Advances in Applied Mechanics Volume 38
Editorial Board Y. C. FUNG DEPARTMENT OF BIOENGINEERING UNIVERSITY OF CALIFORNIA, SAN DIEGO LA JOLLA, CALIFORNIA PAUL GERMAIN ACADEMIE DES SCIENCES PARIS, FRANCE C.-S. YIH (Editor, 1971-1982) JOHN H. HUTCHINSON (Editor, 1983-1997)
Contributors to Volume 38 PAOLO MARIA MARIANO CHANG-LIN TIEN PIN TONG JIAN-GANG WENG THEODORE YAOTSU WU TONG-YI ZHANG MINGHAO ZHAO
A D V A NC E S IN
APPLIED MECHANICS Edited by Erik van der Giessen
Theodore Y. Wu
DELFT UNIVERSITY OF T E C H N O L O G Y DELFT, THE N E T H E R L A N D S
DIVISION OF E N G I N E E R I N G AND APPLIED SCIENCE CALIFORNIA I N S T I T U T E OF T E C H N O L O G Y PASADENA, CALIFORNIA
VOLUME 38
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Contents CONTRIBUTORS PREFACE
vii ix
Multifield Theories in Mechanics of Solids Paolo Maria Mariano I. II. III. IV. V. VI. VII. VIII. IX.
Introduction Configurations and Balance of Interactions Elastic Materials with Substructure Balance in Presence of Discontinuity Surfaces Constitutive Restrictions Evolution of Defects and Interfaces in Materials with Substructure Crack Propagation in Materials with Substructure Latent Substructures Examples of Specific Cases Acknowledgments References
2 9 26 33 38 42 53 68 73 88 88
Molecular Dynamics Simulation of Nanoscale Interfacial Phenomena in Fluids Chang-Lin Tien and Jian-Gang Weng I. II. III. IV. V. VI. VII. VIII.
Introduction Molecular Dynamics Simulation Techniques Liquid-Vapor Interfaces Liquid-Liquid Interfaces Liquid-Solid Interfaces Three-Phase Systems Other Interfacial Phenomena Concluding Remarks Acknowledgments References
96 97 103 122 130 136 139 140 141 141
Contents
vi
Fracture of Piezoelectric Ceramics Tong-Yi Zhang, Minghao Zhao, and Pin Tong I. II. III. IV. V. VI. VII. VIII. IX. X.
Introduction Basic Equations Two-Dimensional Electroelastic Problems and Stroh's Formalism Piezoelectric Dislocation and Green's Function Conductive Cracks Interface Cracks Three-Dimensional Electroelastic Problems Nonlinear Approaches Experimental Observations and Failure Criteria Concluding Remarks Acknowledgments References
148 152 162 186 199 209 220 236 255 274 279 279
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion Theodore Yaotsu Wu I. II. III. IV. V. VI. VII. VIII. IX.
Introduction Subdivisions of Hydrodynamic Theories for Aquatic and Aerial Locomotion Resistive Theory of Aquatic Locomotion Classical Slender-Body Theory of Fish Locomotion A Unified Approach to Nonlinear Theory of Flexible Lifting-Surface Locomotion A Unified Nonlinear Theory of Two-Dimensional Flexible Lifting-Surface Locomotion On Experimental Differentiation between Thrust and Drag in Fish Locomotion Scale Effects in Energetics of Aquatic Locomotion Conclusion and Outlook Acknowledgments References
291 296 300 301
333 338 347 350 350
AUTHOR INDEX
355
SUBJECT INDEX
363
314 316
List of Contributors
Numbersin parenthesesindicatethe pageson whichthe authors' contributionsbegin. PAOLO MARIA MARIANO (1), Dipartimento di Ingegneria Strutturale e Geotecnica, Universith di Roma "La Sapienza," 00184 Rome, Italy CHANG-LINTIEN(95), Department of Mechanical Engineering, University of California, Berkeley, California 94720 PIN TONG (147), Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China JIAN-GANGWENG(95), Department of Mechanical Engineering, University of California, Berkeley, California 94720 THEODORE Y. WU (291), Division of Engineering and Applied Science,
California Institute of Technology, Pasadena, California 91125 TONG-YIZHANG(147), Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China MINGHAO ZHAO (147), Department of Mechanical Engineering, Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
vii
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Preface
The basic intent of Advances in Applied Mechanics, as laid down by founding editors Theodore von K~rm~n and Richard von Mises, is to serve as a forum for informative, expository, and up-to-date accounts of an area of mechanics, for experts and nonexperts alike, interested in their own as well as unrelated fields. Our first concern is that they may be read with interest, profit, stimulation, and perhaps even enjoyment, as put by our former editor, Chia-Shun Yih. This volume of the serial consists of four chapters addressing a variety of topics of basic interest and current activity. An important driving force for advances in solid mechanics is the need to incorporate better and more detailed descriptions of the material behavior. Because engineered materials in particular owe their properties to internal microstructure, much work has been and still is focused on coupling constitutive models with microstructural descriptions. Classically, this is done by introduction of appropriate internal variables, but this is not always sufficient. Therefore, more intricate continuum theories that couple the motion of the solid to other field variables have been developed in recent years. The article by Mariano gives a unified formulation of such multifield, and often nonlocal, theories. It gives a formal treatment of, for example, mechanical stress and configurational forces that drive microstructure evolution. Although written with a certain mathematical rigor, the article is both motivated by practical problems and illustrated by particular known descriptions. Recent advances in studies of nanoscale interfacial phenomena in fluids have opened up a new frontier of fundamental flow science and technology. The exceedingly short lengths and time scales involved present many challenges to experimental and numerical approaches; this is especially true for interfacial phenomena because the surface effects dominate due to the large surface-to-volume ratio in nanoscale configurations. The chapter by Tien and Weng provides an expository survey of the recent development of molecular dynamics (MD) simulations of these phenomena involving all sorts of interfaces, including those consisting of ix
x
Preface
different liquids, vapors, solids, and three-phase systems, in which the physical effects of surface tension, surfactants, diffusive transport, thermal boundaries, and sonoluminescence can all play a role. Through these cases, the success and further development of MD simulation are extensively discussed. Along another frontier, piezoelectric materials made in the form of polycrystalline ceramics are facing rapidly expanded applications to the manufacture of microdevices used in smart structures, microelectronics, and microelectromechanical (MEM) systems. Piezoelectric ceramics have remarkable properties in being chemically inert, immune to ambient conditions, and very quick in conversion between mechanical and electrical energy along mechanical and electrical axes that can be precisely oriented in relation to the arbitrary shape of the ceramics through manufacturing. The article by Zhang, Zhao, and Tong reviews the recent advances in our understanding of the mechanical and fracture properties of piezoelectric ceramics for their significance and importance to mechanical-electrical energy conversion, a process that intrinsically involves comprehensive constitutive and thermodynamic relations, as well as for fulfilling some practical applications to high-tech development. The multidisciplinary subject of aquatic and aerial animal locomotion has renewed strong interest in at least two aspects: achieving a nonlinear theory modeling the hydrodynamic mechanisms underlying the locomotion as observed, and further applications, with the control mechanics incorporated, to developing new robotics, possibly with a biomimetic approach. For both objectives, success in the first is paramount. The article by Wu surveys the advances in hydrodynamic theories modeling aquatic and aerial locomotion at low, intermediate, and high Reynolds numbers, performed by elongated animals and those using lifting surfaces of large aspect ratio for undulatory propulsion. In addition, a new nonlinear theory for evaluating propulsion by a two-dimensional flexible lifting surface moving along an arbitrary trajectory and with motions of arbitrary amplitude is presented. It is intended for general applications to lifting surfaces of large aspect ratio and further extension to three-dimensional configurations. With the publication of the present volume of this serial, it gives us great pleasure to extend our warmest welcome to Professor Hassan Aref of the University of Illinois at Urbana-Champaign, a very distinguished fluid dynamicist and a very highly regarded leader in our profession of Applied Mechanics. Aref will hereby take over T.Y.W.'s responsibility of the co-editorship of Advances in Applied Mechanics. TYW wishes to acknowledge with deep appreciation the benefits of working with John Hutchinson earlier and with Erik van der Giessen more recently for their rewarding editorial teamwork. Warm thanks are due from TYW to the
Preface
xi
members of the present Editorial Board for their valuable help and to the authors for contributing their stimulating articles to this serial during the years of his tenure as co-editor, and to Academic Press for its splendid cooperation. His years with the serial have naturally nourished in him an attachment to it, and seeing that it is now in excellent hands is to him a gratifying comfort. ERIK VAN DER GIESSEN AND THEODORE Y. W u
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A D V A N C E S IN A P P L I E D M E C H A N I C S , V O L U M E 38
Multifield Theories in Mechanics of Solids* PAOLO MARIA MARIANO Dipartimento di lngegneria Strutturale e Geotecnica,' Universitgt di Roma "La Sapienza," 00184 Rome, Italy
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Structure of This Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 2 5
II. Configurations and Balance of Interactions . . . . . . . . . . . . . . . . . . . A. Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Measures of Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Balance of Interactions from the Invariance of Outer Power . . . . E. Effects of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16 21
III. Elastic Materials with Substructure . . . . . . . . . . . . . . . . . . . . . . . . A. Variational Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . B. S o m e Properties of Lagrangian Densities . . . . . . . . . . . . . . . . . . C. Influence of the Substructure on the Decay of Elastic Energy . . .
26 26 27 30
IV. Balance in Presence of Discontinuity Surfaces . . . . . . . . . . . . . . . . A. Interfaces: G e o m e t r i c Characterization . . . . . . . . . . . . . . . . . . . B. Balance at Discontinuity Surfaces . . . . . . . . . . . . . . . . . . . . . . .
33 33 35
V. Constitutive Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Constitutive Restrictions in the Bulk . . . . . . . . . . . . . . . . . . . . . B. Constitutive Restrictions at Discontinuity Surfaces . . . . . . . . . . . VI. Evolution of Defects and Interfaces in Materials with S u b s t r u c t u r e . . A. Configurational Forces in the Bulk . . . . . . . . . . . . . . . . . . . . . . B. Configurational Forces on a Discontinuity Surface . . . . . . . . . . . VII. Crack Propagation in Materials with Substructure . . . . . . . . . . . . . . A. Kinematics of Planar M o v i n g Cracks . . . . . . . . . . . . . . . . . . . . B. Balance of Standard and Substructural Interactions at the T i p . . . C. Effects of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Tip Balance of Configurational Forces . . . . . . . . . . . . . . . . . . . . E. C o n s e q u e n c e s of the Mechanical Dissipation Inequality . . . . . . . E Driving Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. A Modified Expression of J Integral . . . . . . . . . . . . . . . . . . . . . H. Energy Dissipated in the Process Zone . . . . . . . . . . . . . . . . . . .
9 9
11 12
38 39 40 42 43 46 53 54 56 57 59 61 63 65 66
*To M. M., for simple and, at the same time, complicated reasons.
ISBN 0-12-002038-6
ADVANCES IN APPLIED MECHANICS, VOL. 38 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISSN 0065-2165/01 $35.00
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VIII. Latent Substructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Second-Gradient Theories as Special Cases of Latent Substructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Examples of Specific Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Material with Voids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Two-Phase (or Multiphase) Materials . . . . . . . . . . . . . . . . . . . . C. Cosserat Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Micromorphic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Nematic Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E Ferroelectric Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Microcracked Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68 70 73 74 75 76 78 80 81 83
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
I. Introduction A. GENERAL INTRODUCTION The term multifield theories indicates the wide range of models in which some graphic fields must be introduced to describe the influence of material substructures on the gross mechanical behavior of solids. In a certain sense these fields (order parameters, also called phase fields, microstructural fields, microdisplacements, microdeformations, etc., in different special cases) are models of the material structure. Solidification of metal alloys and their possible shape memory, damage states and evolution, and the influence of long chains of macromolecules on the behavior of polymers are examples of physical phenomena of great interest in engineering practice that can be analyzed fruitfully with the help of multifield theories. Usually, in continuum mechanics only the placement within the Euclidean space is assigned to each material patch (volume element), then changes in relative placements are evaluated to measure the crowding and the shearing of material patches (i.e., the deformation). In this way, however, the features of material texture, or, more generally, substructure, are overlooked. The starting idea of multifield theories is to assign to each material patch P at least one pair (x, qo) in which x(P) is the placement of P within the Euclidean space and qo(P) (order parameter l) furnishes information on the substructural configuration of the patch. i Of course, the word order is only conventional, qo can also describe disordered arrangements of macromolecules or crystalline grains. The term orderparameter arises from statistical physics,
Multifield Theories in Mechanics of Solids
3
R e m a r k 1 Many choices of qo can be made; some of them are considered natural, whereas others are of convenience. They depend on the physical circumstances and must be specified each time. 9 The simplest choice is to consider the field qo to be scalar valued. For example, qo may represent the void volume fraction in porous solids or the volume fraction of a certain material phase in a two-phase material, as in the case of austenite-martensite mixtures. Such a choice of the order parameter is not unique for this physical situation; it may be not sufficient, for example, to describe in some detail the directional distribution of grains of austenite or martensite at each point, and tensor-valued order parameters must be introduced. A scalar-order parameter may be also used in the case of mixtures of two fluids or to describe solidification phenomena. 9 qO can be a vector. Liquid crystals are the classical example of bodies modeled by such a special choice of the order parameter. Other situations in which the vector choice is made are direct models of rods and shells. A shell can be represented roughly speaking by the middle surface and a field of unit vectors orthogonal to it in the reference configuration (a Cosserat surface). Analogously, triads of vectors may be used to represent the behavior of cross sections of rods or of stratified rocks, such as gneiss. Vectororder parameters are also useful tools in describing the mechanics of defective crystals (they can be identified with the optical axes of the crystal) or microcrack systems (in this case, they may represent vectors orthogonal to plane microcracks or the perturbation of the displacement field induced by the presence of such defects), ferroelectrics, and magnetostrictive solids. 9 If the material patch is characterized by large molecules undergoing homogeneous deformations, second-order tensor-valued order parameters may be used, a choice that also describes the dipole approximation of some distribution of directional data; for example, the distribution of microcracks. Moreover, such a second-order tensor can represent the Nye's tensor in dislocated continua. 9 Distributions of microcracks can be also described by their quadrupole approximation. A fourth-order tensor must be introduced with the role of order parameter.
in which Landau introduced order parameters to study second-order phase transitions consisting of abrupt changes of symmetry in solids.
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Many other examples can be made, and some are presented in Section IX in a more detailed way. The order parameters quoted in the previous remark are elements of some finite dimensional manifold, here indicated with AA and usually considered compact and without boundary..A4 can be also infinite dimensional, in which case the order parameter field assigns to each material patch an entire distribution function that can represent the geometry of the material substructure or can be the distribution (not necessarily canonical) of different levels of energy within the patch. In the following discussion, I refer the developments presented in this article to the case in which .A/[ is finite dimensional. The characteristic features of each special model depend on the choice of M , and hence on the mathematical properties of it. The physical meaning (or, for instance, interpretation) of these properties should be specified each time, although they are often considered to be convenient devices only. For example, the metric on M determines the quadratic part of the kinetic energy (if any) associated to the order parameter, and therefore to the substructure of the material. In the case of crystalline materials (e.g., such a prominent role of kinetic energy can be recognized only at very high frequencies; however, although the case is rare, such a kinetic energy should be considered in these regimes and determines the metric of M . Basically, the order parameter is considered to be an observable quantity. An external spatial observer must take two different measures to evaluate both the position of each material patch and information on its substructure. In this way, x and ~ together characterize the physical configuration of the solid. Interactions are associated with ~: They are substructural interactions and depend on the nature of the material substructure. These interactions develop explicit power in the rate of the order parameter and perhaps of its gradient, and must be balanced. Consequently, new balance equations arise in addition to those of Cauchy and represent the balance of some sort of generalized momentum and moment of momentum. The latter balance implies an expression of the skew part of Cauchy stress in terms of substructural measures of interaction. The representation of the substructural interactions is a delicate problem. When the gradient of ~o can be evaluated in a covariant way through a connection (hopefully with a physical significance) on A/l, these interactions can be represented by appropriate tensors called microstress and self-force. This terminology is conventional only and evokes special situations in which these tensors are really "perturbations" (in some sense) of the macroscopic stress tensor and also represent some kind of internal forces. In addition, the question of the connection (by which the covariant X7~ can be expressed) is delicate from a conceptual point of view. There are situations in which a physically significant connection can clearly be recognized (e.g., for
Multifield Theories in Mechanics of Solids
5
nematic liquid crystals); however, there are other situations in which this is not so. The choice of the connection, in fact, influences not only the explicit representation of the gradient of r but also the representation of the power. When many connections can be defined indifferently, it is necessary to require the invariance of the power with respect to their choice. Such a requirement allows one to obtain results analogous (in terms of the structure of the balance equations) to the results assured by the existence of a natural and physically significant connection. In general, however, the interactions are represented by general functionals. This situation also occurs when nonlocal effects due to material substructure are considered in both time (memory) and space. Nonlocality can be represented by integrals in time or space. When these functionals must be introduced, retardation (memory) or myopia (space) theorems help develop them (to within some material constant) in terms of differential operators of the fields involved. In this case, one accounts for weak nonlocalities only. Alternatively, one can assign to each material patch P only its placement x(P) in the three-dimensional Euclidean space and decide to introduce internal variables describing microstructural effects. These variables are nonobservable objects by definition; no balance equations are associated with them. The derivatives of the free energy with respect to the internal variables (and, possibly, with respect to their gradients) are in fact not genuine interactions, but are rather only affinities that must satisfy only the second law of thermodynamics, and need not be balanced because they do not develop explicit mechanical power. However, models with internal variables can be derived from multifield theories by appropriate internal constraints. In this case, the substructure becomes latent. Only a possible kinetics (evolution rule) is associated to it. From a mathematical point of view, the assignment of a kinetics to qp, without considering the balance of genuine interactions, is tantamount to take some initial value of qo--say, corresponding to some point qo* on M - - a n d to give a rule selecting elements of the tangent space of M at qa*. Finally, the order parameter field can be chosen to be a stochastic field taking values on the manifold M . In this way, not only the descriptor of the substructure is associated with each material patch, but also the probability that such substructure is really present.
B. STRUCTURE OF THIS ARTICLE The aim of this article is to show that the multifield description of continua is a flexible framework to study many physical situations in which the analysis of substructures is important for both practical and theoretical reasons. The analytical
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tools (and, of course, the difficulties that one tackles when using them) appear to be necessary to describe with certain detail the behavior of material substructures. The general theory is presented first, followed by special cases and applications. The description of configurations and the deduction of balance equations for bodies lacking in discontinuity surfaces are in Section II. Section III is dedicated to the special case in which the behavior of the substructure is elastic, in which case the force exerted on an inclusion in the body by the surrounding medium is deduced from an appropriate version of Noether's theorem accounting for the order parameter field. The influence of the substructure on the axial decay of energy in linear elastic cylinders is also discussed. Section IV deals with the derivation of thermomechanical balance at discontinuity surfaces (interfaces) that are endowed by their own measures of interactions. Surface stress, surface microstress, and self-force are defined on the interfaces. Constitutive restrictions arising from a mechanical version the second law of thermodynamics and involving the measures of substructural interactions are treated in Section V. Section VI is dedicated to the analysis of the influence of substructures on configurational forces that drive the evolution of interfaces. The kinetic equation for interfaces is deduced from the balance of configurational forces and is expressed in terms of a generalized expression of Eshelby tensor. In Section VII, special attention is given to the evaluation of the influence of material substructures on macrocrack propagation. In Section VIII, the case in which the substructure becomes latent in presence of appropriate internal constraints is discussed. Finally, Section IX deals with the application of the general theory to special cases.
BIBLIOGRAPHIC NOTE The "Th6orie des corps d6formables" of Cosserat and Cosserat (1909) is the first known historical example of a special case of multifield theories treated systematically, even though the germinal idea was formulated by Voigt (1887). As is well known, Cosserat and Cosserat's point of view consists of considering each material patch as a rigid body (possibly described by its peculiar triad of vectors) that can rotate independently of the neighboring patches. Couple stresses are associated with these additional degrees of freedom 2 and are balanced. In 1958, Ericksen and Truesdell gave new insight to Cosserat and Cosserat's theory. Following the Cosserats, they consider such a theory to be a suitable tool to describe the 2They are considered "additional" degrees of freedom because, in the classic case of Cauchy materials, each point has only three degrees of freedom.
