grain growth law of Faul & Jackson (2007). These grain growth data are used because they are derived from confined experiments at 300 MPa with no melt; more suitable than unconfined experiments in which it has been suggested that porosity retards grainboundary migration (Evans et al. 2001). There are no satisfactory data for activation volume for grain
263
growth in olivine so a median value of activation volumes for grain-boundary diffusion creep is used as an approximation (Hirth & Kohlstedt 2003). This is reasonable since activation volumes for grainboundary diffusion (which is likely the rate controlling step in grain-boundary diffusion creep) and for grain growth are similar in other phases, for example ferropericlase (Tsujino & Nishihara 2009). This allows data from 300 MPa to be extrapolated to 14 GPa (using equation (17)), resulting in an increase in activation energy of c. 80 kJ mol21. Grain-boundary diffusion is considered to be the dominant form of diffusion creep in this region of the mantle. Calculations using the values for Nabarro –Herring creep given by Weertman (1978) show that, for the conditions considered, Coble creep would dominate. A constant mantle density of 3500 kg m23 is assumed and the phase change is assumed to occur at 14 GPa (Katsura & Ito 1989). Maximum upwelling velocities of between 9.6 × 10210 and 30 × 10210 ms21 were used to bracket observed rates of between 3 cm a21 (Azores; Bourdon et al. 2005) and 30 cm a21 (Hawaii; Sims et al. 1999). Plots of critical grain size, critical time, total strain and total displacement as a function of distance from the centre of a 100 km wide upwelling are shown for different starting grain sizes in Figure 5a–d. The analysis shows that all the parameters investigated are practically independent of starting grain size for grain sizes up to 103 mm. The strain is localized in the outer 10 km of the plume with strain of ,10 over 90% of the plume. The distance travelled by material at the centre of the plume is around 350–400 km. 50% of the material travels between 100 and 150 km in the diffusion-creep-dominated field. For the plume illustrated in Figure 5d this means that, at true scale, there is a rather tall tapered cylindrical region in which diffusion creep dominates. The parameters of Figure 5 are plotted in Figure 6 as a function of the upwelling rate for w ¼ 100 km and d0 ¼ 1 mm (note that the starting grain size, if ,1 mm, does not significantly influence the length of time that the rocks deform by diffusion creep; Fig. 5). The critical grain size does not vary greatly over the majority of the width of the plume or with upwelling rate, varying between 2 and 7 mm. Close to the centre, the critical grain size increases rapidly as the strain rate decreases to 0 in the centre. The critical time is relatively constant for any given upwelling rate over the majority of the plume width and varies by c. 1 order of magnitude over an order of magnitude upwelling rate. There is a slight increase at the centre of the plume and a more significant increase at the edge. The strain achievable before dislocation creep
264
M. A. PEARCE & J. WHEELER
Critical Time (tcrit)
(a)
(d)
–10
10
Total Strain Achievable
–10
10
4
–4 –3 –1
–1
–2
0 –14
1
10
3
10
Bulk Shear Strain Rate, s
Bulk Shear Strain Rate, s–1
–12
10
–12
–16
2 1
–14
10
0 –1
–16
10
10
–2 –18
10
400
d0 = 5μm 500
d0 = 5μm
–18
10
600
700
800
400
900
500
600 700 Temperature, °C
Temperature, °C
(b)
900
–10
10
–3 –1 0
–14
1
10
3
–12
–2
Bulk Shear Strain Rate, s
10
–1
–12
Bulk Shear Strain Rate, s–1
800
(e) –10
10
–16
10
10
2 1
–14
10
0 –1
–16
10
–2 –18
10
400
d0 = 50μm 500
–18
10
600
700
800
400
900
(c)
–3
d0 = 50μm 500
600 700 Temperature, °C
Temperature, °C
800
900
(f) –10
–10
10
10
–12
–1
–12
10
–3 –1
Bulk Shear Strain Rate, s
Bulk Shear Strain Rate, s–1
–3
–2
0
–14
10
1 –16
10
10
2 1
–14
10
0 –1
–16
10
–2 –18
10
400
d0 = 100μm 500
–18
10
600
700
800
900
Temperature, °C
400
–3
d0 = 100μm 500
600 700 Temperature, °C
800
900
Fig. 4. (a–c) Log10 contour plots of critical time in Ma and (d– f) total strain achievable where a ¼ 0.1 for plagioclase deforming under the isostrain-rate boundary condition for starting grain sizes of (a, d) 5, (b, e) 50 and (c, f) 100 mm. Shading shows strains .10.
becomes significant is virtually independent of upwelling rate, as is the displacement. The influence of plume width is investigated in Figure 7 and the critical grain size, time, strain and displacement are plotted as a function of normalized plume width, where 0 is the centre and 1
is the edge. These transects are calculated for a starting grain size of 1 mm and an upwelling rate of 10 cm a21. Critical grain sizes and times only increase by c. 2 mm (or Ma) for an order of magnitude increase in plume width. The strain magnitudes are higher for narrow plumes, but the maximum
DIFFUSION CREEP LIFETIME
(b)
Critical Grain Size
14
17.5
12
15 12.5 10 7.5 5
10 8 6 4 2
2.5 0
0
5
10
15
20
25
30
35
40
Distance, km
(c)
45
0
50
0
5
10
15
100 μm
102 μm
101 μm
103 μm
(d)
Maximum Shear Strain
20
25
30
35
40
45
50
35
40
45
50
Distance, km
Starting Grain Size
30
Displacement
400 350
Displacement, km
25 20
Strain
Critical Time
20
Critical Time, MYr
Critical Grain Size, mm
(a)
265
15 10 5
300 250 200 150 100 50
0
0 0
5
10
15
20
25
30
35
40
45
Distance, km
50
0
5
10
15
20
25
30
Distance, km
Fig. 5. Starting grain size dependence of (pressure-dependent) grain growth in a plume. Graphs show variation of (a) critical grain size, (b) critical time, (c) strain and (d) displacement as a function of distance from the plume centre. An upwelling rate of 10 cm a21, plume width of 100 km, adiabatic temperature of 1500 8C, starting pressure of 14 GPa and constant mantle density of 3500 kg m23 are used for all calculations.
difference is only a factor of 3 –5 for an order of magnitude increase in upwelling rate and corresponds to a difference of 70 km.
Discussion Causes of contour morphology The loop geometry shown by contours of finitestrain magnitude variation are due to the interaction of the temperature-dependent growth rates and temperature- and stress-dependent flow laws. At low temperatures, Ta on Figure 2d, there is no grain growth but diffusion creep is still active so strain is limited by the strain rate and the maximum time that the shear zone is active (i.e. dcrit is never reached). At higher temperatures, the strain rates are higher so larger strain can be accomplished (Tb). However, the grain growth rate also increases so there is less time to accomplish strain before equation (4) is satisfied. With further increasing temperature, the growth rate is much faster than the strain rate so dcrit is reached faster
and the finite strain decreases with increasing temperature Tc. Furthermore, for any given temperature the diffusion creep strain rate increases with stress. Higher stresses therefore accumulate more strain (Td). However, dcrit decreases with increasing stress due to the stress dependence of both the diffusion and dislocation creep laws (Fig. 1); increases in stress will therefore lead to less time before d ¼ dcrit. Eventually the decrease in dcrit leads to a decrease in the maximum strain achievable (Te). Contours of maximum achievable strain under constant strain rate (Fig. 4d –f) also have two parts: an isothermal part which occurs when the deformation time is set to the maximum 50 Ma and a curving part which occurs when tcrit , 50 Ma. This curvature is caused by the balance of tcrit and bulk strain rate. Increased strain rates lead to a decrease in tcrit because faster deformation favours dislocation creep. However, increased strain rates lead to higher finite strains. Generally, the strain-rate increase is such that even though the deformation time is decreasing, higher finite
266
M. A. PEARCE & J. WHEELER
(b)
Critical Grain Size
Critical Time
30
30
25
25
Critical Time, MYr
Critical Grain Size, mm
(a)
20
15
10
15
10
5
5 0
20
0
5
10
15
20
25
30
35
40
45
Distance, km
0
50
0
3cm a–1 9cm a–1 15cm a–1
(c)
15
20
25
30
35
40
45
50
Distance, km
21cm a–1 27cm a–1 33cm a–1
(d)
Maximum Shear Strain
30
Displacement
400 350
Displacement, km
25 20
Strain
5 10
Upwelling Velocity
15 10 5
300 250 200 150 100 50
0
0 0
5
10
15
20
25
30
35
40
45
50
Distance, km
0
5
10
15
20
25
30
35
40
45
50
Distance, km
Fig. 6. Upwelling rate dependence of (pressure-dependent) grain growth in a plume. Graphs show variation of critical (a) critical grain size, (b) critical time, (c) strain and (d) displacement as a function of distance from the plume centre. A starting grain size of 1 mm, plume width of 100 km, adiabatic temperature of 1500 8C, starting pressure of 14 GPa and constant mantle density of 3500 kg m23 are used for all calculations.
strains are achieved at faster strain rates. However, close to the boundary where d0 ¼ dcrit, tcrit decreases faster with increasing strain rate such that the finite strain decreases with increasing strain rate, causing the contour to close with increasing temperature.
Plume transect morphology The shape of the dcrit transects is primarily governed by the strain-rate variations. The centre of the plume therefore has a large critical grain size, decreasing towards the edge as the strain rate increases. Since g˙ = 0 at x ¼ 0, the shape of the transect is governed by ∂dcrit /∂x. This is a function of the reduction in pressure as a result of upwelling, which leads to an increase in grain growth rate and creep rates, and ∂g˙ /∂x which is itself a function of plume width. The critical time curve has concave-up geometry; this is caused by slower strain rates causing a larger dcrit in the centre of the plume and the slow upwelling rates at the edge which inhibit grain growth. However, these times have little effect on the shear strain attained which correlates
with the strain rate that is high at the edge and 0 in the centre. The independence of the displacement profile on upwelling rate is interpreted as being due to scaling between decreased strain rates and longer critical times for slower upwelling rates. Small differences in critical times for different width plumes are responsible for the variation in displacement (leading to variations in grain growth rate) because the maximum upwelling rate is the same. While there are larger displacements for larger plumes, the reduction in width causes higher strains to be accumulated in the narrower plumes because strain is the vertical displacement divided by the width. In summary, if we assume that the initial grain size of olivine formed from spinel in a plume is ,1 mm, our simple model shows that the size and shape of the region of diffusion creep is rather insensitive to that grain size and shows only minor dependence on upwelling rate. There is greater dependence on plume width; a 50 km wide plume shows diffusion creep extending c. 60 –250 km above the spinel field, while a 500 km width plume shows
DIFFUSION CREEP LIFETIME
(a)
(b)
Critical Grain Size 30
Critical Time
50
25
40
Critical Time, MYr
Critical Grain Size, mm
267
20 15 10
30
20
10 5 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0
0.1
0.2
Normalized Distance
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance
Plume Width 50km 100km
Maximum Shear Strain
(c)
500
25
Displacement, km
400
20
Strain
Displacement
(d)
30
15 10
200
200
100
5 0
250km 500km
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Normalized Distance
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance
Fig. 7. Plume-width dependence of (pressure dependent) grain growth in a plume. Graphs show variation of (a) critical grain size, (b) critical time, (c) strain and (d) displacement as a function of distance from the plume centre. A starting grain size of 1 mm, upwelling rate of 10 cm a21, adiabatic temperature of 1500 8C, starting pressure of 14 GPa and constant mantle density of 3500 kg m23 are used for all calculations.
100 –400 km (Fig. 7d). The centres of wide plumes and/or those under thick continental lithosphere may therefore be deforming dominantly by diffusion creep within their olivine-rich upper parts.
Effect of varying a The ‘boundary’ between two deformation mechanisms on a deformation mechanism map is defined as where the deformation mechanisms contribute equally to the strain rate. However, our primary interest was to investigate how much strain could potentially be accommodated following grain-size refinement before the rocks began to develop a crystallographic preferred orientation (CPO) through the activity of dislocation creep. Experiments on analogue materials (Bons & Jessell 1999) suggest that deformed rocks where ,25% of the strain rate was accommodated by dislocation creep may develop strong CPOs. We therefore chose 10% dislocation creep as the upper bound whereby rocks would start developing a CPO; however, this may still be an overestimation.
Analysis of the equations presented for the timedependent diffusion creep allow an assessment of the effect of considering different values of a to be the point at which dislocation creep becomes ‘significant’. For example, considering a ¼ 0.5 as the end-point (this is the conventional boundary between fields on a deformation mechanism map) and assuming isostrain-rate deformation increases the critical stress by 60%, the critical grain size by 30% and the critical time by 100%, doubling the amount of strain achievable relative to a ¼ 0.1. The effect of choosing different values of a can be illustrated graphically using a deformation mechanism map (Fig. 8). Conventional maps (Ashby 1972) show fields labelled according to which deformation mechanism is dominant. While the relative contributions of different mechanisms have been investigated (e.g. Mohamed & Langdon 1974), it is useful to contour them on the map to show, for example, the range of grain sizes for which dislocation creep contributes to ,10% of the bulk strain rate. The solid arrows on Figure 8 show growth paths for initial grain
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Fig. 8. Deformation mechanism map for plagioclase at 600 8C illustrating isostress and isostrain-rate paths and the effect of different a values. The map uses the flow laws of Rybacki & Dresen (2000) for diffusion (grey) and dislocation (white) creep. Varying greyscale illustrates the area where both mechanisms are active, and this is contoured for varying values of a. The conventional boundary (a ¼ 0.5) is represented by the thick dashed line. Also shown are contours of the bulk strain rate (s21).
size of 3 mm to the point where a ¼ 0.1. The dashed arrows show the extension of these paths if a ¼ 0.5. It can be seen that the increase in a has relatively little effect on the length of the path. However, because the growth rate decreases exponentially over time, the critical time is still doubled for this increase in a.
Diffusion creep in the crust Both modelled bounds show large variation in finite strain magnitude as a function of the starting grain size with a reduction in the maximum finite strain achievable from .106 to 103. For starting grain sizes of 5–100 mm, plagioclase-rich shear zones active below c. 650 8C can accommodate very large strains with little influence from dislocation creep. Where extreme grain-size reduction has resulted from metamorphism, a large amount of the finite strain may have accumulated by diffusion creep (possibly g ¼ 10 –100). Once dcrit is reached, the ‘lifetime’ of the diffusion creep shear zone is reached and further strain is accommodated by dislocation creep: there may be no record of the diffusion creep deformation. Once dislocation creep is active, a steady state may be reached where the rate of dynamic recrystallization and grain growth keep the grain size close to the boundary. Under greenschist facies to lower amphibolite facies conditions, grain growth is slow in plagioclase and therefore large strains can be accommodated without any contribution from dislocation
creep. Grain-size refinement under these conditions is often accomplished by neocrystallization driven by excess chemical energy (Stu¨nitz 1998; Ree et al. 2005), producing fine-grained rocks with a grain size between 5 and 50 mm. One such rock was examined by Jiang et al. (2000) who concluded that the pure albite in the rock had deformed by diffusion creep to strains .10 at temperatures between 300 and 450 8C. In this case the rock had nowhere near 50 Ma to deform; however, even limiting the maximum deformation time to 10 Ma, our analysis shows that strains of 100 can be achieved by diffusion creep at 450 8C. The albite grains in this mylonite are somewhat elongate, although they do not reflect the total strain because diffusion creep is always accompanied by neighbour switching (Spingarn & Nix 1978; Wheeler 2009). Numerical models of the effect of deformation on grain growth rate (Kellermann-Slotemaker & De Bresser 2006) show that constantly elongating the grains increases the driving force for grain-boundary migration (growth) and leads to an increase in the growth rate. However, the increased growth rate will have a relatively minor effect in comparison to the other parameters displayed in Figure 2, noting the many orders of magnitude of strain predicted. Generally, strains . 10 cannot be quantified in the field, but Figures 2 and 4 show, in essence, the circumstances under which ‘high’ strains can be accommodated during the lifetime of diffusion creep. Introduction of a second phase (e.g. amphibole in mafic lower crust) into the feldspar-rich regions will serve to prevent grain growth and
DIFFUSION CREEP LIFETIME
prolong the period that the rocks are deforming by diffusion creep.
Diffusion creep in the mantle Upwelling mantle undergoes a phase change from spinel to olivine at around 14 GPa, depending on the temperature. This process results in fine-grained olivine aggregates which, still entrained in the upwelling, continue to deform as well as growing to reduce surface energy. Under the conditions considered in this study, olivine deforms by diffusion creep for anywhere between 1 and 25 Ma. In contrast to the crust, there are no conditions under which grain growth is thermally inhibited and therefore grains in the mantle should always reach a size where dislocation creep becomes significant. However, it should always be recognized that the single-phase mantle considered here is an oversimplification and that, while grain growth to the critical grain size is possible, pinning of olivine boundaries by second phases (e.g. pyroxene or garnet) will probably inhibit the large grain sizes need at low strain rates. Grain growth of two-phase aggregates has been considered for the lower mantle (Solomatov et al. 2002) where growth is by Ostwald ripening and is therefore controlled by grain-boundary diffusion. Under these conditions, it was found that the maximum grain size achievable was between 100 and 1000 mm due to the timescales for diffusion over distances longer than this. The nucleation and growth of spinel after undergoing the olivine– spinel phase change was considered by Riedel & Karato (1997) with reference to down-going slabs. This model considered the nucleation process to take place over time whereas our analysis assumes that nucleation is instantaneous following the passage of material through the phase transition. The length of time for the new phase to become interconnected and therefore accommodate the deformation (Handy 1990) will be controlled by reaction kinetics. If this is fast with respect to the grain growth rate then the phase change can be considered as instantaneous. The relative rates of these processes are related to the driving force and, generally, ‘chemically induced’ driving forces (including phase changes) are much greater than surface-energy induced driving forces (Gottstein & Shvindlerman 1999). There are no experimental data describing the kinetics of the spinel –olivine phase change, but in situ experiments on titanium show that the transformation occurs over a period of minutes whereas subsequent grain growth takes hours to alter the grain size (Seward et al. 2004). In addition, Berger et al. (2010) show that nucleate and subsequent growth is fast relative to coarsening of the resulting
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aggregate. The phase change will therefore occur on a relatively short timescale when compared to the rate at which the aggregate of new grains grows once the phase change is complete. Mantle models of plume formation (van Keken 1997), convection (Van den Berg et al. 1995; Hall & Parmentier 2003) and subduction (Billen & Hirth 2007) incorporate both Newtonian and nonNewtonian rheologies by using a composite rheology which includes both dislocation and diffusion creep flow laws. In such models, grain growth and phase changes are ignored in order to examine the first-order effect of stress and temperature on the strength of the mantle and especially upwelling zones. However, this analysis shows that diffusion creep deformation within upwelling mantle occurs for a finite lifetime. In the centres of the plumes where strain rates are slow, the grains are unlikely to reach the grain sizes needed for the onset of dislocation creep and therefore remain relatively weak. In wide plumes, the centre of the plume may be deforming by diffusion creep as it reaches the base of the lithosphere (c. 300 km displacement) but the majority of the plume will be deforming by dislocation creep. While there is some dependence of strain and displacement on plume width, the insensitivity to starting grain size and upwelling rate means that this model presents an overarching prediction covering many conditions within mantle plumes.
Conclusions By combining time-dependent grain growth with flow laws for dislocation and diffusion creep, it has been shown that large strains can be accommodated by diffusion creep alone before the transition back to dislocation creep. At temperatures ,650 8C, plagioclase flow and grain growth laws suggest that, once the grain size has been reduced (by whatever means), long timescales are needed to reach the dislocation creep field. Isostress and isostrain-rate boundary conditions give similar maximum strains accomplishable, but isostress conditions predict higher strains at higher temperatures. Once grain-size reduction to ,100 mm occurs in amphibolite facies shear zones, strains of 10–100 can be accommodated giving the diffusion creep a ‘lifetime’ of 10 –100 Ma before there is a significant contribution from dislocation creep. In the mantle, upwelling zones which pass through the spinel –olivine phase change show a similar extreme grain-size reduction. Olivine in upwelling mantle can accommodate displacements of 100– 150 km over the majority of the width of the plume before it approaches the dislocation creep field. Critical grain sizes, times, strains and
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displacements are independent of initial grain size and upwelling rate and show a small dependence on plume width over an order of magnitude of widths. Shear zones dominated by plagioclase which control the strength of the middle and lower crust can accommodate large-scale deformation by diffusion creep, and this should be considered when estimating the strength of these regions of the crust. Both the examples illustrated in this study show that, in rocks which have undergone extreme grainsize reduction, grain growth is an important control on the strength and strength evolution. Real rocks with second phases and grain-boundary impurities would have slower grain-growth rates than estimated from the pure anorthite or olivine experiments and therefore stay in the diffusion creep field longer, giving minimum estimates of strains achievable. The graphical framework developed here can be applied to any monophase material where deformation and growth kinetics are well defined and reveal, through the inter-relation of these properties, the lifetime of diffusion creep in the Earth. This work was completed during the tenure of a PhD studentship funded by the University of Liverpool (MP). We thank reviewers J. Warren and J. Gomez Barreiro for their comments.
Appendix 1: Effect of a on total strains achievable before dislocation creep becomes significant For the isostrain-rate case, the critical stress tcrit, critical grain size dcrit and critical time tcrit can be expressed as the ratio of the values for a large (subscript MAX) and small (subscript MIN) value of a. This ratio then illustrates the effect of changing a from 0.1 to some other value. The exact values of the critical parameters can then be calculated from the analysis presented in this paper. The ratio of the critical shear stress is calculated as follows: ⎛
tcritMAX tcritMIN
⎞1/pdis
⎜ ⎟ g˙ bulk ⎜ ⎟ ⎝ ⎠ 1 Adis e(−Qdis /RT) 1+ aMAX = ⎛ ⎞1/pdis ⎟ ⎜ g˙ bulk ⎟ ⎜ ⎠ ⎝ 1 Adis e(−Qdis /RT) 1+ aMIN ⎛ ⎞1/p 1 1 + ⎜ aMIN ⎟ ⎟ =⎜ ⎝ 1 ⎠ 1+ aMAX
dcritMAX dcritMIN
1/mdiff ndiff (1 + aMAX )Adiff tcritMAX e−Qdiff /RT g˙ bulk = 1/mdiff ndiff (1 + aMIN )Adiff tcritMIN e−Qdiff /RT g˙ bulk 1 + aMAX 1/mdiff tcritMAX ndiff /mdiff = 1 + aMIN tcritMIN ⎛ ⎞ndiff /pmdiff 1 1/mdiff 1 + ⎜ 1 + aMAX aMIN ⎟ ⎜ ⎟ = . ⎝ 1 ⎠ 1 + aMIN 1+ aMAX
The values for the ratio of dcrit can then be substituted into the normal growth law to find the critical time ratio p p − d0p )k0 e−Qgg /RT dcritMAX − d0p tcritMAX (dcritMAX = . = p p tcritMIN k0 e−Qgg /RT (dcritMIN − d0p ) dcritMIN − d0p
Following the evaluation of dcritMAX d0 = d and = j, dcritMIN dcritMIN tcritMAX (ddcritMIN )p − (jdcritMIN )p d p − j p = = . p tcritMIN dcritMIN − (jdcritMIN )p 1 − jp For small j, tcritMAX = d p. tcritMIN Using the values for m, n and p, increasing the value of a from 0.1 to 0.5, leads to a 65% increase in tcrit, a 30% increase in dcrit and a 100% increase in tcrit relative to the values show in Figures 3 and 4. A similar argument can be made for the isostress boundary condition, although the total strain has to be evaluated numerically.
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Stu¨nitz, H. & Tullis, J. 2001. Weakening and strain localization produced by syn-deformational reaction of plagioclase. International Journal of Earth Sciences, 90, 136– 148. Tsujino, N. & Nishihara, Y. 2009. Grain-growth kinetics of ferropericlase at high-pressure. Physics of the Earth and Planetary Interiors, 174, 145– 152, doi: 10.1016/ j.pepi.2008.04.002. Urai, J. L., Means, W. D. & Lister, G. S. 1986. Dynamic recrystallization of minerals. In: Hobbs, B. E. & Heard, H. C. (eds) Mineral and Rock Deformation (Laboratory Studies). American Geophysical Union, Washington, Geophysical Monograph, 36, 161–200. Van den Berg, A. P., Yuen, D. A. & Van Keken, P. E. 1995. Rheological transition in mantle convection with a composite temerpature-depedent, nonNewtonian and Newtonian Rheology. Earth and Planetary Science Letters, 129, 249– 260. van Keken, P. 1997. Evolution of starting mantle plumes: a Comparison between numerical and laboratory models. Earth and Planetary Science Letters, 148, 1– 11. Vogt, P. R. 1976. Plumes, subaxial pipe flow, and topography along the mid-oceanic ridge. Earth and Planetary Science Letters, 29, 309–325. Wayte, G. J., Worden, R. H., Rubie, D. C. & Droop, G. T. R. 1989. A tem study of disequilibrium plagioclase breakdown at high-pressure – the role of infiltrating fluid. Contributions to Mineralogy and Petrology, 101, 426 –437. Weertman, J. 1978. Creep laws for the mantle of the earth. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 288, 9–26. Wheeler, J. 2009. The preservation of seismic anisotropy in the earth’s mantle during diffusion creep. Geophysical Journal International, 178, 1723–1732.
Microstructures in deforming – reactive systems BRUCE E. HOBBS1,2* & ALISON ORD1 1
Centre for Exploration Targeting, School of Earth and Environment, The University of Western Australia, Perth, 6009, Australia
2
CSIRO Earth Science and Resource Engineering, Perth, Western Australia, Perth 6102, Australia *Corresponding author (e-mail:
[email protected])
Abstract: The inter-relationships between mineral reactions and deformation are explored with a view to understanding the development of certain mineral foliations and lineations. The following arguments are presented: (i) the processes involved during mineral reactions in deforming metamorphic rocks are described by coupled reaction–diffusion–deformation equations; (ii) these reactions can become unstable producing compositional patterning in both space (metamorphic differentiation) and time (compositional zoning); (iii) the patterns (foliations and mineral lineations) that result from coupled reaction–diffusion–deformation equations are described by surfaces that approximate minimal surfaces (surfaces of zero mean curvature) and an example of such geometry is given; and (iv) the foliations and mineral lineations that form by such processes are controlled by the evolution of the kinematics of the deformation history and not by the finite strain tensor.