Multifield Theories in Mechanics of Solids
7
mechanics of rods and shells, which they represent as lines and surfaces, respectively, endowed at each point by triads of mutually perpendicular vectors (the triads describe the behavior of sections). Ericksen and Truesdell's (1958) seminal paper constitutes a generalization of the Cosserats' (1909) ideas because they consider such vectors (the order parameters) to be stretchable (for other contributions to the general theory of Cosserat and Cosserats' materials, see also Mindlin, 1965a,b; Toupin, 1964; Truesdell and Toupin, 1960; Aero and Kuvshinskii, 1960; Grioli, 1960; Mindlin and Tiersten, 1963; Marsden and Hughes, 1983; Povstenko, 1994; Epstein and de Leon, 1998; for related computational techniques, see also Simo et al., 1992). Such an approach to the mechanics of elastic structures have been used in many works since 1958 (see, for example, Amman, 1972, 1995; Amman and Marlow, 1993; Green et al., 1965; Green and Laws, 1966; DeSilva and Whitman, 1969, 1971; Ericksen, 1970; Naghdi, 1972; Simo and Vu-Quoc, 1988; Villaggio, 1997; for computational and stability aspects, see also Fox and Simo, 1992; Simo and Fox, 1989; Simo, et al., 1988, 1989, 1990). Various suggestions to adopt the Cosserats' scheme to describe dislocated structures in crystalline solids have been discussed. Triads of vector-order parameters have also been used by Davini (1986) to introduce a continuum theory of defective crystals (see also Davini and Parry, 1991). Within different settings, vector-order parameters have also been used by Ericksen to describe the behavior of macromolecules within a body (1960, 1962a) and to begin the modern continuum theory of liquid crystals ( 1961, 1962b,c, 1991) that has been further developed by Capriz (1988, 1994,) Capriz and Biscari (1994), and Virga (1994). Mindlin (1964) considers each material patch to be an elementary cell ("interpreted as a molecule of a polymer, a crystallite of a polycrystal or a grain of a granular material," p. 51) that can deform independently of the surrounding medium. A second-order symmetric tensor-valued order parameter is assigned to each cell. Such a continuum is usually called micromorphic (see also Grioli, 1960, 1990; Mindlin, 1965b; Mindlin and Tiersten, 1963; Eringen, 1992, 2000). The approaches of Ericksen and Truesdell (1958), Grioli (1960, 1990) Mindlin (1964) and Toupin (1964, 1965) are special cases of continua with affine structure (see Capriz and Podio-Guidugli, 1976, 1977; Capriz et al., 1982). Higher-order micromorphic continua have been introduced by Green and Rivlin (1964) and further discussed by Germain (1973). A proposal of a nonlocal theory of micropolar continua can be found in Eringen (1973, 1976). For further studies, see Wang and Dhaliwal (1993). Scalar-order parameters were used in 1972 by Goodman and Cowin for describing granular flows. In 1979, Nunziato and Cowin used an analogous approach to
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Paolo Maria Mariano
introduce a nonlinear theory of elastic porous materials (further studies concerning this topic can be found in Cowin and Nunziato, 1983; Cowin, 1985; Dhaliwal and Wang, 1994; Nunziato and Walsh, 1978; Diaconita, 1987; Fr6mond and Nicolas, 1990; Mariano and Bemardini, 1998). Scalar-valued order parameters have been also used to describe phenomena of recrystallization (Gurtin and Lusk, 1999), general solid-solid phase transitions (e.g., Colli et al., 1990; Fr6mond, 1987; Fried and Gurtin, 1993, 1994, 1999; Fried and Grach, 1997), solidification phenomena (Anderson et al., 2000), and isotropic damage evolution (Markov, 1995; Fr6mond and Nedjar, 1996; Fr6mond et al., 1999). Anisotropic damage has also been studied from the point of view of multifield theories by Augusti and Mariano (1999; see also Mariano and Augusti, 1998; Mariano, 1999, and references therein) by using tensor- or vector-valued order parameters. Models with scalar-valued order parameters can be considered special cases of materials with "spherical" structure (Capriz and Podio-Guidugli, 1981). On the basis of classic Lagrangian dynamics of systems of particles, a first attempt to construct general framework for continua with substructures, at least in the case of holonomic-order parameters, has been proposed (Capriz and Podio-Guidugli, 1983). Capriz (1985) introduced the concept of latent microstructures, proving that some higher-order gradient theories of continua can be considered to be multifield theories with appropriate internal constraints. In 1989, Capriz proposed a general theory that is useful for establishing order parameter-based models of continua with substructure. This work opened the way to many theoretical questions, some of which are discussed in Capriz and Giovine (1997a, 1997b), Binz et al. (1998), Segev (1994, 2000), Capriz and Virga (1994), and Mariano and Capriz (2001). In 1990, Capriz and Virga adapted Noll's axioms on interactions in continuous bodies to account for self-interactions among microstructures described by order parameters in linear spaces. In principle, one may think that a body with a fine distribution of voids (or vacancies or microcracks) can be obtained from a mathematical point of view as a limit of a sequence of bodies. In other words, one takes the region occupied by the body in some configuration and at each step of the sequence considers different sets of discontinuities, assigned (perhaps) with some rules. Then one calculates the limit of the sequence and accepts the limit region of the Euclidean space obtained as a reasonable picture of the original finely fractured body. These limit processes are analogous to those used to reach the optimal shape of bodies under some optimum conditions (Kohn and Strang, 1986a,b,c). In 1993, Del Piero and Owen showed that an appropriate fabric tensor describing the influence of microcracks on the macroscopic deformation can be obtained as a consequence
Multifield Theories in Mechanics of Solids
9
of limit of sequences of bodies and corresponding deformations. In 2000, Del Piero and Owen showed that even the vector-order parameter describing liquid crystals (according to Ericksen's theory of nematic liquid crystalsmsee Ericksen, 1962b,c; Capriz, 1988, 1996) can be obtained with a procedure involving the limit of bodies. General results on the evolution of discontinuity surfaces and related configurational forces in continua with substructures during solid-solid phase transitions or during the evolutions of defects have been obtained (Mariano, 2000a, 2001).
II. Configurations and Balance of Interactions
A. CONFIGURATIONS As suggested in Section I, the complete placement of a material body B is described by mappings of the type K ' B ~ g3 x Ad
(1)
assigning to each material patch P of B the pair (placement, order parameter). Of course, g3 is the three-dimensional Euclidean point space, whereas Ad is the collection of all possible configurations of the substructure and is considered a finite-dimensional differentiable compact manifold without boundary. The mapping
KE3 "B ~ g3
(2)
assigning to each material patch its placement, defines the apparent configuration, (i.e., a representation of the body in which the substructure is forgotten). Moreover, KM'B
--+ .A4
(3)
defines the order parameter mapping. In this way, each K is a pair (Kg3, K.A4). For future use, an apparent reference configuration Kg3 is considered, with I(E3(B) being indicated by/3. It is assumed that B is a bounded connected regular region 3 (a fit region) of the Euclidean space and is endowed with a coordinate system {X}. 3Details about the minimal topological requirements necessary for/3 to develop continuum theories can be found in Noll and Virga (1988) and Del Piero and Owen (2000). For the purposes of this Section, one may thinkmroughly speaking--/3 as a bounded connected set that coincides with the interior of its closure and is endowed with a surface-like boundary with well-defined unit normal to within a finite number of corners and edges.
10
Paolo Maria Mariano
For each K, the placement field is indicated by x(.) = (Ke3 o
~-l E~ j (.)
(4)
and the order-parameter field is indicated by
More precisely, given X 9 13, x(X) is the placement of a material patch P resting at X in/3, whereas qo(X) is a descriptor of the substructure of the same patch. At each X, the order parameter qo(X) is an element of.M, and .M itself is a nonlinear manifold, in most cases. 4 For each K, it is assumed that 9 x(/3) is also a fit region 9 x(.) is a one-to-one mapping of 13 into ~3 and is continuous and piecewise continuously differentiable 9 the gradient of deformation Vx, indicated with F, is such that detF > 0; that is, x(.) is orientation preserving 9r
is continuous and piecewise continuously differentiable on/3
In this way the space of the configurations is the collection s of pairs (x(.), qo(.)), each one deriving from the corresponding K. s may be endowed by the structure of a manifold; its tangent space is indicated by T ff, whereas the cotangent space is indicated by T*s Example 1 A useful example to illustrate the statements presented previously is the direct modeling of plates. To this aim, consider an orthogonal frame of reference in s namely {Oele2e}, with O the origin, and a compact bounded set A in the plane ele2. If {X*} is a coordinate system in ele2, with X* = XTel + X~e2, the complete reference configuration of the plate is given by the set
X*+~elX*eA,~ 9 -~,~
(6)
where h is the thickness of the plate. In this picture, A is the apparent configuration. 4Some authors embed .M in a linear space and work consequently, using the handy properties of linear spaces. Their reasoning is based on the hypothesis that .M is finite-dimensional. Each finite-dimensional manifold may, in fact, be embedded into an appropriate linear space (Withney's theorem). However, such an embedding is not unique, and the question of what embedding is phisically significant is completely open. The only clear general results on this point of view are in Capriz and Virga (1990).
Multifield Theories in Mechanics of Solids
11
To define the deformed configuration of the plate, two fields must be defined on A 9 x(.)" A ~ ~3, identifying the placement in ~3 of the points of the "midplane" of the deformed plate; it maps A onto a surface x(A) in ,f3 9 qO(') assigning to each X* a vector t belonging to the unit sphere S: = {t 6 ~3llt I -- 1}; i.e., ~(-) 9(A) ~ S 2 Therefore, in this case, A// coincides with S 2. It is assumed that x(A) is a regular surface in ~.3 and (xl • x 2 ) . t > 0
(7)
everywhere in A where the fields are defined. In (7), x l(X*) and x z(X*) are tangent vectors of x(A) at x(X*); in particular, x,1 is the partial derivative of x with respect to X~ and X,e has the analogous meaning. Condition (7) imposes that t is never tangent to x(A) and excludes the physically unreasonable situation of infinite shearing deformations. Finally, provided the validity of (7), the current complete configuration of the plate is given by the set
x(X*)+~tlX*eA,
teS 2,~
-~,
(8)
Because t need not be normal to the deformed middle surface x(A), shear deformations are allowed in this description of plates. Moreover, the assumption that t is of unitary length precludes thickness changes and cannot take into account initial variable thickness. To account for thickness stretch and initial variable thickness, it suffices to require only Itl > 0 (not the more stringent condition Itl = 1).
B. MOTIONS
Motions are time-parameterized curves (xt, qPt) in the space of configurations, and at a given instant t ~ [0, d], the current placement and the order parameter, respectively, associated to each X ~ B are given by x(X, t)
~o(X, t)
(9)
Moreover, the velocity fields are given by ~(-, .)
~(., .)
(10)
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Paolo Maria Mariano
where the dot over x and qo means time derivative. Of course, qb belongs to the tangent space of .A4, and the pair (~(., t), ~(., t)) belongs to Tff. In the following, Vel indicates the set of pairs of fields (:~, ~). Consider two different observers differing by a rotation described by a proper orthogonal tensor Q with corresponding vector q (i.e., Q = exp(oq), (where e is Ricci's three-dimensional permutation indicator). These two observers evaluate two different values of qoBfor example, qOq and qp--connected by the following relation:
I
t.pq -- qg) -~- --~-q q=O
q + o(Iql)
(11)
If a time-parameterized family of rotations q(t) is now considered, inserting q(t) in (11) and evaluating the time derivative, it follows that, to within higher-order terms, ~q = d~oq dq
cl
(12)
q=0
where/1 is the angular velocity. The term (dqoq/dq)lq=0 is indicated in the sequel of this article with .,4; it is an operator mapping vectors of R 3 into elements of the tangent space of M . In terms of coordinates, ,4 is of the form .147, in which Greek indices denote (here and in the following discussion) the components of the atlas of coordinates on M , whereas Latin indices denote the coordinates in ,s In its matrix representation, .,4 is a (dim .A4 x 3) matrix (three columns and a number of lines equal to the dimension of AA). In the mathematical parlance, ,4 is the infinitesimal generator of the action of the orthogonal group S0(3) on .A4. With these premises, one can say that velocity fields (10) are rigid (and one indicates them with/~R and ~R) if XR - - c(t) +/1 • (x - x0);
r
-- ,,'d-Cl
(13)
C. MEASURES OF INTERACTION
Granted the possibility of defining a covariant gradient of the order-parameter field, indicated with Vqp, a set Jl (C) whose elements are of the form
(x, F, ~a, XT~a)
(14)
may be built up: in the geometric parlance, it is the first jet bundle on the manifold (space of the configurations).
Multifield Theories in Mechanics of Solids
13
Analogously, J1 (Vel) indicates the set whose elements are of the type
(~5)
(~, F, ~, V~)
The power 79 is defined here as a real-valued functional on ,71 (Vel); that is,
79:Jl(Vel) ~ IR
(16)
and accounts for both ordinary and substructural interactions. In addition, it is assumed that, as usual, 79 is additively decomposed into external and internal contributions, for any part B* of the body T't3, = 79~xt - 79~t
(17)
where with the term part of B indicates any subset of B that is also a fit region. The basic problem is thus the representation of T'~t and T'~ t. Assuming the validity of (16), it is necessary to introduce measures of interaction acting on all the elements of (15) and developing power on them. Taking this into account, the following expressions for T'bxt and 79~t are assumed to hold for any part 13" of B:
~, (t 9i + 7- 9qb) dA
~int
(s.~ +z.
dV + ft~ ( T . F + S . ~7~)dV
(18) (19)
where the measures of interaction in (18) and (19) have the following meaning: 9b
external bulk forces
9 /3
external bulk interactions on the substructure
9
boundary traction
9 7"
generalized boundary traction associated with the substructure
9s
zero stress
9z
internal self-force
9T
first Piola-Kirchhoff stress tensor
9S
microstress tensor
From this point, volume, area, and line differentials (namely dV, dA, and dl) are omitted in the integrals to render the formulas more schematic. Of course the
14
Paolo Maria Mariano
reader will understand immediately the kind of differential he needs to use in developing explicit calculations by looking directly to the set on which the integral is calculated. Cauchy's theorem assures that Tn - t
on 0/3*
(20)
where n is the outward unit normal to the boundary of/3*, indicated with 0/3*; moreover, 5 S n - 7-
on 0/3*
(21)
One can write (19) when it is possible to define a connection by which the gradient of qo may be evaluated in covariant manner. This allows for the decomposition of, the substructural contributions to the power in terms of densities/3, qb, z . qb, and S . V gb. However, such a contribution could be expressed by some general complicated functionals of qo, qb, and their spatial derivatives, or could even disappear, as in the case of internal variable schemes. Note that, if necessary, one may define the power on the second jet bundle of ~s or the third, and so on. In (21), the nature of 7- is that of a generalized boundary traction that should be assigned at the external boundary of the overall body where S n - ~'. However, some microstructures, such as porous or microcracked solids, do not allow prescribed boundary data of the type ~'. A pore (or a microcrack) is determined by the surrounding medium and not by itself; therefore, it does not exist at the boundary (pores and microcracks can be considered "virtual" substructures). Hence the boundary data must be specified through a constitutive prescription, or are obtained as a result of some limit procedure based on shrinking some boundary layer at the boundary of the body. However, 7- can be completely prescribed as boundary datum in the case of liquid crystals or other material substructures. Analogous reasonings hold in the case of Dirichlet data (i.e., when values of the order parameter must be assigned on the boundary of the body). It may be that in some situations boundary layer data must be accounted for; however, the treatment of these situations is almost completely open. 5The existence of the microstress tensor by a Cauchy-like theorem is discussed by Capriz and Virga (1990), but in the case in which it is assumed that the manifold .A,4is embedded in some linear space. Probably a more general proof can be made by following the variational procedure in Fosdick and Virga (1989) for Cauchy continua because such a procedure can underline the need for the existence of the covariant gradient of ~o. A general proof of the existence of the microstress tensor on the basis of geometric measure theory might obtained by using the results of Degiovanni et al. (1999). A generalized Cauchy's theorem on manifolds has been obtained by Segev (2000).
Multifield Theories in Mechanics of Solids
15
Balance equations can be deduced from (17), (18), and (19) by assuming (as axioms) that at the equilibrium 9 the overall power T'B, vanishes for any choice of the velocity fields :~, qb and of any part B* of B 9 the internal power T'~t vanishes for any choice of rigid velocity fields and of any part 13" of/3 To this aim, equations (20) and (21) must be used together with Gauss theorem. In applying such a theorem, it is assumed that discontinuity surfaces, lines, and points for the fields involved in the integrals are absent. After some calculation, the overall power can be written in the following form:
T'B, = ft3, ((b
- s + DivT). i + (/3 - z + DivS). qb)
(22)
The requirement that ~e, = 0 or any choice of the pair (~:, qb) and of the part B* of B implies that b - s + DivT = 0
in B
(23)
/3 - z + DivS = 0
in B
(24)
Equation (23) is the standard Cauchy's balance to within the zero stress that must be formally introduced as a consequence of (16); equation (24) is the balance of substructural interactions, a sort of generalized balance of momentum. General information on the structure of s and T are obtained by exploiting the axiom requiring that T'~t vanishes when it is calculated on rigid velocity fields [defined by (13)] and on any part of B. By applying such an axiom, in fact, it follows that
eTF r = ATz +
(25)
inB
s=0
(vAT)S
in B
(26)
Of course, by (25), equation (23) reduces to the well-known balance b + DivT = 0
in 13
(27)
In addition, note that when the influence of the material substructure is negligible and order parameters are not considered, equation (26) reduces to skw(TF r) = 0
(28)
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Paolo Maria Mariano
which is the standard property of symmetry of Cauchy stress tensor TF T [skw (.) extracts the skew-symmetric part of its argument]. Previous balances hold in general dissipative situations and must be supplemented by appropriate constitutive equations.
D. BALANCE OF INTERACTIONS FROM THE INVARIANCE OF OUTER POWER
The procedure discussed in Section II.C has been exploited in different cases to derive balance equations. One may question, however, if such a procedure needs too many hypotheses that may be relaxed. In particular, one might find a more general procedure in which the expression of inner power follows as a consequence and is not an axiom. This would be desirable because, in principle, it is possible to evaluate the outer power by experiments only. In Cauchy's solids, Noll's procedure requiring the invariance of the outer power with respect to changes of spatial observers allows one to obtain standard balance of forces and the symmetry of Cauchy stress. An analogous procedure can be followed in the case of materials with substructure. It underlines some delicate questions that may emerge when one selects some order parameter and tries to write balance equations. Basically, one writes the outer power in (18), taking into account (20) and (21), as
extf ~. --
(b. ~ +/3.
*
~b) +
~*
( T n . ~ + S n . ~b)
(29)
and requires the invariance of 7J~' with respect to all changes of observer. This is a request of invariance with respect to Galilean and rotational changes of spatial observers. Changes of spatial observers are typically given by ~* -- ~ + c(t) + ~l(t) x (x - x0)
(30)
(p* -- (p + A q ( t )
(31)
where ~* and @* are the fields evaluated by the observer after its change and e(t) is the translational velocity. As is common in multifield theories, in writing (31), it is assumed that rigid translations have no effects on the values of r Such an assumption applies even in the case in which the order parameter is a microdisplacement because it is always a "relative" microdisplacement. Galilean changes of observers are obtained from (30) and (31) with the choice ~ - 0, ~1 - 0, whereas rotational changes of observers are characterized by ~ - 0, ii - 0.
Multifield Theories in Mechanics of Solids
17
The requirement of invariance of (29) under transformations (30) and (31) implies
b + f ~, Tn)+ q. (f, , (x • b +
+ f ~, (x x Tn + AT$n))--0 (32)
for any choice of c and Cl and the part/3". The arbitrariness of c and ~1implies
f b+f Tn-0 *
dB*
(33)
O*
, ] 0 O*
Equation (33) is the standard integral balance of forces. The arbitrariness of/3* and the application of Gauss theorem imply from (33) that b + DivT = 0
(35)
From (34), taking into account the validity of (35), the following relation follows from the arbitrariness of/3* and the application of Gauss theorem: e T F T = .AT/3 + Div(.A TS)
(36)
or, by developing the divergence, e T F T - AT/3 + ATDivS + (VA T) S
(37)
To assure the validity of (37), two conditions must be satisfied. It is necessary that all the elements that are not multiplied directly by AT be equal to the product of A T with some generic element z of the cotangent space T*.A4 of .A4 (/3 and DivS are elements of the cotangent space of.A/l); that is, the existence ofz e T*.A// is necessary such that
ATz = e T F T - ( v A T ) s
in/3
(38)
which coincides with (26) and represents the generalized balance of "couples." Equation (37) can be thus written as
.AT (3 + Div8 - z) = 0
(39)
and is satisfied when the term in parentheses belongs to the null space of the linear operator A T6 /3 + Div8 - z 6 null space of AT
6AT is a matrix with three lines and a number of columns equal to dim .AA.
(40)
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Paolo Maria Mariano
which is the second condition. In general, the null space of .Ar is the space orthogonal to the range of ,,4 (range of ,,4)• = null space of .,4r
(41)
when a concept of orthogonality is available on T.A4. The range of A at each qo is by definition a subset of the tangent space of .A4 (namely, T.A/[) at the same qo or it is coincident with the whole T.A4 at go. When the range of ,4 is coincident with the whole tangent space of .A4 at qo, the elements of its orthogonal in T.A/[ reduce to the singleton {0}; then the term/3 + DivS - z must be equal to the sole element of the orthogonal "of range of A." In other words,/3 + DivS - z vanishes identically /3 + DivS - z = 0
(42)
There are counterexamples in which the range of A does not coincide with the whole tangent space of .A4 at a certain qo. The prominent counterexample is the case in which the order parameter qo coincides with a stretchable vector cl; this happens in models of shells with through-the-thickness shear or of microcracked bodies. In this case, ,A = ocl, with o Ricci's tensor. It is obvious that at 0 = 0, the range of A coincides with the singleton {0}. Another counterexample is the model of porous bodies in which the order parameter is the scalar void volume fraction; in this case, .A4 coincides with some interval [0, a] of the real axis, and A vanishes identically. This follows by considering that for porous bodies, the value of qo remains unchanged by rigid rotations: then qgq = 99 and the derivative (dqgq/dq) Iq=o vanishes identically. Models of microcracked bodies also make use of second-order symmetric tensor-valued order parameters ~ij and the operator ,,4 coincides with (Oijr~rk -1- Eirerjk). Even in this case, at E = 0, the range of A coincides with the singleton {0}. In the case in which the range of ,A does not cover T.A/[ at any qo, the argument leading to (42) is not exhaustive, and one must write the differential inclusion (40) as
/3 + DivS - z' = - z "
(43)
where z' satisfies (38) and z" belongs to the null space of A r (i.e., A r z '' = 0). Consequently, (42) still holds, taking for z the difference z ' - z". This property cannot be derived by using the procedure in Section II.C. Example 2 An example of the occurrence of a term like z" in the balance of substructural interactions can be found in the theory of liquid crystals. The usual order parameter of liquid crystals is a vector d belonging to the unit sphere S 2 in/t~3.
Multifield Theories in Mechanics of Solids
19
A coincides with od, which can be also written as d • In balancing substructural interactions, it is required that the covector/3 + DivS - z at each material patch be parallel to the "averaged" direction d of the rodlike molecules of the liquid crystal at the same patch. Therefore, in this case, the balance of substructural interactions is written as
tic + divSC _ z c = c~d
(44)
with c~ some scalar constant and the operator div is calculated in the current configuration (see Ericksen, 1991). It is evident that otd is - z " because d x d = 0. Of course, (44) is written in the current configuration because one deals with a liquid. Therefore, tic, Sc, and z c are measures of interaction in the current configuration. The request of invariance under changes of spatial observers could perhaps not be conclusive in some cases (thanks to the arbitrariness of z'), and more stringent requests of invariance could be required to justify completely the balance equations. A possible way consists in requiting the invariance with respect to all possible representations of each type of substructure. Suggestions for such an invariance follow from the knowledge of situations in which some of the components of the order parameter can be chosen arbitrarily. If one requires in fact that some scalar function k of ~ - - a s , for example, the substructural kinetic e n e r g y - - b e invariant under the action of the rotation group S0(3), it is necessary that
.A T(O(,,k(~)) = 0
(45)
This is a system of partial differential equations in 2(dim .M) + 1 variables. Following a theorem of Tricomi (1954), it is possible to show that previous system admits 2(dim .M) + 1 - char A independent variables as solutions, where char.A is the characteristic of .,4 because .A is a matrix with three columns and a number of lines equal to dim A/[. Such independent variables can be chosen arbitrarily by an external observer. Under such a suggestion, one could think to map .A4 into other manifolds A/" by Cr-diffeomorphisms (r > 1), indicated with Jr, and call these diffeomorphisms 7 representations of the substructure when dim .M = dim A/'. If two elements on .M are related by a rigid rotation, then the corresponding elements on A/" (through the mapping zr) must be related by a rigid rotation as well. To each Jr : . M --+ A/', a mapping Trr is associated; it maps elements of the tangent space of M (i.e., TA4) into elements of the tangent space of A/" 7Roughly, a diffeomorphism is a one-to-one differentiable mapping such that its inverse is differentiable as well.