A large number of workers have concerned themselves with the interaction or coupling between deformation and mineral reactions, and an extensive review is given by Vernon (2004, pp. 353 –359). Important experimental investigations include Arzi (1978), van der Molen & Paterson (1979), Brodie & Rutter (1985, 1987), Dell’Angelo et al. (1987), Wolf & Wyllie (1991), Yund & Tullis (1991), Rushmer (1995), Rutter & Neumann (1995), Brown & Rushmer (1997), Daines & Kohlstedt (1997), Stunitz & Tullis (2001), De Ronde et al. (2004, 2005), and Delle Piane et al. (2009); insightful discussions are given by Rubie (1983) and Wintsch (1985). In both experimental and field investigations the emphasis is commonly on the effects mineral reactions (including melting) have on the deformation mechanisms (such as grain-size reduction and water infiltration) and mechanical properties. In particular the concern has been whether such reactions produce strain weakening and/or strain-rate weakening (Hobbs et al. 2010), and hence are likely to promote localization of deformation. In this paper we are concerned with the influence mineral reactions have on the microstructures that form during deformation, such as certain foliations (Fig. 1), mineral lineations and compositional zoning, and the information these processes provide for linking the geometry of these microstructures to the processes and kinematics of the deformation. The use of the terms foliation and mineral lineation is restricted to such fabric elements as are defined by compositional heterogeneities, rather
than by the preferred orientation of individual grains such as micas and amphiboles. Evolving metamorphic systems are generally accompanied by finite irreversible deformations while mineral reactions are in progress. The system is commonly considered to be closed to mass transfer although some form of infiltration of H2O and CO2 is proposed in many instances, especially for those systems undergoing retrogression and some granulite facies metamorphic events. Even in systems closed to regional mass transfer, there is some form of mass transfer at the local scale arising from diffusion and from advection associated with local fluid flow (Meth & Carlson 2005). We consider only systems closed to regional-scale mass infiltration in this paper; the consideration of open systems such as those discussed by Korzhinskii (1959) adds a layer of complexity (derived from the dissipation arising from fluid fluxes across gradients in temperature and chemical potentials) that is best left to another paper. Some important contributions that approach the infiltration problem from the same point of view as is presented here are Rusinov & Zhukov (2000, 2008). Even neglecting regional-scale mass transfer, the system is open to heat (i.e. entropy) transfer and energy is dissipated (entropy is produced) within the system, arising from the mechanical and chemical processes that operate during metamorphism. Entropy production is used here as the quantity of heat produced at a point per unit time at the current temperature (Truesdell 1966) by all
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 273– 299. DOI: 10.1144/SP360.16 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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scalar quantity (with units Joules kg21 s21) as is the total dissipation rate F. All of these processes are therefore coupled through the partitioning of dissipation expressed by equation (1). In this paper we explore the thermodynamic framework that links mineral reactions and deformation. We show that deformation is able to induce instabilities in mineral reactions that otherwise would be stable, and that such instabilities can lead to metamorphic differentiation and to compositional zoning. Many, if not all, metamorphic mineral reactions consist of a number of nested or networked coupled mineral reactions where one reaction cannot proceed without another proceeding. Such situations are referred to as ‘cyclic reactions’ by Vernon (2004). The concept originated in the classical paper by Carmichael (1969) who pointed out that many mineral phases behave as catalysts during the reaction process. An example is the reaction: kyanite sillimanite
(2)
which microstructural evidence indicates is better represented as a process by at least one pair of coupled reactions of the form: Fig. 1. Metamorphic differentiation. In both examples the theory developed here would propose that the layering produced by deformation and metamorphism is the result of unstable, coupled reaction– diffusion– deformation equations and forms initially parallel to a principal plane of stretching or deformation-rate. (a) Schist from Rhoscolyn, Anglesey, UK. The primary bedding here is approximately horizontal. (b) Quartz– muscovite schist from Picuris Range, New Mexico (photograph courtesy of Ron Vernon).
the dissipative processes operating in the system. While these processes are in operation and while metamorphic reactions are taking place the metamorphic system is not in equilibrium but must obey the second law of thermodynamics. This states that the total dissipation of energy F is greater than or equal to zero (Truesdell & Noll 1965; Coussy 1995, 2004), the equality holding at equilibrium. The total dissipation is the sum of the specific energy dissipated by mechanical, chemical, thermal and mass-transport processes: F = Fmechanical + Fchemical + Fthermal + Fmass transport ≥ 0
(1)
where F is the total specific dissipation function and Fmechanical, Fchemical, Fthermal and Fmass transport are the contributions to the total dissipation from mechanical, chemical reaction, thermal and masstransport processes respectively. Each of them is a
4 kyanite + 3 quartz + 2K+ + 3H2 O 2 muscovite + 2H+ 2 muscovite + 2H+ 4 sillimanite + 3 quartz + 2K+ + 3H2 O.
(3)
If only concerned with the construction of mineral equilibrium phase diagrams, such microstructural detail is of little concern since the equilibrium phase diagram may be constructed using only equation (2). But if (as in this paper) concerned with the mechanisms of metamorphic processes and the significance of observed microstructures, then the reactions described in equation (3) are of fundamental importance and equation (2) is irrelevant. In this set of coupled reactions, quartz acts as a catalyst in the sense that it appears in both the products and the reactants and muscovite acts as a catalyst in the sense that it participates in the reactions but does not appear in the final product of the reactions: sillimanite. Carmichael (1969) gives other more complicated forms of such coupled reactions and emphasizes the importance of catalytic processes. The concept of coupled or cyclic reactions is further explored by Foster (1977, 1981, 1986, 1999), Yardley (1977), Vernon & Pooley (1981), Vernon et al. (1983), Simpson (1985), Simpson & Wintsch (1989), Guidotti & Johnson (2002), Likhanov & Reverdatto (2002), Vernon (2004), Wintsch et al. (2005) and Whitmeyer & Wintsch (2005). This concept is well known in the literature
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concerning non-linear chemical kinetics (Clarke 1976, 1980; Epstein & Pojman 1998; Ross 2008) where such reactions are referred to as networked reactions. We will show that such cyclic reactions commonly correspond to coupled chemical reactions with multiple non-equilibrium stationary states (Ross 2008) and, as such, have the potential to develop unstable behaviour with the system either oscillating in time (so-called Hopf instabilities; Cross & Hohenberg 1993; Wiggins 2003), forming spatial patterns of mineral distribution (so-called Turing instabilities; Turing 1952) or various combinations of the two (De Wit et al. 1996). These stationary states are not equilibrium states (Ross 2008). It is the formation of spatial instabilities that especially interests us, and we propose that these are the origin of many microstructures in metamorphic rocks including patterns such as metamorphic differentiation, mineral lineations and some patterned distributions of melt in anatexites (Sawyer 2008). We show that many of these structures can occur because of instabilities induced by coupling deformation with the mineral reactions. Such a proposal is not new and was reached in a different but closely related form by Ortoleva (1989, 1994) and coworkers. Similar instabilities, not coupled to deformation, were considered by Fisher & Lasaga (1981). Symbols used in this paper are defined in Table 1 and also when first introduced in the text. In particular, the Einstein summation convention (Nye 1957, p. 7) is used where repeated indices are summed except where otherwise specified, and extensive use of the scalar product between two tensors is ˙ is a scalar used. The term sij 1˙ ij (equivalent to s:1) and is the scalar product of two second-order tensors (the stress sij and the strain rate 1˙ ij ). If the subscript notation is not used then matrices, tensors and vectors are represented in bold format.
Thermodynamic framework The thermodynamic framework followed here derives from Biot (1955, 1958) who introduced the Helmholtz energy for a deforming system (elastic, plastic or viscous) and the concept of internal variables to define the state of a system not at equilibrium. These concepts were further developed by Kestin & Rice (1970) and by Rice (1971, 1975) who showed that the specific Helmholtz energy (defined at each point in the body) acts as a potential function for the strain at that point while the Legendre transform of the Helmholtz energy (the dual to the Helmholtz energy and equivalent to the specific Gibbs energy) acts as a potential for the stress. These concepts have been further refined by Coussy (1995, 2004,
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2010), Coussy & Ulm (1996) and by Houlsby & Puzrin (2006) so that a rigorous basis now exists for the thermodynamic treatment of deforming (inelastic) chemically reacting systems. In this approach, the specific Helmholtz energy (an intensive variable) is a function of spatial position so that the treatment can handle heterogeneous distributions of plastic deformation, chemical potential and processes such as micro-cracking (Rice 1975). This approach is an extension of the thermodynamic approach developed through the 1960s to the 1980s by workers such as Kamb (1959), Larche´ & Cahn (1973) and McLellan (1980) that considered only elastic systems at equilibrium or at local equilibrium. Below we derive the general expression for entropy production rate in a deforming–reacting system and then simplify the treatment by making various assumptions such as constant temperature or no deformation. We show that this leads to systems of coupled reaction–diffusion equations (Cross & Hohenberg 1993) or reaction –diffusion– deformation equations (Ortoleva 1989). These represent convenient expressions of a combination of the first and second laws of thermodynamics. The framework that has evolved since the work of Biot (1955, 1958) proposes that the evolution of a system not at equilibrium can be described using just two potentials: an energy function such as the specific Helmholtz energy C (or, if convenient, the Legendre transform of C i.e. the specific Gibbs energy G) and the dissipation function F defined at each point in the body. The Helmholtz energy is usually the energy function selected because it involves quantities such as the strain or the deformation gradient, the extent of any chemical reactions and the temperatures that arise from the boundary conditions, the initial conditions of the system and any contributions from dissipative processes operating in the system. The Helmholtz energy is convenient rather than the Gibbs energy which involves the stress which, for a given strain or deformation gradient, depends on the constitutive behaviour of the material. We want to develop an argument that does not depend on the constitutive behaviour and applies to elastic, brittle, plastic and viscous solids; an energy function that depends on quantities such as the strain (elastic and/or plastic) or deformation gradient and temperature that are independent of the constitutive behaviour is therefore convenient. Following Rice (1975) we define the specific Helmholtz energy (J kg21) C as C = U − Ts = C (1elastic , T, mK , j M , aQ ij ) ij
(4)
where U is the specific internal energy, T is the absolute temperature, s is the specific entropy, m K is the concentration (kg m23) of the Kth chemical
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Table 1. Symbols used in the text with units Quantity AM A, B, C, D A, B, C, D c cp DK F, G F D, G D G H J JK J K k
k Turing M L˙
mK r1, r2 s s1, s2, s3 t T U Vo, VoA xi aijQ xij, xijd
g , 1plastic 1, 1ij , 1elastic ij ij 1˙ ij , 1˙ dissipative ij
h, hA j M j˙ , j˙ kthermal lTuring m mK r, ro
Description Affinity of the Mth chemical reaction Symbols representing chemical components Concentrations of chemical components Coefficient Specific heat at constant pressure Diffusion coefficient for the Kth chemical component Functions that express the rates of production of chemical components Functions that express the rates of production of chemical components in a deforming environment Specific Gibbs energy Mean curvature Mass flux Flux of the Kth chemical component Jacobian matrix Gaussian curvature Rate constant for chemical reaction
Wavenumber associated with a Turing instability Rate of latent heat production from Mth reaction Concentration of Kth reactant or product Principal radii of a surface Specific entropy Coefficients Time Absolute temperature Specific internal energy Specific volume of the material, specific volume of A Spatial coordinate State variable for process Q Generalized stress associated with plastic strain; generalized stress associated with grain-size reduction Ratio of diffusivities Small strain tensor, elastic strain, plastic strain Strain rate, strain rate arising from dissipative processes Viscosity, viscosity of A Extent of a mineral reaction Rate of a mineral reaction, rate of the Mth mineral reaction Thermal diffusivity Wavelength of the patterning associated with a Turing instability Chemical potential Chemical potential of the Kth chemical component Density, initial density
Units, typical values J kg21 Dimensionless kg m23 Dimensionless J kg21 K21 m2 s21 kg m23 s21 kg m23 s21 J kg21 m21 m s21 m s21 s21 m22 kg12x my21 s21 where x and y depend on the order of the reaction and whether surface processes are important 21 m J kg21 s21 kg m23 M J kg21 K21 Dimensionless s K J kg21 m3 kg21 m Depends on nature of Q Pa Dimensionless Dimensionless s21 Pa s 0 ≤ j ≤ 1;dimensionless s21 m2 s21 m J kg21 J kg21 kg m23 (Continued)
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Table 1. Continued Quantity s, sij fA F Fmechanical Fdiffusive Fchemical Fthermal
x x A, x B C z
Description Cauchy stress Volume fraction of A Dissipation function Contribution to the total dissipation from purely mechanical processes Contribution to the total dissipation from diffusive processes Contribution to the total dissipation from chemical reactions Contribution to the total dissipation from thermal diffusion Taylor–Quinney coefficient Fractions of mechanical dissipation partitioned between reactions that produce A and B Specific Helmholtz energy Variation of fluid content
component, j M is the extent of the Mth chemical reaction and aQ is a parameter (scalar, vector or tensor) that describes the state of some variable Q of interest such as damage, grain-size, crystallographic preferred orientation (CPO) and so on (Houlsby & Puzrin 2006). Quantities that are conjugate to the internal variables expressed in equation (4) (following arguments presented by Coussy 1995, 2004; Coussy & Ulm 1996; Houlsby & Puzrin 2006) are Vo sij =
∂C , ∂1elastic ij
∂C A =− M, ∂j M
Units, typical values
s=− Vo YijQ
∂C , ∂T
=−
V o mK = −
∂C ∂mK
∂C ∂a Q ij
(5)
where sij is the Cauchy stress, mK is the specific chemical potential of the Kth chemical component, A M is the activity of the Mth chemical reaction and Y Q is the ‘generalized stress’ associated with the Qth process and is the ‘driving force’ for the Qth process. Vo is the specific undeformed volume. We emphasize that equations (4) and (5) are defined at each point in the body so that the treatment can handle heterogeneous bodies as well as gradients in the internal variables (Rice 1975). The second law of thermodynamics may be written as an expansion of equation (1): T s˙ = F = Fmechanical + Fchemical + Fthermal + Fmass tranport ≥ 0
(6)
where the overdot signifies differentiation with respect to time t.
Pa Dimensionless J kg21 s21 J kg21 s21 J kg21 s21 J kg21 s21 J kg21 s21 Assumed to be unity; dimensionless Dimensionless J kg21 Dimensionless
The various dissipation functions are (see Coussy 1995; 2004) ˙ K + sT˙ + V o mK m Fmechanical = Vo sij 1˙ plastic ij
Fmass transport
+ Vo YijQ a˙ Q ij ∂mK ∇T = −JK · ∇mK − ∂T M
(7)
M
Fchemical = AM j˙ + L˙
Fthermal = −cp kthermal ∇2 T where summation on K, M or Q is intended and K, M and Q take on the values 1, . . . , ℵ, 1, . . . , < and 1, . . . , Y respectively. In equation (72), · represents the scalar product of vectors; in equation M (73) L˙ is the rate of heat production from reaction M; in equation (74) kthermal is the thermal diffusivity and cp is the specific heat at constant pressure. J K is the mass flux of the Kth component. If we consider isothermal systems (as is commonly done in metamorphic petrology) and neglect processes defined by Q in equation (4), then equation (6) reduces to ˙ K − JK · ∇mK T s˙ = Vo s:1˙ plastic + Vo mK m K K − AK j˙ − L˙ ≥ 0.
(8)
equation (8) is the Clausius –Duhem equation for an isothermal dissipative deforming system with coupled chemical reactions and diffusive mass transfer (Coussy 1995, 2004) and expresses the complete isothermal coupling between deformation, chemical reactions and mass transfer. The inequality on the right-hand side derives from the second
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law of thermodynamics; the equality is true only for equilibrium. The importance of the equation is that it also involves an equality on the left-hand side that enables the entropy production rate to be completely specified. Using the expressions s = −∂C/∂T and Vo mK = −(∂C/∂mK ), equation (71) becomes ˙ K. Fmechanical = Vo sij 1˙ elastic − sT˙ + Vo mK m ij
(9)
If we assume that 1˙ total = 1˙ elastic + 1˙ plastic ij ij ij
(10)
then we can eventually arrive at the Energy equation that expresses the change in temperature arising from all of the dissipative processes and does not include contributions from elastic deformations, that is, ˙K + Vo mK m cp T˙ = xVo sij 1˙ plastic ij mass transport − FK K
−
(11)
Fchemical + Fthermal K
K
where x is the Taylor–Quinney coefficient and represents the proportion of mechanical work arising from dissipative deformation that is available to increase the temperature or to drive diffusion, chemical reactions and structural adjustments such as fracturing or grain-size reduction. At high strains where the energy arising from deformation is stored in crystal defects, x is generally within the range 0.85 ≤ x ≤ 1 (Taylor & Quinney 1934) and here we assume x ¼ 1. For isothermal situations, equation (11) reduces to a set of ℵ coupled reaction –diffusion–deformation equations of the form: K
˙ K = JK · ∇mK + AK j˙ + L˙ Vo mK m − Vo x K s:1˙ plastic
and multiple stationary states. A stationary state is one where the time rate of change of a particular variable is zero. A common example is a simple chemical system where one stationary state is an equilibrium state defined as a state where the entropy production rate is zero. However, even a simple chemical system A + B kk12 C + D has another stationary state where the rate of production of a chemical component is zero but the chemical reactions are still in progress. Such reactions therefore have at least two stationary states: one an equilibrium state and the other a steady non-equilibrium state given through the law of mass action by an equation of the form (Kondepudi & Prigogine 1998): dA = kf AB − kr CD = 0 j˙ = − dt
where j˙ is the reaction-rate, kf and kr are the rate constants for the forward and reverse reactions and A, B, C, D represent the concentrations of the chemical components A, B, C, D. We use concentrations throughout this paper for simplicity, although the discussion can be readily extended to use activities instead of concentrations (Ross 2008, chapter 9). The units of concentration are chosen to suit the problem involved but dimensionless quantities (volume% or mass%) are commonly the most convenient; the rate constants in equation (13) then have the units s21. In coupled chemical reactions where the products of one reaction are used as reactants in another coupled reaction, multiple non-equilibrium stationary states may exist. This can be seen by examining two coupled chemical reactions involving two chemical components, A and B, where the rates of formation of the two components can be written: dA = F(A, B); dt
K
(12)
where summation on K is not intended but K takes on the values 1, . . . , ℵ. x K is the proportion of energy dissipated by the deformation partitioned to the Kth reaction. This means that if a particular reaction involves only strong mineral phases, then more of the energy dissipated is used to enhance that reaction rate than if the reaction only involves weak phases and x K is relatively large.
Stability of networked mineral reaction systems In order to proceed, it is convenient to divide mineral reacting systems into those with single
(13)
dB = G(A, B). dt
(14)
F and G are (normally non-linear) functions of A and B and are defined by the law of mass action. We now need to establish those conditions under which reaction rates are zero. For each chemical component, such conditions define a null-cline which is a curve in composition space that is the locus of all non-equilibrium stationary states; it is the curve of non-equilibrium zero production rate of a chemical component. If we take only the first of these rate equations (equation (141)) and consider a stationary state when dA/dt = 0, then we obtain the curve in (A, B) space for the A null-cline as shown in Figure 2a where we have assumed that the function F is non-linear and something like a cubic. The B null-cline is shown in Figure 2b where we have assumed that the function G is
STRUCTURES IN DEFORMING– REACTIVE SYSTEMS
279
Fig. 2. Compositional phase-space illustrating behaviour of various chemical systems. (a) F null-cline showing positions of three stationary states X, Y and Z. Regions above the null-cline correspond to F , 0 while regions below correspond to F . 0. Arrows show the directions in which the F reaction proceeds if displaced by a small amount from the stationary state. The stationary states are stable if F1 = ∂F/∂A is negative and unstable if positive. (b) G null-cline; arrows show the directions in which the G reaction proceeds if displaced by a small amount from the stationary state. (c) F and G null-clines with both F1 and G2 , 0; hence Tr , 0 and the node is stable. (d) F and G null-clines with F1 . 0, G2 . 0 and F1 , G2; hence Tr . 0 and the node is unstable. (e) F and G null-clines with F1 . 0, G2 , 0 and F1 , G2; hence Tr . 0 and the node is unstable. (f) F and G null-clines with F1 . 0, G2 , 0 and F1 ¼ –G2; hence Tr ¼ 0 and the node is a Hopf bifurcation.
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linear for simplicity. For regions below the nullclines, the respective reaction rates are positive whereas for regions above the null-clines the respective reaction rates are negative. This means that if the system in Figure 2a is displaced by some perturbation to a point above the null-cline, then the concentration of A at constant B moves to the left. Similarly, if the system is displaced to some point below the null-cline then the system moves to the right. Notice that for the A null-cline a fixed concentration of B corresponds to three possible concentrations of A, resulting in three stationary states. Detailed discussions of the stability of various stationary states are given by Murray (1989) and by Epstein & Pojman (1998) and are not repeated here. In general, if ∂F/∂A is positive then the stationary state is unstable; if this derivative is negative then the stationary state is stable. Detailed examination of Figure 2a shows that of the three stationary states shown, two are stable in the sense that a small perturbation in concentration away from that state will return to that state (X and Z in Fig. 2a); for the other stationary state (Y) however, a small perturbation will continue to move the system away from that state. We emphasize that these stationary states are non-equilibrium stationary states since they represent situations where the chemical reactions are still taking place, but the production rates of components are zero. The stability of a stationary state may be derived (Murray 1989; Epstein & Pojman 1998) by examining the eigenvalues of the Jacobian matrix J given by ⎛ ∂F ⎜ J = ⎝ ∂A ∂G ∂A
∂F ⎞ ∂B ⎟ = F1 ∂G ⎠ G1 ∂B o
F2 G2
Table 2. Some behaviours of two component coupled chemical reactions with multiple non-equilibrium stationary states Tr(J ) D
G
Behaviour
,0
.0 .0 Stable node
,0
.0 ,0 Stable focus
.0
.0 ,0 Unstable focus
.0
.0 .0 Unstable node
,0
Saddle point
.0
Hopf bifurcation
Phase diagram
(15) ¼0
where ∂F ∂F , F2 = , F1 = ∂A o ∂B o ∂G G2 = ∂B o
∂G G1 = ∂A
, o
(16)
and the subscript o indicates that the derivatives are evaluated at the steady state A(x, t) = Ao = 0 and B(x, t) = Bo = 0. The possible states of stability and instability are summarized in Table 2 where Tr (J) ¼ F1 + G2, D ¼ det (J) ¼ F1G2 –F2G1 and G ¼ (Tr (J))2 2 4D. Some behaviours of two component systems are also shown in Table 2 and Figure 2c –f. For such systems, extremum principles based on entropy production rates do not exist since the
entropy production rate has no extremum. Ross & Vlad (2005) and Ross (2008, chapter 12) show that, for systems where the relationship between thermodynamic fluxes and forces is linear, an extremum exists for the entropy production rate which corresponds to a stationary state that is not an equilibrium state. This is the state of maximum entropy production rate, commonly referred to in the solid mechanics literature and derived from Ziegler (1963, 1983). The statement is true both ‘close’ and ‘far’ from equilibrium. A rigorous discussion of the terms ‘close’ and ‘far’ is given by Hunt
STRUCTURES IN DEFORMING– REACTIVE SYSTEMS
et al. (1987, 1988). If the relationship between thermodynamic fluxes and forces is non-linear, then only one extremum in the entropy production rate exists and that is an equilibrium state. Ross & Vlad (2005) emphasize that all chemical reactions are non-linear in the sense that the rate of a chemical reaction is not proportional to the affinity of the reaction and that all heat conduction processes are also non-linear. The linear situation is that commonly quoted and discussed by Prigogine (1955), Kondepudi & Prigogine (1998) and a host of others. Ross & Vlad (2005) show that such linear thermal and chemical systems do not exist; equations such as equations (12) and (14) therefore need to be solved in order to understand the evolution of the system. The behaviour of two component systems such as given in equation (14) may therefore be one of the following types. (i) Stable behaviour expressed as either a stable node or a stable focus (Table 2): one possible set of such null-clines is shown in Figure 2c. If these systems also involve diffusion of A and B then the system is defined in terms of reaction –diffusion equations (such as those given in equation (17) below). Then, subject to other conditions considered elsewhere in the paper (equation (23)), spatial patterns known as Turing instabilities can form spontaneously in an otherwise homogeneous material. (ii) Unstable behaviour expressed as an unstable focus, an unstable node, a saddle point or a Hopf bifurcation (Table 2): some configurations of the null-clines involved are shown in Figure 2d– f. The closed ellipse in the representation of a Hopf bifurcation in Table 2 is known as a limit cycle. If the system spirals outwards or inwards to blend with the limit cycle, then it is known as a supercritical Hopf bifurcation. If the system spirals inwards or outwards away from the limit cycle, then it is known as a subcritical Hopf bifurcation (Wiggins 2003, pp. 282, 283). The attractor shown for a Hopf bifurcation in Table 2 is therefore supercritical. In simple systems these kinds of behaviour are relatively easy to explore (Murray 1989; Cross & Hohenberg 1993; Epstein & Pojman 1998) but in highly non-linear reactions, such as are common in metamorphic rocks, the details of such instabilities may be impossible to establish. By highly nonlinear in this context we mean that functions such as F and G in equation (14) are highly non-linear as is exemplified by many of the reactions described by Carmichael (1969), Whitmeyer & Wintsch (2005) and Wintsch et al. (2005). An important theorem
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in this regard is the Poincare–Bendixson theorem (Andronov et al. 1966; Strogatz 1994; Epstein & Pojman 1998), which states that if a two-component system is confined to a finite region of concentration space then it must ultimately reach either a steady state or oscillate periodically. If instability can be demonstrated, then periodic oscillations of the system must exist although it may prove impossible to define these explicitly. Clearly compositional zoning can form by these oscillatory processes as suggested by Ortoleva (1994). If diffusion of A and B is coupled with unstable systems then travelling concentration waves can develop (Murray 1989; Cross & Hohenberg 1993; Epstein & Pojman 1998) so that the system oscillates in chemical composition both spatially and temporally. Carmichael (1969) presents highly non-linear coupled reactions; one such reaction is: 43 biotite + 7K+ + 18(Mg, Fe)++ + 7H2 O O 17 sillimanite + 6 biotite + muscovite + 91 quartz + 43Na+ . In these reactions it will probably prove quite difficult to demonstrate stability or instability and, for such highly non-linear systems, a number of theoretical and graphical approaches have been developed. An excellent summary of these methods is given in Epstein & Pojman (1998, chapter 5). In particular, the network methods developed by Clarke (1976, 1980) deserve special consideration. Other developments are discussed by Schreiber & Ross (2003). Although no overarching principles are yet available to explore the stability of highly non-linear chemical systems, it appears that those involving changes in pH and/or Eh are particularly prone to instability (Epstein & Pojman 1998). Silicate networked reactions, such as that quoted above involving species such as Na+, H+ and multivalent elements such as iron, manganese and titanium, are therefore the top candidates for instability. Some simple mineral systems are examined in the next section.
Mineral reactions with no coupled deformation The thermodynamic framework has indicated that isothermal metamorphic processes involving mineral reactions and deformation can be expressed as a set of coupled reaction –diffusion–deformation equations (equation (12)). At this stage we neglect the deformation and consider only reaction –diffusion equations in which the heat of reaction is also neglected. It is important to note that the following discussion also assumes that the reacting system is
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homogeneous, meaning that the reaction sites are not localized by the microstructure or by localized deformation. This is the common way of treating non-linear chemical reactions (Epstein & Pojman 1998) and clearly metamorphic systems are not homogeneous in this regard. It is useful to discuss homogeneous systems first as a jumping-off point for inhomogeneous systems (Ortoleva & Ross 1974). Consider a pair of reaction –diffusion equations expressed in terms of a one-dimensional spatial coordinate x and two chemical components A and B as: dA ∂2 A = DA 2 + F(A, B) dt ∂x and
dB ∂2 B = DB 2 + G(A, B) dt ∂x
(17)
where t is time, DA and DB are the diffusion coefficients associated with the two chemical components and F(x, t), G(x, t) are functions that describe the rates of chemical reactions involving A and B in terms of space and time. Stability of the system to perturbations is guaranteed (see Table 2) if Tr = F1 + G2 , 0
Fig. 3. Behaviour of the system (Ao, Bo) in the (g, Tr) plane with D . 0 and F1 , 0 both fixed. The system is stable for Tr , 0 and below the curve labelled Turing. Above the curve labelled Turing the system is unstable to spatial instabilities. Above the line marked Hopf (where Tr changes sign), the system is unstable to temporal instabilities that are spatially uniform.