20
Paolo Maria Mariano
(i.e., TA/'). The mapping TJr may be then described by appropriate Jacobian matrices J, such that if qb is an element of TA/[, then Jqb belongs to TA/'. In other words, defining qo' - zr(qo), it follows that qb' - ~ and Jl - dqo'/dqo. The requirement of invariance of T'~xt under all possible representations of the substructure reduces to the invariance under changes qb --+ Jqb. To render T'~xt invariant under changes qb ~ ~ , it is necessary that
(46) The arbitrariness of qb and the application of Gauss theorem reduce (46) to (I)T(/3 + DivS) + (VJT)S = 0
(47)
where (I) = I - J. The validity of (47) implies the existence of some element i of the cotangent space of .A4 such that (I)Ti + (V2 T) S = 0. Thus (47) reduces to (I)v (/3 + DivS - ~) = 0 and the term in parentheses must vanish identically as a result of the arbitrariness of,II, thus of (I). However, this conclusion is only formal because the condition ~T~ + (VjIT)S = 0 may imply that S vanishes identically, as a result of the arbitrariness of dl, so as to obtain a reduced balance of the form /3 = 0
(48)
Among other things, such a conclusion could induce doubts about the expression of the external power 79~xt. There are situations in which nonlocal terms could appear in the expression of 79~,t through some general functional that may be expanded in series by means of some "myopia" theorems formally analogous to the theorems of "fading memory" used in standard continua with memory. In the former case, such theorems reduce the nonlocal influence of material patches, surrounding the one assigned, to a rather weak nonlocality in space, whereas in the latter, theorems of "fading memory" cut the influence of the events in the past on the present event (the nonlocality is thus in time). In any case, the validity of balances (35), (38), and (42) implies the proposition in the following. Proposition 1
By virtue of the balance equations (35), (38), and (42), it follows
that
(49)
and the last integral in (49) takes the name of internal power and is indicated with 79~t.
Multifield Theories in Mechanics of Solids
21
Note that (49) was used in Section II.C as an axiom [see (18) and (19)] to deduce balance equations (24), (26), and (27) by means of the "virtual power procedure." In this section, another procedure has been followed; only the explicit expression of the external power is assumed as an axiom, then the invariance with respect to changes of external observers is requested. This procedure is in essence more general than that in of Section II.C because it underline the need of elements of the type z" and the expression of the internal power has been deduced in this section as a theorem.
E. EFFECTS OF INERTIA One possible way to account for the effects of inertia pertaining to both the macroscopic motion and possible internal vibrations of the substructures consists of decomposing the volume forces b and/3 in their inertial (in) and noninertial (ni) parts as b = b in - [ - b ni
(50)
j~ _. j~ in ..[_ j~n i
(51)
and in assuming that, for any part B* of/3, d {kinetic energy of 13"} +
~ ) __ 0
(52)
rate of kinetic energy + power of inertial forces -- 0
(53)
dt
(b in x + j~in
9
In this way, one assumes the validity of the balance
and may interpret (53) as a constitutive prescription on the explicit expression of inertial terms once an expression of the kinetic energy has been selected. Although the kinetic energy density associated with the macroscopic motion is proportional to I/~l2, in fact, the kinetic contribution of substructures may have some complicated structure that depends on each special model and, in a certain sense, has constitutive nature. The kinetic energy of 13" is given by 1
ft3, (-~pi~. ~ + k(qo, (p))
(54)
where l p/~./~ is the standard kinetic energy density of material particles and
22
Paolo Maria Mariano
k(qo, qb) is the contribution of the substructures of the material to the kinetic energy density. The term k (., 9) is a nonnegative function such that k(., 0) = 0
(55)
02~k -r 0
(56)
where, here and in what follows, 0y means partial derivative with respect to the argument "y". Really, the symbol 0 was previously used before letters indicating sets. When 0 precedes a letter indicating a set (e.g., 0B), it indicates the boundary of B, whereas when 0 precedes any function, it means partial derivative. Moreover, k (-, 9) must be frame indifferent; that is, taking into account (11), it is necessary that k(qo, qb) - k(qo + Aq + o(Iql), qb + A/I + o(Iql))
(57)
for every q. The invariance condition (57) implies O~kA + O(ok(O~oA)(o - 0
(58)
Equation (58) can be obtained by developing in series the right-side term of (57) around k(qo, ~). Substituting (54) in (52) and taking into account that (52) must hold for any part B*, it follows that b in = -p:~ -
~-~-,geX - ,9~X
)
(59) (60)
where X is called substructural kinetic coenergy density and is such that k -- O~X. O -
X
(61)
In the mathematical parlance, the kinetic coenergy is the Legendre transform with respect to ~ of the substructural kinetic energy density k. As a consequence of (59) and (60), the balances (35) and (42) become DivT +
b ni - - p ~
DivS - z + ~ni _
d dt
O(oX -- O~X
(62) (63)
It is noted that inertial effects ~in associated to the substructure of materials are very minute and not perceivable unless the substructure itself oscillates at high frequencies, as indicated by some experiments on the scattering of phonons within the lattices of crystalline materials, or on liquid crystals.
23
Multifield Theories in M e c h a n i c s o f Solids
The explicit expression of k is of constitutive nature; a possible quadratic form, 8 like the simplest one given by 1_ k - ~ D ( ~ , ~)
(64)
determines the metric on .M. In (64), the brackets and the comma (i.e., (.,-)), indicate the scalar product on TAd, the tangent space of Ad, and /) is some appropriate constant chosen to adjust eventually physical dimensions. When k is constitutively prescribed, the kinetic coenergy density can be obtained by solving the partial differential equation (61), whose solution X(qO, ~) is the sum of a special solution of the complete equation Xs and the homogeneous solution Xh, corresponding to k -- 0:
(65)
X = Xs + Xh
The explicit solution of (61) can be found in (Capriz and Giovine, 1997a). If k is homogeneous of second degree (hsd) in qb, then it coincides with ,~s. Finally, the inertial contributions of the material substructures can be written as follows:
[~in ( d d = -
- ~ 04, Xs - O~ X, + --~ Or
-- 0,r Xh
)
(66)
where the term d d-t 0r Xh - 0,r Xh is powerless, that is,
(d
O~Xh - O~,Xh
)
90 -- 0
(67)
(68)
When k = O, the solution of equation (6 l) prescribes that the kinetic coenergy density X from (61) can be, at most, linear in O; in this case,/3 i" can also be, at most, linear in O, and situations of parabolic evolution can arise, as in the case of magnetostrictive solids or ferroelectrics. When one considers a relative velocity Oret of the order parameter and writes Orel + .,4(:1instead of the absolute velocity qb, it is possible to underline the presence 8Quadratic expressions for the substructural kinetic energy density can be found, for example, in the case of direct models of plates (Antman, 1995), multifield descriptions of ferroelectric (Da~i and Mariano, 2001), or microcracked bodies (Mariano, 1999) when microdisplacements are considered as order parameters. More general expressions must be considered when, for example, there is an intrinsic limit velocity for the propagation of perturbations in the materials. For example, in crystalline materials the dislocations cannot propagate with a velocity greater than the velocity of sound.
Paolo Maria Mariano
24 of a centrifugal term
a 2~ X ( ( O~Ail)(Ai~))
(69)
( O2cpX) .,461
(70)
- 2(0~,(.A/I)) TO~g
(71)
an entrainment term
and a Coriolis term
associated to substructural dynamics. 1 When k is a quadratic form in ~b (e.g., k(~, ~b) - 5~b. N~b, with some constant tensor N) and the velocities are only of rigid rotational type (/1 x x and A/l), l the total kinetic energy of the body becomes 5/1. (J -+- H)(1 where J is the standard moment of inertia given by f8 P (]x]21 - x | x) and H - f8 ATN'A" The final expression of the kinetic energy seems not to be compatible with the classical dynamics of rigid bodies. This paradox can be eliminated by imposing that
k(t,p, ~ ) -
k(qo, ~rel)-
Moreover, there are cases in which F is involved in the kinetic energy density. This is the case of liquid crystals when one chooses as order parameter an observerindependent vector given by F -l d, instead of an element d of the unit sphere S2 in R 3. A complete theory of liquid crystals that accounts for this has not yet been developed. Inertial interactions may also involve spatial derivatives of the acceleration fields. In this case, possible values of the order parameter are subjected to some internal constraint, so the substructure becomes latent (see Section VIII). A prominent example is given by models of capillary phenomena described by Korteweg's fluids, in which the order parameter is scalar and is coincident with det F (see Capriz, 1985). R e m a r k 1 Let the invariance of the kinetic energy density be required under Galilean changes of observers. Let also the invariance of the overall energy (sum of internal and kinetic energy) be required under rotational changes of observers. As a consequence, after some calculations not entirely trivial, the inertial contribution of the substructures results in d dtOCpk - O~k
(72)
When k is homogeneous of second degree (hsd) in ~, or when X is so, (72) may substitute the term in parentheses in (60). In this case, the two expressions of the inertial contributions are equivalent to within powerless terms. When the
Multifield Theories in Mechanics of Solids
25
substructural kinetic energy density requires more complicated expressions than hsd, the problem of the equivalence of the two procedures just explained is far from being completely clarified. Cases different from hsd may be necessary when a limit speed of the substructural perturbations (thus of ~) must be emphasized; invariance with respect to rules of changes in observers more general than the action of the rotation group could be necessary, and the consequences of the relevant gauge invariances examined. Remark 3 When cases in which the balance of substructural interactions reduces to/3 = 0 occur, from (51) and (60) it is possible to write only a kinetic equation of the form
t~ni__ d oqx(u q~rel) _ oqx(~o, q~grel) dt O0rel 0r
(73)
An example of absence of microstress in models of bodies with substructure is the theory of liquid with bubbles presented in Kiselev et al. (1999).
BIBLIOGRAPHIC NOTE Balance equations (24) and (26) are Capriz's (1989), their presentation in Section II.D follows that of Mariano (2000a). Mathematical details on the topics in Section II.D can be found in Di Carlo (1996). Additional remarks on the geometric nature of microstresses and self-forces can be found in Segev (1994, 2000) and Capriz and Giovine (1997a,b). Details on the examples before Remark 2 can be found in Capriz (1989, 2000). For a derivation of balance equations, see also Capriz and Podio-Guidugli (1983) and Capriz and Virga (1990, 1994). Discussions on the derivation of balance equations by means of virtual power arguments (involving the assumption a priori of the expression of the inner power)can be found in Germain (1973), but they are referred only to order parameters that are higher-order perturbations of the displacement field, then only first- and higher-order micromorphic materials are treated (see also Maugin, 1990). The procedure to obtain balance equations for substructural interactions on the basis of the invariance of the external power 7~! under changes of observers has been developed in Capriz and Virga (1994), Capriz (2000), and Mariano and Capriz (2001). Noll's classic procedure of invariance of external power can be found in the 1973 article. "Myopia" theorems for spatial nonlocalities and their applications to Cauchy's continua are in Capriz and Giovine (2000), whereas analogous "fading memory" theorems are in Coleman (1971).
Paolo Maria Mariano
26
All details about the relations between the substructural kinetic energy and the substructural kinetic coenergy are in Capriz and Giovine (1997a).
III. Elastic Materials with Substructure A. VARIATIONALCHARACTERIZATION
In this section, only hyperelastic materials with substructure (or, with some abuse, just "elastic") are examined. They are characterized by the existence of an elastic energy density, indicated with w, such that 6f,. w=f,.
(T 9~iF + S 93(Vqo) + z 96qo)
(74)
for all parts B* of/3, where ~ indicates the variation operator. 9 A consequence of this definition is the possibility to write the measures of interactions in terms of F, V qO, qO, once the expression of w has been selected. In general, in fact, one can assume w = t~(F, qo, XTqo)
(75)
Consequently, by calculating the variation of the first integral in (74), it follows that
f ( ( O v w - T). ~F + (~v~w - S)-~(XTqo) + (O~w - z). 6qO) = 0
(76)
Because variations can be chosen arbitrarily in (76), the following constitutive restrictions hold
Remark 4 structure:
0vw = T
(77)
Ov~w = S
(78)
O~ow = z
(79)
Consider the following special case of an elastic material with sub-
9 ~ni vanishes identically 9 t~ in
is only of powerless type and is given by Bqb, with B an appropriate tensor
9Of course, equation (74) can be written in terms of velocity fields.
Multifield Theories in Mechanics of Solids
27
9 w - t~(F, qo), thus the weakly non-local contribution of the order parameter due to the gradient of qo is neglected The balance (42) reduces to
B~o- O~w
(80)
and the self-force 0~,w becomes powerless. The order parameter assumes the character of an internal variable not satisfying balance of interactions that develop explicit power. Equation (80) is coincident with the evolution rule of an elastic internal variable. An analogous reduction of multifield models to internal variable ones can be obtained in nonconservative cases, as is shown in some parts of the following subsections.
B. SOME PROPERTIES OF LAGRANGIAN DENSITIES Before discussing some properties of Lagrangian densities for elastic materials with substructure, it is convenient to introduce a special symbol for a product between tensorial quantities that will be of future use both here and in following sections. This symbol is _,. The product ,_ is here defined as _," Lin(It~3, T.A4) x Lin(I~ 3, T*.A4) -+ Lin(It~3, R .3)
(81)
where Lin (It~3, •,3) is the space of linear forms associating three-dimensional covectors (belonging to the dual of R 3, which is usually indicated with R .3 and identified with R 3) to vectors in ]t~3 and Lin (It~3, T*.A4) is the space of linear forms o n R 3 taking values on the cotangent bundle ~~ T*.A4. Then, taking Vqo and the microstress S [which is a linear form associating elements of the cotangent space of.A//to vectors in ~;~3 thus an element of Lin(R 3, T'A//)], one writes by definition (Vqo r_,S)n 9v - S n . (Vqp)v
(82)
for any choice of vectors n and v. Note that when the order parameter is scalar valued, the product_, coincides with the standard tensor product | On the contrary, when qo is not scalar valued, the meaning of_, is not the one of a dyadic product. For example, if qo is a third-order ij tensor with components ~k ), one has (Vqo T,_S)jl n j 1) l - - Si~nJ(~rqo)i~l)vI. l~ cotangent bundle of.AAis the space of linear forms on the elements of the tangent space of .A.4.
Paolo Maria Mariano
28
Another product, indicated with +, is also of future use. It is defined as ,i, : T*.AA • Lin(R 3, T M )
~ ]~,3
(83)
thus its result is a covector, l~ Consequently, the products z,i,~'qo and/3+~'qo are covectors. For example, if qo is a fourth order tensor with components ~0/,nJ,,)one mn
ij
has (z+Vqo)l = zij (Vq~ Lagrangian densities for conservative dynamics in multifield theories are of the form
s -- s
x, ~, F, qo, qb, ~Tqo)
(84)
Of course, s depends on the metric y on the referential configuration 13 and on the metric on .A4 through the quadratic part of the kinetic energy k. Granted some regularity properties of the Lagrange density function, ~2 the following Euler-Lagrange equations hold: d
dtO~s - Oxs + DiVOFE -- 0
(85)
dtOCps - 0~,s + DivOv~,s - 0
(86)
d
To obtain a more compact form of equations (85) and (86), let the four-dimensional gradient V 4 be introduced. It is defined by
V4 -- ( ~ )
(87)
The four-dimensional divergence is indicated with V 4. , and then, by indicating with H and fI the derivatives 0V4xs and 0v4~,s respectively, equations (85) and (86) can be written as
V 4. lI - Oxl~ = 0
(88)
V 4" 1=I - 0qo,~ --" 0
(89)
I I With the same symbol +, a product + : Lin(R 3, T*.A/[) x Lin(R 3, Lin(R 3, T.M)) --> ]1~,3 is also indicated. It is not an abuse of notation, because the two products have the same structure. In this way, the product S+VV~0 determines a vector of I~3. 12See, for example, Renardy and Rogers (1993) for the discussion of regularity properties for s in the case of Cauchy materials. The extension to multifield theories is left to the reader as an exersize.
Multifield Theories in Mechanics of Solids
29
Let 1) be defined now by I~ -- s 4 -- V4x r FI -- V 4 (~T ,__1=i
(90)
where 14 is the four-dimensional unit tensor. Tensor 1) has physical dimensions of an energy density and is expressed in material coordinates. I) is also of second order, thus of the type ]?s, with indices r, ~ running 0, 1, 2, 3 (the coordinate 0 being the time). In the following, I? indicates the spacelike part of 1), is of the type ]?m with m, n running 1, 2, 3 and is given by P - s Proposition 2
FrT-
Vq~r_,S
(91)
If the equations of motion hold, then 4-r V~Ps + s
-0
(92)
where the comma as subscript represents in (92) the explicit partial derivative with respect to the coordinates. The proof of this proposition is based on the calculation of the derivative O~])~ and on a lemma stating that oy.....z ; - - ~ 1P m"
(93)
The proof of (93) is rather technical and can be found in (Mariano, 2000a) as well as the complete proof of previous proposition. An important Corollary is the following:
When the body is homogeneous and inertial effects can be neglected, for any closed sufficiently smooth surface within it, Corollary 3
fc,
osed surface
Pn - fc,
osed surface
( w I - F T T - vqoT . S ) n -- O
(94)
Of course, n is the outward normal to the surface. The proof of Corollary 3 follows from the simple application of Gauss theorem. The integral (94) is the extended version of Rice's (1968) integral to multifield theories. Its importance in the study of crack propagation is explained in Section VII. The second-order tensor in the integrand, namely ( w I - F T T V qoT*,.q), is a modified version of Eshelby tensor that holds for elastic continua with substructure. If one inserts an inclusion in an elastic homogeneous body with substructure and defines the force ~ exerted on the inclusion by the surrounding medium
30
Paolo Maria Mariano
(following Eshelby, 1975) as the integral on a region 13" containing the inclusion in its interior, namely, -- -- f/3* /~,i
(95)
then, by (92) and (94)), in absence of inertial effects, one obtains - f ( w l - F T T - Vqpr_,S)n Ja B*
(96)
C. INFLUENCE OF THE SUBSTRUCTURE ON THE DECAY OF ELASTIC ENERGY
One of the central results of the linearized theory of elasticity is the proof of the longitudinal decay of elastic energy in cylinders loaded at one base only by equilibrated force systems. This phenomenon is usually known as Saint-Venant's effect. Here the analogous in multifield theories is shown and the influence of the material substructure on such a decay is evaluated. In this section, in which linearized situations are treated, reference and current configurations are identified with each other and the relevant measures of interaction are denoted with T, ,9, 2. The displacement field u - x(X) - X is introduced for convenience, and F replaced by Vu in the constitutive relations. Let (0X1X2X3) be an orthogonal coordinate system and D an open-bounded compact region in the plane X l X2. The body considered here is a semi-infinite cylinder if2 - / ) x [0, +cx~). In the following, n3 indicates the outward unit normal at /) (i.e., n 3 - - e 3 , with e3 the unit vector along X3), whereas nL indicates the outward unit normal at the lateral boundary 0/) • [0, +c~)./)(~'3) indicates the cross section at S(3 E [0, +c~), whereas f2(2"3, l) -/3(2"3) • [Yf3, Y(3 + l]. Moreover, it is understood in the following that
fa (.)=limf (X3)
l'---~~
(-'~3 ,/)
(.)
(97)
provided the existence of the limit. With these premises, the following assumptions apply" 1. External volume forces vanish: b - 0,/3 = 0; then DivT = 0
(98)
Div,9- ~ = 0
(99)
Multifield Theories in Mechanics o f Solids
31
2. The lateral boundary 0 b x [0, + e c ) is traction flee; boundary tractions are applied to b only and are self-equilibrated in the sense that Tn3 - 0
(100)
Sn3 - 0
(101)
x x 'l'n3 + ,,4T8n3) -- 0
(102)
fo fb( Moreover, it is assumed that lim f
J D(X3)
X3 ---~~
(103)
(1"n3 9u + ,Sn3 9q)) = 0
3. The elastic energy w(Vu, q0, Vqo) is a positive definite quadratic form in its variables. By indicating with y the triplet (Vu, qO, Vqo) in a way such that yl
= VII,
Y2 -- go, Y3 = V qo, the elastic energy density can be written as 1 w(Vu, qo, Vqo) -- -~aijYiYj
(104)
where aij is the ijth element of a matrix a expressed by
/
og~
a = (aij) =
Ogumlo 0g~
og .mlo
/
og mlo]
with o indicating a stress-flee state considered to be a natural (or reference) state of the body. Matrix a is such that aij = aji. It is assumed that there exists a > 0 such that aij Yi Yj <_ a ~'i f/i for any pair of constant ~7's. Assumption 3 also implies that w must be sufficiently regular to assure the validity of Taylor expansion around some "natural" state o, at least up to the second order, and that residual stresses are not accounted for; that is, it is assumed that Wo = 0;
8vuWlo = 0;
O~,wlo = 0;
Ov~oWlo= 0
(105)
32
Paolo Maria Mariano
Let U(X3) be the elastic energy of the part of the cylinder given by f2(X3) D(X3) x [X3, nt-oo). It is defined by
U(X3)- [ 113 J~ (x3)
(106)
The following proposition holds. It explains Saint-Venant's effect in elastic materials with substructure and is the main result of this section. Proposition 4
If the fields Tn3, ,.~n3, u, qp are square integrable, then
U ( X 3 ) < U ( 0 ) e x p ( X 3 - 1~ )_ _
(107)
for any 1 > O, X3 > I. The scalar )~ can be interpreted as the lowest non-zero characteristic value of the free vibrations of f2( f(3, l) with quadratic substructural kinetic energy.
When the ratio a/)~ is greater than the analogous ratio in the material considered without substructure, the presence of material substructures within an elastic body has a stabilizing effect and decreases the decay length of the elastic energy. Another kind of decay can be studied: the radial decay (Knops-Villaggio's effect) related to boundary conditions imposing that both u and qp vanish at the lateral boundary. It is possible to prove that the elastic energy of a cylindrical annulus at a given distance from/), of variable height and whose other surface coincides with the lateral surface of the cylinder, decays to zero algebraically at most.
BIBLIOGRAPHIC NOTE Section III.B is based on the first part of Mariano (2000a). With reference to Section III.C, the proof of Saint-Venant's decay in continua with substructure follows basically Toupin's (1965) proof in Cauchy's continua. Analogous proofs in the special cases of Cosserat continua and micromorphic continua have been developed in Berglund (1977) and Batra (1983), respectively. The Knops-Villaggio effect in the case of Cauchy's materials is proved in their 1998 article, whereas its possible discussion within the setting of multifield theories is in Mariano (2000b), along with the proof of Proposition 4.
Multifield Theories in Mechanics of Solids
33
IV. Balance in Presence of Discontinuity Surfaces The presence of discontinuity surfaces (also called interface) is a recurrent phenomenon in solids with material substructure. Interfaces between fluid and solid or solid and solid are evident in solidification phenomena and occur also in second-order phase-transitions between austenite and martensite. This section discusses how interactions can be balanced across discontinuity surfaces and at the junctions among them. Each discontinuity surface may or may not be considered to be endowed with its, own structure. In the former case, surface measures of interaction must be introduced because they take into account classic surface tension and a surface tension as a result of the material substructure.
A. INTERFACES: GEOMETRIC CHARACTERIZATION Within the body in its reference placement/3, an interface E is considered to be an oriented surface defined by E _= {X E cll3, f ( X ) =
0},
(~o8)
where the function f is smooth on/3. The orientation of Z is given with the normal vector field in, defined by m = V f / I V f l (Figure 1). Given any continuous and differentiable vector field e defined on Z, it is possible to define its surface gradient Vz by considering parametrized curves t on E and
V
FIG. 1. Geometric characterization of interfaces.
34
Paolo Maria Mariano
applying the chain rule e -- (Vze)i where i is the derivative of t with respect to the parametrization of the curve. The curvature tensor L of E is defined as the opposite of the surface gradient of the normal vector field, namely, L = -Vzm
(109)
whereas the overall curvature E is the trace of L KS = - t r ( V z m ) = - D i v z m
(110)
The previous formula allows one to define the surface divergence as the trace of the surface gradient. On each subsurface of E enclosed in a genetic part B* of B, a vector field u on the curve OB*NE is defined; it belongs to the tangent space of E at OB* A E m that is, v is the normal to 0B*N E in the direction of the tangent of E (see Figure 1). Given any field e that is continuous on B \ E, the limits e+ = lime(X + em, t), e---~0
X 6E
(111)
define the jump [e] through [e] = e + - e -
(112)
when the difference makes sense. Given two such fields--for example, el and ezmthen, with some significance assigned to the product, [ele2] = [el](e2) + (el)[e2]
(ll3)
where (e) is the average given by (e)-
1 ~(e + + e - )
(114)
If the fields F and V q0 have the properties just mentioned for e, then a discontinuity surface E is called coherent if [F](I - m | m) = 0
(115)
[Vqo](l - m | m) = 0
(116)
where I is the second-order unit tensor, whereas (I - m | m) is the projector on E and condition (116) is taken here only for convenience in the sequel of this article. In the following, it is convenient to indicate with F the surface gradient of x, namely, F = V~x - (F)(I - m | m)
(117)
Multifield Theories in Mechanics of Solids
35
while with N the surface gradient of qo, namely N = Vzqo = (Vqo)(I - in | m)
(118)
Interfaces across which the order parameter is continuous are considered only in the sequel of this article; discontinuities are accounted for the gradients of the order parameter itself. Cases in which the order parameter itself can be discontinuous across E may occur. However, because qO is in general the element of a nonlinear manifold, the possible difference qo+ - qo- may not make sense. For example, if .A4 coincides with the unit sphere in the example in Section II.A, the difference of two arbitrary elements of such a sphere may not belong to the sphere itself. However, one can define the jump [qo] of qo even on a nonlinear manifold .A// when it is possible to find some group that acts on .A// as the translation group; that is, there exists on .A4 a given element @ such that it is possible to reach any other element of .A/[ by multiplying @ by elements of the group. In this case, the jump [qo] can be defined through the difference of the elements of the group. Alternatively, one may embed .A4 into a linear space by Whitney's theorem, even if the embedding is not unique. When qo is continuous across E, the jumps [~] and [V~o] are always defined because both qb and XTqoare elements of linear spaces.
B. B A L A N C E AT DISCONTINUITY SURFACES
1. Discontinuity Surfaces without Own Structure The simplest way to characterize interfaces is to imagine them only as purely mathematical surfaces free of any form of own structure. Balances of measures of interaction can be derived easily in this case. First, one must consider the integral form of (35) and (42) and for a generic part B*, crossed by the discontinuity surface E (i.e., B* A E -r 0), one writes
fBTn-0. b+f ~.
fB(~--Z)+] *
f ,J0 B*
(119)
Sn--0
(120)
In equations (119) and (120), the bulk interactions b,/3, and z are assumed to be continuous throughout B*, whereas stress measures T and S may suffer jumps at E. Balances at E; can be obtained from (119) and (120) in different ways. Probably the simplest one consists of shrinking B* at the discontinuity surface E with a limit procedure prescribing that B* --+ B* A E. In making this, the bulk integrals vanish as a result of the continuity of their integrands, whereas the integrals on the boundary OB* reduce to integrals on B* A E of the jumps of their integrands.
36
Paolo Maria Mariano
By translating the previous words into symbols, it follows that
f,
ft3 0 b--+ , ,
0
as/3* ~ / 3 * n yz
(121)
Consequently, (119) and (120) reduce to f
IT]in - 0
(122)
[31m - 0
(123)
~*NZ
f
13*nz
The arbitrariness of/3* implies [T]m = 0
on Z
(124)
[S]m = 0
on Z
(125)
Proposition 5 In absence o f standard and substructural surface tensions on a discontinuity surface ]E, the standard and generalized tractions (t = Tm and 7" = 8 m , respectively) are continuous across Z. As a consequence of the invariance of the lower, equation (125) should be written as .,4 T [S]m -- 0. Thereof the assumption in Proposition 5 that surface tensions are absent implies that even surface self-forces of the type ~" such that .AT~" -- 0 vanish identically. 2. Discontinuity Surfaces with Own Structure
More complicated is the case in which ~ is endowed with its own structure. Then ]E may posses standard surface tension measured by means of a surface referential stress T and a surface tension induced by the material substructure and measured through a surface referential microstress S. By definition, T(I - in | m ) = T, S ( I - m |
m ) = S.
Let/3* be a genetic part of/3 intersected by the discontinuity surface ~:, as in Section IV.B.1, T v and St,, are, respectively, standard and generalized tractions at the curve 0/3" n Z (i.e., at the intersection of Y: with the boundary 0/3" of/3"). Vector u has been defined in Section IV.A: At each X belonging to 0/3* n Z, u(X) is the normal at the curve 0/3" N Y: at X in the direction of the tangent of Z at X. With these premises, the external power is written here as t3, --
*
(b. ~ + / 3 . ~) +
f
~*
(Tn. ~ + S n . ~)
+ f ( T u . :~+ + •u. ~b+) da /3*OZ
(126)
Multifield Theories in Mechanics of Solids
37
By shrinking/3* at the discontinuity surface E as in Section IV.B. 1, 79~xt reduces to the external power "Dext ~*NE developed on E namely, r
T'~,t -+ T'~,tnr.
as 13" -+ 13" n E
(127)
where, thanks to the previously mentioned hypotheses of continuity of bulk interactions b and/3, 79~;'n$ =fB*n~: ([Tm /']
+ [Sm" ~]) + L t3* n:c (Tu
9~+ + ~u 9~ ~:)
(128)
Balance equations at the discontinuity surfaces can be obtained from (128) by applying the same procedure of Section II.D, that is, by requiring the invariance of 79ext B*NE with respect to all changes of observers After some calculations in which one accounts for the arbitrariness of k, ~,/3*, and uses Gauss theorem on the surface E, one finds the necessity of the existence of an element 3 of the cotangent space of .AA, called surface self-force, such that
ATz
=
eTF
T -
(vzAT)s
on E
(129)
and the validity of the following balance equations: [T]m + Divx T = 0 [S]m + DivES - 3 = 0
on E on E
(130) (131)
Equation (129) represents a sort of generalized "balance of couples" on E, whereas (130) and (131) are balances of interactions on E. The surface self-force must be considered to be the difference of two terms 3' and 3" such that 3' satisfies (129) and 3" belongs to the null space of .AT, that is.,
A r3'' = O. R e m a r k 5 Note that, in absence of surface substructural interactions ,9 and z, i.e. in absence of substructure, the standard surface Cauchy stress TF T is symmetric, i.e. ~F T = F~ T.
Proposition 6 By virtue of balance equations (35), (38), (42), (129), (130), and (131), it follows that
f (b 9 + ~ 9~) + f
~ (Tn 9k + Sn 9~)
BNE
+ f (/T/m. r*J+/S/m. ~e~)+ f(T. F + ~. N +~. e~)
(~32)
Paolo Maria Mariano
38
The last integral of equation (136) is thus the internal power "Dint in presence of a discontinuity surface endowed with its own structure within the body.
BIBLIOGRAPHIC NOTE Balances of interactions at interfaces free of own structure have been obtained in Capriz and Virga (1994). Special forms of balances (124) and (125) suitable for micromorphic (or higher-order micromorphic) continua are in Germain (1973). General balances of interactions at interfaces endowed with own structure have been obtained in Mariano (2000a). The matter of discontinuity surfaces in standard Cauchy continua is treated extensively in scientific literature (see, for example, Abeyaratne and Knowles, 1990, 1991, 1994; Pence, 1992; James, 1983; Gurtin, 1995, 2000, and references therein). Of course, in the case of Cauchy continua (see references mentioned previously), equation (131) is not present, and equation (129) reduces to the requirement of symmetry of the surface Cauchy's stress TF r. The scientific literature on discontinuity surfaces in standard Cauchy continua includes cases in which surface measures of interaction are considered as well as cases in which they are not. For example, the derivation of equation (130) and detailed comments on it can be found in Gurtin (1995, 2000).
V. Constitutive Restrictions Usually, in multifield theories one postulates that the first second laws of thermodynamics can be written in a form analogous to the one of Cauchy continua once one considers the extra power developed by the substructural measures of interaction. First, one formally assumes for any part 13" of 13 at each instant t the existence of the internal energy E1 (/3"), the entropy Ht(]3*), the heat flux Qt(]3*, ~,e), and the entropy flux QtI-I(B*, B *e) exchanged by B* with its exterior B *e. After the "natural" assumption that E1 and Ht are time differentiable, one then writes for the first law of thermodynamics d
d t E t ( B *) - Qt(13* ' 13*e) + "l)ext13* ,-
(133)
and for the second law d
d t H t ( B , ) >_ QH(B, ' ~,e)
(134)
(heat sources and entropy sources are neglected here for the sake of simplicity).
Multifield Theories in Mechanics of Solids
39
By introducing Helmoltz free energy density as {free energy density} = {internal energy density} - {temperature} {entropy density}
(135)
the second law may be written in terms of free energy, and is a tool that allows one to derive constitutive restrictions on the measures of interaction once the external power ~S)ext has been substituted by the internal power "pint. Then one must represent explicitly all the elements of (133) and (134). Basically one introduces bulk densities for internal energy and entropy; however, surface internal energy and entropy densities (together with surface heat and entropy fluxes) can be introduced in the presence of interfaces when relevant to describe different physical phenomena. Here thermodynamic phenomena related to variations of temperature are not treated (no details on the various explicit expressions of (133) and (134) are then given) and an isothermal version of the second law of thermodynamics is considered. It is called here mechanical dissipation inequality and prescribes that d
m dt
{free energy of 13"} - "Dint ,-t3, -< 0
(136)
A. CONSTITUTIVE RESTRICTIONS IN THE BULK In the bulk, it is assumed that {free energy of B*} - ft3, ~
(137)
where 7t is the bulk free-energy density. Consequently, the mechanical dissipation inequality becomes d--~ , ~ -
,(T'~'+z'q~+S'V~b)-<0
(138)
Usually, one assumes for ~ a constitutive structure of the form - ~(F, ~o, V~o)
(139)
By calculating the time derivative of the free energy in the mechanical dissipation inequality and collecting terms, it follows that
ft3((Ov~ - T)-~" + ,
1
z). q, +
- S)- V~b) _< 0
(140)
In principle, given any triplet (F, ~o, V~o) representing the isothermal state of any material patch placed at X, one can arbitrarily choose velocity fields F, ~b and
40
Paolo Maria Mariano
~'q~ from (F, qo, Vg~). Consequently, since the integrand of (140) is linear in the velocity fields, the following proposition follows. Proposition 7 interaction:
The following constitutive restictions hold for bulk measures of T = 0v~(F, ~, Vq~)
(141)
z = 0~p(F, qa, Vqp)
(142)
$ = 0v~,~p(F, qo, Vqo)
(143)
Other expressions for #r could be assumed; for example one may choose #r = ~(F, ~, Vqo, F, ~b, Vgb). By inserting this expression in (138) and calculating the time derivative, one obtains t~,((0r~ - T). ~" +
- z).
+
+
- S ) . Vqb) 0
(144)
dr3 ,
Since one can arbitrarily choose in principle both velocity and acceleration fields, and the integrand is linear in the acceleration fileds, one obtains 0~.1# = 0, 0~ ~p = 0, 0v~ ~P = 0 that reduce ~p to the expression (139).
B.
C O N S T I T U T I V E R E S T R I C T I O N S AT D I S C O N T I N U I T Y S U R F A C E S
When a discontinuity surface E endowed with its own structure crosses the body, a surface excess energy must be accounted for. It assures the stability of the surface itself, in some sense its existence as a "thin layer" within the body. In this case, one writes {free energy of t3*} - f~, o + f B * n z 4~
(145)
where 4~ represents a surface free-energy density (i.e., the excess energy at the interface). Consequently, the mechanical dissipation inequality changes as
dt
*
V, +
*NZ
~
)
-
*
( T . ~" + z. ~, + S . V~,)
, ~ (
m. [~l + <S>m . [ ~ l ) - f ~ , ~ (T 9F + ~ 9Iq + 3 9~,•
_< 0 (146)
Multifield Theories in Mechanics of Solids
41
where, in writing (146), one uses the result in Proposition 6. By developing the time derivative of the integral ft3, ~P and taking into account the results of Proposition 7, the interfacial mechanical dissipation inequality follows:
afu
dt
*n~
fu *nz (
- { (ql". I~ + S. fil + 3" ~• Jr3*NE
< 0
(147)
Usually, one may accept for 4~ the following constitutive structure:
4, - ~(F, ~, N)
(148)
underlining in this way that the excess energy at the interface depends on the bulk deformational and substructural state immediately next to it. By developing the time derivative of 4~, the interfacial mechanical dissipation inequality (147) reduces to
f~
*NZ
((aF4, - 7 ) . I~ + (a~4, - 3)" r177+ (aN4, - s). lq)
- f ((T)m. [k] + (S)m. [~]) < 0 Jr3*NE
(149)
Given any triplet (F, qa+, N) representing the isothermal state of any material patch at the interface, it is possible in principle to choose arbitrarily velocity fields F, ~+, Iq from (F, qo+, N). Because the integrand of previous inequality is linear in the velocity fields, the following proposition holds.
Proposition 8 of interaction:
The following constitutive restrictions hold for surface measures qI' = OF4~(F, qo, N)
(150)
3 = 0~,#~(F, qp, N)
(151)
S = 0N4~(F, q#, N)
(152)
The arbitrariness of E implies also that (T)m. [k] + (S)m. [qb] > 0
at E
(153)
Once the explicit expression of the surface free energy has been selected, the surface measures of interaction follow from (150) to (152). Representation theorems for such energy expressions would be desiderable; they could assure that the derived measures of interaction possess the properties assigned by definition.
42
Paolo Maria Mariano
A research program with the aim of describing (or better, representing) energies such as 4~(F, qg, N) for interfaces is still far from being developed.
BIBLIOGRAPHIC NOTE
A general discussion on the expression of the principles of thermodynamics in multifield theories together with the derivation of bulk constitutive restrictions in absence of interfaces is in Capriz (1989). Standard references on thermodynamics of continua are Truesdell (1984), Coleman and Owen (1974), Ericksen (1998a), Silhav3~ (1997). The validity of a general mechanical dissipation inequality such as (136) has been proved in Mariano (1998) by requiting the boundedness from below the action functional along state transformations.
VI. Evolution of Defects and Interfaces in Materials with Substructure
When a body B is subjected to standard deformations, a material patch at X is mapped into x(X, t), the placement in the current configuration ~t at the instant t. Conversely, by the inverse motion x-l(X, t), one may associate the actual placement x of the material patch with its original place X. The picture becomes more complicated when the integrity of the body is altered by, for example, the evolution of defects or phase interfaces. When one observes such an evolution, one may describe it by taking time-varying parts of the reference configuration B. In the following, R(t) indicates any time-varying part of B. The loss of integrity of the body due to the evolution of defects or phase interfaces is basically considered as an additional "independent" kinematics in the reference configuration; this kinematics is associated to R(t). The time variation of R(t) generates new interactions between R(t) itself and the rest of the body; they drive or obstruct the motion of R(t) and are measured by means of a set of additional forces called configurationalforces. These forces are material forces in the sense that they live (or better, they are generated) in the reference configuration. In the standard picture of deformative processes, configurational forces do not exist. The first Piola-Kirchhoff stress T and the referential body forces b (as well as the referential measures of substructural interaction/3, z, S) are only mathematical pictures in B of the real interactions induced by deformation in the current configuration. They are obtained, in fact, by means of pull-back of the measures of interaction arising in the current configuration into B, which is considered fixed permanently.
Multifield Theories in Mechanics of Solids
43
When some part of 13 varies in time, the configurational forces generated by its evolution are measured through second-order tensors and internal forces. These tensors are introduced first as undeterminate terms that develop extra power in the kinematics of R(t), then are expressed explicitly in terms of standard and substructural measures of interaction. Inertial effects are neglected in this section for the sake of simplicity.
A. CONFIGURATIONAL FORCES IN THE BULK
It is assumed that the boundary 0 R(t) of R(t) is a regular surface in R 3, endowed with a outward unit normal n, and parametrized by a pair of parameters Ul and u2 in a way such that
X ~ OR(t)----> X -
X(ul,
U2,
t)
(154)
The velocity v of 0 R(t) in the reference configuration is ^
u -- 0 t X ( U l , //2, t)
(155)
Only the normal component V -- v 9n of v is independent of the parametrization (Ul, u2, t), and this property is crucial in the following. Let e be any continuous field of the time t and the place X. The time derivative of e following 0 R(t) is indicated with e ~ and defined by eO = ~ d e(]~(u i, u2, t'), t')lt,= t
dt'
(156)
and is the time derivative of e along trajectories crossing 0 R(t) at t' = t. When the previous definition is applied to x(X, t) and qo(X, t), it follows that x~ =
~ + Fv
(157)
qo~ = qb + (Vqo)v
(158)
The configurational interactions generated by the motion of R (t) in the reference configuration/3 are measured by means of 9 a
second-order tensor ~' representing the configurational stress
9 an internal configurational body force g 9 an external configurational body force e
44
Paolo Maria Mariano
The balance of bulk configurational interactions and their expression in terms of standard and substructural measures of interaction are shown in the following discussion. To obtain these results, first consider material observers that measure events in the reference (material) configuration. In evaluating the power -,pext - R ( t ) developed during the migration of R (t) in/3, a fixed material observer writes
ext_f R(t)
(b. z~+ / 3 . ~) + (t)
(Tn. x ~ + S n . ~p~ + ~ n . v)
(159)
R(t)
Forces g and e develop no power because they act on bulk points of/3 that are considered fixed when evaluated by a fixed material observer. Bulk interactions b and/3 develop power because they are mathematical pictures in/3 of interactions that in the current configuration act on points that move with respect to the material observer through x (X, t). When one considers a material observer moving with constant velocity v*, such an observer sees g and e as migrating with velocity v* relative to him; so g and e ,pext, mov. obs. evaluated by develop an extra power density (g + e) 9v*. The power --R~t) the moving observer can be thus written as R(,)mo
.
--
(b. ~ + / 3 . ~) + (t)
f.
(g + e). v* (t)
+ f (Tn. x ~ + S n . ~ ~ + ~ n . (v + v*)) J~ R(t)
(160)
However, a classic axiom of mechanics requires that the power be independent of the observer, namely, j~)ext
R~t)
~
,E)ext, mov. obs.
--R~t)
VV*
(161)
To satisfy this requirement, taking into account the arbitrariness of v*, it is necessary and sufficient that
f
R(t)
n+f, (g+e)-0 (t)
(162)
which is the integral balance of configurational forces on R (t). The arbitrariness of R (t) and the application of Gauss theorem imply DivY + g + e = 0
in 13
which is the local balance of configurational forces.