In both cases the critical wavenumber k Turing for the onset of instability when Tr ¼ 0 is given by
and
D = F1 G2 − F2 G1 . 0
(18)
where we have defined Tr ¼ F1 + G2 as the trace of the Jacobian J and D is the determinant of J. Murray (1989) and Epstein & Pojman (1998) show that instability is possible if D2 − (DB F1 + DA G2 )k2 + DA DB k4 , 0
(19)
which is possible if F1 + g 2 G2 . 2gD
(20)
√
where g = DA /DB . Equations (19) and (20) can only be simultaneously satisfied if F1 and G2 have opposite signs and g = 1. There are therefore two cases: F1 , 0 and G2 . 0:instability occurs if
1 g. D + D2 − F1 G2 . 1 G2
(211 )
F1 . 0 and G2 , 0:instability occurs if
1 −1 g . D + D2 − F1 G2 . 1. F1
(212 )
k
Turing
=
2p
lTuring
1 D 4 = g2
(22)
where lTuring is the critical wavelength for the instability. In general, the wavelength that develops is constrained by the size of the domain in which the pattern develops (Murray 1989, section 14.3); for a domain which can be taken to be very large with respect to the wavelength of the pattern, the wavenumber is given by equation (22). Following Dawes & Proctor (2008) we plot regions of stability of (Ao, Bo) in (g, Tr) space in Figure 3 for fixed D . 0 and fixed F1 , 0. A Hopf bifurcation comprising a spatially uniform instability that oscillates in time occurs when Tr ¼ F1 + G2 changes sign. For large g, the Turing instability occurs before the Hopf instability as Tr is increased. For small g the Hopf instability occurs first. Thus, for metamorphic systems in which the component A is taken to be (say) silicon (or oxygen or an alkalimetal) and component B is taken to be aluminium, then DAl ≪ DSi (Carmichael 1969) and so g . 1. For such systems Turing instabilities are expected to occur before Hopf instabilities. This is the same situation that occurs in the development of dislocation patterns in deformed metals (Walgraef & Aifantis 1985a, b, c, 1988) and that has been postulated for the development of joint patterns in deformed rocks (Ord & Hobbs 2010). The situation in the vicinity of the conditions where the Hopf
STRUCTURES IN DEFORMING– REACTIVE SYSTEMS
and Turing instabilities occur simultaneously is described by De Wit et al. (1996). In summary, the five conditions for a Turing instability to form in a two-fold coupled mineral reaction system, together with the wavenumber k Turing that ultimately develops in an infinite domain, are given by: (i) Tr = F1 + G2 , 0; (ii) D = F1 G2 − F2 G1 . 0; (iii) F1 is opposite in sign to G2; (iv) DA /DB . 1 for F1 , 0; (v) DB F1 +DA G2 .2[DA DB (F1 G2 −F2 G1 )]1/2 .0. (23) The resulting wavenumber is kTuring =
2p
lTuring
=
F1 G2 − F2 G1 DA /DB
1/4 . (24)
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Example 1: networked mineral reactions involving redox changes Networked chemical reactions that involve changes in redox state are particularly susceptible to instabilities. One reason for this seems to lie in the fact that many redox reactions can be expressed in an autocatalytic manner. The oxidation of Fe2+ can therefore be written: Fe2+ + 2Fe3+ 3Fe3+ + electron. We take an example where a mineral A reacts to produce another mineral B, that is, A B. In the process A dissolves in an inter-granular fluid to contribute Fe2+ to that fluid. An example could be biotite reacting to produce garnet. A set of
Fig. 4. Results of autocatalytic system involving redox reactions of equation (25). In this example the reaction involved is Fe2+ Fe3+ but any reaction involving a change in redox state would behave in a similar manner. The parameters for the model in (a) and (b) are k1 ¼ 0.008, k2 ¼ 0.02, k3 ¼ 1.0 and k4 ¼ 1.0; in Figure (c) the parameters are k1 ¼ 0.008, k2 ¼ 0.02, k3 ¼ 0.5 and k4 ¼ 0.8 The model is based on results from Boyce & DiPrima (2005). (a) Plots of the reactant A and the product B (both divided by 200) together with the intermediate species Fe2+ and Fe 3+, both of which oscillate in time after passing through a Hopf bifurcation. Note that equilibrium is far to the right of the diagram. (b) Zoom of part of (a). (c) Another model of the reaction with regions marked that correspond to zonation systems that develop in a growing grain in (d).
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networked reactions that describes the redox part of the reaction is: k1′
Reaction of A to produce Fe2+ : A − Fe2+ k2′
Non-catalytic step:
Fe2+ − Fe3+
Autocatalytic step:
Fe2+ + 2Fe3+ − 3Fe3+
k3′
k4′
Production of B from Fe3+ : Fe3+ − B
(25)
where the ki′ are the reaction rates for the four reactions. The set of equations that describes the coupling between these reactions is: ∂A = −k1 A ∂t ∂Fe2+ = k1 A − k2 Fe2+ − k3 Fe2+ [Fe3+ ]2 ∂t ∂Fe3+ = k2 Fe2+ + k3 Fe2+ [Fe3+ ]2 − k4 Fe3+ ∂t
∂B = k4 Fe3+ ∂t
(26)
where the ki are functions of the ki′ and of the concentrations of any other components that may be involved in the reactants. Solutions to these equations are shown in Figure 4a– c where the system oscillates in time with respect to the concentrations of Fe2+ and Fe3+ in solution. The resulting compositional zonal sequence within a growing grain is illustrated in Figure 4c, d.
Example 2: Diffusion with redox changes Fisher & Lasaga (1981) consider a classical networked reaction (the Brusselator) described by the coupled reaction –diffusion equations: dA ∂2 A = DA 2 + a + A2 B − (b + 1)A dt ∂x dB ∂2 B = DB 2 + bA − A2 B. dt ∂x
(27)
Here the A 2B term could arise from autocatalytic reactions of the form equation (253) and so could be
Fig. 5. Development of instabilities for the Brusselator system described by Fisher & Lasaga (1981). (a) Initial heterogeneous distribution of mineralogy. The fluctuations are up to 1022 of the steady-state background. (b) Metamorphic layering produced after a dimensionless time of 1.0 defined by Fisher & Lasaga (1981). (c) Instabilities developed in the reaction described by equation (27) for parameters where temporal oscillations occur. The model assumes instabilities in the redox state of Fe, but any element such as Mg or Ca that substitutes for Fe would give similar bursts with time. The plot shows oscillations about an initial homogeneous background set at zero. (d) The model shown in (b) shortened 80% normal to the layering assuming Maxwell constitutive behaviour. Initially present passive markers develop crenulations.
STRUCTURES IN DEFORMING– REACTIVE SYSTEMS
relevant in many networked metamorphic reactions. In fact, the reactions described by equation (27) are similar to equation (25) except that both the reactant and the product in equation (27) involve Fe3+ (Epstein & Pojman 1998, p. 39). Fisher & Lasaga (1981) treat this system in detail and the reader is referred to their paper for details. In particular, they derive the conditions under which compositional differentiation will arise, and the expression for the rate of growth of such layering from an initial compositional distribution that consists of small perturbations g(x) above a background steady-state composition co, defined by a Fourier series: g(x) = co +
npx . an sin L n=1
N
(28)
These initial perturbations might represent initial fluctuations in the spatial concentrations of micas and of quartz. In equation (28) L is some length scale associated with the system and an is the initial compositional fluctuation associated with the nth wavenumber. The spatial evolution of the composition c of the system is given by npx exp (wn t) an sin c(x, t) = co + L n=1 N
equation (27) using values of wn derived by Fisher & Lasaga (1981) is shown in Figure 5a–b. The system of equations (equation (27)) is also capable of undergoing oscillations in time so that as the system passes upwards across the line Tr ¼ 0 for high values of g in Figure 3, the system passes from one characterized by the Turing instabilities shown in Figure 5b through a Hopf bifurcation to one characterized by oscillations in time as shown in Figure 5c. The oscillations in this case comprise a series of bursts, producing the spiked zonal patterns observed in the concentrations of Ca and Mg in some garnets (Chernoff & Carlson 1997).
Example 3: the muscovite – quartz reaction We consider a simple set of coupled mineral reactions: k1′
quartz ˘ quartz in solution
where wn is the amplification of the nth mode and is discussed by Fisher & Lasaga (1981). The twodimensional evolution of a system described by
(301 )
k2′
k3′
muscovite ˘ 3 quartz + 3K+ + H2 O k4′
k5′
(29)
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+
(302 )
AlO− 3
muscovite ˘ 3 quartz in solution k6′
+
+ 3K + H2 O +
(303 )
AlO− 3
and represent the concentrations of quartz, quartz in solution and of muscovite as Q, Qs and M,
Fig. 6. Two possible mechanisms involved for reaction (30). (a) Muscovite breaks down reversibly to produce quartz. Quartz may then ‘dissolve’ to produce quartz in solution and can re-precipitate as quartz at a different rate to which it dissolves, or can be removed from or added to the system. In this reaction quartz in solution acts as a parameter; the concentration of quartz in solution acts as a control switch which decides whether the reaction is a stable node or a stable focus. (b) Alternatively to the situation in (a), quartz in solution can also react to produce muscovite and muscovite can react to produce quartz once more at a different rate.
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respectively. k1′ and k2′ are the forward and reverse rate constants for reaction (301) while k3′ , k4′ and k6′ , k6′ are the equivalent rate constants for reactions (302) and (303), respectively. We consider two possibilities for these reactions, the first of which is illustrated in Figure 6a. This effectively neglects reaction (303) and proposes that the only processes involve muscovite forming quartz and vice versa and the dissolution/precipitation of quartz. Quartz in solution may be removed (or added) to perhaps precipitate locally as quartz veins. This corresponds to some models of ‘pressure solution’ (Worley et al. 1997). The second possibility involves all three reactions in equation (30) so that quartz formed by breakdown of muscovite can also be involved in dissolution and precipitation (Fig. 6b). This corresponds to a closed system. The rate equations for quartz and muscovite involving only reactions (301) and (302) are: dQ = −k4 Q3 − k1 Q + k2 Qs + k3 M dt dM = −k3 M + k4 Q3 dt
(311 )
k2 Qs k1
and
Mo =
k4 k23 3 Q k3 k13 s
(32)
and F1 = −k1 − G1 =
3k4 k22 2 Qs ; k12
3k4 k22 k12
Q2s ;
F 2 = k3
G2 = −k3
(33)
from which Tr = −k1 − k3 −
3k4 k22 2 Qs , 0 k12
and
(34)
D = k1 k3 . 0. (Tr − 4D) is positive or negative depending on the magnitude of 4k1k3 relative to (Tr)2. Thus, from Table 2, the behaviour of this system is either a stable node or a stable focus depending on the magnitude of 4k1k3 relative to (Tr)2. Note that the situation is not as simple as first appears because the relative magnitudes of 4k1k3 and (Tr)2 depend on the magnitudes of Qs and the concentrations (activities) of K+, H2O and AlO2 3 ; the system behaviour in a locally open system, where these components might be removed from or added to the local 2
√
1/2 k1 (k1 + k3 ) + k1 k3 . k2 3k4 These components act as control switches between the two types of behaviour. Either way, the conditions Tr , 0 with D . 0 are two of the conditions for a Turing instability to form. In addition, for a Turing instability to develop we require that F1 has the opposite sign to G2; this is never the case for the muscovite–quartz reaction, so the coupled mineral reactions given by equations (301) and (302) in a homogeneous system or in the absence of other influences such as deformation can never develop Turing-type spatial patterning. The same conclusion arises if, in addition to reactions (301) and (302) we add reaction (303). In the following section, we explore the conditions under which reactions (30) can become unstable and produce Turing instabilities.
(312 )
in which k1 = k1′ , k2 = k2′ , k3 = k3′ and k4 = k4′ · K+ · H2 O · Al2 O′ 3 . The reactions have steady-state solutions: Qo =
system, could therefore be a stable focus or a stable node depending for instance on whether Qs is larger or smaller than
Mineral reactions with coupled deformation If we consider the reactions described by equation (14) to be occurring in a deforming material where the local mechanical dissipation drives the chemical reactions, then the system is described by equation (12). In essence, this means that equation (14) is replaced by dA = F(A, B) + 1A H(A, B) dt dB and = G(A, B) + 1B I(A, B) dt
(35)
where H and I are functions of A and B defined by heterogeneities in the system and/or by deformation and 1A, 1B define the strength of the contribution of H and I to the steady-state production rates of A and B. This means that the null-clines defined for F and G are shifted (and distorted) by H and I as shown in Figure 7, the amount of shift and distortion being controlled by the magnitudes of 1A and 1B. For the moment we neglect the diffusion terms and the rate of production of latent heat and assume that the constitutive law for deformation is Newtonian viscous of the form sij = h1˙ ij where h is the viscosity. The partitioning of the viscous dissipation between the two reactions in equation (14) can then be achieved by writing h = hA fA + hB fB where fA and fB are the volume fractions of A and B and fA + fB = 1. The volume fractions are related to the concentrations of A and B by
STRUCTURES IN DEFORMING– REACTIVE SYSTEMS
287
Fig. 7. (A– B) compositional phase space showing progression of system behaviour with progressive changes in a dissipative process that affects reaction rates. In this diagram the effect on the reaction rate that produces A (the F null-cline) is large compared to the effect on B (the G null-cline). (a) Intersection of the F and G null-clines results in temporal stable behaviour, but such that Turing instabilities are possible. (b) The dissipative process has moved the F null-cline relative to the G null-cline such that a Hopf bifurcation now occurs. (c) Further relative movement of the two null-clines results in oscillatory temporal behaviour. (d) Summary of a system where the proportion of total mechanical , moves the F null-cline relative to the G null-cline from an dissipation partitioned to the production of A, x A sij 1˙ dissipative ij intersection at P 1 (unstable in time) to P 2 (Hopf bifurcation) to P 3 (Turing instability).
fA ¼ AV A and fB ¼ BV B where V A, V B are the specific volumes of A and B. For a general set of coupled reaction –deformation equations involving two chemical components deforming at a strain rate 1˙ , we then have (in 1D): F (A, B) = F(A, B) D
− x A (hA AV A + hB (1 − AV A ))˙12
GD (A, B) = G(A, B) − x B (hA (1 − BV B ) + hB BV B )˙12 (36) where F D and G D are the rates of formation of A and B in a deforming environment whereas F and G are these same rates in a non-deforming environment. Similar expressions hold for power-law materials.
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x A and x B are the fractions of the energy dissipated by the deformation partitioned between the F and G reactions, respectively, and play the same roles as 1A and 1B in equation (35); we also have x A + x B = 1. We then obtain F1D = F1 − x A V A (hA − hB )˙12 ; F2D = F2 + x A V B (hA − hB )˙12 B A A B 2 GD 1 ; 1 = G1 − x V (h − h )˙ B B A B 2 1 . GD 2 = G2 + x V (h − h )˙
(37)
It follows that Tr D = Tr − (x A V A − x B V B )(hA − hB )˙12 DD = D − (hA − hB )˙12 (V A (x B F2 − x A G2 ) +V B (x B F1 − x A G1 )).
(38)
Thus, Tr D ¼ 0 when Tr = (x A V A − x B V B ) (hA − hB )˙12 and DD ¼ 0 when D = (hA − hB )˙12 (V A (x B F2 − x A G2 ) + V B (x B F1 − x A G1 )). This means (see Table 2) that a system that is stable in a non-deforming system can become unstable (including a Hopf bifurcation) when coupled to a deformation. In a general system, the transition from one form of behaviour to another is strongly dependent on the strain rate. Transitions of this type are illustrated in Figure 7 where (in Fig. 7d) the A and B null-clines coupled to deformation are labelled F D and G D, respectively. As a specific example, consider the quartz – muscovite system described by equations (301) and (302); then Tr D = −3k4 Q2o − k1 − k3 − (xQ V Q − x M V M )(hQ − hM )˙12 .
(39)
Since V Q . V M and we assume hQ . hM, Tr D is always negative for the deforming muscovite– quartz reacting system if more of the energy dissipated by the deformation is partitioned to the quartz reaction than to the mica reaction. This is the case if quartz-rich domains are stronger than mica-rich domains. However, DD = k1 k3 − (hQ − hM )˙12 (k3 V Q − (3k4 Q2o + k1 x M )V M )
(40)
so that DD is positive or negative depending on the magnitude of k1k3 relative to (hQ − hM )˙12 (k3 V Q − (3k4 Q2o + k1 x M )V M ), which again is clearly dependent on the strain rate. DD can also be zero which means the system is unstable (corresponding to a saddle point) and, by the Poincare –Bendixson theorem, must oscillate in time. In addition, although F1D is always negative, GD 2 can be positive or negative depending on the relative M M Q M 2 magnitudes of k3 and x V (h − h )˙1 . Thus for positive GD 2 , corresponding to relatively fast strain rates, Turing instabilities become possible. The behaviour of the deforming muscovite– quartz system can be summarized as follows: (i) TrD is always negative and hence the system (without coupled diffusion) is stable unless DD , 0; in this case the system oscillates. (ii) DD is positive unless k1 k3 ≤ (hQ − hM )˙12 (k3 V Q − (3k4 Q2o + k1 x M )V M ); in this case the system is unstable. Since Tr D = 0 this instability corresponds to a saddle point and the system oscillates in time. (iii) F1D is always negative but GD 2 is positive whenever x M V M (hQ − hM )˙12 . k3 . Thus if diffusion is coupled, this corresponds to the development of Turing instabilities with a wavelength lTuring given by
lTuring =
√
DD 1/4 2p 2 g
(41)
and so depends on the strain rate.
Discussion Extension to more complicated coupled reactions We have only considered the simplest of coupled metamorphic reactions in this paper, namely some simple redox reactions and the muscovite –quartz reaction. If this last system is considered to be homogeneous it is always stable against perturbations even if coupled to diffusion. However, if the system is inhomogeneous with localized deformation sites, it becomes unstable and metamorphic differentiation can develop. The common networked reactions in metamorphic rocks are far more complicated (Carmichael 1969; Whitmeyer & Wintsch 2005) than the quartz–muscovite reaction and commonly involve redox reactions. In order to understand how these reaction networks behave in non-equilibrium situations, it is necessary to classify them according to Table 2. In particular,
STRUCTURES IN DEFORMING– REACTIVE SYSTEMS
for networked reactions to develop Turing instabilities it is necessary that the conditions outlined in equation (23) be satisfied. Given the complexity of the reaction networks proposed by Carmichael, Wintsch and others this involves protracted algebraic manipulations resulting in considerable algebraic opacity; other ways of looking into these systems need to be investigated. A range of such approaches have been described by Epstein & Pojman (1998, chapter 5) and some other developments are discussed by Schreiber & Ross (2003). However, given that coupling between deformation and mineral reactions is capable of altering the nature of the stability that characterizes a particular network in the undeformed state (Fig. 7), we expect that most deforming networked metamorphic reactions in deforming metamorphic rocks are capable of behaving somewhere in the spectrum between systems that oscillate in time and Turing instabilities; compositional zoning and segregation or differentiation of mineral assemblages arising from these processes should therefore be common. Identical conclusions were reached by Ortoleva (1989, 1994). We attribute most foliations and mineral lineations and compositional zoning patterns within grains that are observed in metamorphic rocks to the development of instabilities of this type.
Metamorphic differentiation The formation of compositional patterns (mineral foliations and lineations) in rocks of initial homogeneous mineralogical composition is a widespread phenomenon in deformed metamorphic rocks (Darwin 1846; Williams 1972; Worley et al. 1997; Vernon 2004). In many instances, especially at low metamorphic grades, the process of differentiation is attributed to ‘pressure solution’ with removal of quartz from the rock to some neighbouring site (Williams 1972; Worley et al. 1997). In other instances (Vernon 2004) the process is attributed to ‘recrystallization’ and is iso-chemical in the sense that no material is removed to some site outside of the system. In many instances the process is intimately associated with the development of crenulation cleavage although the general development of schistosity and foliation, and particularly mineral lineations in high-grade rocks, may have little obvious association with crenulation cleavage. In general, there would seem to be agreement that metamorphic differentiation is associated with deformation. In this paper we have attempted to develop a theory that explores the development of mineral patterns in terms of reaction –diffusion equations or some modification of these. Other attempts at a quantitative explanation of the phenomenon have been made by Robin (1979), Fletcher (1977,
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1982), Fueten & Robin (1992), Ortoleva (1994) and Fueten et al. (2002). It is not intended here to review the extensive literature on metamorphic differentiation, but rather to indicate where the above studies make contact with the present work and how future developments may evolve. The essence of the above approaches is to couple the kinetics of mineral dissolution and/or reaction to the stress dependence of the chemical potential of material dissolved in a fluid, and follows the earlier work of Gibbs (1906) and Kamb (1959). In addition, some approaches propose that the kinetics of dissolution is enhanced by the presence of grain contacts between dissimilar minerals (Ortoleva 1994) or by stress concentrations at the ends of propagating fractures (Robin 1979; Fueten & Robin 1992; Fueten et al. 2002). The most comprehensive approach is that of Ortoleva (1994). He and co-workers propose that the classical reaction– diffusion equations are modified by coupling to some deformation process that is concentration and/or stress dependent. In this respect, the approach developed here is identical to that of Ortoleva except that Ortoleva proposes quite specific coupling models rather than the general approach proposed here based on thermodynamics. The two approaches are completely compatible. Ortoleva proposes that the kinetics of the mineral reaction processes are influenced by one or all of (i) concentration and/or stress-dependent grain growth; (ii) changes in local mean stress that in turn is controlled by the concentration of weaker or stronger phases in the neighbourhood (this in the terminology of Ortoleva is a texture control on the mean stress); and (iii) changes in the stored energy of reactants controlled by changes in mean stress, in turn influenced by the texture. These feedback processes result in otherwise stable coupled reacting mineral systems becoming unstable and developing spatial differentiation of mineral phases. The processes appear identical to the processes proposed here but they differ in detail; for instance, Ortoleva describes spatial differentiation in systems where the diffusivities of chemical components are equal (Dewers & Ortoleva 1989). This cannot occur in a strict Turing-unstable system. Clearly, there may be several ways of arriving at the same structure. However, the fundamental basis for all these processes is identical in that feedback between deformation processes and networked mineral reactions leads to instabilities in both space and time that may not otherwise occur. This is equivalent to moving the relative position of null-clines by some coupled dissipation process, as illustrated in Figure 7. A numerical example of the progressive development of metamorphic differentiation in association with deformation is presented in
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Fig. 8. The deformation of initially layered materials with Maxwell constitutive behaviour. Strain rate is 10212 s21. Matrix material has a viscosity of 5 × 1017 Pa s. Strong layers have a viscosity of 1020 Pa s. Differentiated layers have a viscosity of 1017 Pa s. The fold mechanism is identical to the Biot mechanism. Initial distribution of composition is given in Figure 5a. By the time folding starts, the metamorphic layering has developed to that shown in Figure 5b. (a) The overall view of a three-layer system with horizontal shortening of 40% and a dextral shear strain parallel to the axial plane through 308. (b) Zoom of right-hand side of (a). (c) Same model shortened 40% with no shear parallel to the axial plane. (d) Natural example (image courtesy of Tim Bell).
Figures 5 and 8. The initial undeformed and undifferentiated material is shown in Figure 5a. It is assumed that the material has Maxwell constitutive properties and undergoes chemical reactions that are described by equation (26), so that the spatial instabilities discussed by Fisher & Lasaga (1981) develop. These reactions could, for
example, be some of the networked reactions discussed by Carmichael (1969) where sillimanite, biotite, muscovite and quartz form by the breakdown of biotite. The structure that develops would be biotite layers alternating with layers comprised of sillimanite, muscovite and quartz. As the material shortens, this layering develops
STRUCTURES IN DEFORMING– REACTIVE SYSTEMS
as in Figure 5b. If no initial layering such as bedding is present, the final result of the deformation (80% shortening in Fig. 5d) is a strong differentiated layering with fine crenulations of initial passive markers. If stronger layers are present initially, as in Figure 8, then folds develop with the metamorphic layering approximately axial plane in orientation. The metamorphic layering undergoes shearing as shown in Figure 8a– c with displacement of thin initial layers and the development of a crenulated differentiated cleavage. These structures are quite similar to naturally deformed systems (Figs 1 & 8d). The progressive development of this ‘axial plane’ structure and the compositionally dependent strains/displacements associated with the structure therefore depend critically on the kinematic history. In this instance, the metamorphic layering develops rapidly with respect to the rate of fold development. Future work will involve an exploration of the effects of changing the relative rates of fold growth and metamorphic differentiation. The coupling between mineral reaction rates and stretching expressed by equations (12) and (36) and subsequent discussion indicates that instabilities only develop if the deformation rate (the stretching) in a particular direction is non-zero. Otherwise, the mineral reaction is stable and no spatial patterning develops in that direction. These considerations predict that these fabric elements evolve in relation to the principal axes of stretching (not strain). Similar conclusions were reached by Ortoleva et al. (1982), who show that the compositional patterns developed through deformation-driven instabilities are controlled in orientation by the stress tensor.
Compositional zoning Ortoleva & Ross (1974) discuss the development of wave-like instabilities in chemically reacting systems when heterogeneities exist in the distribution of reaction sites. Depending on the nature of the equivalent reactions in a homogeneous situation, a number of types of chemical waves can originate at the individual reaction sites. These include planar waves of concentration, oscillations of concentration and spiral compositional waves; spiral waves are commonly restricted to the site of a chemical reaction. These waves travel with a phase velocity that is invariably different to the transfer of material by diffusion or advection alone and is a function of the non-linear kinetics of the chemical reactions. This means that, in some instances, the influence of a wave may be restricted at any instant to a small part of the system; in other instances the influence of the wave may extend throughout the system so that
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simultaneously all parts of the system undergo the same change in composition. A growing grain interface that incorporates chemical components from these waves will preserve a record of the nature of these waves: a dominantly monotonic change in concentration arising from planar waves, oscillatory changes in concentration arising from oscillatory waves and spiral-shaped changes in concentration arising from spiral waves. The above three different types of compositional zoning within porphyroblasts have been reported in the literature. A review of observations up until 2004 is presented by Vernon (2004). The three types are: (i) more or less monotonic increases or decreases in the concentration of an element from the inferred site of nucleation of a porphyroblast towards the rim with some discontinuities in the case of Ca and Mg in particular (Chernoff & Carlson 1997); (ii) oscillatory zoning superimposed on variable trends in concentration of an element towards the rim (Schumacher et al. 1999; Yang & Rivers 2002; Meth & Carlson 2005) and (iii) spiral zoning of the concentration of an element (Yang & Rivers 2001). In most cases, the zoning is interpreted in terms of variations in the flux of nutrients towards the growing porphyroblast either by diffusion or advection in a fluid. The interpretation in terms of variations in the supply of nutrients remains if the Ortoleva and Ross approach is adopted, but the origin of these variations is now proposed as specifically arising from competition in the production and consumption of nutrients in nearby unstable chemical reactions. The scale on which these compositional phase waves occurs is controlled by the phase velocity of the chemical waves, and it may therefore be dangerous to base arguments of the scale of equilibration in such situations on diffusivities alone. Given the above discussion, a transition within a zoned grain that reflects the progressive increase in strain rate as the deformation increases in intensity is expected. One scenario is as follows. At very small strain rates, as the metamorphic event begins, the reaction network is expected to be stable with no variations or oscillations in chemical composition. As the strain rate increases, compositional zoning should initiate for some chemical components. At the critical strain rate where Hopf bifurcations are no longer possible, Turing instabilities nucleate and compositional zoning stops. In this particular scenario, the initiation of zoning should therefore coincide with the inception of a critical strain rate and the initiation of metamorphic differentiation should coincide with the cessation of compositional zoning. Certainly there are aspects of the compositional zoning documented by Meth & Carlson (2005) that fit with such a scenario, in that the inception of compositional zoning is correlated
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with the onset of a deformation. At a more detailed level, if the spacing between compositional zones increases outwards from a periodic pattern in a grain towards a region where zoning no longer exists, then this would correspond to a subcritical Hopf bifurcation. If the spacing decreases, this would correspond to a supercritical Hopf bifurcation. The detailed examination of compositional zoning patterns in conjunction with the associated deformation history in deformed metamorphic rocks offers considerable insight into the processes that operate in deforming–reacting systems and of the history of that system.
Minimal surfaces De Wit et al. (1992, 1997), Mecke (1996, 1997), Leppanen et al. (2004), Alber et al. (2005) and Glimm & Hentschel (2008) have pointed out that the spatial segregation of phases into Turing instabilities forms interfaces separating the segregations that approximate minimal surfaces (a minimal surface being one where the mean curvature is zero). The mean curvature H (with dimensions per unit length) at any point is defined as H = 0.5(1/r1 + 1/r2 ) where r1 and r2 are the principal curvatures of the surface at that point (LopezBarron & Macosko 2009). In addition, the types of interfaces that develop for reaction –diffusion equations are saddle shaped or hyperbolic and are characterized by values of the Gaussian curvature K (with dimensions per unit length2), that are less than zero (hyperbolic) or zero (planar). K is given by K = 1/r1 r2 . Aksimentiev et al. (2002) and Lopez-Barron & Macosko (2009) show that, during the annealing of immiscible polymer blends, the microstructures can evolve in both selfsimilar and non-self-similar manners and that the evolution along various paths is represented in the shapes and standard deviations of plots of the mean and Gaussian curvatures. Paths towards minimum energy configurations are characterized by decreases in the standard deviations of both H and K, a transition in the mean value of K ranging from zero to a distribution with a skewness towards negative values indicating a transition to a dominance of hyperbolic or saddle-shaped topologies. In metamorphic rocks, quartz-rich and mica-rich domains commonly interweave in the system while individual grains of other phases such as garnet and feldspar are embedded in this topology. An interweaved interface separates these domains; in S-tectonites these are flattened lozenges while in L-tectonites these domains are strongly elongate and perhaps approximate flat elongate ellipsoids. A typical example of such domainal structure is the ‘millipede’ structure of Bell (1981) in three
dimensions. Reaction –diffusion theory proposes that the topology of this interface is or evolves towards that of a minimal surface. It should be emphasized that much of the literature on chemical instabilities is involved with homogeneous systems and the discussion is normally in terms of coupled reaction –diffusion equations where differences in diffusion coefficients of the various components exist (Cross & Hohenberg 1993; Epstein & Pojman 1998). The metamorphic environment is far from homogeneous and is characterized by localized reaction sites, where one reactant mineral phase meets another, and by localized dissipation sites. Ortoleva & Ross (1974) pointed out that, in such systems, instabilities can arise where in a homogeneous system only stability prevails. Moreover, coupling reacting systems to deformation (Ortoleva 1989) also leads to spatial patterning even in the simplest of chemical systems such as A B (Ortoleva et al. 1982), where no networked relations exist. The outcome of this analysis is that spatial patterning is to be expected in most if not all deforming–reacting mineral systems leading to a variety of forms of metamorphic differentiation such as metamorphic layering in gneisses, mineral lineations and differentiated crenulation cleavages. As part of this argument a number of authors including in particular De Wit et al. (1997), Leppanen et al. (2004), Alber et al. (2005) and Glimm & Hentschel (2008) have pointed out that the iso-concentration interfaces produced by reaction –diffusion equations are minimal surfaces or close to minimal surfaces. In particular, Alber et al. (2005) and Glimm & Hentschel (2008) have shown that the family of minimal surfaces (or surfaces close to being minimal) that develop from reaction–diffusion equations if one is close to the Turing bifurcation are represented parametrically by u = s1 cos x + s2 cos y + s3 cos z
(42)
where u is a parameter; changing the value of u produces a new surface in the family. These surfaces are not strictly minimal surfaces but approximate minimal surfaces. Alber et al. (2005) show that there are only three possible cases: (i) s1 = 0, s2 = s3 = 0 leading to sheet or lamellae structures which would be represented in rocks as foliation planes, (ii) |s1 | = |s2 | = 0, s3 = 0 leading to cylindricallike structures which would be represented in rocks as lineations and (iii) |s1 | = |s2 | = |s3 | = 0 leading to undulating surfaces that can encompass a sphere or ‘nodule’ which would be represented in rocks as foliation surfaces encompassing
STRUCTURES IN DEFORMING– REACTIVE SYSTEMS
293
Fig. 9. Surfaces that arise from reaction–diffusion equations: (a) lamellar structures; (b) cylindrical structures; (c) undulating surfaces; (d) Scherk surface.
lozenge-shaped regions, perhaps occupied by porphyroblasts. The solutions are shown in Figure 9a–c respectively. The case s1 cos x + s2 cos y + s3 cos z is very close to a Schwarz P-surface which is a triply
periodic minimal surface (Aksimentiev et al. 2002) (although only part of the surface is shown in Fig. 7c). These types of structures are common in polymers (Aksimentiev et al. 2002; LopezBarron & Macosko 2009). Another possible minimal surface is shown in Figure 9d and is
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given by z = log [ cos (cx)/ cos (cy)]/c which corresponds to a Scherk surface (Scherk 1834). If the system is far removed from the Turing bifurcation point, then the iso-concentration surfaces can be curved and evolve towards a steady structure. As this evolution occurs, defects in the geometry occur (Cross & Hohenberg 1993, pp. 898– 922) and these defects can also develop as minimal surfaces. The Scherk surface is an example (De Wit et al. 1997; Glimm & Hentschel 2008). The evolution of a Turing system with the formation
of defects can be visualized at http://www.cmp. caltech.edu/~mcc/Patterns/Demo4_3.html where the Swift –Hohenberg equation has been used to study the progressive changes in the pattern as it evolves. Glimm & Hentschel (2008) show that curved surfaces such as those shown in Figure 9 represent conditions for maximum diffusive flux normal to the surface if there are gradients in the chemical reaction-rate field (defined by the functions F and G in equation (17)) and are surfaces of minimum diffusive flux if the chemical reaction-rate field is
Fig. 10. Some of the foliation surfaces imaged by Bell & Bruce (2006, 2007). Data supplied by M. Bruce.
STRUCTURES IN DEFORMING– REACTIVE SYSTEMS
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Fig. 11. Values of mean and Gaussian curvatures for synthetic and natural surfaces. (a) The surface u ¼ cos(x) + cos(y) predicted as arising from reaction–diffusion equations by Glimm & Hentschel (2008) shaded according to the mean curvature. Light is high positive mean curvature and dark is negative. (b) Histogram of the values of mean curvature for the surface in (a). (c) Histogram of the values of Gaussian curvature for the surface in (a). (d) The Bell and Bruce surface shown in Figure 8a shaded according to mean curvature. Light is high positive mean curvature and dark is negative. (e) Histogram of the values of mean curvature for the surface in (c). (f ) Histogram of the values of Gaussian curvature for the surface in (c). Note that many of the values in the tails of the histograms arise from artefacts at the edges of the surfaces that we have not removed. The vertical scale in the histograms is the number of nodes, normalized to 1, on the triangulated surface that has a given curvature. Images produced using Meshlab (MeshLab, Visual Computing Lab–ISTI– CNR, http://meshlab.sourceforge.net/).
constant. Gabbrielli (2009) shows that minimal surfaces represent conditions for minimizing stress concentrations in a loaded aggregate. If the schistosity and/or lineation that form as interfaces between quartz and biotite concentrations in a quartz –
biotite schist develop as instabilities arising from reaction –diffusion– deformation reactions, then these interfaces are therefore expected to define or approximate minimal surfaces. To date there has been relatively little study of such surfaces in
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metamorphic rocks, but examples perhaps are the 3D images generated in anatexite by Brown et al. (1999) and in a garnet – staurolite schist by Ketcham & Carlson (2001). The opportunity exists to analyse the microstructures in metamorphic rocks using the concepts developed by the above workers and by Aksimentiev et al. (2002). Figure 10 shows a portion of the dataset collected by Bell & Bruce (2006, 2007) for the shapes of foliation surfaces in porphyroblastic schists. The surfaces are characterized by regions of double curvature similar to the surface shown in Figure 9c that results from reaction –diffusion equations. The Bell and Bruce surfaces are also similar to the computer-simulated surfaces that arise in the lamellar phase of surfactant systems (Holyst 2005). Figure 11a shows the surface u ¼ cos(x) + cos(y) proposed by Glimm & Hentschel (2008) as arising from reaction –diffusion equations shaded according to the mean curvature and Figure 11b, c shows the corresponding histograms of the mean and Gaussian curvatures across the surface. By comparison, Figure 11d shows one of the Bell and Bruce surfaces (Fig. 10a) shaded for values of the mean curvature while Figure 11e, f shows the corresponding histograms for mean and Gaussian curvatures. The topological resemblance of the Bell and Bruce surface to surfaces predicted by reaction –diffusion theory is sufficient to warrant further investigation. The observations that, for the Bell and Bruce surface, the mean value of H is zero, the standard deviation of H is less than that for the theoretical surfaces of Glimm & Hentschel (2008) and K is negative supports the suggestion that the Bell and Bruce surface is part of an evolution towards a minimal surface. Demonstration of the widespread existence of surfaces that approach minimal surfaces would be a good test of the proposition that the microstructures form in deformed metamorphic rocks by reaction– diffusion processes.
Concluding remarks We have presented a discussion that proposes that metamorphic differentiation develops in response to spatial instabilities in coupled mineral reactions. A parallel temporal response gives rise to compositional zoning in metamorphic minerals. The instabilities that arise here are a direct response to the networked or cyclic mineral reactions described by authors such as Carmichael (1969), Vernon (2004) and Wintsch et al. (2005) and, in some instances, are an expression of the influence of deformation on reaction–diffusion equations. In other cases, autocatalytic redox reactions may play an important role. The interfaces between different
mineral concentrations that form as foliations and lineations by these processes should evolve as they grow to ultimately become minimal surfaces. We have given one example in this paper. The control on chemical instability by the deformation expressed by equation (12) states that the shape and orientation of mineral foliations and lineations that result from such instabilities reflect the shape and orientation of the ellipsoid that represents the deformation rate tensor. An identical conclusion was reached by Ortoleva et al. (1982); it is therefore the kinematics of the deformation that control the shape and orientation of these types of fabrics. We thank R. Vernon, K. Regenauer-Lieb, G. Hunt and H. Muhlhaus for many years of inspired discussion that have ultimately led to this paper. They of course may or may not agree with our conclusions. We are particularly indebted to P. Bourke for assistance in measuring the curvatures of the Bell and Bruce surfaces and to M. Bruce for allowing us access to the datasets for these surfaces. Our understanding of the issues in this paper was enhanced by reviews from J. Wheeler and J. Becker and by comments from E. Rutter.
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Biases in three-dimensional vorticity analysis using porphyroclast system: limits and application to natural examples D. IACOPINI1*, C. FRASSI2, R. CAROSI2 & C. MONTOMOLI2 1
Geology and Petroleum Geology Department, University of Aberdeen, Aberdeen AB24 3UE, UK 2
Dipartimento di Scienze della Terra, Universita` di Pisa, via S. Maria 53, 56126, Pisa, Italy *Corresponding author (e-mail:
[email protected]) Abstract: A description of the systematic errors associated with the measurement of the vorticity number from poryhroclasts in natural systems is presented and discussed. We show that strong biases and systematic errors could derive both from some erroneous physical (i.e. no slip across clast/matrix boundary, homogeneity within the matrix) as well as geometrical assumptions (i.e. the radius ratio and angular measurements carried out in two dimensions on outcrop surfaces and thin section). By comparing natural datasets of porphyroclast shape preferred orientation (SPO) with different theoretical curves plots, we suggest that at least one of the Jeffery physical assumptions can be tested when applying vorticity techniques. The comparison of different possible sources of systematic errors indicates that, for medium-to-low vorticity numbers (Wm , 0.8), vorticity data are strongly biased and that a minimum systematic error of 0.2 should be taken into account. Finally, we use data from natural shear zones from the Southern Variscan Belt in Sardinia to test and discuss the starting assumptions of the Jeffery model.
One of the main challenges of modern structural geology is to try to obtain information about the rheology and kinematics of flow occurring in the deepest crust and upper mantle. Several microstructure fabrics (e.g. fabric asymmetry) associated with mylonites represent potentially important sources of information to obtain kinematic data as well as for estimation of deformation parameters (see Passchier & Trouw 2005 for a review). In particular, porphyroclast systems have been shown to be a sensitive indicator of flow parameters and can be used to constrain the relative ratio of combined pure and simple shear (i.e. the vorticity of the flow). Several theoretical and practical studies indicate that syn-kinematic recrystallization of mantles (s- or d-shaped tails) around porphyroclasts and shape preferred orientation (SPO) of porphyroclasts may store information on the amount of shear strain. With the aim of developing a vorticity gauge, theoretical and experimental models for study of the rotation and behaviour of elliptical porphyroclasts in a fluid have been proposed and developed. The first attempt began with the pioneering works of Jeffery (1922) and Eshelby (1957) that found solutions for rigid objects immersed in a viscous and linear-elastics fluid, respectively. Bretherton (1962), Ghosh & Ramberg (1976) and Bilby & Kolbuszewski (1977) further adapted the basic Jeffery analytical solution to material science and structural geology. Subsequent investigations used the previous theoretical calculation to develop efficient strain and vorticity gauge techniques capable of quantifying the flow parameters within shear
zones (Passchier 1987; Wallis 1992; Simpson & De Paor 1993; Holcombe & Little 2001). Three main analytical techniques (Passchier 1987; Wallis 1995; Holcombe & Little 2001) are commonly employed to characterize flow within shear zones using rigid porphyroclats (Jessup et al. 2007). All these techniques use the measurement of the axial ratio of porphyroclast R (the shape factor; Fig. 1) and the angle of long axis porphyroclast with respect to foliation or stretching direction h (Fig. 1) and the distribution of the cross-plotted curves to define the mean vorticity number Wm. These curves define a threshold number Rc, below which the porphyroclasts continuously rotate and above which they record stable sink position. However, as many authors (Passchier 1987; Marques & Coelho 2001) have already noted, the Jeffery model assumes: (1) rigid particles immersed in a non-confined viscous flow, (2) a perfect coupling on matrix/particle interface and (3) no interference between particles during deformation. Use of the Jeffery model implies that these physical assumptions are met, and/or need to be, at least preliminarily, tested. Although techniques to estimate the vorticity of flow have become routinely used on sheared rocks in a variety of tectonic settings (e.g. Klepeis et al. 1999; Xypolias & Doutsos 2000; Xypolias & Koukouvelas 2001; Bailey & Eyster 2003; Law et al. 2004; Carosi et al. 2006, 2007; Iacopini et al. 2008; Sarkarinejad et al. 2008, 2010; Frassi et al. 2009; Larson & Godin 2009), relatively few detailed analysis of error sources have been proposed and a critical discussion
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 301– 318. DOI: 10.1144/SP360.17 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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Fig. 1. Box describing and defining the main parameters of a porphyroclast system.
vorticity information. Over the last 20 years, the mathematical model proposed by Jeffery (1922) has provided the theoretical basis for much of the models developed for rigid porphyroclast behaviour in high-strain shear zones and for all the rigid grainbased vorticity techniques available. Ghosh & Ramberg (1976) first applied the model to a structural geology problem demonstrating that, with an increasing contribution of the pure-shear component (0 , Wm , 1), porphyroclasts will either rotate in the direction of the simple shear or rotate backwards until the main axes reach a sink position u that is unique to each R and Wm value. Based on the fundamental relationship between Wm, R and u, three main vorticity techniques using rigid porphyroclasts have been proposed so far (Passchier 1987; Wallis 1995; Holcombe & Little 2001) and are described in the following sections.
Method I: Passchier (1987) of data has not yet taken place (see Tikoff & Fossen 1995; Bailey et al. 2007; Mulchrone 2007a; Iacopini et al. 2008). In this paper, after describing some commonly used vorticity techniques, we discuss and compare some of the main systematic errors resulting from measurement of the aspect ratio R and main orientation axes of porphyroclasts within 2D thin sections or outcrops. We then suggest an empirical test to check at least one of the Jeffery assumptions, and illustrate and discuss its application to selected samples from the Variscan belt in northern Sardinia Island (Italy).
Theoretical base of the vorticity techniques The kinematic vorticity number Wk has its origin in fluid dynamics and records the amount of rotation relative to the amount of stretching at a point in space and at an instant in time (Malvern 1969). It has been introduced into geological literature (Means et al. 1980) because it represents basic flow parameters and is able to describe flow kinematically, that is, to distinguish between pure and simple shear within shear zones. Assuming a steady-state deformation, its application into geology has been facilitated by the use of Mohr circle strain enabling the correlatation of Wm to the velocity gradient tensor and the deformation matrix (Passchier 1986). Thanks to several numerical and analogue models, observed structures developed in and around rigid objects immersed in a viscous fluid have greatly aided our ability to understand geological deformation and so far represent key microstructures to obtain reliable strain and
This method is preferably applied to mantled porphyroclasts. The measurements of the angle h, between the long axis of porphyroclast and the foliation (or stretching lineation), and the shape factor B* (Fig. 1) are plotted in order to separate the field of continuous rotation, defined by d-shaped tails clasts (that often show quite scattered h values), from the fields of stable sink rotation, theoretically defined by s tails clasts (Fig. 2a). The curves that divide the two fields are defined by the equation:
h = 12 sin−1 Wm {(1 − Wm2 )
1/2
− (B∗2 − Wm2 )1/2 }
(1)
where B* is defined as B∗ = (Mx2 − Mn2 )/(Mx2 + Mn2 )
(2)
where Mx and Mn are the major and minimum axes of porphyroclast, respectively (Fig. 1). If the porphyroclasts measured follow the Jeffery assumptions, the curves defined by equation (1) should fit well with the shape preferred orientation (SPO) distribution of stable porphyroclasts in a B* versus h plot (Fig. 2a). The geometrical parameters B* and h are directly measured onto rocks or thin sections, while Wm is obtained directly from the Rc parameter which is defined as the threshold number that separates the field where porphyroclasts continuously rotate from those that show a stable sink position in the best fit of the theoretical curves defined in equation (1) (Fig. 2a).
Method II: Wallis (1992) Wallis (1992) adapted the Passchier method to tailless porphyroclasts using a simple aspect ratio R
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The critical threshold Rc is roughly estimated on the plot by separating the fields where the porphyroclast population shows scattered and random values of h from the population with a more limited orientation (Fig. 2b). In order to define these two fields (i.e. to better define the SPO porphyroclast orientation) an enveloping surface should be created (Fig. 2b; see Law et al. 2004). However, to unify the Wallis and the Passchier methods and to reduce ambiguities on measuring Wm, a graphical device called Rigid Grain Net has been proposed by Jessup et al. (2007).
Method III: GhoshFlow (Holcombe & Little 2001) This method uses the freeware GhoshFlow software (Holcombe & Little 2001) that, based on the Ghosh & Ramberg (1976) analysis, utilizes syn-tectonically rotated porphyroblasts containing an internal foliation Si (Ghosh 1987) (Figs 1 & 2c). The degree of rotation of the porphyroblasts can be directly measured from the orientation of internal helicitic inclusions Si with respect to the external foliation Se (Fig. 1). For steady-state monoclinic flow, the amount of internal rotation is a function of (1) pure- and simple-shear components (or Wm) and (2) the initial orientation of the porphyroblast long axis with respect to the shear direction (Ghosh & Ramberg 1976). In two dimensions, the sinusoidal theoretical curves that describe the angular velocity of the long axis of an elliptical rigid particle are:
m˙ =
Fig. 2. Vorticity techniques using SPO porphyroclasts. (a) Passchier method. The curves are defined in equation (1). (b) Wallis method for tailless clasts using envelope curves. (c) Holcombe & Little (2001) method. The curves are defined by equation (4).
versus h plot (Figs 1 & 2b). Vorticity estimates cannot be determined directly from the plot, but need to be calculated using the relation Wm = (R2c − 1)/(R2c + 1).
(3)
˙ 2 m − (R2 − 1)˙1sin2m g˙ R2 sin2 m + gcos R2 + 1
(4)
where R represents the aspect ratio, m the orientation of the clasts long axis with respect to the normal at the boundary wall and g˙ and 1˙ are the far field bulk simple shear and pure shear, respectively. The method consists of compiling a cross-plot of the orientations of the long axis of porphyroclasts versus the orientation of their internal foliation Si (Fig. 2c). In this case, both the parameters should be measured with respect to the direction normal to the shear plane. The distributions are compared to theoretical curves described by (4) (Fig. 2c). The three described techniques assume that the SPO parameters (aspect ratio and orientation of the long axes of porphyroclasts) can be determined easily by means of field and microstructural analysis. However, as with geological field measurements, they are intrinsically biased by systematic errors which are mainly induced by the difficulty in translating the observed 2D fabric into real 3D structures.
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In order to obtain meaningful vorticity data, a careful choice of rock samples suitable for the analysis, a deep understanding of the possible systematic errors affecting the parameters and a statistically robust measurement of the SPO parameters are therefore necessary. These potential biases and systematic errors affecting the measurement need to be taken into account.
Preconditions for the methods We propose that a reliability test of the assumptions of the Jeffery method is required before application to a natural sample. As a consequence, (1) the porphyroclast should behave as an isolated rigid inclusion and (2) the deformation should not localize either through slipping or through a fluid film around the porphyroclast/matrix interface, or within the surrounding matrix. For this reason, we present and discuss in detail (1) the isolation factor (i.e. the absence of interference between porphyroclasts during deformation), (2) the presence of unconfined flow in the matrix and (3) the slipping effect (i.e. the presence of perfect coupling at the matrix/particle interface) and the localization within soft material in the porphyroclast.
Isolation factor As suggested by Passchier (1987), a necessary condition to apply the Jeffery model is that the porphyroclasts should not interact or perturbate their respective motion during the rotation. How far and which distance should be considered is a matter for debate but Ildefonse et al. (1992) showed that, for equally sized particles, the interaction effect becomes significant if the distance between individual clasts is smaller than their maximum long axes. This implies that in order to reasonably exclude the interaction between clasts, a careful investigation of the relative distribution of porphyroclasts should be performed in each thin section or outcrop. However, even if clasts strongly interact with each other or produce imbricate fabrics during the flowing event (they will not respect the Jeffery conditions and consequently will not fit the curves defined by equations (1) and (2)), the data distribution in a R versus h or h versus Si plot can still give interesting information about strain and vorticity (see Tikoff & Theyssier 1994). If the interacting porphyroclasts show an aspect ratio ,2 their final SPO could be strain insensitive, inducing clasts to reach a stable and sink orientation independently of the strain registered by the flowing matrix (Piazolo et al. 2002). As a consequence, a careful critical analysis of their mean distance, percentage of imbrication and
aspect ratio is necessary before applying Jefferyderived curves to the observed SPO porphyroclasts.
Flow confinement effect Analogue material experiments (Bons et al. 2003) and numerical simulations (Marques et al. 2005a, b) demonstrated that the confinement effect (i.e. confined shear zones) could produce a stable and strain-independent SPO of rigid clasts that differ from that predicted by the Jeffery model. This effect could be expected within heterogeneous, strongly layered and localized shear belts defined by micro/mesoscopic shear bands in which the ratio between the shear-zone thicknesses and the short axis of clast is ,3 (Marques et al. 2005a).
Slipping effect Recent analogue, numerical as well as outcrop studies of porphyroclast systems (Ildefonse & Mancktelow 1993; Pennacchioni et al. 2001; Marques & Bose 2004; Schmid & Podladchikov 2004; Mulchrone 2007a) indicated that slip on the boundary between the rigid clast and the matrix could significantly influence the behaviour and the final SPO distribution of porphyroclasts. Different h versus B*, h versus R or m versus Si internal foliation plot distributions have been proposed or suggested (Mulchrone 2007a, b) but all of them predict a different behaviour with respect to the classical distribution or, worse, a strain-insensitive behaviour (Pennacchioni et al. 2001). In the authors’ opinion, one of the limits of the techniques derived from the Jeffery model is that occurrence of slip between matrix and particle cannot be easily tested, demonstrated or ruled out a priori. As a consequence, if the data points are quite scattered in the h versus B*, h versus R plot or m versus Si internal foliation plots, the straightforward application of one of the theoretical curves proposed in those plots (Passchier 1987; Holcombe & Little 2001; Jessup et al. 2007) through a best-fitting approach could produce erroneous solutions that do not represent the real SPO distribution. A qualitative analysis of the prophyroclast/ matrix boundary could help to better elucidate how the porphyroclasts interacted with the surrounding matrix but, at the same time, does not solve or reduce the uncertainty. The presence of (1) recrystallized mantle made by a weaker mineral phase with respect to the clast and/or oxide partial melted fluid or film as well as (2) thin rims of mica and/or quartz and/or fine-grained feldspar produced during the syn-kinematic rotation could indicate that a slip or lubrication effect has occurred at the matrix/clast boundary. Such evidence can suggest that the SPO of porphyroclasts
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do not fit with the prerequisites of the Jeffery model. In all of the cases described so far, the real effect of the lubrication is to hamper the rotational behaviour of the clast. This implies that the net simple-shear component is under-estimated and, consequently, that the vorticity estimates point to a pure-shear component higher than that which has really affected the porphyroclast system. Strain localization at the clast margins or within an embedding sheared viscous matrix can strongly compromise the estimation of vorticity number. As recently shown with numerical analysis by Johnson et al. (2009), the critical shape factor cut-off is highly sensitive to the degree of the clast/matrix coupling. Mica in the matrix or at the clast surface plays a key role in lubrication of a clast because, behaving as a weak component, it dramatically affects the rates of diffusion producing slipping along the (001) mica faces (Ten Grotenhuis et al. 2003). As a consequence, increasing modal matrix mica can violate the requirement of homogeneous matrix deformation by promoting discrete zones of strain partitioning. This implies that strain, and the related shear component (e.g. vorticity), is partitioned within the clast/matrix system (Ten Grotenhuis et al. 2003; Johnson et al. 2009). Marques et al. (2005a) suggest that the measurements should be carried out into high-strain shear zones that exhibit clasts immersed in a thick recrystallized and homogeneous matrix not affected by internal anisotropies (i.e. S –C fabrics or networks of small shear zones), that mechanically control the entire system. To obtain sensitive and meaningful vorticity estimation in a similar context, a robust number of measurements are required. Otherwise, the vorticity number obtained is simply the result of a random sampling of clasts chosen along the shear zone that are unlikely to be representative of the strain complexities observed across the whole shear zone. This is particularly true if the study aim is to obtain kinematic results at a regional scale. To avoid such limitations, rigid clast vorticity measurements have to be performed in an area where a large distribution of porphyroclasts are ideally distributed and their results have to be compared to other vorticity estimates obtained using methods based on different starting assumptions.