(163)
Multifield Theories in Mechanics of Solids
45
By taking into account the explicit expressions (157) and (158) of x ~ and qo~ the power "l:3ext --R(t) can be also written as 'Qext--fR R(t)
(t)
(b./~+/3.~)+f
R(t)
(Tn.~+Sn.qb)
+ f (~ + F r T + Vqor.S)n 9v Jo R(t)
(164)
As anticipated at the beginning of this section, only the component of v normal to OR(t) is independent of the parametrization of OR(t), whereas the tangential component of v depends on the parametrization itself. Physical plausibility requires that -"lgext - R ( t ) be independent of the parametrization of 0R(t). To satisfy this requirement, it is necessary that (~ + F r T + ~7qor*S) n be simply a vector normal to 0 R(t). In this way, it takes only the normal component of v when it is multiplied by v. Consequently, it is necessary and sufficient that + FTT + ~7~or,_S = wl
(165)
where co is a scalar function that is undetermined at this stage and I is the unit tensor. To determine co, the mechanical dissipation inequality is used. It requires here that md
"19ext ; ~---n(t)-
(166)
This inequality is conceptually analogous to the inequality (136). The sole difference is that here the part R of 13 varies in time. As a consequence of the timevariation of R, the following transport theorem holds:
"f,
-'~
(t)
(t)
++f
(167)
R(t)
where V is the normal component of v: V - v- n. By taking into account (165) and (167), the mechanical dissipation inequality may be written as
f,
(t)
R(t)
(t)
(b.zt+/3.@)-f~
R(t)
(Tn.~:+Sn.qb)<0
(168) This inequality must hold for any mechanical process involving R(t), that is, it must hold for any choice of velocity fields. Thus, the following identity must hold c o - 7z
(169)
Paolo Maria Mariano
46
Finally, this implies from (165) that ?-
fl-
F T T - VqoV._S
(170)
In this way it is shown that the configurational stress IF' can be expressed in terms of the free energy density, 7t and the stress measures T and S. When the setting is conservative, 7t reduces to the strain energy density w and has thus the expression of the generalized Eshelby tensor obtained in Section III. Here, the way followed to obtain the explicit expression of ~ assures that such an expression holds even for dissipative mechanical processes. This result guarantees the possibility of using the expression of ~ to describe dissipative processes like the evolution of macrocracks. As a result of the validity of the balance of configurational forces and standard and substructural balances of interactions, the expression of IP implies that g - -VTz + T 9V F + S+VVqo + z+Vqo
(171)
e - - F T b - Vqo T+/3
(172)
B. CONFIGURATIONAL FORCES ON A DISCONTINUITY SURFACE Let E(t) be a moving interface within R (t) of normal m (t), as in Section IV. The boundary 0 R(t) N E(t) of R(t) n E(t) is a curve that can be parametrized by some parameter u in a way such that X 6 0 R(t) n E(t) is given by X - X(u, t). Thus, the velocity field u(X, t) of the curve OR(t) n E(t) is given by u(X, t) -- OtX(u, t)
(173)
Because E(t) moves within the body, points of it are described by X 6 E(t) so that X - X(vl, v2, t), where (vl, v2) is a parametrization different from (ul, u2) on OR(t). Consequently, the velocity ~ of E(t) may be defined as f(X, t)
-- OtX(Ul,
1)2,
t)
(174)
The attention is focused in the following discussion on normal velocities -~ - U m . Note that u-
~ + U0v
(175)
where, as in Section IV, u is the normal of the curve OR(t) N E(t) directed along the tangent to E(t) at OR(t) n E(t) and Uo = u . u. Two different time derivatives may be defined: the first is the time derivative following the curve 0 R(t) n E(t), and the second is the time derivative following
Multifield Theories in Mechanics o f Solids
47
E(t). Let e be any continuous field of the place X and the time t; the former type of time derivative is indicated with e ~, the latter with e -~. Velocity fields following the curve 0 R(t) N E(t) are given by x ~ =/~+ + F•
= (/~) + (F)u
qo~" : ~ + + Vqo+u : (~> + (Vqo)u
(176)
(177)
Velocity fields following E(t) are given by x ~ = ~+ + F+~ = (~) + (F)~
(178)
qo-~ = qb+ + V q o •
(179)
(~) + (Vqo)~
The following relations may be derived easily from previous definitions after some algebra: x ~" = x -~ + UoFu
(180)
qo~ = qo~- + Uo N u
(181)
The case in which a moving interface E(t) crosses R(t) introduces the need for additional configurational forces strictly related to the peculiar kinematics of the interface. They represent the forces driving or obstructing the interface: their balance allows one to derive a complete evolution equation for the interface itself. Surface configurational interactions are measured by means of 9 a surface configurational stress C 9 a surface internal configurational force gz The stress C is a second-order tensor that maps vectors tangent to E(t) into vectors of R 3. In evaluating the power "19ext --R(t) on R(t) taking into account the structure at the interface, a fixed material observer measures an extra power due to C v . u and then writes in the present situation
extf.
R(t)
:
(t)
( b . x --}-/~. ~ ) --}-
( T n . x ~ + S n . q o ~ + IPn. v) R(t)
f + ] ( C u . u + ql'u. x c> + S t , . qoc>) Ja R(t)AE(t)
(182)
Taking into account the definitions of x ~ qo~ x ~, qo~', and the results on bulk
Paolo Maria Mariano
48
TQext may --R(t)
be written as
:~ + / 3 . ~) +
fOR(t)(Tn./~
configurational forces,
~ )R(t) e x t-f R (t)(b.
+ S n . qb + ~OV)
+ f ( T u . ~+ + S u . qb+ + (C + F+r• + Vqo+r_,S)u 9u) da R(t)AE(t) (183) Because TQext --R(t) must be independent of the parametrization of E as previously accepted in treating bulk configurational forces, here it needs to be independent even of the parametrization of the curve 0 R (t) 71 E (t). In other words, if t represents the tangent field to 0 R(t) @ E(t), because the component o f u tangential to 0 R(t) N E(t) depends on the parametrization of the curve, to assure independence of the parametrization of the curve 0 R(t) N E(t), it is necessary and sufficient that f~
(C + F+rqF + V ~ +r_,s)u 9t - 0
R(t)AE(t)
(184)
However, C may be decomposed into the sum of its tangential component Ctan and normal component m | e to the interface" Ctan maps tangent vectors to E into vectors tangent to the deformed interface, whereas e - Cm. Thanks to the arbitrariness of R(t), (184) reduces to (C + F+rqI ' + Vqp+r*S)u 9t - 0. In other words, the projection of the vector (C + F+rqr + Vqp + r * s ) u on E must vanish; in symbols, one writes that (I - m @ m)(C + F + r 7 + Vqp + r * g ) u - 0, which yields
Ctan + [::T~ + NT_,S __ coz(l - m | m)
(185)
with cozc a scalar function that is undetermined at this stage. Consequently, it follows that C + (F)r'II' + (Vqo)r_, S - cor~(I - m | m) + m | c
(186)
where c - (C + (F)rql" + (V~)r_,S)m. The following formula connects vectors e and c" e - c - T r (F)m - S(Vqo)m
(187)
When one writes the power as perceived by a moving material observer one must consider the contribution of the internal force gr.. As in the case of bulk configurational forces, the requirement of invariance of the power with respect to changes from fixed to moving observers and vice versa implies the validity of the following integral balance of configurational forces that holds in presence of
Multifield Theories in Mechanics of Solids
49
moving interfaces within the body: 0 R(t)
(t)
R(t)nz(t)
(t)nZ(t)
It is assumed that g and e are continuous on/3, whereas IP may suffer jumps at the interface. Consequently, by shrinking R(t) to R(t) n X(t), the arbitrariness of R(t) implies the following balance of configurational forces at the interface: on ]E
[I?]m + DivzC + gz = 0
(189)
The normal component of the interface balance of configurational forces plays an important role: It allows one to write the evolution equation for the interface itself. To obtain such a normal component, it is necessary only to multiply previous balance by the normal in at ]E. First one must consider that DivzC = Div~Ctan + Divz(m | e) and that 9 m.
DivzCtan
Cta n 9t - -
--
T Divz (Ctan m)
-
Ctan 9V z m
- D i v z ( C r ( I - In | m)m) +
Cta n . t
9 m . Divz (m | c) = Divz c As in Section IV, L is the curvature tensor of Z such that its trace/C = tr L is the overall curvature. With these premises, the normal configurational balance of forces can be written as Ill. [~]nl
+ Ctan " t -+-
Divz;z + gz = 0
on E
(190)
where gz = m . g z . To obtain an explicit evolution equation for the interface, it is necessary to exploit some consequences of the mechanical dissipation inequality written here considering the contribution of configurational forces as [cf. eq. (146)]
'(f,
dt
(t)
(t)nz~t)
vext < 0
(191)
Standard transport theorems prescribe that
dt dt
(t)
(t)
R~t)
(t)NZ(t)
(192) (193)
(t)nz(t)
(t)nZ(t)
where 4~-~ is the time derivative of 4~ following Z(t).
R(t)NZ(t)
PaoloMariaMariano
50
As a tool for developing future calculations, note that from (175), (178), (179), and (185), it follows that fa R(t)NE(t) (Cv. u + T v .
-~
R(t)nE(t)
X t>
coEua+f
+
R(t)nE(t)
~o~') (Cu.~+Tu.x-~+~;u.qo
-~)
(194)
By substituting (183), (192), (193), and (194) into (191), the mechanical dissipation inequality reduces to
fR(t)~-~(t)n~(t,[~P]U+L(t,nr~(t)(4~--c~ICU) R(t)nE(t)
(t)
R(t)
R(t)nE(t)
(195) This inequality must hold for any choice of the velocity fields, and for any choice of Ua. This implies as a first result wE -- 4'
(196)
The identity (196) allows one to express the tangential part of the surface configurational stress explicitly in terms of standard and substructural measures of interaction and the surface free energy, namely, Ctan - -
4~(I - in | m) - FTT - NT,_S
(197)
By shrinking R(t)to R(t) N E(t), as a result of the regularity properties assumed (and declared) in previous sections and to (196), in the limit R(t) --+ R(t) N E(t), the inequality (195) reduces to ([Tin. 5r + [ S m . ~]) (ONE
-{
Jo R(t)NE(t)
(t)nz(t)
(t)NY,(t)
(Cu. ~ + Tu. x -~ + Su. qo-~) _< 0
(198)
To reduce the inequality (198) further, one may use the following lemma, which is valid under the assumptions declared in this section up to this point.
Multifield Lemma
Theories
in M e c h a n i c s
51
of Solids
9
f
( C u . ~ + T u . x -~ + S t , . ~ o -<)
R(t)nE(t) f
(ql'. F -~ + S . N -< + 3" (~o) -< - [ T m . ~] - [,Sm. ~])
f
([~p]U + U g z
dR(t)nE(t) dR(t)NlE(t)
+ wzU/C + c . m -~)
(199)
The proof of this lemma is rather long and somewhat tedious. It is based substantially on the definitions (176)-(179), the use of Gauss theorem, and the auxiliary results 9 V z ~ - - V z ( U m ) -- - m | V z U + U V z m -- - m | m -~ - UL; 9
x -~ 9Divz'~' -- - [ T m .
~] - U m .
[FTT]m;
9
~o-~ 9D i v z ~ - 3" (~o) -~ - [ S m . ~b] - U m .
[V~pr,_S]m;
9 ql'- V z x -< = T . (F) -< - (T T ( F ) m ) . m -< - u(FTqr) 9L; 9 S . Vzqo -< = S . (V~o) -< - (S r m) 9m -< - u(NT_,s) 9L. The results in this list are consequences of previous definitions in Section VI and the balances at the interface derived in Section IV. By introducing (199) into (198) and taking into account (196), one obtains the integral dissipation inequality at the interface
L(t)nE(t) ~)~- -~-L(t)nz(t) UgZ + f
dR(t)nz(t)
(c. m -~ - T . F -~ - 3" (~o) -~ - ~" N -~) 5 0
(200)
The arbitrariness of R ( t ) implies the following local version of the mechanical dissipation inequality, namely, ck <- + e . m <- - T . F <- - 3 . (~o) ~- -
S . N <- + U g z
<_0
(201)
Two results follow from (201). First, it is possible to generalize the results of Proposition 8 to take into account the possible anisotropy of ~2(t). To this aim, one may choose a constitutive relation of the form 4~ = ~(m, F, q0, N)
(202)
52
Paolo Maria Mariano
where the dependence on the normal m accounts for the anisotropy of the interface itself. In further steps, one calculates the time derivative 4~-~ of 4~ following E(t), inserts it into (201) (by collecting analogous terms) and observes that, given any state (m, F, qo, N), velocity fields m -~, F -~, (qo)-~, and N -~ may be chosen arbitrarily from (m, F, qo, N). Consequently, because (201) must hold for any choice of velocity fields, the following relations must hold:
"IP - aFt(m, F, ~, N)
(203)
,~ = a ~ ( m , F, ~, N)
(204)
,~ -
aN~(m,
(205)
c -
- a m P ( m , F, ~ , N)
F, qo, N )
(206)
where, relative to the results in Proposition 8, there is here an explicit dependence on the normal of the interface and vector c represents the force generated by the anisotropy of E during the motion. Consequently, as a result of (203)-(207), the mechanical dissipation inequality reduces to gz U _< 0
(207)
gz - - ~ z ( m , F, ~o, N, U)U
(208)
A solution of (207) is given by
where gz is a positive scalar function of arguments that are the same as in (203)(206) plus the normal velocity U. Note that gz is the normal component of the dissipative force gz driving the interface [inequality (207) states the dissipative nature of gz]. Because gz is dissipative, it depends on the velocity U and may depend on the other state variables: gz must be assigned by a constitutive prescription. When one assigns ~z, one determines the evolution of E(t). The evolution equation of E(t) is, in fact, the normal configurational balance (190) once one substitutes in (190) the explicit expression of ~ [identity (170)], the expression of Cta n [identity (197)], the constitutive restriction (206) [taking into account (187)] and finally (208), which is the fundamental ingredient for writing the evolution equation. Remark 6 With the previous results, it is possible to prove that the tangential component of the configurational force balance (189) is identically satisfied once gz is purely normal" namely, once gz(I - m | m) - 0.
Multifield Theories in Mechanics of Solids
53
BIBLIOGRAPHIC NOTE A rather general treatment of the influence of material substructures on the expression of bulk and surface configurational forces can be found in Mariano (2000a). Some of the proofs discussed here are developed following different methods in some aspects from the ones of the previously mentioned paper. Results on the influence of scalar order parameters on the influence of bulk configurational forces can be found in Fried and Gurtin (1993, 1994, 1999) and Fried and Grach (1997). A detailed treatment of configurational forces in standard Cauchy materials can be found in Gurtin (1993, 1995, 2000), Gurtin and Struthers (1990), Angenent and Gurtin (1989), Abeyaratne and Knowles (1991), Gurtin and Podio-Guidugli (1996), Maugin (1993, 1995), and references therein. The term configurational is due to Nabarro (1985) (see Ericksen, 1998). The seminal ideas on this subject are in Eshelby (1951, 1975).
VII. Crack Propagation in Materials with Substructure One application of the framework of configurational forces discussed in Section VI is the analysis of the propagation of macrocracks. Evolving cracks may be considered to be evolving curves (for two-dimensional bodies) in the reference configuration. By inverse motion, in fact, one could pull back the actual opened and deformed crack into a curve in the reference configuration, which is a picture of the crack taken as closed. Of course, when the crack evolves in the actual configuration, its picture in the reference configuration evolves in time: configurational forces driving the crack must be considered. By using some of the results presented in the previous discussion, it is possible to discuss the influence of material substructures on the force driving the crack tip. Experiments show the influence of material texture on crack propagation (see Herrmann and Roux, 1990). When one models some material as a Cauchy continuum, such an influence is considered only indirectly, through constitutive equations, and often models of the local cohesive structure at the tip of the crack are proposed. They offer interesting special solutions; however, these models can be difficult to manage when one considers the crack propagation to be a global bifurcation problem, because in this case it is often necessary to know the overall distribution of material texture rather than the situation near the crack tip. In the framework of multifield theories, one may derive the driving force at the crack tip in a way accounting for the explicit influence of the perturbations induced
Paolo Maria Mariano
54
on standard interactions by material substructures. This is the result shown in this section, in which for simplicity, all calculations are restricted to two-dimensional bodies.
A. KINEMATICSOF PLANAR MOVING CRACKS
In this section, the body B is considered as an open compact region of the two-dimensional point space ,re. A crack in B is identified as a regular curve C represented by a function r : [0, ~] ---> B (with [0, ~] some bounded interval of the real axis ~) defined as follows:
{r
' ( 0 , ~ ] --+ B r(0) ~ 0B
(209)
This definition states that one considers the crack as a curve in B starting from the boundary and ending within/3, without intersecting further on the boundary of/3. The crack tip is, of course, r(g), and is indicated by Z; it is a point in oc2 belonging to the interior of/3. The curve C is also characterized by its tangent vector t - 0~r, 5 ~ (0, g) and the lateral normal m (thus such that t. m = 0). When the crack evolves in a time interval [0, d], C is a function C(t) of time, defined as
C(t) = {r(s, t) E 1315 E [0, ~], t E [0, d], r(0, t) ~ 0B, r(0, t) -----r(0, 0), Vt} (210) with the condition
C(tl) c C(t2), u
< t2
(211)
It is assumed that C evolves without intersecting further on the boundary of B: the tip Z becomes a function of time Z(t) and Z(t) 6 B, whenever t 6 [0, d]. In other words, one imagines studying the crack propagation during a time interval in which the crack does not completely cut the body B (Figure 2). The velocity of the crack tip Vtip is defined as dZ Vtip = dt
(212)
When one assigns the tangent of C at the tip Z--namely, t (Z(t))--one assigns the direction of propagation of the crack. It is thus possible to write Vtip = f'(t)t (Z(t)). From this definition, it follows that m(X) 9Vtip ~ O, as X --+ Z(t). Special parts of B are of future use; in particular, it is necessary to consider a disc D centered at the tip. The boundary 0 D of the disc intersects C at only one point
Multifield Theories in Mechanics of Solids
55
11
B t(z(o) FIG. 2. Geometric characterization of a cracked body. denoted with X A . The outward unit normal at 8 D is indicated by n, whereas the radius of D is indicated by O. When the crack evolves, the disc is considered to be time varying to follow the evolution of the crack; thus D and XA become D(t) and XA (t). Let f be the velocity of the curve C(t). The velocity of XA (t) can be written as ra(t) -~ Xa(t) -- Ua(t)ta(t), where/~a is the scalar amplitude of the velocity and ta is the tangent of C(t) at the point XA. Because t (Z(t)) is prescribed, it is assumed that ta(t) ~ t (Z(t)) Cta(t) - ~ V(t)
as O ~ 0 as O --~ 0
(213) (214)
Let e represent any continuous field of the place X and the time t. The time derivative of e following the crack tip is defined as e~(X, t) = 8te(Y, t)lv=x-z(t)
(215)
where Y denotes a genetic point in s By applying this definition to x and qo, the velocity fields following the tip can be written as
x <>- ~ + Fvtip qD<>= ~ + (V ~)u
(216) (217)
Assume now the existence of a tip velocity field ~tip and a tip order parameter ~lti p when the crack is deformed. To be more precise, first assume that if X 6 C and x(X, t) is the position of X at the instant t, then x(X, t) ~ x(Z(t), t) and qo(X, t ) ~ qo(Z(t), t), as X ~ Z(t) uniformly in time. The assumption of the existence of Vtip and Vitip i s tantamount to requiting that x<>(X, t) ~ Vtip and 9:,<>(X, t) ~ @tip, as X ~ Z(t) uniformly in time. rate
Paolo Maria Mariano
56
The boundary of the time-varying disc D(t) may be parametrized by some parameter v in such a way that X 6 0 D ( t ) =~ X - fK(v, t). The velocity vo ofthe boundary 0 D(t) of the disc is given by VD(X, t) = OtfK(v, t)
(218)
Only the normal component of v D - - n a m e l y , / 2 -- VD.n--is independent of the parametrization of 0 D(t). Velocity fields x ~ and ~~ following OD(t) may be defined in the same way as in (157) and (158) as X~ -- ~ + FVD
(219)
qa~ - qb + (Vqp)VD
(220)
The physics of the crack imposes a kinematical condition, namely a condition of impenetrability" [x] 9m >_ 0
(221)
In other words, when the body deforms, the margin of the cracks do not penetrate one into another. In presence of a crack, the mapping x(.) is no longer pointwise one-to-one and becomes a piecewise bijection" x(.) is one-to-one to within the curve C; that is, a set of zero volume measure.
B. BALANCE OF STANDARD AND SUBSTRUCTURAL INTERACTIONS AT THE TIP The following assumptions on standard and substructural measures of interaction apply: 9 bulk measures of interaction b and/3 are continuous on/3 9 the self-force z is continuous on/3 9 stresses T and S may be singular at the crack tip and suffer jumps across C; they are also continuous and continuously differentiable outside C 9 surfaces stresses 7i" and S and surface self-force 3 are not considered Two new measures of interactions are assumed acting at the crack tip; they represent external actions acting at the crack tip as "concentrated f o r c e s "
Multifield Theories in Mechanics of Solids 9 tip force
57
btip
9 tip substructural measure of interaction
~tip
The previous assumptions guarantee that balance equations (35), (38), and (42) hold in the bulk, and that balance equations (128) and (129) hold on C. Additional balance equations hold at the tip; they may be derived by writing the integral balances of interactions with respect to the disc D and shrinking the disc to the tip. Integral balances accounting for tip interactions are
fo
b+f
D
(Tn) + btip -- 0
fD(~ -- Z)-l- fo (Sn)-~- ~tip --
(222)
(223)
To write them, it is necessary only to add btip and Otip to (119) and (120). Because T and 8 may be singular at the crack tip, by shrinking the disc to the tip--that is, letting O ~ 0 u t h e s e balances reduce to
where
btip + ftip Tn - 0
at tip
(224)
/3tip + ftip 8n - 0
at tip
(225)
ftip must be interpreted as limo_,o fo"
Remark 7 The assumption that the pair (x <>, ~o<>)tends to (Vtip, ~Ntip) a s X ~ Z(t) implies that
retip) - 0
(226)
Sn . (~<> - fVt~p) = 0
(227)
L T n . (x ~ -
fp
C. EFFECTS OF INERTIA
To evaluate the effects of inertia on standard and substructural balances one may follow the technique adopted in Section II. To this aim, first assume the following
Paolo Maria Mariano
58
decompositions of tip interactions into inertial and noninertial components: bti p - blnp -~- btnip
(228)
fl tnip
(229)
fl tip -- fl iTp +
These decompositions must be used together with relations b = b in + b ni and f l _ flin+ flni. The noninertial part bt'~/p of btip is a possible standard external force applied at the tip of the crack, whereas flt~iipcan be interpreted as a possible external substructural action at the tip. The terms bt'~/pand fltnip are not essential, and are quoted here only for the sake of completeness. On the contrary, the inertial terms may be identified explicitly. To obtain this identification, it is first necessary to evaluate the production of standard momentum and the production of substructural momentum, indicated with pr and pr., respectively, during the evolution of the D(t). Because the disc D evolves in the reference configuration, there is an inflow of momentum across OD(t). The inflow of standard momentum is -faD(t)pi~bl, whereas the inflow of substructural momentum may expressed as -f~o(t)OcpX Lt, where L/ is the normal velocity of OD(t) (i.e., /d = yD. n). Consequently, one may write by definition
dfo p -f
prpr.
~
---
(t)
(t)
(230)
D(t)
(t)
D(t)
The basic step to identify bitp and fli~ is to consider a balance analogous to (53) and write
blnp +
fo
(t)
bin -- --pr
I' I~ITP + I t~in -- --]Or. dD (t)
(232)
(233)
By shrinking D(t) to the tip, the assumptions of regularity of bulk measures of interactions b and fl and the regularity properties of motions imply
fD(t) bin; fo(t) flin; m
d f
dt Jo(t)
p~:
d -~fD(t) ~ --fo(,)~
all tend to0
asD--+Z
(234)
Multifield Theories in Mechanics of Solids Because the previously listed terms vanish in the limit D ~ identifications hold because v o 9n = U --+
Vtip 9n)
bi~p - ft/p(pfc)(vt/p, n) =
as D ~
ftip(p~ |
59 Z, the following
Z
n)Vtip
~'~p : ftip(~X)(Vtip "n) ~ ~.p(~,x Q n)Vtip
(235)
(236)
As a consequence, the tip balances (224) and (225) reduce to
Remark 8
btn.p+ ftipTn : - ftt.p(px | n)vtip
at tip
(237)
tip "3t- Sn -- -- fti p (O(oX Q n)Vtip t~niftl.p
at tip
(238)
Physical plausibility suggests that standard and generalized tractions
Tn and Sn are bounded by up to the tip as D ~ Z. When bt~i%a n d fltnp vanish identically (as usual), the hypothesis of boundedness of the stresses implies fi Tn -- O ==~fi n| P
P
. Sn=O==~ fi n| p
(239)
P
D. TIP BALANCE OF CONFIGURATIONAL FORCES
When the crack evolves, its mathematical picture C varies in time and configurational forces intervene to obstruct or drive the crack. The following assumptions apply to the framework developed in Section VI:
9 ~ may be singular at the crack tip and suffer jumps across C; it is also continuous and continuously differentiable outside C 9 g and e are continuous and on I~ 9 C is now a vector c and is continuous along C 9 gz is continuous on C
Paolo Maria Mariano
60
New configurational forces are associated with the tip and are peculiar of tip singularity. They are 9 internal tip configurational force gtip 9 external tip configurational force etip of inertial nature The integral balance of configurational forces (188) on D must be written here accounting for the tip configurational forces. It may be deduced from work invariance arguments such as those, discussed in, Section VI and is expressed by
D(t)
IPn+ fo (t)
(g+e)+fo
(t)NC(t)
(gz)
+ gtip -k- etip -- CA - - 0
(240)
By shrinking D at the tip of the crack, equation (240) reduces to (241)
gtip + etip -- Ctip + ftip ]?n -- 0
which is the configurational tip balance. It is now necessary to characterize the tip configurational forces explicitly. First, the attention is focused on etip, which is of inertial nature. To derive an explicit expression for it in terms of velocity fields, one may follow a procedure analogous to that discussed in Section II.A and may derive etip by requiring that the rate of kinetic energy E of D(t) plus the power of all inertial forces on D(t) vanish identically. In symbols, one writes
d ~ + fo 0 -- dt
(bin "x + t~in "(P) + btip" in reap + ,~tip" in ,Tvtip+ etip 9Vtip
(t)
(242)
Because D(t) varies in time, in computing the rate of the kinetic energy it is necessary to consider its inflow through the boundary of D(t), because this inflow is due to the motion of D(t) itself. Then the rate of kinetic energy of D(t) is given by dt
~
fo(1 (t)
)
2(P~" ~) + k(~, @) -
D(t)
(p~. ~) + k(~, @) H (243)
The next step is the introduction of (243), (235), and (236) into (242) and the shrinkage of D(t) at the tip Z. Note that f~o(t)(.)H ~ ftip(.)(vtip, n) because
Multifield Theories in Mechanics of Solids
61
D(t) ~ Z(t). At the end of calculations, one finds that etip " u
-- u
" ftip (~(,o~, " x)-'F- k(qo, qo)) n
- vti p
f
Vtip)n-- u
. ftip(OCpX . Wtip)n
(244)
To reduce (244) further, one defines the relative kinetic energy ~.rel as ~.rel-1 1 1 "~plX-~ltip 12 and observes that ~.rel- -~p~Itip " Vtip -- "~pX" X - pX" reap. Moreover, Ytip PVtip " Vtip -- limos0 fad Pretip " ~r but retip is independent of spatial coordinates, then lim0~0 fad pretip 9retip -- lim0~0 retip 9r4tipfao P -- 0 because the density of mass p is a continuous function. By taking into account these auxiliary results on the relative kinetic energy, and because (244) must hold for any choice of the velocity of the tip, it follows that
etip--ftt.pl~reln-+-ftipk(qo,@)n-ftip(O(oX.f~Ctip)n
(245)
The identity (245) characterizes completely the external configurational force etip and specifies its inertial nature.
E. CONSEQUENCES OF THE MECHANICAL DISSIPATION INEQUALITY To give some characterization of the internal configurational tip force gtip, it is necessary to exploit the mechanical dissipation inequality, which is now written as
d re{free energy of D(t)} - {power developed on D(t)} < 0
dt
(246)
A line free energy density 4~ along the crack C is considered besides the bulk free energy density ~p. The line free energy accounts for surface tension c along the faces of the crack. It is assumed that ~p is continuous on/3 and may suffer jumps at C, whereas 4~ is continuous along C. The line free energy density 4~ does not depend on the time t. From (203) to (206), 4~ may depend only on the normal m of C because standard and substructural surface measures of interaction are not considered here. However, m does not depend on t because there is no motion of C along its normal. Possible phenomena of aging are not considered; thus 4~ does not depend explicitly on t. With these assumptions, d--td{free energy of D(t)} -- --~
(t) ~ + -~
(t)nc(t) 4~
(247)
62
Paolo Maria Mariano
Because the normal velocity of C is zero, the transport theorem (167) holds and is written here as --
(t)
dt
r =
(t)
r +
D(t)
eL/
(248)
whereas for the line free energy one finds -dt
(t)NC(t)
~) -- ~tip ~/ -- ~)A bl A
(249)
because the integral in (249) is a line integral. By inserting (248) and (249) into (247) and writing explicitly the power developed on D ( t ) , it follows that fD
(t)
~-s .qt_ f
-fo
(t)
D(t)
(!~r~[) '~ ~)tip (/ -- ~)A~IA
(b"/~ +/3" r
-- btip 9~r
fo
D(t)
(Tn. k + S n .
r + CU)
- ~tip " fVtip - etip 9Vtip + U AIA 9CA < 0
(250)
The last integral in (250) is an alternative manner to write f
D(t)
(Tn.
Xo
+ Sn.qo + I?n. vo) o
and the expression used in (250) can easily be obtained by using (219), (220), and (170) [see also the analogous integral in (183)]. A first result can be obtained directly from (250). Because the inequality (250) must hold for any choice of the velocity fields and--among othersmfor any choice of/~a, as a result of the arbitrariness of D ( t ) , the following identity must hold:
= t. c
(251)
it characterizes completely the tangent part of the configurational stress c: this tangential part is the surface tension along the faces of the crack. By shrinking D ( t ) at the tip, one obtains from (250) the tip mechanical dissipation inequality: ~)tip V -- btip " Vtip - ~tip " Wtip - etip " u
- ft, Tn. ~ - ft/pSn 9~, _< 0
-
lP(Vtip . n )
(252)
63
Multifield Theories in Mechanics of Solids
The use of the tip configurational balance (241) and the identity (251) allow one to reduce the tip mechanical dissipation inequality (252) to
ftip 1 .fti,o fti o o
gtip " Vtip -- btip
" Vtip -- tOtip "~ltip
--
l[f(Vtip
"
n)
Note that from (170) it follows trivially that
so that inequality (253) reduces to gtip 9vtip - btip 9Vtip - ,Otip 9Wtip - ftip T n . (~, + Fvtip) - ftip,..~n . ( ~ -4c-(~Tqo)Vtip) < O
(255)
From (216) and (217) and the assumption previously stated that (x<>, qo<>) (Vtip, wtip) as X ~ Z, it follows that ft.pTn'(~'+Fvtip)+fttipSn'(~(Vq~
Sn
(256) By substituting (256) into (255) and using the tip balances (232) and (233), the mechanical dissipation inequality reduces to gtip " Vtip < 0
(257)
which characterizes the dissipative nature of the internal configurational force gtip. It is necessary to prescribe constitutively only gtip rather than the energy release rate (as usual in technical literature). The constitutive prescription of gtip is subjected only to the condition (257).
F. DRIVING FORCE
Previous results allow one to derive an expression of the driving force at the tip of the crack, which accounts for the presence of material substructures and the interactions they generate (and consequently the expression of the energy release rate during crack propagation). To simplify the developments in the following, it is first assumed that btnp and/3t]p vanish identically.
64
Paolo M a r i a M a r i a n o
By using the identities (170) and (245), the tip balance of configurational forces can be written as ftip((lP "at- ~.rel "-[- k ( ~ , (a) - O~X 9qVtip) I - F r T - ( V ~ ) r * S ) n - (.tip -- -grip
(258) The integral in (258) is indicated here with jm for compactness of notation, jm represents the tip traction exerted by the material on an infinitesimal neighborhood around the tip of the crack. The component of the tip balance of configurational forces along the direction of propagation of the crack is obtained by multiplying the balance (258) by t (Z(t)) t ( Z ( t ) ) . jm -
t (Z(t))
9 C,ip -
-gtip.
t (Z(t))
(259)
By denoting with Jm the product t (Z(t)) 9jm and using (251), equation (259) may be written as (260)
Jm -- Ck,ip -- --grip" t (Z(t))
In equation (260), the term grip 9t(Z(t)) is the internal force exerted by molecular bonds that opposes motion of the tip, whereas f - Jm -dPtip -- O(oX " W t i p )
t (z(t)) 9f,/p((~ +
ereZ +
~:(~,, ~)
I - FTT + ( v ~ ) T * S ) n -
~)tip
(261)
is the driving force at the tip accounting for the influence of material substructures. Because Vtip = ~'(t) t (Z(t)), the internal dissipation inequality (257) and the tip balance (258) imply f'~' >_ 0
(262)
which represents a version of the internal dissipation inequality. From (262), when the crack grows (i.e., when V > 0), 9 the driving force must be nonnegative f >_ 0
(263)
9 the tip traction must form an acute angle with the direction of propagation
t(Z(t)), j,, >_ 4~tip > 0
(264)
Multifield Theories in Mechanics of Solids
65
The results (263) and (264) coincide with the analogous results in Cauchy continua.
G. A MODIFIED EXPRESSION OF J INTEGRAL
As a consequence of (239), the tip traction jm reduces to jm --
(( ~ + ~Pls 1 2 + k(qo, ~) ) I - F T T - ( V ~ ) T,,S ) n
(265)
Consequently, it follows that Jm V -- jm " Vtip = jm " V ( t ) t (Z(t))
= ftip ( T n . i~+Sn. ~ + (Tz + ~ Pl~lZ + k(~p, (o)) (Vtip . n))
(266)
The product Jm V is the flow of energy into an infinitesimal neighborhood of the crack tip. Jm is thus the dynamic energy release rate accounting for substructures because it has the physical dimensions of an energy. When the influence of substructures is not considered, Jm coincides with t(Z(t)).
' )
~p + ~Pl~l 2 I - FTT
)n
which is the standard dynamic energy release rate in Cauchy continua (see, e.g., Freund, 1990; Gurtin, 2000; Maugin, 1992). If inertial effects are absent, the tip traction jm reduces to its "quasi-static", counterpart jm,qs given by
jm,qs -- ftip(~l - FTT-(Vqo)T ,__S)n -- ftip~n
(267)
If the body forces b and ~ are absent, the material is homogeneous [in particular, ~ = ~(F, qo, Vqo), so ~p does not depend explicitly on X], the faces of the crack are free of standard and substructural tractions (T• = 0; S • = 0), the "quasi-static" energy release rate Jm,qs - t (Z(t)) . jm,qs i$ path independent; that is, Proposition 10
Jm,qs -- t
(Z(t)) . fr Pn
(268)
where F is any closed, regular, nonintersecting path beginning and ending at the crack.
66
Paolo Maria Mariano To sketch the proof of previous proposition, let intF denote the closed region
of,f2 with boundary 1-'. By Gauss theorem, fr ]?n = fintF Div]]3) + fintI'nC []l~]m" By using the configurational force balance (163), Div? may be substituted by the sum - (g + e). The absence of body forces implies e = 0 [see (172)], whereas the homogeneity of the material implies g - 0 [one calculates the gradient of 7t, inserts it into (171), and uses Proposition 7]. In addition, by using (170), one realizes that the hypothesis concerning the faces of the crack implies that [/~]m = [~]m. As a consequence,
t (Z(t)) 9f~trnc [~p]m -- f~trnc [~p] t 9m - 0 and the validity of Proposition 10 is proved.
H. ENERGY DISSIPATED IN THE PROCESS ZONE
When a crack propagates in an elastic-plastic material, a critical zone around the crack tip occurs: the process zone. It is highly unstable (in the sense that any increment of the loads may alter its coherence even drastically), in a certain sense "fragmented," so that one may doubt that basic axioms of continuum mechanics (e.g., the continuity of the material) do not work well within it. When one evaluates the energy dissipated during the evolution of the crack, hence of the process zone, one realizes that J integral is no longer sufficient to describe the energy dissipated into the process zone and other path integrals must be introduced. Let P indicate the process zone around the crack tip. Assume that the boundary 0 P of P is a closed regular curve without self-intersection that admits normal n. During the evolution of the crack, P is considered time dependent [i.e., P = P(t)]. The curve 0 P(t) is described by some function X = X(vp, t), with Ve an appropriate parameter along 0 P(t). The velocity Vp of 0 P is given by ve(X, t) -OtX(ve, t). In a local frame {X*} of coordinates centered at the tip Z, the velocity Vp may be written as V p = Vp,tr
+ CI X X* + a ' X * + a
where Vp,tr and/1 are the rigid translational and rotational components, respectively; a'X* is the component of the velocity associated to the self-similar
Multifield Theories in Mechanics of Solids
67
expansion of P(t); and d is the component associated to the distortion of the process zone. Of course Vp,tr, (~, and a* are independent of the space coordinates. Proposition 11
The energy dissipated into the process zone, ~(P), is given by dp(p)
--
YP,tr
" jm(P)
+/1" L + a*M + I
(269)
where i n ( P ) --
1 (( ~pl:~l 1 2 ) 1 (( 1 ) ( ( ,~pl:~l) 2 + k(~o, ~) I 1 (( 1 ) r +
L =
!/r + ~PlRI 2 + k(qo, qb) I - FTT - (gqo)r_,s
P
M =
r +
n x X*
- FTT - (Vqo) T_,S n . X*
P
I =
) ) ) )
+ k(cp, ~) I - F r T - (Vqo)~_,S n
P
r + ~plfr 2 + k(qo, qb) I - F r T - (Vqo)r_,S
P
- f (Tn. ~+Sn-~b) aa P
(270) (271) (272)
n-a (273)
Note that when the results of this subsection are applied to the direct modeling of plates (i.e., to Cosserat surfaces) it is possible to express both J integral and the basic laws of the evolution of cracks (224) and (225) directly in terms of normal and shear stresses and bending moments. This result allows one to obtain a strong reduction of the computational burden in numerical calculations involving cracks that cut the thickness of the plate completely.
BIBLIOGRAPHIC N O T E
This section is based mainly on some unpublished notes of the writer. Proposition 11 may be proved by adapting to the present situation the general results of Proposition 5 in Mariano (2000a); a special case of Proposition 11 can be found in Mariano (1995). In the case of Cauchy materials, the evolution of cracks has been treated with the framework of configurational forces in Gurtin and Podio-Guidugli (1996), Gurtin (2000) and Maugin (1992), and some classic results on crack propagation (see Freund, 1990) on fracture have been reobtained within such a theoretical setting.
68
Paolo Maria Mariano
Detailed discussions on the process zone around the crack tip can be found in Aoki et al. (1981, 1984), Curtin and Futamura (1990), Hutchinson (1987), Freund and Hutchinson (1985) and Lam and Freund (1985). The concept of energy release rate has been introduced in Atkinson and Eshelby, (1968) and Freund (1972), whereas the original motivation of the J integral can be found in Rice (1968).
VIII. Latent Substructures Material substructures are called latent when there is a set of holonomic or anholonomic constraints relating the order parameter to the descriptors of the macroscopic motion and deformation. In defining the concept of latence, Capriz (1985) writes: "I say that the microstructure is latent when, though its effects are felt in the balance equations, all relevant quantities can be expressed in terms of geometric and kinematic quantities pertaining to apparent placements" (p. 49). First, one assumes that 9 there is no substructural inertia: k(qp, qb) - 0 9 substructural bulk interactions are absent:/3 = 0 An immediate consequence is that the balance of substructural interactions (63) reduces to DivS = z
(274)
Consequently, the generalized balance of couples (38) changes in eTF T = Div(.A TS)
(275)
whereas the density of internal power of substructural interactions changes in z. ~ + S . ~'~b - Div(S r qb)
(276)
It has a divergence form only and the product sT ~b can be interpreted as a substructural flux of power that corresponds to the interstitial work flux, which is necessary to consider in the special case of higher gradient elastic materials (as shown in the following). Another crucial assumption is the following: 9 the substructural flux of power is objective
Multifield Theories in Mechanics o f Solids
69
This condition requires that ST ~b must not change when qb changes into ~ + A~I (i.e., S T ~ -- S r (qb + .A~I) for any choice of the rigid rotational velocity el)- This implies that ST A = 0
(277)
Condition (277) further reduces the balance of couples to TF T = FT T
(278)
which is the standard symmetry condition of symmetry Cauchy stress TF T. The last assumption that defines completely latent substructures is the following: 9 the order parameter is constrained by a set of frictionless holonomic and anholonomic constraints that express it in terms of the deformation gradient F and, perhaps, of its gradients R e m a r k 9 When Div(S ~ ~b) -- 0 for some special choice of the order parameter, substructural interactions become powerless. In this case, the order parameter appears on constitutive equations only, and an evolution equation for it must be considered instead of the balance of substructural interactions. This is another case in which multifield theories reduce to internal variable schemes. R e m a r k 10 The balance of substructural measures of interactions gives rise to evolution equations for the order parameter even in situations more general than those occurring in the case of latent substructures. The constitutive prescriptions of Proposition 7 are peculiar of thermodynamic equilibrium, or of a reasonable 13 neighborhood of it. In principle, nonequilibrium parts of the interaction measures, depending on the velocity fields, could be considered along nonequilibrium thermodynamic processes. This happens, for example, when viscosity phenomena occur. An interesting case occurs when one considers a decomposition of the self-force z into its equilibrium and nonequilibrium components. The equilibrium part of z is given by (142), whereas the nonequilibrium part z ne may depend on the rate of the order parameter, that is, z -- 0~o~(F, qo, Vq~) + z"e(F, qa,Vq~; qb)
(279)
The nonequilibrium part of the self-force z ne is dissipative and is such that z "e. qb > 0 13The physical meaning of reasonable is currently a matter of open discussion.
(280)
70
Paolo M a r i a M a r i a n o
A solution of (280) is
(281)
Zne --zne~o
with i ne an appropriate definite positive tensor (possibly scalar in some special model) such that ine is a function
i n e : zne(F, qo, Vr
~0)
(282)
In common special cases, a decomposed free energy of the form
-- ~l(V, ~) + @2(qO,V~) [with 61(I, qg) -- 0, I the unit tensor] may be selected and
(283)
lp2 chosen as
1
~ 2 = -~b V qo . V qo + cr(qo)
(284)
with b an appropriate constant and cr(qo) a double-well coarse-grained potential, as in cases of solidification or solid-to-solid phase transitions. When this happens, the bulk balance of substructural interactions (274), as a result of (143), changes into
A ~ - b A q o - 0~0o'(~o) - 0~o~l(F, ~)
(285)
which is a generalized form of the Ginzburg-Landau equation. When, in fact, both A and qO are scalar valued and the body does not undergo deformations, (285) reduces to A~b = bAqo - 0~ocr(~o)
(286)
which is the standard Ginzburg-Landau equation with kinetic coefficient A.
A. SECOND-GRADIENT THEORIES AS SPECIAL CASES OF LATENT SUBSTRUCTURES An important case of latence is characterized by the internal frictionless constraint = ~(F)
(287)
which expresses the order parameter as a function of the macroscopic gradient of deformation F. From (287), by time differentiation one obtains that along the motion qb = ( ~ ) F "
= (~)(grad~)F
(288)
Multifield Theories in Mechanics of Solids
71
where grad indicates the gradient calculated with respect to x. When the velocity i is rigid [see (13)] gradi = e/l
(289)
@R --- e (OqF~) Fc 1
(290)
and (288) changes into
consequently, from (12) it follows that (291)
A = e(o~)F
This relation reduces (277) to (292)
F = o
eS r (~)
which implies that the third-order tensor A, whose elements are given by (293)
t~AB C -- ( s T ) ~ A ( O F ~ ) i B F i C
is symmetric in the last indices, that is, (294)
AABC = AACB
In the standard treatment of internal constraints [such as (287)] in Cauchy continua, it is necessary todecompose the stress T in its "active" and "reactive" parts, indicated with T and T, respectively (as standard in scientific literature on internal constraints). The latter is assumed to be powerless. Analogously, here it is prescribed that T=
r
+T;
a
z--z+z;
r
S=
~
+
(295)
~.F+~.~+~.V~=0 v~', r
(296)
and
By substituting (288) into (296), one finds two conditions. The first condition is that r
+ ~(aF~ + S(V(aF~)) = 0
(297)
The second condition is that third-order tensor S(0v~) is symmetric in the last two indices
(298)
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Paolo Maria Mariano
Another basic assumption here is that the free energy density 7r has the following structure: A
~p - ~p (F, V F )
(299)
When one uses the mechanical dissipation inequality to obtain constitutive restrictions on the measures of interaction, one finds that (140) reduces to
f
,
+ +
v + (~vv~P
vv) _< o (300)
Given any state (F, V F ) , velocity fields ~" and V~" can be chosen arbitrarily from (F, VF). This arbitrariness implies that a
a
I" + ~ + V(SOF~) -- (VS)OF~ -- 01~
(301)
a
S Ov~ - Ovv ~
(302)
By inserting (297) and (301) in (299) and using (302), one proves the validity of the following proposition"
Proposition 12 (Capriz, 1985) -- Ov~ - Div(Ovr~) - Div(F skw(OvF~F-l))
(303)
or, in components, Tiaa -- OFiAff/ -- (O(VF)iA, O ) , , --(Fis(O(VF)j,c ff/ Faj' -- O(VF)jBAO ) F ~ I ) , c
(304)
where capital indices refer to the reference configuration, whereas the other indices refer to the current configuration. Note that (302) is the standard constitutive restriction of second-gradient elastic materials, which are thus special cases of continua with latent substructure (i.e., special cases of multifield theories).