Suggestion for a preliminary test of the Jeffery boundary conditions All the previously described preconditions strongly restrict the applicability of the Jeffery model. As a consequence, deformed areas resulting from a shearing event need to be carefully analysed before applying the vorticity techniques described in the previous section. However, microstructural
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descriptions will never definitely rule out, or truly assess, if the boundary conditions have been respected previously. The most practical and efficient way to obtain information on the behaviour of the porphyroclast system is to test the SPO strain sensitivity of porphyroclasts using the R versus h plots. If the theoretical assumptions described so far are correct, the porphyroclast distribution will follow at least one of the crossplot functions described in the literature (Ghosh 1987; Passchier 1987; Pennacchioni et al. 2001; Mulchrone 2007a). The function elaborated by Mulchrone (2007a) is based on a 2D analytical solution of rotating rigid elliptical particles with a slipping interface. Mulchrone (2007a) published a series of curves showing the relationship between mean vorticity number Wm, aspect ratio R and main axes orientation h. Equation 8 of Mulchrone (2007) has been rewritten as follows to obtain a relationship between the vorticity number Wm, aspect ratio R and the orientation of the main clast axis angle h (as for the Passchier 1987 formula): ⎛ (R − 1)Wm2 + R − Wm2 − 1 ⎜ ⎜ 2 2 2 −1 ⎜ (Wm − 1)(1 + R ) − 2(Wm + 1)R + 1 . h = cos ⎜ ⎜ 2(R + 1) ⎝ (5) The theoretical curves obtained from equation (5) (black lines in Fig. 3) represent the expected distribution of porphyroclasts that reach a stable
Fig. 3. Passchier curves (grey line) and Mulchrone curves (black line) on aspect ratio R versus angle h plot. All the curves are strongly confined to a value of angular orientation h lower then 0.4 rad (c. 308). This composite diagram represents a necessary test to distinguish strain-sensitive ‘slipping’ clasts from strain-sensitive ‘non-slipping’ clasts. Squares a, b: see description in the text.
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sink position in the presence of a clast/matrix slipping interface. The comparison between the Mulchrone and the Passchier curves (black and grey lines respectively in Fig. 3) indicate that, if SPO clasts are strain sensitive, the slip component results in a quite different distribution of the porphyroclasts that reach a stable sink position. The Mulchrone curves do not vary too much in terms of the function of the vorticity Wm with respect to the Passchier curves (Fig. 3). High values of angular orientations are not expected. Due to the smoothness of the curve, it is more difficult to constrain a critical aspect ratio Rc (i.e. vorticity estimates) that separates the field of continuous rotation from that where the porphyroclasts reach a stable sink position. The overlap of Passchier and Mulchrone curves in a cross-cut plot can represent a possible preliminary test to investigate if the porphyroclast system is either compatible with the Jeffery physical assumptions or not. The Passchier and Mulchrone curves show a distinct behaviour for low- to medium-aspect ratio values (see square a in Fig. 3); for large vorticity values and aspect ratios, however, the slipping clasts are expected to be distributed within a more broad angular field with respect to non-slipping clasts (see square b in Fig. 3). The main limitations of this approach are due to (1) the close range of Mulchrone curves at different vorticity values and (2) the indistinguishable crossplotting distribution of the slipping and non-slipping curves for large aspect ratio. As the Mulchrone curves are definitely less sensitive to the vorticity number, in order to obtain precise vorticity estimates a statistically robust dataset of clasts (number of clasts ideally .1000, see Mulchrone 2007b) is needed; this is rarely possible in natural mylonitic rocks. If not, and the dataset is poorer and the distribution scattered, the distinction between slipping and non-slipping clasts should be determined from the low- to medium-ratio clasts. The vorticity estimate will be defined by a range of possible best-fitting curves with error bars. Finally it is worth noting that, if the curve comparison can tell us information about the interface behaviour and eventual strain sensitivity, it does not give any information about the physical property of the matrix/clasts discontinuity or type of slip behaviour. More theoretical investigations on the SPO of porphyroclasts at different clast/matrix relationship (elastic, visco-elastic, plastic, viscous) should be carried out (see Samanta & Bhattacharyya 2003 for a detailed discussion) to rule out or eventually infer the deformation mechanisms. The crossplot comparison should be performed anyway, as it represents a practical test to verify if the porphyroclast system appears to follow a strain-sensitive rotational behaviour or not.
Analysis of some systematic source errors The vorticity techniques described so far use the geometry of rigid porphyroclasts (i.e. aspect ratio, long axes orientation and inclusion trails orientation) to estimate the kinematic vorticity number Wm. All the parameters are measured in thin section (or along outcrop surface) oriented both parallel to the stretching lineation and orthogonal to the main foliation (Passchier & Trouw 2005). These data are then plotted onto several diagrams and compared to theoretical curves obtained by equations (1), (2), (4) and (5) in order to obtain vorticity estimates. However, before using the obtained Wm values to assess a kinematic interpretation, we should consider the effect and the influence of some error sources introduced during the several steps of the measurement procedure. In this view, we must consider the error source introduced during (1) the measurement of aspect ratio and radii (especially in the absence of information about the structure in the third dimension), (2) the recognition of the real vorticity axis orientation and its related maximum sectional vorticity plane and (3) the recognition of the real strain distribution in the third dimension. Finally, we have to take into consideration the difficulty in assessing possible dilatancy or contraction effects during the deformation event within the shear zones (i.e. equivalent to an underestimatation of a pure-shear component in the vorticity calculation; see Ebner & Grasemann 2006). In addition to the constraints discussed above, the implications of these measurement biases are exacerbated by the fact that porphyroclast axes can experience both a rotational and a revolutional movement. These rotations are dependent on the flow type (simpl- shear deformation, monoclinic or triclinic geometry) as predicted by all the analytical solutions (Passchier 1987; Jiang 2007), as well as on flow perturbation due to initial heterogeneity (Passchier et al. 2005). An intuitive consequence of the roto-revolutional movement is that the porphyroclasts and their vorticity axes remain statistically slightly misoriented with respect to the mean stretching direction (assumed as a reference direction), even for large strain accumulations that theoretically would produce a strong shape preferred orientation (Jiang 2007). Keeping this in mind, an analysis of systematic errors must be produced during each vorticity analysis. In order to understand and quantify the real effect of the three main systematic errors described above, we performed some thought experiments assuming that porphyroclasts are represented by (1) a biaxial ellipsoid and (2) a horizontal stretching lineation.
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Aspect ratio measurement The error source associated with the radius measurement is due to the impossibility of detecting exactly the 3D orientation of the main axes of porphyroclasts in sampled rocks (Fig. 4). For this reason, an estimation of the effect of such a misorientation is necessary in order to evaluate the systematic error introduced during the vorticity analysis. Assuming: (1) biaxial and strongly oriented clasts (i.e. deformation under perfect plane strain geometry), (2) clasts with the same aspect ratio and (3) a data point distribution perfectly fitting the theoretical curves defined in equation (1), we compiled a diagram that describes the effect of the error in radius measurement on the vorticity estimation (Fig. 5). Figure 5 shows how, in an ideal situation, measurement errors will compound to produce an error in the ratio estimation. Assume a real-aspect ratio value of 2 measured along a perfectly oriented section (ideally orthogonal to the vorticity axis with a ¼ 908): different section orientations across the sample (with a , 908) will obviously register a smaller and hence wrong aspect ratio. For each a (corresponding to sections deviating 908 – a from the ideal section), the curve in Figure 5 indicates the expected radius to be measured. As Figure 5 indicates, sectional vorticity planes with angles a of between 808 and 908 will result in measured aspect ratios of between c. 1.9 and 2.0 (i.e. error % c. 5– 0%).
Fig. 4. The effective aspect ratio R of a porphyroclast idealized as a biaxial ellipsoid, measured at different misorientations with respect to the maximum sectional vorticity plane. The angle a represents the angle between the porphyroclasts vorticity axes and the ideal vorticity vector or, in other words, the orientation of the angle between the sectional plane and the vorticity vector. Sections with a ¼ 0 represent planes orthogonal to the vorticity axes.
Fig. 5. Diagram showing how sectional planes differently oriented with respect to the maximum sectional vorticity plane (MSVP) could produce apparent values of aspect ratio. SPO measurements carried out on planes different from MSVP produce lower aspect ratio values that consequently lead to underestimates in the kinematic vorticity number.
The consequences of such results in a B* versus h plot in the simplest scenario of a systematic and constant error in the radius estimation are depicted in Figure 6a, b. In Figure 6a, an example of a real clast dataset, the B* versus h plot measured along a shear zone is shown (we will discuss later in detail the real example) where s and w clasts are represented by white and grey circles, respectively. The grey Passchier curves represent the best fitting overall separation zones between s and w clasts. The diagram shows that the critical ratio based on the distribution of data points defines a vorticity value between 0.2 and 0.3. If the data points have been measured in a sectional plane misfitting by c. 108, the real distribution to be considered is the one shown in Figure 6b (white and grey triangle) where the wrong estimation of aspect ratio should produce a translation of points in the distribution (horizontal dashed lines in the boxes of Fig. 6b, c). The consequence in this case is that the best fit of the Passchier’s curves (grey curves in Fig. 6b) will produce a new value of Wm between 0.3 and 0.4. In practice, if we don’t know exactly the true orientation of the maximum vorticity plane and if we introduce an error of 108 in the estimation of a, Figure 6b suggests that a minimum error of 0.2 has to be taken in account. As defined by equation (1), this type of error affects the vorticity measurement producing an error of at least 5–10% in the vorticity estimation (compare vorticity estimates in Fig. 6a, b). If we consider a pessimistic situation where the aspect ratios do not have the same values and the clasts do not show a strong SPO, the systematic error will have a more scattered distribution (with every clast shifted in a different way; see dashed
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horizontal lines in the box of Fig. 6c). As shown in Figure 4, wrong sectioning will not only produce false aspect ratios but also wrong angular inclination h for the section analysed. A similar consideration is proposed for the angle of inclination (h) of the sectional main axes. In a B* versus h diagram this will also impose an error along the vertical (y) direction. In this case, the interpretation of the inferred best-fit curves obtained is far more problematic and the risk of mismatching the real vorticity number is high. As a consequence, a stronger incertitude in the recognition of the vorticity number values is expected. The second more realistic scenario suggests that, during the vorticity analysis for a small misfit of 108, the estimation of the systematic error of 15% (derived by the simple and ideal assumption of misoriented rigid clasts with an aspect ratio ¼ 2) represents only a minimum error value that has to be assumed if the error estimates cannot be directly made sample by sample. Similar estimations could be made for different aspect ratios and vorticity numbers.
Maximum sectional vorticity plane estimation and measurement of h Assuming we are able to individuate the true aspect ratio of the clast, whether the angular measurements are made along a sectional plane truly orthogonal to the vorticity plane for every clasts system (Fig. 4) remains at the limit of understanding. Figure 7 shows the errors that we can expect if we cut the porphyroclast with a misfit of a ¼ 08 –908 at different vorticity values. The calculations are performed
Fig. 6. (a) Example of aspect ratio R versus angular orientation h crossplot diagram. (b) and (c) Effect of systematic misorientation.
Fig. 7. Diagram showing the distribution of the mean kinematic vorticity number Wk in relation to a simpleshear component calculated for the maximum (black line) and misoriented sectional vorticity planes (grey lines). The sectional vorticity planes are oriented 58, 108, 158, 208 and 258 from the maximum sectional vorticity plane. Squares a– d: see description in the text.
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assuming the errors come from a wrong estimation of the h value. Important biases are expected at medium vorticity numbers (Wm ¼ 0.40 to 0.90) affected by low to medium simple-shear component. At very low vorticity numbers and with a low simple-shear component (square a in Fig. 7) and at very high shear strain component and high vorticity number Wm1 (square d in Fig. 7), the curves become closer to each other and the misorientation of the sectional vorticity plane will not affect the vorticity measurements. In fact, as shown within the squares a and d in Figure 7, a misfit of 10– 508 induces an error of 0.1 –0.2 in the vorticity estimation. On the other hand, larger errors are expected for medium vorticity numbers (square b and c) where the same misfit from 10– 508 induces larger errors (.0.3). However, for geometric reasons a measurement along an erroneous vorticity plane will automatically produce a wrong factor ratio enhancing the error effect during the measurement process (Fig. 4). This implies that the effect of the two errors (those induced by the aspect ratio measurement and the maximum sectional vorticity plane estimation) are in reality strictly related to each other and the estimation of the first cannot exclude the consideration of the other. At best, assuming the revolution component of the vorticity axes to be negligible (as shown by Piazolo et al. 2002), the more the shear zone is affected by high shear strains the more the porphyroclasts are well oriented along the stretching direction and lowest error is expected.
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where the rocks behave homogeneously and the extra dilatancy (defined as the pure-shear component) is partitioned into an extruding component: Figure 8 describes the real effect of some dilatant shear zones with different simple-shear components, and indicates that the dilatancy effect is strongly dependent on the pure/simple-shear ratio. As some authors pointed out (Ebner & Grasemann 2006) most of the extruding flow inducing a dilatancy will be partitioned within the core of the shear zones and the boundary walls. The effect is similar to an extra component of pure shear distributed across the whole shear zone. The rotational component measured cannot take into account or be sensitive in itself to the effect of the pure-shear component if the shear zone is a priori assumed homogeneous and of constant volume. In the large square in Figure 8, it is shown how at fixed pureshear values ¼ 1, the dilatancy parameter becomes an important factor worth consideration during the vorticity estimation (at least for values of g lower then 5). The error biases expected are within the range of 20% or higher. The dilatancy effect becomes negligible for high-strain shear zones (vorticity number bigger then 0.9).
Final considerations If we are dealing with shear zones characterized by general shear flow with vorticity numbers ≤0.9, a systematic analysis of errors affecting fundamental parameters such as radius ratio, vorticity plane and dilatancy estimation should be taken into account.
Third-dimension extrapolation A third systematic error source derives from the fact that the analyses, usually performed in two dimensions (Tikoff & Fossen 1995), underestimate the possible stretching effect along the third strain component. If such a stretching component is not zero, it induces an overestimation of the actual vorticity number by at least 0.05 as suggested by Tikoff & Fossen (1995) for sections perfectly orthogonal to the vorticity vector. However, for a complete analysis of the error effect on different shear kinematics regimes, we refer to the paper of Tikoff & Fossen (1995).
Dilatancy effect Another possible error that should be taken into account derives from the underestimation of the dilatancy effect registered by shear zones within certain extruding tectonic contexts. This parameter could not be precisely measured especially in retrogressed shear zones where the fluid or mass expulsion are overprinted by the last exhumation and deformative events. Assuming a simple condition
Fig. 8. Diagram showing the dilatancy effect on the mean vorticity number Wk for fixed simple-shear rate values. The effect of dilatancy on vorticity numbers are very low and could be neglected for dilatancies of 1.5–2 and for low shear strain values. Square: see description in text.
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Before applying the techniques discussed in the previous sections, it is worth testing or checking if the shear zones follow the main geometrical requirements of the Jeffery model (no distribution of localization, no important confinement). Assuming the shear zones respect these preconditions, the diagrams in Figures 5– 7 indicate that a minimum bias of 0.2 or 0.3 has to be expected, especially if the long axes of the porphyroclasts are not perfectly oriented parallel to the main stretching directions. As a consequence, a careful sampling strategy and knowledge of the misfit angle w are required and their possible tectonic meaning should be taken into an account. The message conveyed is that vorticity measurements performed within shear zones strongly affected by pure-shear component (e.g. transpressive shear zones) contain a certain component of incertitude that limits an accurate measurement or estimation of the vorticity number. A fast and improved practical implementation of the techniques and the proposed considerations can be obtained by using existing automated X-ray tomography techniques (Ketcham 2005). If matrix and clasts comprising the sample show good contrasting absorption parameters, this technique will allow both a good 3D visualization of the geometry and the main orientation of clasts through the rock sample; this significantly reduces the described source of biases.
Case history: the D2 shear zones in the Variscan basement in northern Sardinia (Italy) The Sardinian Variscan shear zones documented in Nurra and southwest Gallura regions (Fig. 9) present a superb example to test the biases and systematic errors discussed. The existing geological knowledge and the great amount of vorticity data collected at the sites provides an opportunity to explore some of the limits of the vorticity methods exposed. The tectonometamorphic evolution of the areas is well known and assessed (Carmignani et al. 1979, 1994; Carosi & Oggiano 2002; Carosi & Palmeri 2002; Carosi et al. 2004, 2005, 2009; Ricci et al. 2004) and strain and vorticity analyses have recently been performed with the aim of evaluating the kinematic, strain geometry and deformation mechanisms (Iacopini et al. 2008; Frassi et al. 2009).
Geological setting The inner zone of the Variscan chain in Sardinia is characterized by a metamorphic grade transition (toward the northeast) from greenschist to amphibolite facies metamorphic (hereafter named low- and medium-grade metamorphic complex: LMGMC) to migmatitic complex (hereafter named high-grade metamorphic complex: HGMC) (Carmignani et al.
Fig. 9. Geological sketch map of Sardinia and the location of study areas.
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1979, 1994; Franceschelli et al. 1982, 2005; Carosi & Oggiano 2002; Carosi & Palmeri 2002; Carosi et al. 2004, 2005, 2009; Ricci et al. 2004) (Fig. 9). This transition is documented by a prograde Barrovian metamorphic assemblage, interpreted as resulting from a collisional stage, that rapidly changes northwards from oligoclase + garnet to garnet + staurolite + kyanite (Oggiano & Di Pisa 1992; Di Pisa et al. 1993; Carosi et al. 2004, 2005, 2009). Late sillimanite crystals grew during the late postcollisional decompressive stage in the HGMC (Cruciani et al. 2007; Carosi et al. 2009; Frassi et al. 2009). A late HT/LP metamorphic assemblage (andalusite + sillimanite) overprints the older Barrovian suite in Nurra –Asinara and Anglona regions. Structural analysis carried out in the Nurra and southwest Gallura regions reveals the same structural evolution. The first deformation phase (D1), connected to the continental collisional, was associated with south-verging folds and ductile to brittle shear zones with a top-to-south and southwest sense of shear (Carmignani et al. 1979; Montomoli 2003). After the areas were affected by a dextral transpressive regime (D2), a kilometre-thick shear belt named Posada– Asinara Line (PAL; Elter et al. 1990) responsible for most of the exhumation of the medium- and high-grade metamorphic rocks (Carosi & Oggiano 2002; Carosi et al. 2004, 2005, 2009; Iacopini et al. 2008; Frassi et al. 2009) was produced at the boundary between LMGMC and HGMC. The final stage of the evolution produced the collapse of the belt with an extensional deformation phase and the emplacement of Variscan granitoids (Carmignani et al. 1994; Carosi et al. 2004, 2005, 2009).
Geometry and kinematics of D2 shear zones In the Nurra region, the D2 shearing developed as discrete mylonitic bands within garnet-bearing paragneisses. They show an S2 crenulation cleavage (from zonal to discrete) that strikes WNW –ESE and dips strongly to moderately towards the southwest (Carmignani et al. 1979). The main S2 foliation is marked by quartz, muscovite, chlorite, oxides, garnet and albite/oligoclase porphyroblasts. Quartz veins, pre-, syn- and post- the shearing event (Simpson 1998), injected the S2 foliation. As a consequence, we can infer that a small component of dilatancy has affected the basement even if the amount cannot be properly quantified. The mylonitic belt is several hundred metres in width and does not show clear boundary walls. It is defined by zones of higher deformation and shear localization coupled to less sheared zones that recorded the main deformation events (Carosi & Oggiano 2002) and acted as boundary walls. Stepping
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structures, mica fish and mantled porphyroclasts with sigmoidal and symmetric w structures, with a top to the west –northwest sense of shear and compatible with a monoclinic flow geometry, have been observed in sections parallel to the lineation and orthogonal to the foliation plane (Iacopini et al. 2008). The presence of sheath folds in northern Nurra confirms the presence of an important strain accumulation during the D2 deformation phase (Carosi & Oggiano 2002). In the southwest Gallura region, the D2 shear zones show a peculiar geometry and kinematics. In addition to the classic dextral kinematics documented along the entire PAL, this sector of the chain recorded earlier sinistral kinematics developed exclusively in the HGMC that, in some outcrops, is further reworked by dextral ductile and brittle shear zones. Although sinistral shear zones shows the same northwest–southeast orientation and northeast-dipping D2 (dextral) mylonites documented in the LMGMC, they recorded slightly different strain geometries, deformation conditions and flow vorticity (Frassi et al. 2009). The S2 mylonitic foliation trends northwest–southeast and dips 30 –408 toward the northeast. A mineral lineation is highlighted by streaking of muscovite and biotite aggregates, by alignment and micro-boudinage of feldspar crystals that locally have lobate grain boundaries and by quartz ribbons and elongate quartz-feldspar aggregates. The S2 foliation trends NNW –SSE, plunging mainly 20 –408 toward the southeast. The main kinematic indicators are centimetre-sized feldspars with d- and s-type, bookshelf structures in feldspar, S –C′ fabrics, mica-fish, asymmetric myrmekites on feldspar and local shape preferred orientation in quartz. The sinistral sense of shear has been also confirmed by quartz c-axis fabric measurements (Frassi et al. 2009). Quantitative vorticity analyses performed in the sinistral mylonite that are not further reworked by the dextral shear zones indicate that deformation within the sinistral shear belts involved general noncoaxial flow with dominant simple shear, contemporaneous with deformation (Frassi et al. 2009).
D2 shear zones Preconditioning Flow confinement effect. A deformed basement preserving a complex deformation history bounded the analysed shear zones. In the Nurra region, the shear zones generally show a core that change in thickness from 10 to 100 m. If we compare the thickness T of the shear zones to the short axes of rigid clasts Mn, a mean confinement T/Mn of c. 1000 can be measured. However, where shear band fabrics (Passchier &
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Trouw 2005) are well developed and clearly behave as a local confining border or localization zones dissecting the mesoscopic shear zones (see Marques et al. 2005a), the thickness ratio could abruptly reduce to values of 2–3. A slightly different scenario can be observed in the southwest Gallura region. As the sinistral mylonites reach a minimum thickness of c. 200 m (Frassi et al. 2009), the ratio of thickness of shear belt T to short axis of porphyroclast Mx should be c. 2000. In some outcrops, the sinistral shearing produced bands (thickened from 2–3 to 20 –30 cm) where S–C′ fabrics and rare centimetre-thick ultramylonites developed. Inside S–C′ mylonites, C′ planes are spaced less than 2 cm apart and less than 10 cm in length. In this case a lower ratio of c. 20–200 is expected. However, in both Nurra and southwest Gallura study areas, the main measurement has been performed across the large mesoscopic shear zones with a high confinement value (thickness of shear zone/short axis of porphyroclast ratio), avoiding the specific zones affected by strong development of S –C and S –C′ fabrics. Isolation factor. The rigid clasts used for the vorticity measurements are represented by 1– 5 mm large feldspar and garnet porphyroclasts in Nurra (Fig. 10a) and by 0.5 –2 mm length feldspar porphyroclasts in southwest Gallura region (Fig. 10b). The porphyroclasts do not show any evidence of intracrystalline deformation and are generally immersed within a fine-grained quartz-micaceous matrix with a good isolation factor. The measurements have been carried out on sectors where neither interaction between grains nor imbricate and bookshelf structures exist. As a consequence, if the actual porphyroclast distribution partly represents the true spatial distribution experienced
by the clasts during the full deformation history, it could be assumed that clast –clast interaction effects do not represent the main bias for the final shape fabric observed. Slipping effect. In the Nurra shear zones the porphyroclasts are mainly defined by plagioclase, feldspar and garnet developed during the pre-D2 deformation phase. They show complex matrix/ clast relationships: most of the garnet and feldspar grains have a sharp boundary with the matrix, while other clasts are surrounded by fine-grained recrystallized aggregate of quartz and mica or by strain shadows of quartz. In plot R versus h the porphyroclasts show a clear angular dependent distribution (Fig. 10a, b). In a multifunction plot in which the Passchier (1987) and Mulchrone (2007a) curves are plotted against the measured porphyroclasts distribution (Fig. 11c), we can observe that the SPO of the clasts shows a distribution with an angular variability much wider than what is expected by the full range of Mulchrone prediction curves. This strongly suggests that the porphyroclasts behaved in a strainsensitive way and, for at least part of their rotational history, roughly followed a Jeffery mechanical behaviour rather than Mulchrone’s model. Mulchrone curves can approximate only part of the overall clast distribution, and the best-fitting curve for part of the data points is represented by the simple-shear (Wm ¼ 1.0) curve. However, a value of Wm¼1 would clash with the finite strain measurement calculated using separate techniques and fold structures (Iacopini et al. 2008), indicating that an important component of pure-shear strain was present during the main shearing. In Figure 11 the Passchier curves related to vorticity values of Wm ¼ 0.2–0.4 provide a better match for the overall clast distribution. As a consequence, it is
Fig. 10. Photomicrographs of porphyroclasts used in (a) the GhoshFlow method and (b)Wallis method (Grt: garnet; feld: feldspar; S2: S2 foliation).
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Fig. 11. Vorticity analysis from Nurra region. (a) Data points of tailed clasts with best-fit functions using Passchier curves. (b) Data points and best fit using Holcombe & Little (2001) curves. (c) Data points plotted in an R versus h diagram with Mulchrone and Passchier curves. (d) Data points versus biased points in an R versus h diagram with Passchier curves.
reasonable to consider the Passchier curves within values of 0.2 and 0.4 as better envelopes for approximation of the SPO clast distribution observed in the study area. In the southwest Gallura shear zones, the vorticity measurements have been carried out on feldspar grains crystallized mainly during the later stage of the D1 deformation phase (Fig 12a). At the microscopic scale, the grains immersed in a fine-grained recrystallized quartz matrix show sharp and slightly rough boundaries without either recrystallized mantles of quartz or feldspar or rims of oxides or phyllosilicates. As a consequence, we can expect that slip along the clast/matrix boundary could have played a marginal role in the SPO distribution during the D2 event. To estimate the effective role of the slipping component, we have overprinted the Mulchrone curves onto the R versus h plot (Fig. 12b). In Figure 12b it is clear that, whereas the measured clasts show a strong h-sensitive distribution and cover all the possible orientations (+908), the Mulchrone curves can only match a
limited h range distribution (+308) (Figs 11c & 12b). The enveloping curves in Figure 11b can instead describe most of the clast measured distribution. This data, coupled with the observation of a lack of soft rim material around the clasts, allows us to suggest that the porphyroclast system has roughly followed the Jeffery mechanical assumption. As a consequence, the vorticity estimates (Wm ¼ 0.83 –0.85; Fig. 11a) obtained using the Wallis method, modified by Law et al. (2004), point to a dominant non-coaxial deformation. The reliability of this assertion is corroborated by vorticity estimates provided by a technique that uses quartz petrofabric measurements (i.e. the presence of oblique grain shape in quartz and the quartz c-axis fabrics; Wallis 1995; Frassi et al. 2009). In this way, it is possible to determine the strain geometry active during the D2 deformation event (i.e. plane strain) and, at the same time, obtain vorticity estimates from the 3D distribution of quartz c-axis. However, as the technique uses the maximum angle of the oblique foliation, it could
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provide vorticity data representative only of the later D2 episode of shear.
Error estimation and vorticity analysis results: Nurra region
Fig. 12. Vorticity analysis from the SW Gallura region. (a) Data points of tailless clasts in an R versus h plot represented using the Law et al. (2004) graphical approach. (b) Mulchrone curves plotted on an R versus h plot. The point distribution does not fit with these curves testifying that, at least in part, the porphyroclast system follows the non-slipping Jeffery assumption. (c) If the SPO parameters have been measured on a different plane from the MSVP, the vorticity estimates underestimate the effective Wm recorded by the rock. Assuming that the SPO parameters measured and projected on Figure 12b have been collected on a plane oriented 108 away from the MSVP, the ‘real’ vorticity estimate is going to be higher (Wm ¼ 0.85–0.86).
As shown in Figures 11a and 12b, the point distribution in the R versus m plot shows a strong variability in the axial inclination and an abrupt change of distribution at ratio values c. 0.2–0.4. In Figure 11b, the distribution of clasts with helicitic inclusions (Fig. 10a) plotted in the GhoshFlow diagram yields a similar result. As for most of the thin sections, we did not have a true picture of the 3D geometry and orientation of porphyroclasts and it was difficult to quantify the amount of misorientation of the main vorticity vector; and as consequence, the effective real error induced into the radius measurement. The deformation geometry registered by the different samples has a monoclinic symmetry (Iacopini et al. 2008). This implies that the sections cut orthogonal to the foliation and containing the main stretching lineation should not have any asymmetric geometry. However, some of the porphyroclasts observed along sections orthogonal to the lineation and the foliation show a clear indication of asymmetric tails or helicitic inclusions. This suggests that the vorticity axes orientation inferred from the mean fabric of the samples is misoriented with respect to the chosen sectional vorticity plane. The minimum misorientation we can presume in order to observe by eye such misorientation is over 5 degrees. This does not sound unreasonable as the pure-shear component observed over the area strongly suggests that the elongated porphyroclasts cannot be perfectly aligned and their vorticity axes are not expected to be always orthogonal to the main foliation plane of analysis as in a simpleshear zone. Using the diagram of Figure 5 to produce the visible clast asymmetry, we can expect an error factor of c. 5 –10%. This error will propagate through the distribution of points in a R versus h plot producing, at best, a translation of the measured distribution. A correct estimation of the radius and of the vorticity axes misorientation strongly depends on the number of serial sections at different orientations that can be made, as well as by randomness in clast orientation. In this example, the analysis performed suggests that a minimum of a 0.1–0.2 error should be taken into account with the vorticity number. This implies that the vorticity values of 0.2 to 0.4 inferred from the distribution in Figure 11a, b, affected by biases of 0.1 –0.2, can possibly vary by between 0– 0.1 and 0.5– 0.6.
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Error estimation and vorticity analysis results: southwest Gallura region The vorticity estimates of the southwest Gallura region presented and discussed in this paper (Fig. 12a) are obtained from the vorticity measurements carried out on four samples collecting by Frassi et al. (2009) in two correlated and close outcrops of fine-grained gneisses affected by sinistral shearing (Sector B in their Fig. 1b). In an R versus h plot (Fig. 12a), the clasts clearly show a bimodal distribution. Most of them show h values widely spanning 0 –908 and an aspect ratio lower than 3.0. A lower number of grains have higher aspect ratio (.3) (i.e. they are strongly elongate) and show low h angle (,308) (i.e. their main axes are oriented subparallel to the main foliation). Using the graphical approach of Law et al. (2004) we are able to individuate the threshold Rc value that discriminates the field of continuous rotation from that in which the clasts reach a stable sink orientation (Fig. 12a). In detail, vorticity estimates, additionally supported by quartz petrofabric measurements and microstructures (Frassi et al. 2009), indicate a predominant component of simple shear that reaches at least 60% of total D2 deformation. Information from quartz c-axis fabrics indicate that the shear deformation occurred with a monoclinic geometry during plane strain conditions (Frassi et al. 2009). As a consequence, we could assume that the porphyroclasts developed a SPO with long axes roughly parallel to the stretching direction. Aspect radii R not measured on the main sectional vorticity plane (MSVP) are lower than the correct one measured along the MSVP (Fig. 5). Even if we are not able to assess the effective error induced by radius measurement into the vorticity estimates, we can infer that the vorticity values estimated on natural samples always underestimate the effective kinematic vorticity number (Fig. 12c). For example, assuming that the plane where the SPO parameters were collected is oriented at a maximum 108 away from the MSVP, we can infer the error associated with this misorientation (c. 5%; see Fig. 5) and the effective vorticity estimates (Wm ¼ 0.85–0.86; see Fig. 12c).
Final consideration The shear zones and porphyroclast analysis performed in the two different natural shear zones indicate how difficult it can be to obtain clear and effective information on both the main source errors as well as on the crucial geometrical parameters: thickness, isolation factor and strain localization. Natural shear zones do not provide continuous outcrop and the investigated geometrical properties
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can change quite fast parallel to the trend of the shear belt. This strongly limits the analysis of possible errors and biases affecting the vorticity analysis. With this contribution we demonstrate that with careful in situ outcrop analysis, some of these limits can be overcome. The most easy (but also timeconsuming) parameter that should be tested is the orientation of the main vorticity vector and the related MSVP by use of several thin sections, cut parallel and orthogonal to the main stretching lineation and orthogonal to the mylonitic foliation. Once obtained, a gross estimation of the main systematic error affecting the principal geometric parameters (e.g. aspect ratio) used to define the distribution of the SPO curves can be made. Using large and good outcrops (better if they have an element of 3D exposure), a good constraint for the confining factor can be obtained using an estimation of the minimum thickness of the shear zone. The dilatancy is extremely difficult if not impossible to investigate and estimate, especially without using geochemical techniques to balance the mass and fluid expulsion across and along the shear zones. Finally, even if the strain localization effect can be easily inferred from micro- and meso-scale observations of the main anisotropic features (i.e. S –C and S– C′ fabrics), it can rarely be quantified. In good outcrops that preserve the best microstructures, all these analyses and in situ measurements should however be attempted in order to better constrain the kinematics and mechanical properties of the shear zone and hence estimate the error sources that may affect vorticity measurements. The pre-test using the R versus h plot combining the Mulchrone and Passchier curves is quite efficient for assessing the effective strain sensitivity of the measured SPO distribution, ruling out the possible slip effect. This test can be routinely used as a preliminary test before discussing the meaning of the vorticity numbers obtained. All the considerations and measurement techniques presented so far suggest that in situ field analyses and measurements can further constrain flow properties before systematically applying the vorticity techniques. Moreover, we indicated that most of the limits and biases basically come from the inability to visualize the real 3D framework of the main geometric parameters (i.e. aspect ratio, clast orientation) measured to achieve the Wm values. As a consequence, the next research step should focus on upgrading the existing tools for 3D visualization through rocks (i.e. X-ray tomography; see Ketcham 2005) or using 3D visualization software traditionally developed to reconstruct consistent 3D geometry from serial sections in seismic interpretation (e.g. 3D move, Midland Valley; Petrel, Schlumberger).
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Conclusion A theoretical discussion of the possible systematic error distribution, associated with the measurement of the geometrical parameters used in the rigid grain vorticity techniques, and a preliminary test to verify at least one of the Jeffery starting assumptions have been presented. A test of all the factors affecting the kinematic analysis has been proposed and discussed and applied to natural shear zones cropping out in the northern Sardinian Variscan Belt (Italy). The theoretical discussion highlights the necessity of testing, in situ, the Jeffery conditions before attempting to estimate the vorticity values. To do this we suggest (1) conducting a preliminary test using cross-plot functions h versus R and (2) estimating both the isolation and confining factors. In the case of the scattered h versus R point dataset, we suggest applying slipping and nonslipping vorticity plots as best-fitting curves. This will fine tune the vorticity analysis and check for slipping effects in the porphyroclast–matrix system. From the proposed theoretical considerations, we expect to obtain ambiguities in vorticity number measurements especially if shear zones have registered transpressive or pure-shear-dominated deformation; in these cases, clast axes never perfectly parallel the main stretching direction. These ambiguities are mainly due to the errors induced during the sectional radii estimation and recognition of the maximum vorticity plane. We thank D. Healy for reading proofs of this paper and K. Mulchrone for useful comment on his cross-plotting curves. English revision and final comment from C. Bond and the editor D. Tatham were greatly appreciated. We thank S. Wallis and D. Grujic for constructive comments.
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Diffusion-creep modelling of fibrous pressure shadows II: influence of inclusion size and interface roughness J. R. BERTON1, D. W. DURNEY1* & J. WHEELER2 1
Department of Earth & Planetary Sciences, Macquarie University, NSW 2109, Australia
2
Department of Earth & Ocean Sciences, Liverpool University, Liverpool L69 3GP, UK *Corresponding author (e-mail:
[email protected]) Abstract: This paper extends previous work by us to gain a fuller appreciation of the physical factors that affect polycrystal diffusion-creep simulations of fibrous pressure-shadow growth around a pyrite inclusion. The earlier work dealt with the effect of diffusion ratio or diffusional conductance of the inclusion/matrix interface. The new work also examines the effects of inclusion geometry: a smaller inclusion of similar smoothness to the original, a regularly serrated inclusion the same size as the original and a coarse irregular inclusion of the same size. The results show: (1) significant enhancement of fibrous pressure-shadow growth and change of matrix strain pattern with decreased inclusion size, similar to an increase in diffusion ratio; (2) approach towards a maximum fibrous pressure-shadow growth at high diffusion ratios in the small-pyrite model; (3) little influence of the model serrations; (4) significant sliding on the interface at low diffusion ratios in all of the models; and (5) enhanced sliding in the irregular-pyrite model at low diffusion ratios. The results are qualitatively consistent with diffusion creep of a single grain interacting with a deforming medium. They demonstrate factors that may influence development of the natural structures under similar conditions in rocks.
Martin Casey has been thanked in print for ‘stimulating argument’ (Wheeler 1987) and this, part of his modus operandi, was much appreciated by his colleagues over many years. In many respects, the present article is a continuation of those discussions, and also reflects his interests in illuminating geological questions through numerical modelling. We adopt Martin’s approach of tackling challenging problems this way: in this case, a problem of coupled chemical transport and deformation around a rigid body. Our study follows work begun by Wheeler (1987, 1992) and Hazzledine & Schneibel (1993) which led to the program ‘DiffForm’ for numerical simulation of grain-boundary diffusion-creep in a single-phase aggregate (Ford et al. 2002). In the first of our series on the subject, Berton et al. (2006) applied this method to a problem of growth and dissolution at the interface between an inert pyrite inclusion and a matrix of diffusion-creeping calcite grains. (Problems arise when both phases are soluble, Ford & Wheeler 2004, but are not of concern here because the ‘DiffForm’ program can be deployed to model the behaviour of a two-phase polycrystal aggregate when the second phase is insoluble.) The purpose of Berton et al.’s modelling was to simulate growth of natural fibrous ‘pressure shadows’ or ‘strain fringes’ around pyrite. As discussed by them, this had been a longstanding problem in natural rock deformation and is a
phenomenon of interest for its ability to record strain history in rocks (Ramsay & Huber 1983; Passchier & Trouw 2005). The driving force for diffusion in the models is the normal-pressure dependence of chemical potential of crystals at grain boundaries and interfaces (Coble creep, ‘pressuresolution’ creep or ‘stress-induced solution-transfer’ creep) and the individual grain-boundary and interface segments are allowed to slide perfectly. Berton et al.’s polycrystal assembly was deformed under plane strain pure-shear velocity conditions to small finite strain in 10 increments of 30 time-steps each and produced ‘pressure-shadow’ structures similar in size to natural structures. The previous study examined the effect of varying what we call the diffusion ratio (DR), which is a measure of the relative diffusional properties of the pyrite/matrix interface and matrix/ matrix grain boundaries. Generally, these properties are represented by what is known as the diffusional conductance (Wheeler 1987, 1992) of the boundary, which is the product Lw of the Onsager diffusion coefficient L of the component in the boundary and the boundary width w. DR is the ratio of Lw of the interface to Lw of the grain boundaries, or the ratio of the two L’s if w is the same (Berton et al. 2006). A high diffusion ratio therefore means that the material diffuses much more readily along the interface than along the grain boundaries.
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 319– 328. DOI: 10.1144/SP360.18 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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The previous study found that the diffusion ratio strongly affected whether a pressure shadow grew, its size, the amount of rotation of individual matrix grains and the patterns of macroscopic strain in the matrix around the inclusion. In particular, it was found that the amount of pressure-shadow growth increased significantly with increasing diffusion ratio, and the strain axes in the matrix changed from bowing outwards around the inclusion at low DR to bowing inwards at high DR. As noted at the time, these findings have consequences for microstructural interpretation of foliation patterns around inclusions and relative amounts of pressure shadow and matrix strain. That study was conducted for a nearly circular pyrite of fixed size. Natural pyrite inclusions of the framboidal type, such as those illustrated by Beutner & Diegel (1985, plates 1 and 2), Koehn et al. (2001, fig. 4) and Berton et al. (2006, fig. 14), often comprise globular clusters of small pyrite grains that give the inclusion a rough surface. The matrix around the inclusions is usually fine grained and the inclusions in a given matrix may vary in size. It is therefore important to investigate whether these factors affect fibrous pressure-shadow development and patterns of strain. Berton (2009) examined this question by testing the model sensitivity to variations in size and surface roughness of the pyrite. All other model parameters, including size of the matrix grains, were kept the same or nearly the same as in the original so that the results would mainly reflect differences in the geometry of the inclusion. Results for those tests are described in this paper and are compared to the previous results. One of the model variants has a smaller pyrite grain of similar smoothness to the original, while two other variants have pyrite grains the same size as the original that are roughened with asperities of different wavelength and amplitude. These models were run to about the same finite strain in the same pure-shear mode with the same material constants as before, and were analysed for macroscopic finite deformation in the matrix as described by Berton et al. (2006). The data reported here are for grain movement, finite deviatoric strain and signed kinematic rotation number in the matrix as well as measurements of apparent pressure-shadow finite strain as observed from the displacements of a matrix grain on the extension axis of the model.
Model variants The starting configurations of the original model and the three model variants are shown in Figure 1. Dimensions and strains are listed in Table 1.
The model at top left in the figure (Fig. 1a) is the original or basic pyrite model with respect to which the variations have been made. The pyrite in this model comprises a series of flat segments that approximate a circle with very low-amplitude asperities due to the corners between consecutive flats. The small pyrite model (Fig. 1b) has a pyrite grain that is nearly smooth, like the basic one, but 62.5% of the size. It has 12 sides around the whole inclusion instead of 16. The grains in the interior of the matrix are similar to those of the basic model, but there are 12 additional grains adjacent to the pyrite (two full grains and two half grains in the first quadrant). The purpose of this network is to test the effect of a smaller pyrite size on the diffusion-creeping matrix. The purpose of the serrated pyrite model (Fig. 1c) is to examine the effect of roughness of the pyrite boundary for a pyrite grain of approximately the same size as the original. Advances in the diffusion-creep program (DiffForm version 1.4) carried out by JW included a provision for internal two-boundary nodes to allow grain boundaries to be divided into two segments. This permitted construction of angular serrations around the interface that are the width of the adjoining matrix grains. In this way, it was possible to achieve a regular geometry simulating a rough
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Table 1. Dimensional characteristics and final strains of the four models (lengths in arbitrary model units)
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19.5 23 1.4 0.18 0.17 5 4.94 1.93 0.05 0.39 3.95 0.01
19.5 23 1.4 0.18 0.17 8 7.55 1.97 0.45 0.26 2.58 0.06
20 23 1.38 0.18 0.16 8 7.61 2.03 0.6 0.27 2.63 0.08
pyrite boundary without altering the size of the matrix grains. In the irregular pyrite model, the pyrite boundary was constructed with the usual internal threeboundary or triple-junction nodes. This variant uses the same nodes and grain sizes as the original model but different node positions on the pyrite boundary (Fig. 1d). Consequently, the flats of the serrations are the same length as the adjoining matrix grains. There are therefore fewer serrations in this model (six main ones); they are irregular in shape and are more than twice the wavelength and greater in amplitude than those of the serrated model.
Results for the model variants Small pyrite model This model was run for the three diffusion ratio cases used earlier for the basic model (DR 10, 100 and 1000) plus an additional case for DR 10 000. Final ‘ghost-grain’ outlines, which represent the displaced positions of the original grains after the ten deformation increments, are shown in Figure 2. The gaps between the two lowermost matrix ghost grains and the pyrite boundary, representing what appears to be pressure-shadow growth, are greater relative to pyrite grain size in each of the diffusion ratio cases than those in the basic model. This is illustrated by extension strains or ‘gap strains’ (e1x) obtained by dividing the gap between the lowest ghost grain and the pyrite by pyrite radius on the x-axis (Table 2). Pressure-shadow growth therefore appears to be significant in this model, even at the lowest diffusion ratio DR 10. There is also more sliding and dilation of grain boundaries and less rotation of matrix grains near the pyrite; these are features associated
with increased apparent pressure-shadow growth, noted previously by Berton et al. (2006). The extra diffusion ratio case DR 10 000 was performed to test the relationship between gap strain and DR at high DR. This case produced only a small increase (Table 2) and correspondingly little change in configuration of the ghost grains. Overall, the results for this series show that the increase in gap strain decreases rapidly with increase in diffusion ratio, suggesting an approach to a maximum limit characteristic of the particular model. In this model, the gap strain approaches a maximum of just over 0.6 or about 3.5 times the model extension strain (c. 0.18, Table 1). The distribution of total finite strain in the matrix has been determined for diffusion ratios DR 10, 100 and 1000 in the small pyrite model. These strains were obtained from a triangular mesh of displaced ghost-grain centroids (Fig. 2), as described by Berton et al. (2006). The three quantities discussed here are: (a) total finite strain major principal axes, (b) natural deviatoric total finite strain 1s = 12 ln [(1 + e1 )/(1 + e2 )]
(1)
where e1 and e2 are major and minor principal strains, and (c) signed two-dimensional kinematic rotation number ∗ = v/1s Wk+
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(‘finite kinematic vorticity’ in Berton et al. 2006), where v is the finite rotation component of the local shape transformation matrix, measured in * radians and counted positive anticlockwise. W k+ was chosen as an approximate finite-strain counterpart of the conventional instantaneous kinematic vorticity number applicable to small finite deformations, but also includes the sign of the rotation.
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(It is called here a rotation for consistency with its finite nature and is shown with a ‘+’ as a reminder that it may be positive or negative.) The magnitude * therefore shows the local type of flow in of W k+ * |1 * | ¼ 0 for pure shear, |W k+ the matrix (|W k+ * | ≫ 1 for nearly pure for simple shear and |W k+ rotation) while the sign shows the sense of rotation or shear. Major axes of total-strain in the small pyrite model are shown at the sample points between grain triplets (Fig. 3). The axes are deflected around the top and bottom of the pyrite in diffusion ratio case DR 10 and bow into the pyrite in cases DR 100 and 1000. These patterns are somewhat like Table 2. Pressure-shadow apparent gap strains (e1x) on the model x-axis: small pyrite and basic pyrite models (data for basic model from Berton et al. 2006, figs 6–8) Diffusion ratio
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those of the larger basic pyrite model, except that the outward deflection is much less at DR 10 and the axes bow inwards instead of being nearly horizontal at DR 100. As inward bowing is characteristic of fluid flow around a weak viscous inclusion (Ramsay & Lisle 2000, fig. 39.23), this suggests an overall shift towards weaker behaviour of the interface in the small pyrite model. This correlates with the greater strains observed around the inner edge of the matrix in this model. The tendency is greatest in case DR 1000 (Fig. 3c), so that case may be approaching the kind of behaviour observed around an inviscid inclusion or a hole. Contoured strain quantities are given in Figures 4 and 5. The positions of sample points for the contours are shown by the strain bars in Figure 3. As discussed by Berton et al. (2006), contours around the outer edges of the matrix are unreliable due to a lack of data and so have been blanked off here. There is also an appreciable gap in the data near the ends of the pyrite (Fig. 3); contours in this area should therefore also be discounted. In addition, the coarse matrix/pyrite grain-size ratio (Table 1) means that resolution of strain variations near the pyrite will be poorest in this model.
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Fig. 4. Small pyrite model total natural deviatoric strain: (a) DR 10, (b) DR 100 and (c) DR 1000.
Despite the lower resolution, natural deviatoric strain in small pyrite case DR 10 (Fig. 4a) shows regularly alternating highs and lows around the pyrite, qualitatively similar to basic case DR 10. The pattern is however weaker and more like an intermediate case between basic cases DR 10 and 100. At DR 100 (Fig. 4b), the alternation near the pyrite has disappeared while lows on the axes at intermediate distances from the pyrite have begun to intensify, similar to basic case DR 1000. These effects are further reinforced at DR 1000 (Fig. 4c).
In other words, there has been a shift towards behaviour more typical of higher diffusion ratios (as for the strain axes). The kinematic rotation number fields (Fig. 5) are also more like those of higher diffusion ratios in the basic model. At DR 10, the rotation number is negative or clockwise near the pyrite on the diagonal of the first quadrant. It is however much weaker (20.2 as against 22 in the basic model) and is replaced by an expanding positive or anticlockwise field by DR 100 instead of by DR 1000. A feature in the small
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pyrite model which is not present in the basic model (and is not well understood) is a band of negative values near the y axis, implying a vertical dextral deformation there. This band persists throughout all of the diffusion ratio cases but weakens progressively with increasing DR.
Serrated pyrite model The results for this model are very similar to those of the basic model. Data are therefore presented only
for case DR 10 where the serrations should have the greatest observable influence on sliding and growth at the interface. Final ghost-grain outlines for case DR 10 (Fig. 6a) show a strong preference for grain rotation over grain-boundary sliding in matrix grains near the pyrite and very little apparent pressure-shadow growth, as in the basic model. Ghost grains adjacent to the pyrite have undergone appreciable sliding tangent to the mean circle of the inclusion surface to positions where the corners of the grains that once formed projections on the matrix boundary (Fig. 1d) now overlap the projections on the pyrite surface (Fig. 6a). This has been achieved by diffusion-creep removal of matrix on the backwards-sloping flats of the serrations and deposition on the forwards-sloping flats, which is a grain-scale process known as diffusionaccommodated sliding (Raj & Ashby 1971). In this case, the process has taken place at the scale of individual serrations on a compositional interface and is therefore an example of diffusion-accommodated interface sliding. Figure 6a also illustrates a type of grain movement near the interface where Grain 2 has moved down towards the model extension axis and in so doing has contributed to the separation of Grain 1 from the interface. This behaviour is an example of progress towards neighbour switching, where two grains originally not in contact (in this case Grain 2 and a symmetrically equivalent grain moving upwards) eventually establish a mutual boundary. Such movement is an integral part of diffusion creep but is particularly prevalent at this point at low diffusion ratios (compare Fig. 2a) and much less in evidence at higher diffusion ratios (compare Fig. 2d). The movement in Figure 6a has been achieved mainly by sliding and rotation of Grain 2 at the interface. The gap strain recorded by Ghost Grain 1 is 0.18 and, clearly, is mostly due to this movement rather than growth at the pyrite/matrix contact. Similar behaviour and gap strain is observed in the basic model (Berton et al. 2006, fig. 6b; Table 2) and suggests that gap strain overestimates pressure-shadow growth at low diffusion ratios. An opposite effect occurs to some extent near the model y axis (Fig. 6a) where Grain 3 has been squeezed out along the interface by convergent movement of Grain 4. The total strain axes (Fig. 6b) are deflected markedly outwards over the top and bottom of the pyrite, as in basic case DR 10. The total deviatoric strain and finite rotation number fields (Fig. 6c, d) are also very similar, with simple patterns of strong highs and lows around the pyrite. (Maximum values are c. 0.35 for deviatoric strain and 3.5 for rotation number compared to just over 0.4 and 2 in the basic model.) Moreover, the combination of
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Fig. 6. Serrated pyrite model case DR 10: (a) final ghost-grain configuration, (b) total-strain axes, (c) total natural deviatoric strain and (d) finite kinematic rotation number. The dashed lines show data limits near the edges of the model.
low deviatoric strain and high negative rotation number along the diagonal of the first quadrant implies almost strain-free clockwise rotation in that sector, consistent with sliding around a circular surface. The conclusion from these observations, surprisingly, is that the asperities have little influence on the behaviour of this model. At DR 10, the dominant interface behaviour is sliding as for the smooth pyrite of the basic model.
Irregular pyrite model To further test the role of asperities, we present final ghost-grain plots for the irregular pyrite model (Fig. 7). At DR 10 (Fig. 7a), this model shows large amounts of grain rotation and sliding at the pyrite/ matrix interface (similar to but greater than the serrated and basic models). Ghost grains at the interface have slid almost one full grain width around the interface compared to little over half that amount in the other two models. Intrusion is also greater and almost completely accounts for the interface gap on the model x axis. The proportion of grain rotation to grain-boundary sliding in grains near the pyrite declines and the amount of dissolution and growth at the interface increases as
the diffusion ratio increases (Fig. 7b, c) so that, by DR 1000, the ghost-grain configuration is very similar to the basic model. These observations indicate that interface sliding and associated strain accommodation in the matrix are even more important in the irregular model at DR 10 and appear to have resulted in less pressureshadow development than in the serrated model. Increased sliding occurred despite the larger asperities. The reason for this is probably the subhexagonal shape of the irregular pyrite which has a nearly straight inclined segment that would be in a position to promote excess sliding. With increasing diffusion ratio, the boundary shape has less and less influence and the observed displacements of matrix grains come to resemble those around the smooth inclusion.
Summary of observations from the model variants The following conclusions are made from observations of the model variants. (a) Reduction of the pyrite size significantly increases pressure-shadow growth and affects grain behaviour and strain in the matrix in ways similar to an increase in diffusion ratio.
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the matrix behaves as if the pyrite/matrix interface were a hole. (c) Asperities have no appreciable influence on model behaviour in the serrated pyrite model. The larger asperities of the irregular pyrite model enhance sliding on the interface and reduce pressure-shadow growth at low diffusion ratio, but this can be attributed to the special geometry of this inclusion which has a nearly straight section of the interface. (d) All of the models are dominantly affected by sliding around the pyrite at low DR and show appreciable pressure-shadow development at high DR.
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Discussion Reduced inclusion size and increased diffusion ratio
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Fig. 7. Irregular pyrite model final ghost-grain configurations: (a) DR 10, (b) DR 100, (c) DR 1000.
At low diffusion ratios, the change of strain patterns in the matrix produced by a times 0.625 size reduction is about half that of a times 10 increase in diffusion ratio. (b) The effects of reduced pyrite size and increased diffusion ratio tend towards a limit beyond which further increases in diffusion ratio have little or no influence. As this limit is approached, pressure-shadow development approaches a stable maximum amount and
A reduced inclusion size reduces the path-length for diffusion around the interface and should therefore increase the rate of transfer of material by diffusion along the interface. This would explain the greater apparent pressure-shadow strains in the small pyrite model and why the effect is similar to an increase in diffusion ratio or diffusional conductance of the interface (which also promotes greater pressure-shadow growth). Qualitatively, this explanation agrees with predictions of boundary diffusion creep for single grains (e.g. Coble 1963; Rutter 1976; Wheeler 1992) where strain rates depend inversely on the cube of grain size and directly on boundary conductance. However, the increases observed in the models are much smaller (e.g. apparent pressure-shadow strain in the models changes by no more than about two-fold for a ten-fold increase of conductance whereas strain rate in a single grain changes ten-fold; Table 2). To understand why this should be so, it is necessary to consider interactions between the pyrite/matrix interface and the flowing matrix. If the interface is thought of as having certain stiffness due to interface diffusion creep, it may be compared with the surface of a viscous inclusion in a viscous medium of different viscosity. The strain rate of a cylindrical viscous inclusion increases non-linearly with decreasing inclusion viscosity and approaches a maximum when the viscosity vanishes (Bilby et al. 1975). This concurs better with the model observations, where increases in apparent boundary strain decline rapidly and approach a maximum strain with increasing diffusion ratio (and thus decreasing diffusion-creep stiffness of the interface). Strains are not analysed quantitatively in this paper because the apparent pressure-shadow strains
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observed in the models are known to overestimate actual growth strains of the boundary, at least at low diffusion ratios. This question is the subject of a separate investigation to ascertain macroscopic growth and slide strain in the models, one which the writers are currently undertaking.