BIBLIOGRAPHIC NOTE
This subsection is based on Capriz (1985). Many other remarks on latent substructures can be found in Capriz (1989), whereas the special case of smectic liquid
Multifield Theories in Mechanics of Solids
73
crystals is treated in Capriz (1994). The standard theory of second-gradient Cauchy materials can be found in Dunn and Serrin (1985), where the necessity of the introduction of a rate of supply of mechanical energy, called interstitial working in the balance of energy, is proved to be necessary to eliminate the incompatibility with the second law of thermodynamics shown in Gurtin (1965) for these models of materials. The fundamental result of Capriz (1985) is that the interstitial working is not an object whose existence is assumed without any explicit reference to some types of interactions; rather, it is a consequence of the existence of substructural interactions due to a substructure that generates the oscillations of deformations that are measured by VF.
IX. Examples of Specific Cases The framework discussed in previous sections allows one to describe many material substructures. Detailed special theories can be found in the references listed at the end of Section I. Here some prominent examples are summarized briefly. To build up any special multifield theory describing some particular phenomenon, one must 9 choose a suitable order parameter and then M to model the substructure of the material 9 choose an appropriate form of the free energy ~p 9 evaluate the possible occurrence of latence of substructures induced by the need of some internal constraints motivated by the physical experience After these steps, the constitutive equations for the measures of interaction follow from Proposition 7 and, in the presence of discontinuity surfaces, are supplemented by the results in Proposition 8. In this way, one can write field equations (35), (39), and (46) [with the addition of (129), (130), and (131) in presence of interfaces] in terms of x(.) and qo(-) and then attempts to solve them. In particular, when one chooses to solve (35), (39), and (43) by means of finite element schemes and then must analize integral (relaxed) forms of the field equations, one obtains stiffness matrices more articulated than those of Cauchy continua and an array containing both the components of the placement (or the displacement) of each material patch and the components of the order parameter.
74
Paolo Maria Mariano A. MATERIAL WITH VOIDS
The model of materials with voids is the simplest multifield model. When pores are finely distributed throughout the body, one way to describe them is to choose the order parameter qo as a scalar that associates to each point X the void volume fraction of the material patch at X. In this case, .Ad reduces to the interval of the real axis [0, 1] and A is identically zero [see discussions before (43)]. Substructural bulk measures of interaction/3 and z reduce to scalar, whereas the microstress S becomes a vector as a consequence of Proposition 7. An interesting case is the one of linear elastic materials with voids (or elastic porous materials), with perhaps some damping effects in the pores. By indicating with e the infinitesimal strain tensor e = symVu (u is the displacement), linear constitutive equations relevant for this case (and written with respect to a reference state free of stress) can be obtained by taking for the free energy ~p a quadratic form in e, qO, and Vq9 and applying Proposition 7. In addition, one can consider small viscous effects due to the surface tension at pores by using the procedure discussed in Remark 8 before (283). At the end of calculations, one obtains for elastic porous materials the following constitutive relations with damping: g-,(1)
,,-,(2)
1"ij --Cijhk~hk + ,-.ijkq),k + Gij (D
(305)
-- C(3)~ - C(4)~ - Ci(5)~ij - C~ 6)q),i
(306)
t.~i _ ,,-,(7) c ij qg,i -Jr-_(8) I-.ijk~jk -Jr- C~9)q)
(307)
where 1", ~, and S represent (as in Section III) linearized measures of interaction; Cijhk is the usual stiffness tensor; and C (i) are appropriate constitutive constants. Of course, ~0,i denotes the derivative Ox, q). In the isotropic case, (305)-(307) reduce to
Tij = ~.~ij~,hh + 21zeij -I- ~ ( l ) ~ i j
(308)
= _ ~ ( 2 ) ~ _ ~:(3)q9 _ ~(l)6h h
(309) (310)
~--~i = ~(4)q9,i
where ~ij is the unit tensor and X and # are the standard Lam6 constants and are related to the other constants by the following inequalities: /z>O; 3X + 2/z ~> O;
~(4)/>0;
~(3)>~0
(3X + 2/z)~ <3) >~ 12~
(311) (312)
Multifield Theories in Mechanics of Solids
75
which assure uniqueness and weak stability of solution of the dynamic balance equations. When one analyzes the dynamics arising from (62) and (63) with constitutive equations (308)-(310), one finds the validity of the proposition in the following. Proposition 13 (Cowin and Nunziato, 1983) Transverse waves propagate at a constant speed without affecting the porosity of the material and without attenuation. Longitudinal acoustic waves are both attenuated and dispersed as a result of the changes in material porosity that accompany the wave. Note that the damping term in (306) arises from a nonequilibrium component z ne of the self-force. In this sense, these linear materials are not purely elastic.
BIBLIOGRAPHICNOTE The nonlinear theory of materials with voids has been introduced by Nunziato and Cowin (1979). Proposition 13 is discussed in Cowin and Nunziato (1983). Further studies on this topic can be found in Cowin (1985), Dhaliwal and Wang (1994), Nunziato and Walsh (1978), Diaconita (1987), Fr6mond and Nicolas (1990), and Mariano and Bernardini (1998).
B. TwO-PHASE (OR MULTIPHASE)MATERIALS In two-phase materials, possible phase transitions may change one phase into another. A scalar-order parameter may be used to associate to each point X information on the phases; that is, it may associate to X the volume fraction of one phase, or it may be the indicator of same phase. In the latter case, ~0 is zero if there is not a certain phase and is equal to 1 if the contrary is true. Then interfaces (across which phase transition occurs) are the boundaries of sets that admit ~0 as an indicator function. When ~0 is interpreted as the fraction of a certain phase, .A4 reduces to [0, 1], and the interfaces are considered to be smeared throughout the whole body. Transition layers are identified throughout the body as the thin layers in which V~o undergoes large oscillations. The use of such a kind of order parameter allows one to approximate the nonconvex variational problems arising in the evaluation of two phase bodies. If order parameters are not introduced, one manages, in fact,
76
Paolo Maria Mariano
potentials that have two wells (one well for each phase). During the evolution of phase transitions, the order parameter is ruled by an evolution law, such as (186), which derives from the balance of substructural interactions. In the case of multiphase materials, one may choose as order parameter a list ~ = (991 . . . . .
qgN)
(313)
whose entries take values on [0, 1] and are subjected to the constraint N Z(/9 i -
(314)
1
i-I
thus only N - 1 are independent. In this case, .A4 becomes the cube [0,1] x . . . x [0,1]
Ntimes
(315)
BIBLIOGRAPHIC NOTE
Two-phase materials have been treated with the help of scalar-order parameters in Penrose and Fife (1990), Colli et al. (1990) Fr6mond (1987), Fried and Gurtin (1993, 1994, 1999), and Fried and Grach (1997), and references therein; the original idea of using order parameters to describe phase transitions is from L. D. Landau.
C. COSSERAT CONTINUA
Each patch is considered to be a rigid body that can rotate independently from the neighboring patches. Such a rotation can be described by using as order parameter an orthogonal tensor Q (X, t). In this case, M reduces to the space of orthogonal tensors Orth + such that QTQ = I and det Q = 1. Simple calculations show that z" is always zero because .A4 coincides with Orth +. The strain tensor E, which in Cauchy solids is given by I ( F T F - I), is here given by E = 2MTM- M T - M
(316)
where 1
M - :(QTFZ
I)
(317)
Multifield Theories in Mechanics of Solids
77
In the setting of small deformations, the infinitesimal strain tensor e becomes = V'u - eq
(318)
with q a vector describing the rotation. If one chooses the free energy as a function of F, Q, VQ, by applying Proposition 7, one realizes that z is a second-order tensor, S a third-order tensor, and, consequently,/3 is a second-order tensor. Moreover, after a straightforward calculation, one realizes that ,4 is a third-order tensor given by .A - - e Q ( . A i j k eijh Qhk), and then (39) becomes -
e(jOQ r - zQ T + (DivS)Q r)
-
0
-
(319)
It is then possible to put 1
- ~e(/3Q T)
(320)
-- ~e(zQ r + S V Q v)
(321)
1
,.~- IoSQT 2
(322)
where/3 and ~ are vectors whose components are 1
~i - -~Oijkl~jlO T
(323)
Zi = ~Oijk (Zjl O~ + Silm Qrlm,k)
(324)
1
whereas S is a second-order tensor given by 1
8ij -- -~OijkSklm O/rm
(325)
The generalized balance of couples (38) reduces to 1 e T F r = e~ 2
(326)
which eliminates the self-force from the balance of substructural interactions. Consequently, the balance of substructural interactions becomes /3 - 1 T F r + Div,~ - 0 2
(327)
To account for inertial effects, one must consider the right-side term of (63), in which the kinetic coenergy X density must depend only on the product Q v Q to be objective. Simple algebra allows one to write tensor Q T 0 in terms of a vector
78
Paolo Maria Mariano
a -- e ( Q TI)), and X may depend thus only on a. In this way, the explicit dynamic expression of (327) becomes
/ 3 - ~ T1 F T + D i v , ~ -
( p OX(a))" Oct
(328)
The framework of Cosserat materials is widely used to model structural elastic elements such as beams, plates, and shells. In addition, some proposals for a Cosserat plasticity have been developed with the aim of analizing certain aspects of localization of deformations.
BIBLIOGRAPHIC NOTE
A detailed list of references on Cosserat materials is given at the end of Section I. Here I followed some notes in Capriz (1989), Ericksen and Truesdell (1958), Antman (1972), and Villaggio (1997). For Cosserat plasticity, see Steinmann, (1994).
D. MICROMORPHICMATERIALS The model of micromorphic materials is useful to describe the influence of large molecules on gross mechanical behavior, as happens in some polymers. Each material patch is considered to be a cell that may undergo additional deformations independently of the neighboring patches. The order parameter qo is a second-order invertible tensor (the gradient of microdeformations) and A4 reduces to Lin + [i.e., the space of linear forms between vector spaces], with positive determinant. By using Proposition 7, one realizes that the self-force z becomes a second-order tensor whereas the microstress S is a third-order tensor. Nonlinear measures of deformation are (cf. Capriz, 1989)
E--~I(FTF-I); M- ~(qoVF-l); ~-qo -~Vqo
(329)
By assuming that r = Qg~ after rigid changes of spatial observers (where Q is an orthogonal tensor describing the rotation of the observers), one obtains that A is a third-order tensor given by eg~, or in indices, ~ i j k = eijl~Olk. Of course, because ~o ~ Lin § it never vanishes; then z" in (43) is identically zero, as in the case of Cosserat continua.
Multifield Theories in Mechanics of Solids
79
An interesting aspect of micromorphic material appears evident in the setting of small deformations. It is possible to introduce the relative infinitesimal strain ~r as er = V u -
qo
(330)
and associate to each patch the state (e, er, Vqo). In this way, the internal power becomes f13 (T " ~ "~- Z " ~r -'~"S " V qo)
(331)
,
If one assumes an internal constraint prescribing that ~r = 0
(332)
the inner power (329) reduces to
fB
(T.e+S.
VVu)
(333)
,
which is the inner power that one writes when dealing with second-gradient materials. From the multifeld scheme of micromorphic materials with the internal constraint (332), strain gradient plasticity has been formulated to account for length scale-dependent effects in plasticity, such as the Hall-Petch effect (the strength of polycristalline aggregates increases with decreasing grain size) or the experimental results showing strong size effects in the case of torsion tests on copper wires. Basically, one realizes that the presence of VV u in (333) accounts for weak nonlocal effects as a result of the latence induced by the constraint (332). However, each multifield theory is "weakly nonlocal" in nuce as a result of the presence of the gradient of the order parameter in the list of constitutive variables.
BIBLIOGRAPHIC NOTE
Micromorphic materials have been introduced in Mindlin (1964; see also Grioli, 1960, 1990; Bofill and Quintanilla, 1995; Mindlin, 1965b; Mindlin and Tiersten, 1963; Eringen, 1992, 2000; Capriz and Podio-Guidugli, 1976). For the strain gradient plasticity, see Fleck and Hutchinson (1997).
Paolo Maria Mariano
80
E. NEMATICLIQUID CRYSTALS Nematic liquid crystals are characterized by the presence of rodlike molecules smeared throughout the liquid. A commonly accepted choice of the order parameter is a second-order tensor Do representing a second-order approximation of an unknown orientation distribution function 14 of the molecules 9 In some sense, DD(X) represents the macroscopic average of the orientation of the molecules in the patch at X. Do is also such that
DD -- DTD
trDo = 1
(334)
and has nonnegative eigenvalues. Usually, the deviatoric part D of Do is adopted instead of Do, and one has D = Do - lI. Nematic liquid crystals are characterized by the coincidence of two of the three eigenvalues of D, which is represented in the form
D=~' (
d|
')
9 d.d-1
9
2' < ~ < 1
(335)
The scalar ~"is interpreted as degree of orientation. Vector d (which is the averaged direction of the molecules of each material element) and the parameter ~" are taken as order parameters. Consequently, one has from Proposition 7 measures of interaction z0 and So associated to d (which is the averaged direction of the molecules of each material element) and measures of interactions zc and Sc associated to ~'. There are two manifolds A/l; the first manifold coincides with the unit sphere in It~3 and the second manifold is the interval [ - 21, 1]. Two different substructural balances of the same form of (42) must be satisfied. As pointed out in Example 2 in Section II.D, because one deals with a liquid, one must have measures of interaction in the current c c C configuration and denotes them with/3~, z~, $~,/3~, z~, S~. In this way, one writes two balances of substructural interactions [because one has at his disposal two order parameters; see (44)] /3~ - z~ + divS~ - otd
(336)
/3~ - z~ + divS} - 0
(337)
the Cauchy balance of standard interactions divT C + b' = 0
(338)
where T r is Cauchy stress and b c denotes bulk forces in the current configuration, and only one balance of torques because r on [ - ~1, 1] vanishes identically skw (T ~ + z~ | d + (gradd) rS~) - 0 14An analogous choice can be made in the case of microcracked bodies.
(339)
Multifield Theories in Mechanics of Solids
81
where grad is calculated with respect to coordinates in the current configuration. _ct! m 0. Note that in (337), zc The free energy depends on cl and its gradient and ot and its gradient, in addition to the measures of deformation. The use of Proposition 7 allows one to obtain constitutive prescription after pushing forward the relevant constitutive variables. The degree of orientation is not the sole characteristic parameter that can be derived from D. To describe the possible emergence of optic biaxiality of the nematic, two other parameters must be used and may be derived from D. In particular, one realizes that the tensor Do generates an ellipsoid. Then the degree of prolation dg and the degree of triaxiality dt are useful parameters to describe the nematic distribution and are defined by I
dg -- -~
(340)
(3)~i - 1 ) I
dt =
6~/-3
I)~i - ~.i+11
(341)
i=1
where ~-i a r e the eigenvalues of Do, i = 1, 2, 3. They can also be considered to be order parameters.
BIBLIOGRAPHIC NOTE
The theory of liquid crystals with variable degree of orientation has been introduced by Ericksen (1991) and the derivation of balance equations as well as the constitutive restrictions can be found in his paper. The degree of prolation dg and the degree of triaxiality dt have been introduced by Capriz and Biscari (1994).
F. FERROELECTRIC SOLIDS
Some crystalline materials as barium-titanate experience spontaneous polarization associated to the rearrangements at the crystalline temperature called Curie temperature. To describe polarization, one may choose as order parameter a vector p of ~3 such that [Pl < Ps with Ps a material constant. In this way, M becomes the ball {p 6/~3 I Ip[ < Ps}. In writing the overall power 79 , one must consider not only the external power j'Qext and the inner power TAint of mechanical interaction, but also a self-power TAself of electrical nature. Then, 79 = j-Qext _ ~')int _ TAself. In particular, with reference to
Paolo Maria Mariano
82
any part B~' of the current configuration ~t of the ferroelectric body, one writes
pext(B'~)-faB: ( b C ' v + j O c ' o ) + f
B~ ( (
Tc -- 2(P" 1
n)2 ) n . v +
SCn 90 ) (342)
/,
~s)int(]3t) -- [
dB
79self(Bt) --
(T C. gradv +
(pC(grade)p. v + ;
S c.
gradl~ + z c. 1~)
fie. P)Jr f
l
~ ( p - n ) 2 n 9v B;
(343) (344)
where v is the velocity ~, e is the electric field, and the superscript c indicates that the relevant fields are taken in the current configuration. Moreover, one may disregard gravitational forces and consider the noninertial parts of bulk interactions b c and/3 c to be measures of the electrostatic interactions of B t with its exterior. Taking into account such a hypothesis, the procedure of Section II.C allows one to obtain balance equations in the current configuration divT C + pC(grade)p = divSC - zC +
PCe --
pC~r
Prop + B p
(345) (346)
where Pm is a material constant and B is an appropriate second-order tensor such that B p . p = O. In the reference configuration, the balances (344) and (345) become DivT + p(grade)p = p~
(347)
DivS - z + pe - (det F)(pmli -+- BI~)
(348)
T = (det F)TCF - r
(349)
S = (det F)SCF - r
(350)
z = (det F)z c
(351)
F)p c
(352)
where
p = (det
To derive constitutive equations one chooses in this case ~ / - - it(F, p) + W ( p ) + 1 ff2Vp 9V p
(353)
where 9 is a material constant and s is such that Lt(Q, .) = 0
(354)
Multifield Theories in Mechanics of Solids
83
for any orthogonal tensor Q. Taking into account (353), Proposition 7 allows one to derive constitutive equations. During the polarization, the walls of ferroelectric domains evolve; in addition, the coupling of mechanical deformation and electric polarization can cooperate and generate localization of domain walls on some "macroscopic" discontinuity surface that may evolve and can be considered as an "emerging" domain wall. To study the evolution of standard and emerging domain walls, one can use the general results of Section VI, taking into account that here the configurational stress I? assumes the form ~ = ~pl - F r T - (Vp) TS. In the case of rigid ferroelectrics, I? reduces to
I~rigid--
W ( p ) + ~ff V p . Vp
I-
ff V p ) r V p
(355)
and the evolution equation (190) reduces to an equation describing an isotropic motion by curvature (see Dav] and Mariano, 2001) which fits, for example, the experimental results on barium titanate.
BIBLIOGRAPHIC NOTE
The model of ferroelectric materials summarized in Section IX.F is discussed in Dav] (2001) and in Dav] and Mariano (2001) and allows one to deduce expressions for the evolution rules for domain walls and comers in accord with experimental evidences. Basic remarks on ferroelectrics can be found in Jona and Shirane (1962), Little (1955), Hwang et al. (1995), Rosakis and Jiang (1995), and references therein.
G. MICROCRACKEDMATERIALS In bodies free of microcracks, any given deformation maps each patch at X into a point x(X) of the Euclidean space and the displacement u(X) from X to x(X) is u(X) = x(X) - X. When microcracks are diffused within the body, under the same given deformation the patch at X undergoes a displacement u + d, where d is the kinematical perturbation due to the enlargement or the closure of microcracks. At each x, d(x) is a relative displacement; it is the difference between the real placement x'(X) of the patch at X and the theoretical placement x(X) occurring at X if microcracks were absent i.e. d = x' - x. It is thus possible to take the vector d as order parameter; therefore, A//reduces to I~3. By using Proposition 7, one realizes that the self-force z is a vector and the microstress S is a second-order tensor.
84
Paolo M a r i a M a r i a n o
The standard gradient of deformation F is given by F - I + Vu (where I is the unit tensor), whereas the overall gradient of deformation Ftot is given by Ftot F+Vd. Overall measures of deformations associated to Ftot are the overall right CauchyGreen Ctot given by Ctot --FtotFtot T and the overall deformation tensor Etot = 1 ~ (Cto, - I). Two linearized measures of deformation may be derived by taking IVul << 1 and IVcll << 1. They are indicated with eu and ed and defined by eu - symVu
ed -- symVd
(356)
Balance equations may be deduced in the manner indicated in Section II. When one considers the equilibrium of a microcracked body without the help of order parameters, one writes the energy and tries to minimize it over a domain endowed with an unknown set of discontinuities for the displacement field: the microcracks. Then one must find the solution of the variational problem
minlftsw(X,Vu)+fG(ts)r(X,[u]| }
(357)
where w(., .) is the bulk strain energy density, r(., .) is the surface energy density of microcracks, and nm is the normal to each microcrack face. Problem (357) has two unknowns: the displacement field u(. ) and the set G(/3) of microcracks. Thus the equilibrium problem becomes a free boundary variational problem and is very difficult to solve. By using the just described order parameter, one may regularize this free boundary problem and reduce it to a more simple one. In particular, it is possible to "approximate" (357) with the problem
min{fw(x, vu, ct,vcl)]
(358)
where/3 is considered to be free of microcracks and cl represents the kinematical perturbation induced to u by microcracks. When irreversible behavior occurs and the microcracks may grow, one considers the free energy r instead of w and may use the procedure in Section II.D and Proposition 7 to write balance equations of the form Div(OvCr) + b = 0
(359)
Div(OvdTr) -- 0d~ = 0
(360)
s k w ( O v ~ F r + OdTZ|
cl + Vd T0vd~) = 0
(361)
where Ov~ is the Piola-Kirchhoff stress T, 0vo r is the referential microstress S, and Od~/ris the self-force z'. A basic problem is thus to find an explicit expression for ~r (i.e., constitutive equations for the measures of interaction). One way to obtain
Multifield Theories in Mechanics of Solids
85
constitutive equations is to use an identification procedure based on the equivalence of the density of power in the continuum and the power of a cell of a periodic discrete model of the microcracked body. This procedure may always be used when the order parameter has the physical meaning of a displacement, a rotation, or a deformation, provided a suitable choice of the discrete model. The physics of microcracked bodies suggests the adoption (at least in the case of elastic behavior) of a discrete model made of two periodic lattices connected by elastic links. The first lattice (macrolattice) is made of indistinct material points (spheres) and describes the material at molecular level. The second lattice (mesolattice) is made of empty shells (lakes) connected to one another by elastic links and describes the material at the mesoscopic level of microcracks; each shell is forced to deform along a plane orthogonal to its major axis only. This is a case of "virtual" substructures because the stiffness of each shell is the one of the material surrounding each microcrack. Because the lattice is assumed to be periodic, the attention is focused on a cell of the lattice: a representative volume element (RVE). Chosen the RVE of the discrete system, it has N spheres and M shells; in the macrolattice of the RVE there are L links: LN is the number of interlattice links, and LM is the number of links in the mesolattice. It is assumed that the elastic links in the RVE can carry only axial forces: ti denotes the force in the ith link of the macrolattice, zt the force in the lth interlattice link, Zj the force in the jth link in the mesolattice, and the force on the shell located at h is indicated with z0h. Spheres located at a and b in the RVE undergo displacements u a and u b, whereas a shell located at h (or at k) undergoes a relative displacement clh (or clk) of its margins along the plane of deformation. In this way, appropriate measures of deformation in the discrete RVE are Oh, clh - clk, Illa -- d h, and u a - u b, where the last three measures of deformation make sense only when the spheres at a and b and the shells at h and k are connected. There are two steps to obtain constitutive equations for the measures of interaction in the continuum from the discrete model in the case of infinitesimal deformations.