Increased surface roughness The increased surface roughness of the serrated pyrite model was expected to reduce interface movement but did not produce any appreciable difference. The effect of the serrations should be two-fold. First, they should tend to retard interface sliding because of the asperity obstacles and the need to overcome them by local diffusional transport around the asperity corners (Fig. 8a). Intuitively, retarded sliding may retard pressureshadow growth because the two movements are linked to the one boundary deformation. Second, the path-length for diffusion along a rough interface is greater than for a smooth interface of the same mean arc length (Fig. 8b). This would reduce rates of interface mass transport and thus further reduce the rates of pressure-shadow growth and sliding. To get an order-of-magnitude idea of how much the model serrations would affect sliding, it is noted that Raj & Ashby’s (1971) formulae for rate of boundary-diffusion-accommodated sliding depend on the inverse square of asperity amplitude. In contrast, rates of boundary-diffusion growth on the same boundary would depend on the inverse square of inclusion radius. The asperity amplitude in the serrated model is 0.06 of the radius of the pyrite (Table 1); sliding would be c. 103 times easier than growth and therefore not noticeably different from perfect sliding. The asperity sides in the serrated pyrite model deviate from the mean circle by some 308 on average. This would increase the path-length for diffusion by sec (308) or about 1.15 times and make pressure-shadow growth harder by a factor of 1.1522 or 0.75 compared to a smooth inclusion. It is therefore inferred that the serrations of the serrated pyrite model have no practical influence
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Fig. 8. (a) Sliding on a serrated interface, accommodated by interface mass transfer (small arrows show diffusion currents) and (b) relative lengths of a serrated surface and a smooth surface.
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on sliding. They should reduce pressure-shadow growth, but the model data were not sufficient to resolve this difference.
Conclusions Many rocks appear to have been deformed naturally by processes akin to diffusion creep, often denoted ‘pressure-solution’ (Ramsay & Huber 1983) or ‘solution-transfer’ (Durney 1972; Vernon 2004) processes. These involve chemical transport around grains, the greatest driving force for which is generally the effect of differences in normal pressure on chemical potential at the surface of the grains (Wintsch & Dunning 1985; Wheeler 1987). It is therefore important to investigate this effect not only as a mechanism by which many rocks may deform but also in mineral reactions, which may be governed by the same transport coefficients (Wheeler 1987). The numerical algorithm used in this study and the earlier study of Berton et al. (2006) employs this driving force. It is also an advance on most previous attempts at modelling fibrous pressureshadow structures in that it satisfies coupling of chemical transport, stress and grain movements at the grain scale, similar to what would be expected in a natural polycrystalline material. The results of the model variations tests are thus able to provide insights into the way different factors affect rates of growth and dissolution at the interface with a rigid inclusion. The earlier results of Berton et al. showed that increasing diffusion ratio (increasing interface conductance) promotes faster interface diffusion and more growth at the interface. This has been reaffirmed in the present study, which furthermore shows that reduced size of the inclusion relative to the matrix grains also increases these rates. The effects are qualitatively like those of boundary conductance and grain size in single-grain boundarydiffusion creep, but are moderated by interaction with the deforming matrix in such a way that they approach a maximum at the highest diffusion ratio, similar to a weak inclusion or a hole. Diffusion-accommodated interface sliding is important in the models and dominates interface behaviour at low diffusion ratios. Serrations should retard this movement and growth at the interface, but this was not detected in the model that was designed to test it. The reason put forward is that diffusion accommodation is a relatively fast process because it depends on asperity amplitude, which is small compared to inclusion size. On the other hand, the irregular pyrite had a special shape that actually favoured faster interface sliding than the other models.
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Overall, the model tests show that interface conductance and inclusion size are the main factors affecting growth and dissolution at the interface. Similar but numerically different effects might therefore be expected at interfaces of sub-spherical natural inclusions. Our model results suggest that these should be quantifiable, at least on a relative basis, by comparing pressure-shadow strain variations in different homogeneous rocks that host inclusions with a variety of sizes. The studies have nevertheless been carried out for a previously unexplored application and there is scope for further development, both in model networks and in observations of natural fibrous pressure-shadow behaviour. Finer networks would be needed to resolve details of strain variations and growth forms close to the inclusion, and edge effects near the model boundaries may warrant further examination. Apparent pressure-shadow strains have been reported as observable ‘gap strains’ at the interface, and the matrix-strain contributions to these need to be differentiated for applications that require quantitative data on strain of the macroscopic matrix boundary. One asperity geometry with a path-length ratio of 1.15 was examined in the serrated model. This factor could have a greater influence in asperities with higher pathlength ratios. Finally, it is not known to what extent the model assumption of perfect sliding on boundary segments applies to natural examples. This needs to be assessed through comparisons of predicted behaviours, particularly the amount of interface sliding, with natural cases. The study uses data acquired by JB with the help of facilities at Macquarie and Liverpool Universities and Matlab software, assisted in part by tenure of an Australian Postgraduate Award, which are gratefully acknowledged. D. Koehn and D. Dysthe are thanked for constructive comments that led to improvements, and we are grateful to D. Prior, E. Rutter and GSL Books for advice and guidance through the online process.
References Berton, J. R. 2009. Numerical modelling and analysis of fibrous pressure-shadows around pyrite. Unpublished PhD thesis, Macquarie University, New South Wales. Berton, J. R., Durney, D. W., Wheeler, J. & Ford, J. M. 2006. Diffusion-creep modelling of fibrous pressure-shadows. Tectonophysics, 425, 191–205.
Beutner, E. C. & Diegel, F. A. 1985. Determination of fold kinematics from syntectonic fibers in pressure shadows, Martinsburg slate, New Jersey. American Journal of Science, 285, 16– 50. Bilby, B. A., Eshelby, J. D. & Kundu, A. K. 1975. The change of shape of a viscous ellipsoidal region embedded in a slowly deforming matrix having a different viscosity. Tectonophysics, 28, 265 –274. Coble, R. L. 1963. A model for boundary diffusion controlled creep in polycrystalline materials. Journal of Applied Physics, 34, 1679– 1682. Durney, D. W. 1972. Solution-transfer, an important geological deformation mechanism. Nature, 235, 315–317. Ford, J. M. & Wheeler, J. 2004. Modelling interface diffusion creep in two-phase materials. Acta Materialia, 52, 2365–2376. Ford, J. M., Wheeler, J. & Movchan, A. B. 2002. Computer simulation of grain boundary creep. Acta Materialia, 50, 3941–3955. Hazzledine, P. M. & Schneibel, J. H. 1993. Theory of coble creep for irregular grain structures. Acta Metallurgica et Materialia, 41, 1253–1262. Koehn, D., Aerden, D. G. A. M., Bons, P. D. & Passchier, C. W. 2001. Computer experiments to investigate complex fibre patterns in natural antitaxial strain fringes. Journal of Metamorphic Geology, 19, 217–231. Passchier, C. W. & Trouw, R. A. J. 2005. Microtectonics. 2nd edn. Springer-Verlag, Berlin. Raj, R. & Ashby, M. F. 1971. On grain boundary sliding and diffusional creep. Metallurgical Transactions, 2, 1113– 1127. Ramsay, J. G. & Huber, M. I. 1983. The Techniques of Modern Structural Geology, Vol. 1: Strain Analysis. Academic Press, London. Ramsay, J. G. & Lisle, R. J. 2000. The Techniques of Modern Structural Geology, Vol. 3: Applications of Continuum Mechanics in Structural Geology. Academic Press, San Diego. Rutter, E. H. 1976. The kinetics of rock deformation by pressure solution. Philosophical Transactions of the Royal Society of London, A, 283, 203–219. Vernon, R. H. 2004. A Practical Guide to Rock Microstructure. Cambridge University Press, Cambridge. Wheeler, J. 1987. The significance of grain-scale stresses in the kinetics of metamorphism. Contributions to Mineralogy and Petrology, 97, 397–404. Wheeler, J. 1992. Importance of pressure solution and Coble creep in the deformation of polymineralic rocks. Journal of Geophysical Research, 97B, 4579– 4586. Wintsch, R. P. & Dunning, J. 1985. The effect of dislocation density on the aqueous solubility of quartz and some geologic implications: a theoretical approach. Journal of Geophysical Research, 90, 3649– 3657.
Rock mechanics constraints on mid-crustal low-viscosity flow beneath Tibet E. H. RUTTER, J. MECKLENBURGH* & K. H. BRODIE Rock Deformation Laboratory, School of Earth, Atmospheric and Environmental Sciences, University of Manchester, Oxford Road, Manchester M13 9PL, UK *Corresponding author (e-mail:
[email protected]) Abstract: It has been inferred from various types of geophysical data that the Tibetan middle and upper crust is detached from the underlying lower crust and mantle by a weak, mid-crustal zone involving partial melting at about 30–35 km depth. Previous modelling of the flow has used an arbitrary mid-crustal rheology to match the constraints imposed by the overall flow regime. Here we show that extrapolation of experimental rock mechanics data for solid-state flow of a quartz-dominated Tibetan middle and upper crust, plus flow of partially molten synthetic ‘granitoid’, are consistent with the geophysical constraints and provide an experimentally constrained basis for the modelling of crustal rheology involving partially molten rocks.
A prime purpose of high-temperature rock mechanics experiments on crustal rocks has always been to obtain constitutive flow laws that can form a basis for modelling the mechanical behaviour of the continental crust. Extrapolations of laboratory data, obtained from experiments on both natural and synthetic hot-pressed samples, to inferred natural conditions of temperature, deformation rate, confining pressure and activities of vapour phase components have been reported and summarized by many experimentalists (e.g. Paterson & Luan 1990; Kohlstedt et al. 1995; Rutter 1995; Rybacki & Dresen 2000; Hirth et al. 2001; Rutter & Brodie 2004; Mecklenburgh et al. 2006, 2010). Published flow-law data have also been seized upon by geologists and geophysicists to help explain field-based observations (e.g. Meissner & Strehlau 1982; Kusznir & Park 1986; Ranalli & Murphy 1987; Jackson 2002; Afonso & Ranalli 2004). For approximate modelling of the mechanical behaviour of the ductile (middle and lower) part of the continental crust, it has commonly been assumed that flow is by solid-state intracrystalline plasticity or by grain-size-sensitive flow (if grain size is sufficiently fine), dominated either by carbonate minerals, quartz or feldspars, usually inferred to dominate particular depth ranges. Relatively little has yet been done to model solid-state flow of polymineralic silicate rocks. There are a few generalized flow laws that have been published for natural polymineralic silicate rocks (e.g. Wilks & Carter 1990; Mackwell et al. 1998; He et al. 2003). Given the propagation of errors of measurement in the extrapolation of laboratory data, and the fact that application to a particular geological situation
necessarily involves comparing a specific experimental microstructure and mineralogy with only vague generalizations about the rock types in crustal sections being modelled, it must be remembered that such extrapolations are at best mere approximations to expected behaviour (Rutter & Brodie 1991). When parts of a crustal section are expected to be partially molten, it has not previously been possible to carry out the above type of modelling because suitable experimental data on partially molten rocks of granite-like compositions has been lacking. The determination of a constitutive flow law requires that a series of experiments be carried out in which a single dependent variable is changed while all others are kept constant. For example, in the case of a partially molten rock, we might wish to determine how flow stress varies with temperature while melt fraction, melt composition, water content (hence melt viscosity), strain rate and grain size are all held constant. With a partially molten natural rock this is not usually possible because, for example, melt fraction and composition change with temperature. We recently reported a set of experimental data on a synthetic partially molten ‘granitoid’ (Mecklenburgh & Rutter 2003; Rutter et al. 2006) designed to overcome these practical problems, so that a flow law could be obtained expressing strain rate as a function of temperature, flow stress and melt fraction for a fixed solid-phase grain size and a fixed water content in the melt. In previous experimental studies of partially molten granites, these controls could not be achieved (e.g. van der Molen & Paterson 1979; Rutter & Neumann 1995). At the present time the influence of changing the
From: Prior, D. J., Rutter, E. H. & Tatham, D. J. (eds) Deformation Mechanisms, Rheology and Tectonics: Microstructures, Mechanics and Anisotropy. Geological Society, London, Special Publications, 360, 329– 336. DOI: 10.1144/SP360.19 # The Geological Society of London 2011. Publishing disclaimer: www.geolsoc.org.uk/pub_ethics
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grain size of the solid phase has not been determined experimentally. The purpose of this short note is to demonstrate how this experimental data can be extrapolated to geological conditions to impose constraints on geophysical modelling. We have used the example of the frequently inferred existence of a zone of partially molten rock in the mid-crust beneath the Himalaya –Tibetan Plateau orogenic system.
Inference of crustal structure beneath the Himalaya – Tibetan Plateau A great deal has been written about crustal structure and processes beneath the Himalaya –Tibetan Plateau (e.g. Hodges 2006), but several groups of geological and geophysical observations made in the region have led many to the conclusion that an anomalously weak zone or zones exists in the middle-crust of the region. The Tibet Plateau is the most extensive and highest (average 5 km) plateau region on Earth, yet lacks the surface relief typically associated with high mountain region (Fig. 1). The inability to support substantial lateral variations in topography suggests that at some depth the crust is too weak to support lateral variations in load (Bird 1991; Fielding et al. 1994). Such an effect is seen on Antarctic ice, where large subglacial lakes such as Lake Vostock (Siegert et al. 2001) lie beneath regions of flat, featureless ice. Another comparable high-plateau region is the Altiplano–Puna Plateau of Andean South America, for which the geophysical characteristics can be interpreted in a similar way as Tibet (Beck & Zandt 2002; Tassara et al. 2007). Klemperer (2006) provides a detailed summary of geophysical data pointing to a weak layer in the thickened (totalling 70–80 km Priestley et al. 2008) crust beneath Tibet. Such evidence includes the occurrence of high heat flows (Wang 2001) and the fact that 95% of located earthquakes develop at depths less than 17 km, implying that crustal rocks at greater depths are more prone to flow than to fracture. In southern Tibet a few lowercrustal/upper-mantle earthquakes have been reported at depths of c. 65– 85 km. A broad belt of lowered seismic S-wave velocities at mid-crustal depths has also been found, implying reduced elastic stiffnesses that might relate to rocks approaching or exceeding their solidus temperatures (Hung et al. 2010). Deep electrical soundings (e.g. Unsworth et al. 2005), indicating high conductivity, have been interpreted to be caused by partial melting in the 30–40 km depth range over almost the whole of southern Tibet. There have been many modelling studies of the regional tectonics of the Himalayan–Tibetan
region. One of these (Shen et al. 2001) interprets the north-northeast-directed measurements by GPS (global positioning satellite) of the regional flow field in terms of relative northwards motion of a ‘strong’ (constant 1021 Pa s viscosity, h0 ) Tibet Plateau of thickness z0, over a stiff lower crust and upper mantle, with a low viscosity channel in the middle crust decoupling the two stiffer layers. The weak detachment layer was arbitrarily modelled as one of Newtonian viscosity h, exponentially decreasing with depth z beneath z0 (¼30 km) according to
h = h0 exp
z − z 0 . a
(1)
Viscosity decay parameter a is chosen so that the cumulative displacement rate matches the GPS constraint for an arbitrarily selected thickness for the weak zone. Many viscosity models could have been chosen for the weak zone such that the cumulative displacement rate constraint was satisfied, but none would have been constrained by experimental rock deformation data. Other workers have modelled the early development of the Himalaya –Tibetan orogen (e.g. Beaumont et al. 2001, 2004, 2006) and also used arbitrary weakening functions that are not constrained by experimental data to describe the weakening due to partial melting in the lower crust. These models are different from this study and the models of Shen et al. (2001) in that they are not modelling current flow of the orogen, but rather the early development of the orogen involving channel flow and crustal extrusion of the lower crust toward the south. In this report, we therefore pose the question: are these types of models (e.g. Shen et al. 2001) consistent with the results of relevant rock mechanics data, including recent experiments on the rheology of partially molten granitic rocks?
Application of experimental rock mechanics data Here we simplify the Tibetan crust to consist of an upper layer, c. 30 km thick, whose rheology is presumed to be dominated by that of quartz, underlain by c. 5 km of partially molten granitoid, with the melt fraction increasing as temperature increases with depth. If rock chemistry remained unchanged with increasing depth, the melt fraction would increase via a sequence of melting reactions until the rock was completely molten. We have therefore placed a more refractory zone beneath 36 km depth with a rheology corresponding to solid-state flow of a mafic rock (dolerite: parameters shown in Table 1; Mackwell et al. 1998).
LOW-VISCOSITY FLOW BENEATH TIBET
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Fig. 1. Image showing the topography of the Tibetan Plateau. The length scale corresponds to the scale parallel to lines of longitude. The elevation data used to generate this image are taken from the ETOPO2v2 (2006) database.
Various displacement patterns in the low viscosity channel have been considered previously (Klemperer 2006), including Poiseuille flow between rigid boundaries, Couette flow induced by relative shearing motion between the upper and lower slabs and combinations of both. We assume simple shear to be induced in the weak layer as a result of northwards displacement of the upper slab relative to a static lower part of the section. Our overall conclusions are not affected by the particular assumptions made in this respect, however. The horizontal component of shear stress from midcrustal depths downwards is assumed constant, and
of a magnitude that gives rise to the surface displacement rate observed from GPS measurements of c. 50 mm a21 (Chen et al. 2000; Shen et al. 2001; Gan et al. 2007; Jime´nez-Munt et al. 2008) relative to an assumed fixed lowermost crust and upper mantle. For modelling purposes a linear geothermal gradient of 24 8C km21 was assumed, so that melting begins at c. 32 km depth (760 8C).
Quartz flow laws From recent experimentally constrained flow laws published for quartzite, it is recognized that the
Table 1. Flow-law parameters and experimental conditions from rock mechanics experiments Material Quartzite Quartzite Quartzite Average synthetic quartzite Partially molten ‘granite’ Dry Columbia dolerite Dry Maryland dolerite
logA (MPa2n s21)
H (kJ mol21)
N
Reference
24.88 + 2 28.72 + 0.6 22.45 + 0.4 27.2 21.39 2.27 + 0.25 0.9 + 0.22
223 + 56 135 + 15 242 + 24 150 + 2.0 220 + 65 485 + 30 485 + 30
4.0 + 0.9 4.0 2.97 + 0.3 3.1 + 0.6 1.8 + 0.3 4.7 + 0.6 4.7 + 0.6
Gleason & Tullis (1995) Hirth et al. (2001) Rutter & Brodie (2004) Luan & Paterson (1992) Rutter et al. (2006) Mackwell et al. (1998) Mackwell et al. (1998)
All quartz experiments were in axisymmetric shortening, performed in gas-medium testing machines at 300 MPa confining pressure except for Gleason & Tullis (1995) which was at 1500 MPa. Their quartzite flow-law parameters are recast to 300 MPa water fugacity assuming a fugacity exponent of unity. The flow law of Hirth et al. (2001) combines both experimental data and microstructural observations of naturally deformed rocks to constrain the flow parameters. Water fugacity in the granitoid melt is estimated at 70 MPa at 900 8C. Mackwell et al. (1998) reports two different log A values corresponding to two different dolerite types. Confining pressure was 500 MPa. Samples were tested dry with f O2 controlled between the nickel –nickel oxide and iron –wu¨stite buffers. Parameters uncertainties are shown where they are reported.
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flow strength depends on water fugacity of the pore fluid. We have used four flow laws to bracket quartzite behaviour (Table 1; Luan & Paterson 1992; Gleason & Tullis 1995; Hirth et al. 2001; Rutter & Brodie 2004). Water fugacity varies with depth in nature, but for the present calculations it is assumed constant at 300 MPa. The form of the flow law is therefore
g˙ = Atn exp
−H RT
(2)
in which g˙ is shear strain rate (s21), t is shear stress (MPa), H is activation enthalpy for flow (J mol21), T is temperature (K), R is the gas constant and A is a dimensionally consistent constant. Values of material parameters A, H and n are listed in Table 1. Grain size is assumed sufficiently large such that no component of grain-size-sensitive flow develops. The plastic flow laws were obtained in axisymmetric loading and hence in terms of differential stress (s1 – s3) and axial longitudinal strain rate 1˙ . These can be recast in terms of shear
stress and shear strain rate (e.g. Hill 1950) using √ g˙ = 1˙ 3
and t =
s1 − s3 √ . 3
(3)
Figure 2 shows how the apparent viscosity of quartzite is expected to decrease as temperature increases with depth at a constant shear stress. The shaded area indicated shows the range of viscosities (for a constant flow stress) predicted at a given depth by the different flow laws. The greatest range is encountered at shallow depths, and the variations are smallest at the onset of melting.
Rheology in the melting interval Study of progressive dehydration melting of pelitic schists with particular reference to the origins of Himalayan leucogranites (Patino Douce & Harris 1998; Harris et al. 2000) showed that melt fraction increases approximately linearly with temperature above the onset of melting at 760 8C and 32 km depth, attaining 35% melt at about 860 8C. To
Fig. 2. Calculated viscosity profile for the crustal section modelled, subjected to a constant horizontal shear stress of 0.10 MPa (+0.03 MPa in the partially molten region) that produces a horizontal velocity difference of 50 mm a21 between top and bottom surfaces of the section. At less than 32 km depth, rheology is that of solid-state flow of quartz with the bounds shown for the four flow laws used. From 32 to 35 km depth, melt fraction varies from 0 to 35% and viscosity decreases accordingly. Below 35 km depth, rheology is inferred to correspond to that of dry, melt-free basalt. Also shown is the exponential variation of viscosity with depth used in the modelling of Shen et al. (2001) with viscosity decay parameter a ¼ 0.4 km and the viscosity profile used by Beaumont et al. (2004). The Shen et al. (2001) profile is roughly coincident with the extrapolation of the experimental rheological data because the parameters are chosen to give the same surface relative displacement velocity of 50 mm a21. In all cases the lower bounding refractory layer is taken to begin at a depth of 35 km.
LOW-VISCOSITY FLOW BENEATH TIBET
obtain a flow law (Mecklenburgh & Rutter 2003; Rutter et al. 2006) in which the melt fraction was not determined by temperature, we mixed synthesized per-alkaline albite-quartz glass of 2.5 wt% water content in various proportions (f ¼ 0.1, 0.2, 0.25 and 0.3 volume fraction) with crystalline quartz of 50 mm mean grain size and deformed it in axisymmetric compression at 900 and 1000 8C and 300 MPa confining pressure. The flow law obtained was 1˙ = 10
−1.39
(s1 − s3 )
× exp (192f3 ).
1.8
−220, 000 exp RT (4)
Use of a single solid phase prevented melt fraction increasing with temperature through reactions between solid phases, and no significant reaction between quartz grains and melt occurred. Microstructural studies together with the small value of the stress exponent suggested that flow rate was governed by a combination of diffusive mass transfer coupled with grain-boundary sliding that involved periodic rupture and sliding of the sintered contacts between solid grains. The parameter that most sensitively affects flow rate is the melt fraction. There is some indication from the experimental data that the stress exponent (here 1.8) may tend to decrease as strain rate (or flow stress) decreases. In this respect, equation (4) predicts strain rates that are likely to be upper bounds. An important parameter missing from this formulation is effective grain size. The flow law (equation (4)) above was determined for only one grain size: 50 mm. It is technically difficult to work with a wide range of solid-phase grain sizes in experiments on synthetic partially molten granitoids. At finer grain sizes the lower matrix permeability inhibits flow of the relatively viscous melt, whereas larger grain sizes are prone to impingement fracturing during the initial hot isostaticpressing stage. In experiments in which much lower viscosity basalt melt was used in a matrix of crystalline olivine (e.g. Scott & Kohlstedt 2006) it was demonstrated that flow rate decreased with increasing grain size, as is found for solid-state grain-size-sensitive flow in fine-grained rocks. When diffusive mass transfer is involved in the flow process strain rate is expected to vary as d 2m, where d is grain diameter. The parameter m is not very well constrained but is expected to lie between 2 and 3, according to the relative importance of dislocation creep in the solid phases, grainboundary sliding and diffusion creep. We therefore used this factor with a conservative m ¼ 2 to scale the predictions of the flow law (equation (4)) to a
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larger grain size of 0.5 mm, more likely to be representative of lower crustal partially molten rocks. Choosing a larger grain size or, perhaps more importantly, intergranular contact diameter increases the shear stress required to maintain the overall displacement velocity unchanged. Average intergranular contact size may increase with dwell time between grain sliding events. After casting the flow laws in terms of horizontal shear stress and shear strain rate, vertical profiles of shear displacement rate and viscosity were calculated assuming a constant horizontal shear stress. Due to the non-linear rheologies of the components involved, viscosity differences are best appreciated when the horizontal component of shear stress is held constant (Fig. 2). This leaves the only unknown to be the horizontal shear stress and we find that a shear stress of 0.1 MPa yields a surface displacement rate of 50 mm a21.
Discussion and conclusions A model based on experimental rock mechanics data, and consistent with the observed surface displacement rate from GPS measurements of c. 50 mm a21 (Chen et al. 2000; Gan et al. 2007) and experimentally determined melting relations, gives rise to an average shear stress of 0.1 MPa in the partially molten layer. This shear stress leads to a corresponding variation of effective viscosity from 1021 to 1015 Pa s across the partially molten layer thickness at this shear stress (Fig. 2). The surface displacement rate is dominated by the contribution from the partially molten layer. We have no other way to independently estimate the shear stress in the partially molten layer. The shear stress can arise from the far-field plate tectonic stresses plus the tendency for the topographic height of the Himalayas to drive gravity spreading towards the north. The solid-state flow in all of the rock column above the weak layer accounts for less than 1% of the total shear displacement rate because we have arbitrarily assumed the horizontal shear stress to remain constant, but it could become much larger in the strong middle and upper crust if large stress differences were allowed to arise from horizontally directed forces applied at the margins of the plateau. Such larger stress differences would be required to account for upper-crustal seismicity, for example. Beneath the weak layer, at depths greater than 35 km, it is assumed that the mechanical behaviour reverts to solid-state flow in a more refractory pyroxene + feldspar rock type with viscosity increasing markedly to c. 1028 Pa s immediately beneath the interface.
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Figure 2 also shows the exponential variation of viscosity with depth below 30 km assumed by Shen et al. (2001) to model the behaviour of a weak layer in the Tibetan mid-crust. In this case, parameters were chosen to result in the same surface displacement rate and total thickness of the weak zone as expected from melting constraints. Given the insignificant contribution of intracrystalline plastic flow of the middle and upper crust at the same shear stress as in the weak layer, it was inevitable that the same surface displacement rate would require a similar variation of viscosity over the melting interval, but in this case the viscosities could not be shown specifically to be consistent with laboratory-derived rheological data. Studies aimed at modelling channel flow and lower crustal extrusion during the early formation of the Himalaya–Tibetan orogenic system (Beaumont et al. 2001, 2004, 2006) used the wet quartz flow law of Gleason & Tullis (1995) for the upper crust and then assume a partially molten layer which weakens linearly from the viscosity of Gleason & Tullis (1995) at 700 8C to 1019 Pa s at a temperature of 750 8C. Like Shen et al. (2001), these models do not use an experimentally derived flow law for the rheology of the partially molten layer. In Figure 2 we show what the strength profile used in one of the models of Beaumont et al. (2004) would look like in our model. The extrapolation of laboratory-derived experimental data for the flow of a synthetic partially molten ‘granitoid’ supports the inference of the existence of a highly localized weak layer from the geophysical data reviewed above, and that it may be caused by partial melting of crustal rocks. In the same way that caution is required in the extrapolation of laboratory data to construct mechanical models for the solid-state plastic flow of an assumed rheologically stratified crust, it is important to bear in mind the limitations of the above model. First, we are making a generalization about a single weak layer at a constant depth interval beneath the whole of Tibet, and this is no doubt a gross oversimplification. On the other hand, at present it is beyond the scope of this article to attempt to refine the description of the crustal structure. We recognize that a weak zone does not have to be at the same depth everywhere and that it may vary in thickness, petrological and strength characteristics from place to place (Klemperer 2006). Second, practical experimental considerations forced us to employ a simplified synthetic partially molten aggregate of granular quartz plus quartzofeldspathic melt in order to control variables sufficiently to obtain a constitutive flow law. So far, these experiments have been performed only for one grain size for the solid phase; to extrapolate behaviour to coarser grain sizes we have had to appeal to
the nature of grain-size sensitivity found in the flow of olivine + basalt melt experiments. The form of the grain-size sensitivity and uncertainty over possible variation in the most applicable stress exponent means that estimated shear stress and hence viscosity might range about one order of magnitude above or below the values shown in Figure 2. Segregation of the melt-creating, melt-enriched and melt-depleted zones could have a profound affect on the rheology of the individual zones, but may not significantly alter the bulk rheology of the partially molten layer. The key point for present purposes is that a weak zone produced by partial melting of crustal rocks beneath the Tibetan Plateau can produce flow characteristics that can be consistent with geophysical constraints and with the cautious extrapolation of the results of laboratory rock mechanics experiments. This work was supported by UK NERC grant GR3/13062. We are grateful for critical comments from M. Searle. R. Law and J. Renner are thanked for their constructive reviews which helped improve this contribution. The data for Figure 1 comes from the US Department of Commerce, National Oceanic and Atmospheric Administration, National Geophysical Data Center, 2006 2-min Gridded Global Relief Data (ETOPO2v2). D. Irving is thanked for his help in preparing Figure 1.
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Index Page numbers in italic denote Figures. Page numbers in bold denote Tables. Acebuches metabasites 80, 81, 82 CPO 84– 85, 86–87, 88, 90– 93 dynamic recrystallization 91 foliation orientation 80, 82 grain-boundary migration 82, 83, 91–92 mechanical twinning 82, 83, 91 micro-kinking 82, 83, 91 misorientation 84, 85, 88, 91 petrofabric 82– 84 plagioclase deformation 88– 93 seismic properties 84, 85, 90 slip-system transitions 88– 90 aggregate crystal physical properties 175–191 polyphase, anisotropic properties 186 –190 Almonaster cross-section 80, 81 Alpine Fault Zone, New Zealand, mylonites 33–46 amphibole CPO, and deformation 18–20 foliation 26 seismic properties 8, 9, 22, 28, 29 P-wave anisotropy 26 P-wave velocity 26, 28 shear-wave velocity 27, 28 VTI 26, 27 amphibolite Acebuches metabasites 80, 82, 83 CPO 84, 84–85, 86– 87 seismic properties 90 ‘banded’ middle-to-lower crust analogue 11 seismic properties 22– 24 seismic anisotropy, antigorite 97 seismic properties 90 anhydrite CPO 239, 242, 244 –245, 248, 252 crystallography 238–240 deformation mechanisms 251, 252 EBSD 240, 242, 243–245, 248 grain-boundary pinning 251 grain-boundary sliding 251 grain-shape 242, 243– 244 grain-size 242, 243– 244 and CPO 250– 251 intra-grain distortion 249, 250 microstructure 241, 242, 243– 245, 251 misorientation 244– 245, 246– 247, 250 rheology 239– 240 slip systems 238– 240, 250 tilt boundary model 250 torsion experiments 239 Triassic Evaporites 237–238 anisotropy, polyphase aggregates 186– 190 anisotropy, seismic augen orthogneiss, Nanga Parbat Massif 57–58, 59 continental crust 9– 10, 16, 22 ‘banded’ amphibolite 23– 24, 26 mica 26 mylonites, Alpine Fault Zone 40–42, 43
olivine, CPO 113, 123–125 pyroxene granulite 24–25, 26 western USA 27 antigorite 97 serpentine mantle CPO 114, 121, 123–125 microstructure 118, 119 SPO 119, 120 serpentinite mylonites 98–110 CPO 99, 100, 103– 106 Zermatt-Saas(U)HP metamorphic zone serpentinite 132, 134–136, 140, 141, 142, 144 aspect ratio measurement 307– 308, 315 augite pyroxene, seismic properties 8 azimuth Acebuches metabasites 85 and seismic properties 16– 17, 22, 25 Balsam Gap Dunite 212 torsion experiments 212– 221 bassanite 250 biotite mica Alpine Fault Zone mylonites 37, 41, 42 Nanga Parbat Massif 57 seismic properties 8, 9, 10, 21, 22, 27 bulge nucleation see grain-boundary migration Burgers vector 154, 155, 156, 167 C-plane mica 38– 40, 42, 43 calcite, seismic properties 8 Carrara marble, deformation experiments 201 Casey, Martin (1948-2008), life and work vii-x, xi cataclasis 257 Cauchy stress 277 chlorite 132, 133, 134, 136, 140 Christoffel tensor 185 –186 chrome spinel 118, 119 chrysotile 97, 118, 119 cleavage, crenulation 140, 142, 289, 291, 311 clinopyroxene 11, 132, 136, 137, 138, 142 Coble creep 144 compressibility linear 185, 186 scalar volume 184 continental crust ductile deformation 7, 9 low viscosity flow, Tibet, rock mechanics 329 –334 lower, P-wave velocity 9– 10 middle, P-wave velocity 9–10, 20– 22 Nanga Parbat Massif 49– 75 P-wave anisotropy 22 seismic properties 7 –30 modelling 20–25 Couette flow 331 creep see diffusion creep; dislocation creep; dissolution precipitation creep creep deformation, and grain size 200– 201 creep flow laws 258
338 critical resolved shear stress, Torridon quartz mylonite 158, 159, 161– 165, 167– 168 crystal orientation, mathematics 178–179 crystal reference frame, tensors 177– 178, 179– 181 crystallographic preferred orientation (CPO) 1, 10, 199– 200 Acebuches metabasites 84– 85, 86–87, 88, 90– 93 anhydrite 239, 242, 244– 245, 248, 252 grain-boundary sliding 251 and anisotropy 1–3 antigorite, serpentinite mylonites, CPO 99, 100, 103–106, 110 ‘banded’ amphibolite 23 and deformation 17–20 and dislocation creep 267 mantle olivine 121– 125, 215– 216, 218, 220–221 mica 9 mylonites, Alpine Fault Zone 33– 46 Nanga Parbat Massif 51, 55–75 olivine, seismic anisotropy 113 quartz mylonite 151–171 evolution 159–166 and seismic properties 25 serpentinite 132, 133, 139– 140, 142– 144 shear zones 202– 203 VPSC 156, 158– 166, 167– 168, 170, 199–200 crystals, single aggregate 175– 191 anisotropic properties 181– 186 geometry, and tensor mathematics 176– 181 ODF 175, 187, 188– 190 orientation 178– 179 deformation D2 Higashi-Akaishi peridotite 115–116 Sardinian Variscan shear zones 310 –316 Southern Iberian Shear Zone 80 ductile, continental crust 7, 9 grain growth 257–270 constant temperature and pressure 259–260 isostrain-rate 259, 262–263, 264 isostress 258 –259, 261 time-dependent pressure change 260, 263–265 Torridon quartz mylonite 158– 159, 169– 170 deformation determination, Nanga Parbat Massif orthogneiss 66– 69, 70 deformation mechanisms 3 anhydrite 251, 252 plagioclase 79 Acebuches metabasites 88– 93 values of a 158, 267–268, 270 deformation partitioning, Nanga Parbat Massif orthogneiss 61, 62 deformation-reaction systems, effect on microstructure 273– 296 differentiation, metamorphic 289– 291 diffusion creep anhydrite 239, 240, 251 and grain growth 257–270 crust 268 mantle upwelling 260, 263–265, 266–267, 269 grain-boundary 263 isostrain-rate deformation 259, 262–263, 264
INDEX isostress deformation 258–259, 261 modelling, fibrous pressure shadows 319 –328 diopside 132– 133, 134– 136, 138, 139, 140, 142 –143 dislocation, quartz mylonite 154–156, 157, 166– 167 dislocation climb, Torridon quartz mylonite 167 dislocation creep 91, 144– 145, 231, 258–259, 261, 265, 267 –268, 270 anhydrite 239, 240, 251 and CPO 267 dislocation density 227–234 dislocation glide, Torridon quartz mylonite 166– 167 dissolution precipitation creep 129, 130, 143–145 dolomite, in anhydrite 242, 243, 251 dunite Higashi-Akaishi peridotite 115, 116 microstructure 116– 119 olivine CPO 121– 125 torsion experiments 211–221 CPO 215– 216, 218 dynamic recrystallization 218, 219, 220 microstructure 215 dykes, mafic CPO 21 rock-recipe modelling 11– 13, 21 seismic properties 14, 15 sheared 11 elastic properties 40, 184– 186 antigorite 97–98, 104, 106 electron backscatter diffraction (EBSD) anhydrite 240, 242, 243 –245, 248 clinopyroxene porphyroclasts 137 dunite CPO 214, 219 fine-grained polycrystalline olivine 227, 228 mica, Alpine Fault Zone 34, 37, 40–46 orthogneiss, Nanga Parbat Massif 55–75 plagioclase, Acebuches metabasites 82– 84 Torridon Shear Zone 152, 153 entropy 273, 275, 276, 278, 280– 281 equilibrium states 278 Euler angle reference frame 179–181 European Variscan Chain 80, 81 evaporites, Triassic 237– 238 exhumation Nanga Parbat Massif 51, 53, 55 (U)HP metamorphic rocks 129– 130 fabric-recipe modelling 13 feldspar CPO, Alpine Fault mylonites 41, 43 seismic properties 8, 9 Flinn diagrams 17– 18, 19 flow, mid-crustal low-viscosity, Tibet, rock mechanics 329 –334 flow confinement effect 304, 311–312 focus stable 280, 281, 286 unstable 280, 281 folding, Zermatt-Saas(U)HP metamorphic zone serpentinites 140, 142, 144 foliation 273 Torridon quartz mylonite 152, 154 Zermatt-Saas serpentinites 133, 140
INDEX foliation orientation Acebuches metabasites 80, 82 Alpine Fault mylonites 37 amphibole 26 Nanga Parbat Massif orthogneiss 59– 61, 73 and seismic properties 13–16, 21, 22 serpentinite mylonite 104 forsterite 132– 133, 134– 137, 138– 142, 144 forsterite olivine, seismic properties 8 Fourier coefficients 189, 190– 191 Fry plots 243, 244 Gallura, Sardinia Variscan shear zones 310– 316 vorticity analysis 315 garnet, seismic properties 8, 11 Gaussian curvature 276, 292, 296 GhoshFlow vorticity technique 303 –304, 312 ghost-grains irregular pyrite model 325, 326 serrated pyrite model 324, 325 small pyrite model 321, 322 Gibbs energy 275, 276 gneiss see orthogneiss grain distortion, anhydrite 249, 250 grain growth 257– 270 constant temperature and pressure 259– 260 law 258 pressure-dependent, mantle upwelling 260, 263–265, 266–267 grain-boundary bulging 258 grain-boundary diffusion creep 263 grain-boundary migration 197– 198 Acebuches metabasites 82, 83, 91–92 mafic schist 82, 83, 91–92 Torridon quartz mylonite 167 grain-boundary pinning, anhydrite 251 grain-boundary sliding, anhydrite 251 grain-shape, anhydrite 242, 243 grain-size anhydrite 242, 243 and CPO 250– 251 evolution 194– 199, 200– 202 reduction 257 granulite, pyroxene see pyroxene granulite gypsum 250 Happo ultra-mafic complex, serpentinite mylonite 98 Helmholz energy 275, 277 Hida Marginal Tectonic Belt 98 Higashi-Akaishi peridotite 114–115 deformation 115– 116 P-T evolution 115–116 Hill average 188–190 see also Voigt-Reuss-Hill average Himalayan-Tibetan Plateau, crustal structure 330 Himalayas, Nanga Parbat Massif 51–75 Hopf bifurcation 280, 281, 282, 287, 288, 292 hornblende Acebuches metabasites, CPO 85 rock-recipe modelling 11– 13 hornblende amphibole, seismic properties 8, 9, 27 hypersthene pyroxene, seismic properties 11, 27, 28
339
inclination, angular, vorticity analysis 308– 309, 315 intra-grain distortion, anhydrite 249, 250 isolation factor, vorticity analysis 304, 312 isostrain-rate deformation 259, 262–263, 264, 270 isostress deformation 258 –259, 261 isotropy see vertical transverse isotropy Jacobian matrix 276, 280, 282 Jeffery model 301, 312–313 boundary condition test 305–306 preconditions 304– 306, 310, 311–312 kinematic indicators, Alpine Fault mylonites 37–40, 42–43 kink bands Zermatt-Saas(U)HP metamorphic zone serpentinites 140, 144 Kohistan Arc Terrane 50, 51, 52, 53 lattice preferred orientation see crystallographic preferred orientation (CPO) layering see differentiation, metamorphic Liachar Shear Zone 52, 53 augen orthogneiss 54 Liachar Thrust 52, 53 lineation, mineral 273 lizardite 97, 118, 119 low-velocity zone, Nanga Parbat Massif 71 Main Mantle Thrust 50, 51, 52 mantle, olivine, CPO, seismic anisotropy 113 –125 mantle upwelling, pressure change, grain growth 260, 263–265, 266–267 mathematics, tensor 176–181 melting, partial, Tibetan crust 332 –334 metabasites, Acebuches see Acebuches metabasites metamorphism reactions and deformation 273–296 compositional zoning 289, 291–292 differentiation 289–291 UHP 129– 145 mica CPO 9 Alpine Fault mylonites 33– 46 and deformation 18–20 Nanga Parbat Massif 43, 59 seismic properties 8, 9, 21, 22, 27, 28 P-wave anisotropy 26, 40– 41, 59 P-wave velocity 25–26, 28 VTI 27 micro-kinking, plagioclase, Acebuches metabasite 82, 83, 91 microscopy see transmission electron microscopy; transmitted light microscopy microstructure 3 –4 anhydrite 241, 242, 243–245, 251 and deformation-reaction systems 273–296 evolution 1 fine-grained polycrystalline olivine 227 Higashi-Akaishi peridotite, dunite 116–119 mantle olivine 117–119, 215 and rheology, shear zones 193–194, 200–205 serpentinite mylonite 99– 100
340 microstructure (Continued) Torridon quartz mylonite 152, 154, 155– 156 Zermatt-Saas(U)HP metamorphic zone serpentinite 132–140 minerals lineation 273 reactions, and deformation 273– 296 seismic properties 8, 9, 12, 13 stationary states 278 misorientation Acebuches metabasites 84, 85, 88, 91 anhydrite 244– 245, 246– 247, 250 fine-grained polycrystalline olivine 229, 231, 233 porphyroclasts 307 MTEX 175–176 crystal orientation 179 crystal reference frame 178, 180, 181 elastic properties 185– 186 tensors 177, 182–183, 184, 189 visualization 186 Mulchrone curves 305 –306, 312 –313 muscovite CPO, Alpine Fault Zone 37, 39, 41, 42 Nanga Parbat Massif 57 muscovite mica, seismic properties 8, 10, 21 muscovite-quartz reaction 285– 286 mylonite Alpine Fault Zone, New Zealand 33–46 mica CPO fabrics 37–40 anisotropy modelling 40–42 quartz, Torridon CPO 151–171 evolution 159– 166 CRSS 158, 159, 161– 165, 167– 168 deformation 158 –159, 169 –170 dislocation 154– 156, 157, 166–167 foliation 152, 154 microstructure 152, 154, 155–156 simple-shear model 159, 160– 165, 169– 170 slip systems 158–166, 167–168, 169–170 TEM 154–156, 157, 159, 166–167, 170 VPSC 156, 158– 166, 167– 169, 170 serpentinite antigorite-bearing 98–110 microstructure 99– 100 seismic anisotropy 103 seismic velocity 101 –103, 106 –111 Nanga Parbat Massif 50, 51– 53 augen orthogneiss 53–55 CPO 55–75 deformation partitioning 61, 62 foliation orientation 59–61, 73 seismic properties 57–58 seismic reflection coefficients 61– 62, 64 and controlled source surveys 64– 66 deformation determination 66–69, 70 seismic wave propagation 61, 63 exhumation 51, 53, 55 low-velocity zone 71 mineralogy 58–59 rock-recipe modelling 58–59 seismology 69– 71, 72 shear-wave splitting 59–61, 69–71, 73
INDEX tectonics 71, 73 pure-shear v. simple-shear model 73–75 tectonites, mica 43 neocrystallization, plagioclase 258, 268 node stable 280, 281, 286 unstable 280, 281 non-equilibrium states 275, 278– 280 nucleation see grain-boundary migration; subgrain rotation nucleation Nurra, Sardinia Variscan shear zones 310– 316 vorticity analysis 314 olivine CPO 121–125, 215–216, 218, 220– 221 dynamic recrystallization 218, 219, 220 microstructure 117– 119, 215 seismic anisotropy 113, 123– 124 shape preferred orientation 119, 120 torsion experiments 211 –221 diffusion creep 269 fine-grained polycrystalline EBSC 227, 228 grain size 227, 228 microstructure 227 pole figure analysis 227, 229 slip systems 225– 234 olivine porphyroclasts, serpentinite mylonites, CPO 99– 100, 105– 106, 110 orientation density function 175, 187, 188– 190 orthoclase feldspar, seismic properties 8 orthogneiss augen, Nanga Parbat Massif 53–55 felsic, middle crust analogue 11, 21, 22 orthopyroxene, seismic properties 11 Ossa-Morena zone 80 P-wave anisotropy Alpine Fault mylonites 40– 42 continental crust 22 determination 10 P-wave velocity Acebuches metabasites 85 continental crust 9– 10, 14–16, 20– 22, 24, 25–27 determination 10 Nanga Parbat Massif 57– 58 pyroxene granulite 24– 25, 26, 28 serpentinite mylonites 101–103, 107–111 palaeopiezometers 3 Passchier curves 305 –307, 312–313 Passchier vorticity technique 302, 303 peridotite see Higashi-Akaishi peridotite plagioclase Acebuches metabasites 80 CPO 84– 85, 86– 87 deformation mechanisms 88–93 dynamic recrystallization 91 EBSD 82–84 mechanical twinning 82, 83, 91 micro-kinking 82, 83, 91 misorientation 85, 88, 91 petrofabric 82– 84 slip-system transitions 88– 90
INDEX deformation 79 neocrystallization 258, 268 rock-recipe modelling 11– 13 plagioclase feldspar, seismic properties 8, 21–22 plasticity, crystal 130 Acebuches metabasites 88 antigorite 104 plume, mantle see mantle upwelling Poincare´-Bendixson theorem 281, 288 Poiseuille flow 331 Poisson ratio 185 polarization, shear-wave splitting 26 Nanga Parbat Massif 57–58 pyroxene granulite 25 pole figure analysis fine-grained polycrystalline olivine 227, 229 Torridon Shear Zone 153, 154 pore fluid pressure, anhydrite 251– 252 porphyroclast systems vorticity analysis 301–316 angular inclination measurement 308– 309, 315 aspect ratio measurement 307–308 dilatancy effect 309, 311 Sardinian Variscan Shear zones 311– 316 systematic errors 306 –310, 311–316 third-dimension extrapolation 309 vorticity plane estimation 307–309 pressure shadows, fibrous diffusion creep modelling 319– 328 irregular pyrite model 320, 325 serrated pyrite model 320, 324 –325, 327 small pyrite model 320, 321–324, 326 protomylonites, CPO, Alpine Fault Zone 34, 41 Puerto de Veredas cross-section 80, 81 pure-shear model, Nanga Parbat Massif 73– 75 pyrite, diffusion creep modelling irregular model 320, 325 serrated model 320, 324–325, 327 small model 320, 321– 324, 326 pyroxene granulite lower crust analogue 11 seismic properties 21–22, 23 P-wave anisotropy 26, 27 P-wave velocity 26, 27, 28 shear-wave velocity 27, 28 quartz CPO, and deformation 17–18 flow laws 331– 334 mylonite CPO 151– 152 see also mylonite, quartz, Torridon rock-recipe modelling 11– 13 seismic properties 8, 9 anisotropy, Alpine Fault mylonites 41 see also muscovite-quartz reaction Raikhot Fault 53 reaction-diffusion equations 281– 286 reaction-diffusion-deformation equations 278, 286–288 reactions coupled 274–275 behaviour types 280, 281 muscovite-quartz 285–286
341
non-equilibrium stationary states 278– 280 null-cline 278, 279, 280, 286, 287 redox change 283– 285 cyclic 274–275 and deformation, effect on microstructure 273– 296 stability 278– 281 recrystallization, dynamic 197– 198 dunite torsion experiments 218, 219, 220 plagioclase, Acebuches metabasites 91, 92 redox change, reactions 283– 285 Reuss average 98, 104, 105, 106, 107, 108, 109, 110, 188–190 rheology and microstructure, shear zones 193– 194, 200– 205 subduction zones 130 rock mechanics, mid-crustal low-viscosity flow, Tibet 329–334 rock-recipe modelling 11–13, 21 Nanga Parbat Massif 58– 59 S-C-C0 mylonite 38–40, 41, 42, 43 S-plane mica 38– 40, 42, 43 Sachs model 200, 225–226 saddle point 280, 281 San Carlos olivine 226–234 Sardinia D2 shear zones 311–316 Variscan shear zones 310–311 scaling relationships 198– 199 schist mafic, Acebuches metabasites 80 CPO 84, 85, 86–87 grain-boundary sliding 82, 83, 91– 92 quartzofeldspathic, Alpine Fault Zone 34–46 schistosity, serpentinites 132, 133– 136, 140 Schmid factor, olivine slip systems 225, 229, 230, 232, 233 seismic modelling 13– 14 Nanga Parbat Massif 58– 69 seismic properties Acebuches metabasites 84, 85 continental crust 7 –30 foliation orientation 13–16, 21 modelling 20–25 western USA 27– 29 determination 10–11 mantle olivine, CPO 113–125 Nanga Parbat Massif orthogneiss 57–58 seismic reflection coefficients Nanga Parbat Massif orthogneiss 61–62, 64 controlled source surveys 64–66 deformation determination 66–69, 70 seismic stratigraphy modelling 22–25 seismic wave propagation, Nanga Parbat Massif orthogneiss 61, 63, 68 serpentinite mylonite 97–111 see also, mylonite, serpentinite viscosity 130 Zermatt-Saas(U)HP metamorphic zone 130 –145 CPO 132–133, 139, 142– 144 microstructure 132–140 SPO 132 –133, 139, 142– 144
342
INDEX
shape preferred orientation (SPO) mantle olivine 119, 120 porphyroclasts 301, 302 –304, 305–306, 312–313, 315 Zermatt-Saas(U)HP metamorphic zone serpentinite antigorite 132, 133, 139, 142–144 shear modulus 185 shear stress see critical resolved shear stress shear zones D2 Sardinia 311–316 dilatancy 311 error estimation and vorticity analysis 314–315 geometry and kinematics 311 preconditioning 311–314 flow, porphyroclasts 301 microstructure and rheology 193–205 CPO 202–203 grain size 200–202 mechanical evolution 203–204 shear-wave anisotropy 22 shear-wave splitting 13, 14–16 anisotropy, mylonite 40– 42 Nanga Parbat Massif orthogneiss 59– 61, 69–71, 73 polarization 26, 40–42 shear-wave velocity 16, 27, 28 amphibole 27 determination 10 serpentinite mylonites 101– 103, 107– 110 simple-shear model Nanga Parbat Massif 73–75 Torridon quartz mylonite 159, 160–165, 169–170 slip systems anhydrite 238– 240, 250 fine-grained polycrystalline olivine 225–234 effect of rheology 229, 231 Torridon quartz mylonite 158– 166, 167– 170 transitions, Acebuches metabasites 88– 90 slipping effect, porphyroclast systems 304– 305, 312 sol-gel olivine 226–234 South-Portuguese zone 80 Southern Alps see Alpine Fault Zone, New Zealand Southern Iberian Shear Zone, Acebuches metabasite deformation 80, 81, 82, 84–85, 89, 91– 92 spinel-olivine phase change 269 stationary states 278 strain shadows, prismatic 132, 133, 136–138, 142 subduction channel model 129 subduction zones, rheology 130 subgrain rotation nucleation 197– 198, 258 surfaces, minimal 292–296 talc, elastic properties 186, 187 Tato Ridge, Liachar Shear Zone 52, 53 Taylor model 200, 225 –226 Taylor-Bishop-Hill model 158, 159, 226 Taylor-Quinney coefficient 277, 278 Tectonics, Nanga Parbat Massif 71, 73 pure-shear v. simple-shear model 73–75 ‘tectonic aneurism’ model 71, 73 teleseismic receiver functions 9 tensors 176–177 average 187– 190 Christoffel 185–186
crystal reference frame 177– 178 elasticity 183 –184 fourth-rank 183 mathematics, and crystal geometry 176–181 second-rank 181–184 texture see crystallographic preferred orientation (CPO) thermodynamics 274, 275, 277– 278, 280– 281 Tibet, mid-crustal low-viscosity flow, rock mechanics 329 –334 tilt boundary model, anhydrite 250 titanoclinohumite 132, 135, 140, 142 Torridon Shear Zone 152, 153, 154 deformation 169– 170 see also mylonite, quartz, Torridon transmission electron microscopy, Torridon quartz mylonite 154– 156, 157, 159, 166 –167, 170 transmitted light microscopy 241, 243 tremolite 106–107, 132–133, 136, 139, 140, 143, 144 Triassic Evaporites 237– 238 Turing instability 276, 281, 282, 286–289, 292, 294 twinning anhydrite 239 Carlsbad 91 Dauphine´, Torridon Shear Zone 154, 159, 170 mechanical, plagioclase 82, 83, 91 ultramylonites, CPO, Alpine Fault Zone 34, 41 Upper Loch Torridon, quartz mylonite 152–171 upwelling see mantle upwelling USA, western, seismic properties 27– 29 Val d’Aosta, serpentinite 131, 132 Valtournanche, serpentinite 132 Variscan chain, Sardinia D2 shear zones 311–316 geology 310– 311 geometry and kinematics 311 velocity, seismic see P-wave velocity; shear-wave velocity Veredas cross-section 80, 81 vertical transverse isotropy 18, 20, 25, 26, 27 viscoplastic self-consistent modelling (VPSC) 156, 158 –166, 167–169, 170, 199– 200 viscosity, serpentinite 130 visualization, MTEX 186 Voigt average 98, 104, 105, 106 –110, 187 –190 Voigt-Reuss-Hill average 98, 104, 105, 106–110 vorticity analysis, porphyroclast systems 301–316 angular inclination measurement 308–309, 315 aspect ratio measurement 307 –308 dilatancy effect 309, 311 Sardinian Variscan Shear zones 311–316 systematic errors 306– 310, 311– 316 techniques 302–304 third dimension extrapolation 309 vorticity plane estimation 307– 309 Wallis vorticity technique 302– 303, 312, 313 wave velocity, elastic 185– 186 Young’s modulus 185, 186 Zermatt-Saas(U)HP metamorphic zone 130 –132 serpentinite 130 –145 zoning, compositional 289, 291 –292
This collection of papers presents recent advances in the study of deformation mechanisms and rheology and their applications to tectonics. Many of the contributions exploit new petrofabric techniques, particularly electron backscatter diffraction, to help understand evolution of rock microstructure and mechanical properties. Papers in the first section (lattice preferred orientations and anisotropy) show a growing emphasis on the determination of elastic properties from petrofabrics, from which acoustic properties can be computed for comparison with in-situ seismic measurements. Such research will underpin geodynamic interpretation of large-scale active tectonics. Contributions in the second section (microstructures, mechanisms and rheology) study the relations between microstructural evolution during deformation and mechanical properties. Many of the papers explore how different mechanisms compete and interact to control the evolution of grain size and petrofabrics. Contributors make use of combinations of laboratory experiments, field studies and computational methods, and several relate microstructural and mechanical evolution to large-scale tectonic processes observed in nature.