1. First one equalizes the density of the internal work in the continuum with the internal work developed in the discrete RVE and in the case of linearized kinematics writes T . Vu + ~,. cl + , S . V d -
1 Vol(RVE)
LN
M
l=l
h=l
i=1
LM j=l
ti 9(u a - u b)
) \
(362)
Paolo Maria Mariano
86
where L is the number of links of the macrolattice in the RVE, LN is the numbers of interlattice links, M is the number of shells, and LM is the number of links in the mesolattice. 2. Then one must write the measures of deformation in the discrete RVE in terms of the ones of the continuum. A reasonable choice is to take U a --" U ( X ) +
Vu(x)(a - x)
(363)
d h -- O(X) +
Vd(x)(h
(364)
-
x)
with x a point in the RVE chosen such that ua
-
u b =
O h -- O k : Ua -- O h :
Vu(x)(a
Vu(x)(a
-
(365)
b)
Vd(x)(h - k) -
x) -
(366)
By inserting (363)-(367) in (361), one obtains
1
Vol(RVE)
i=1
ti | (a - b) + Z
1=1
1
1 Vol(RVE)
h=l
zt | (a - x)
M
Vol(RVE) ~
(367)
V d ( x ) ( h - x)
z~
z0~ | (h - x ) + ~ zj | (h - ~ j=l
)
(368)
(369)
~ z, | (h - x~ t=l
(370) At this point, it is necessary only to choose appropriate constitutive relations for the links in the RVE to find explicitly the measures of interaction in the linearized kinematies T, ~, ,S in terms of Vu, d, Vcl. The simplest choice is ti - IK(ua - ub), Z~ - - Q d h, z l - ]~-]~(Ua - dh), and zj - - D l d h l l d h l ( ( h - k ) / I h - k12), with K, Q, H, D appropriate stiffnesses. Note that the first three constitutive relations are forms of the Hook's law, whereas zj (the interaction between two neighboring shells) is nonlinear is due to the fact that two neighboring microcracks interact like neighboring dislocations located at a distance ( h - k ) / I h - kl 2 from each other. When one inserts the previous constitutive relations into (368)-(370), one finds explicit constitutive equations for the measures of interaction in the continuum. The simplest form, which is valid in the case of linear elastic behavior
Multifield Theories in Mechanics of Solids
87
(microcracks do not grow) and when the material is centrosymmetric, is the following: T = AVu + A'Vd - Cd S - A'Vu + GVcl
(371) (372) (373)
where the fourth-order tensors A, A', C, G have major symmetries and can be computed explicitly by the just described procedure (a detailed calculation of them can be found in Mariano and Stazi, 2001). Consequently, balance equations (359) and (360) become, in the case of linear elastic behavior of the material, b + div(AVu + A'Vd) = 0
(374)
div(A'Vu + GVd) - Cd = 0
(375)
When one solves with standard finite element procedures the equilibrium problem with constitutive equations (371)-(373), one finds as a basic result the occurrence of localization phenomena of the deformation. This result cannot be obtained by using standard linear elasticity in the setting of Cauchy continua and has an immediate experimental evidence (to make an experiment, the reader can take any rectangular sheet of paper, load it with a traction in the middle point of one of its sides and fix only the opposite side; localization occurs). These localization phenomena can be obtained from (374) and (375) as a result of the presence of the self-force Cd and of the gradient of the order parameter. Basically, I conjecture that one may find localization phenomena of the order parameter (i.e., of the substructure) in a wide range of possible multifield models as a result of the analytical structure of the balance of substructural interactions. From a physical point of view, I interpret these localization phenomena of the order parameter as the description of cooperative patterns of material substructures.
BIBLIOGRAPHIC NOTE
Details about this multifield model of microcracked bodies can be found in Mariano (1999), Mariano and Stazi (2001), and Mariano et al. (2001); localization phenomena of deformation within the setting of linearized constitutive equations are described in Mariano and Stazi (2001) and Mariano et al. (2001).
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Paolo Maria Mariano
Acknowledgments I t h a n k G i u l i a n o A u g u s t i , G i a n f r a n c o Capriz, and E r i k van der G i e s s e n for p r o f o u n d and d e t a i l e d d i s c u s s i o n s a b o u t the topics in this article. M y friend and f o r m e r student F u r i o L o r e n z o Stazi was f u n d a m e n t a l in m a k i n g c o r r e c t i o n s and t y p i n g this article.
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Nunziato, J. W., and Cowin, S. C. (1979). A nonlinear theory of elastic materials with voids. Arch. Rat. Mech. Anal. 72, 175-201. Nunziato, J. W., and Walsh, E. K. (1978). On the influence of void compaction and material nonuniformity on the propagation of one-dimensional acceleration waves in granular materials. Arch. Rat. Mech. Anal. 64, 299-316. Pence, T. J. (1992). On the mechanical dissipation of solutions to the Riemann problem for impact involving a two-phase elastic material. Arch. Rat. Mech. Anal. 117, 1-52. Penrose, O., and Fife, P. C. (1990). Thermodynamically consistent models of phase field type for the kinetics of phase transitions. Physica D. 43, 44-62. Povstenko, Y. Z. (1994). Stress functions for continua with couple stresses. J. Elast. 36, 99-116. Renardy, M., and Rogers, R. C. (1993). An introduction to Partial Differential Equations. Springer Verlag, Berlin. Rice, J. R. (1968). Mathematical analysis in the mechanics of fracture. In Fracture 2 (H. Liebowitz, ed.), pp. 191-311, New York, Academic Press. Rosakis, P., and Jiang, Q. (1995). On the morphology of ferroelectric domains. Int. J. Eng. Sci. 33, 1-12. Segev, R. (1994). A geometrical framework for the static of materials with microstructure. Math. Mod. Meth. Appl. Sci. 4, 871-897. Segev, R. (2000). The geometry of Cauchy's fluxes. Arch. Rat. Mech. Anal. 154, 183-198. Silhav2~, M. (1997). The Mechanics and Thermodynamics of Continuous Media. Springer Verlag, Berlin. Simo, J. C., and Fox, D. D. (1989). On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization. Comp. Meth. Appl. Mech. Eng. 72, 267-304. Simo, J. C., and Vu-Quoc, L. (1988). On the dynamics in space of rods undergoing large motions--a geometrically exact approach. Comp. Meth. Appl. Mech. Eng. 66, 125-161. Simo, J. C., Fox, D. D., and Hughes, T. R. J. (1992). Formulations of finite elasticity with independent rotations. Comp. Meth. Appl. Mech. Eng. 95, 277-288. Simo, J. C., Fox, D. D., and Rifai, M. S. (1989). On a stress resultant geometrically exact shell model. II. The linear theory: Computational aspects. Comp. Meth. Appl. Mech. Eng. 73, 53-92. Simo, J. C., Fox, D. D., and Rifai, M. S. (1990). On a stress resultant geometrically exact shell model. III. Computational aspect of the nonlinear theory. Comp. Meth. Appl. Mech. Eng. 79, 21-70. Simo, J. C., Marsden, J. E., and Krishnaprasad, P. S. (1988). The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods and plates. Arch. Rat. Mech. Anal. 104, 125-183. Steinmann, P. (1994). A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity. Int. J. Sol. Struct. 44, 465-495. Toupin, R. A. (1964). Theories of elasticity with couple-stress. Arch. Rat. Mech. Anal. 17, 85-112. Toupin, R. A. (1965). Saint-Venant's principle. Arch. Rat. Mech. Anal. 18, 83-96. Tricomi, E G. (1954). Lezioni sulle Equazioni alle Derivate Parziali. Gheroni, Torino. Truesdell, C. A. (1984). Rational Thermodynamics. Springer-Verlag, Berlin. Truesdell, C. A., and Toupin, R. A. (1960). Classical field theories of mechanics. In Handbuch der Physics. Springer-Verlag, Berlin. Villaggio, P. (1997). Mathematical Models for Elastic Structures. Cambridge University Press, Cambridge. Virga, E. G. (1994). Variational Theories for Liquid Crystals. Chapmann & Hall, London. Voigt, W. (1887). Teoretische Studien tiber Elasticit~itsverh~iltnisse der Krist~ille. Abh. Ges. Wiss., Gottingen, 34. Wang, J., and Dhaliwal, R. S. (1993). On some theorems in the nonlocal theory of micropolar elasticity. Int. J. Sol. Struct. 30, 1331-1338.
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A D V A N C E S IN APPLIED M E C H A N I C S , V O L U M E 38
Molecular Dynamics Simulation of Nanoscale Interfacial Phenomena in Fluids CHANG-LIN TIEN and JIAN-GANG WENG Department of Mechanical Engineering, University of California, Berkeley, California 94720 I. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. M o l e c u l a r D y n a m i c s S i m u l a t i o n T e c h n i q u e s . . . . . . . . . . . . . . . . A. Classical M o l e c u l a r D y n a m i c s S i m u l a t i o n . . . . . . . . . . . . . . . . B. N o n c l a s s i c a l M o l e c u l a r D y n a m i c s S i m u l a t i o n . . . . . . . . . . . . .
96 97 97 103
III. L i q u i d - V a p o r Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. P l a n a r Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Spherical Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. C y l i n d r i c a l Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103 104 113 119
IV. L i q u i d - L i q u i d Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. P l a n a r Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Spherical Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. M i x t u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122 124 128 128
V. L i q u i d - S o l i d Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. P l a n a r Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Spherical Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. C y l i n d r i c a l Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
130 130 134 135
VI. T h r e e - P h a s e S y s t e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. P l a n a r Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Spherical Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. S p r e a d i n g Wetting and C o n t a c t L i n e R e g i o n . . . . . . . . . . . . . . VII. O t h e r Interfacial P h e n o m e n a . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. S o n o l u m i n e s c e n c e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. C o n c l u d i n g R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
136 136 137 138 139 139 140 140
Acknowledgments ...................................
141
References .........................................
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95 ISBN 0-12-002038-6
ADVANCES IN APPLIED MECHANICS, VOL. 38 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISSN 0065-2165/01 $35.00
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Chang-Lin Tien and Jian-Gang Weng
I. Introduction
Molecular dynamics (MD) simulation, the computational method that simulates detailed material behavior by simultaneously solving the equations of motion for a system of molecules interacting with given potentials, has been applied in various disciplines. Such molecular level understanding provides scientists and engineers with tremendous opportunities for the exploration of material properties in different length and time scales. The past several decades have witnessed an explosive growth in literature of MD simulation. Chou et al. (1999) and Maruyama (2000) provide excellent and comprehensive reviews of MD simulation in microscale and nanoscale studies. Interfacial phenomena have attracted researchers' attention for a century. Research in this field has made a significant impact on many industry applications, ranging from macroscale (e.g., power generation and material processing) to microscale and nanoscale (e.g., thermal inkjet printing and liquid microcoolers). However, many fundamental problems still remain unsolved, which may hinder further understanding of interfacial phenomena. For example, insufficient understanding of the curvature effect on surface tension makes it difficult to predict the nucleation temperature in homogeneous nucleation, which has significant influences on condensation, thin film growth, and many other heat transfer and material processing applications. In the past, researchers applied statistical mechanics and statistical thermodynamics methods to solve interfacial problems. These approaches treat the interface as a zero-thickness region in which abrupt changes of material properties occur. Because the interface thickness is in the nanometer range, such simplification is reasonable in studying macroscale phenomena, but it becomes highly questionable in studying nanoscale phenomena. In studies of nanoscale mass, momentum, and energy transport processes, detailed information on molecular structures is often required and MD simulation can become an effective tool to provide that. This chapter reviews recent developments of MD simulation on mechanical aspects of interfacial phenomena in fluids (e.g., surface tension and interface stability), discusses some unsolved problems, and proposes several future directions. All interfaces, based on the different phases involved, are grouped into liquidvapor interfaces, liquid-liquid interfaces, liquid-solid interfaces, three-phase systems, and other interfacial phenomena (i.e., sonoluminescence and surfactants). Because simulation of liquid-vapor interfaces has the longest history, it is therefore emphasized here. In several groups, interfaces are further divided into three subgroups according to their geometric shapes: planar, spherical, and cylindrical. Concluding remarks are then given at the end of this chapter.
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II. Molecular Dynamics Simulation Techniques An attractive feature of MD simulation is that its techniques are simple and it imposes fewer assumptions than other numerical methods. In classical MD simulation, the only material property to be specified beforehand is the intermolecular potential function. In other cases in which quantum effects are important, such as in electron transport or photon absorption, empirical potential functions may be unavailable. Therefore, nonclassical or Quantum Molecular Dynamics (QMD), such as the ab-initio method, is developed. This section reviews basic MD simulation techniques related to the study of interfacial phenomena. The first subsection discusses classical MD simulation and the second subsection discusses nonclassical MD simulation. For more general information on MD simulation techniques, the reader is referred to the books written by Allen and Tildesley (1987) and Haile (1997).
A. CLASSICALMOLECULAR DYNAMICS SIMULATION
Classical MD simulation methods can be divided into equilibrium MD (EMD) and nonequilibrium MD (NEMD). In the past, most work on MD was based on EMD, and NEMD has not been treated thoroughly. However, because it can handle calculation of transport coefficients easily NEMD will receive increasing attention in the future. This subsection reviews several issues on classical MD simulation, including intermolecular potentials, the finite difference method, initial conditions, boundary conditions, equilibrium criteria, simulation ofthermophysical properties, and simulation of transport coefficients.
1. Intermolecular Potentials In classical MD simulation, a molecule moves according to imposed forces on it, which can be expressed by Newton's classical equations of motion
d?)i
mi--d~ - E
Fij
(1)
J
where mi is the mass of a molecule i, ~i the velocity vector, and Z j Fij the summation of all force acting on the molecule i from other molecules. Here, Fij is determined by differential of the intermolecular potential
-. ddp(rij ) Fij --- - drij t'ij
(2)
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Chang-Lin Tien and Jian-Gang Weng
where ~(rij) is the intermolecular potential, rij is the intermolecular distance, and ?ij represents a unit vector pointing from a molecule j to the molecule i. Much MD work uses the Lennard-Jones (LJ) potential, which is a simple twobody potential form and known to be suitable for inert monatomic molecules such as neon, argon, and krypton
_ (~j.j)6] (~(rij ) -- 4e
m rij
(3)
where e is the energy parameter and 0. is the length parameter; these two parameters vary for different types of molecules. MD simulation adopts non-dimensionalized variables to make simulation results more general. Such non-dimensionalization is based on 0., e, and m, so that the reduced parameters (denoted with an asterisk) of temperature, length, time, mass density, pressure, and surface tension are T* = k s T / e , r* = r/0., t* = t(e/m0"2) 1/2, p* = p0"3/m, P* -- P0"3/e, and y* -- v0"z/e, respectively. In this sense, it does not matter which inert monatomic molecules are simulated in the program; however, for ease of physical understanding, most studies assume the "LJ molecules" to be argon. Other types of potential functions are also available to simulate different materials. In simulation of liquid water, one may use ST2, SPC, TIP4P, MYC, and CC potential functions. Tersoff and SW potential forms can model materials with covalent bonds such as carbon and silicon. Refer to Maruyama (2000) for a more detailed description of these models. Other examples include the shell model for ionic solids and the embedded atom method for metals. If several types of molecules exist in the system, the potential parameters between unlike molecules are usually different from those between like molecules. For example, if both types of molecules are modeled by the LJ potential, the potential parameters between molecules of type 1 and type 2 can be expressed as
612 --- ~:~/611e22
(4)
O'12 ~" 70.5(O"11 + 0.22)
(5)
where the subscript 12 refers to the interaction for unlike molecules, subscripts 11 and 22 refer to like molecules, and s~ and 71 are constants. According to LorentzBerthelot combination rules, ~ and r/are 1.0, but different values have also been used (Mecke et al., 1999). Other models are also available to make interaction between unlike molecules different from that of like molecules, such as by introducing a parameter to modify intermolecular attraction, the second term in the LJ potential, between two unlike molecules (Dfaz-Herrera et al., 1999). The potential function for polymers is another interesting example. In simulation of polymers, all
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99
pairs of molecules interact with one potential (e.g., the LJ potential), and bonded neighbors in a chain further interact with the Finitely Extensible Nonlinear Elastic (FENE) potential, expressed as
~FENE(rij ) --
2
Ro J
where k equals 30 and R0 equals 1.5 in a non-dimensionalized system (Kremer and Grest, 1990). MD simulation uses the common convention of a cutoff radius rcut in the calculation of intermolecular potentials and forces, and rcut is defined as the distance beyond which the interaction between molecules is neglected. In a homogeneous system, the choice of the cutoff radius is not very critical because long-range forces from all directions cancel each other. However, in nanoscale interfacial regions where the local density changes by two orders of magnitude over a few nanometers, the choice of the cutoff radius becomes important. Generally, the cutoff radius should be sufficiently large to avoid possible long-range force correction, but it should also be small enough to keep computational time reasonable.
2. Finite Difference Method There are many types of algorithms to advance the differential equations of motion in MD simulation, but the "velocity Verlet" algorithm is usually preferred. Let At be the duration of each time step. During At, the "velocity Verlet" algorithm includes the following four steps:
7i(t + At) = ri(t) -k- At. vi(t) + vi t + ~
-- ~)i(t)-~
2
At2 ~'i(t) 2 mi
(7)
mi
Evaluate Fi(t + At) based on molecule positions at t + At
( 2t) AtFi(t+At) ~)i(t + A t ) = ~)i t + ~ +--2mi
(9) (10)
For other methods, such as the Gear predictor-corrector method, refer to Allen and Tildesley (1987) and Haile (1997).
1O0
Chang-Lin Tien and Jian-Gang Weng 3. Initial Conditions
Here, only the initial conditions for liquid and vapor phases are considered. The solid phase is usually treated as one of the boundaries in interfacial studies (except in studies of melting, in which the solid phase is modeled as a crystalline structure). Initial conditions for liquid and vapor phases can be: (1) a crystalline (solid) structure that melts and further vaporizes as the temperature exceeds the melting point; (2) a combination of different-phase samples that have equilibrated individually; and (3) the final condition from a previous simulation. Generally speaking, for EMD simulations, as long as a final equilibrium state can be reached, different choices of initial conditions lead to the same results.
4. Boundary Conditions Due to the limitation of current computational capacity, the classical MD technique can only simulate a system with about 106 molecules in a period of several nanoseconds. There is generally nothing one can do with the short time scale except avoid simulation of slow reactions. The small length scale limitation, however, can be overcome by implementation of periodic boundary conditions in which the computational domain is replicated throughout space to form an infinite system. According to such periodic boundary conditions, any information outside the computational domain can be inferred from the information within the domain, and thus direct simulations of bulk material properties are possible. It should be noted that this scheme is unable to simulate any phenomena that occur in a scale larger than the domain size (e.g., a wave with a very long wavelength). If solid walls exist in the system and interact with liquid (or gas) molecules, they are sometimes modeled differently from liquid and vapor molecules. In case that dynamics of solid molecules can be ignored, Abraham (1978) presented four models to simulate walls: (1) solid molecules are put into crystalline lattice sites and not allowed to move; (2) the potential between the liquid (or gas) and the wall is infinite when liquid (or gas) molecules are in the wall plane, and zero otherwise; (3) the solid wall is simulated as a semi-infinite medium with a uniform density, so that interaction forces acting on a liquid molecule from wall molecules are integrated throughout the wall and the liquid molecule feels the force only in the direction normal to the wall surface (z direction); and (4) the third model is improved by considering the possibility of finding a solid molecule in a certain distance z.
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101
In many cases, dynamics of solid molecules should be taken into account. To simulate a wall at constant energy, the simplest method is to follow Abraham's first model, but allow molecules to move according to imposed forces. For a wall at a constant temperature, which is frequently encountered in real applications, three simulation methods are available: (1) perform velocity scaling once every few steps; (2) add Langevin noise and frictional terms to the equations of motion (Allen and Tildesley, 1987); and (3) use the "phantom molecules" model (Tully, 1980; Maruyama, 2000), in which the first several layers of solid molecules interacting with liquid (or gas) are modeled as regular molecules, and a layer of "different molecules" (phantom molecules or ghost molecules) represent the other wall molecules that interact with the surface molecules. 5. Equilibrium Criteria In EMD methods, it is important to make sure that the system is at the equilibrium condition before the commencement of statistical calculation of properties. As stated previously, it is impractical to simulate a macroscopically long period. Some criteria, therefore, must be chosen to determine whether the system is at equilibrium. Unfortunately, such choices are still arbitrary in the literature. Maruyama et al. (1994) chose the criteria that both the temperature and the number of vapor molecules settle to constant values in simulations of droplets. Weng et al. (2000) chose the requirement that the local temperature and normal pressure component be constant in liquid film simulations. 6. Simulation of Thermophysical Properties MD simulation programs yield position and velocity of each molecule and intermolecular forces at each time step. Statistical methods must be applied to convert such detailed molecular level information to profiles of thermophysical properties such as local density, pressure components, and local temperature or concentration (if any). Three steps are needed to achieve this: (1) the computational domain is first artificially divided into small parallel slabs (for planar interfaces) or concentric shells (for curved interfaces); (2) properties within these slabs (shells) are calculated at each time step; and (3) at the end, time averaging is performed on these instantaneous values to get local properties. For example, the local temperature can be calculated as -
--
mivZi
(11)
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Chang-Lin Tien and Jian-Gang Weng
where nk is the total number of molecules in slab (or shell) k, kB is the Boltzmann constant, and the brackets ( ) denote a time average of the instantaneous values. Note that the slab or shell should be divided so in a manner that every point within the slab or shell is statistically equivalent.
7. Simulation of Transport Coefficients Two classes of methods, EMD and NEMD, are available to calculate transport coefficients such as viscosity, compressibility, diffusion coefficients, and thermal conductivity. In the first class of EMD methods, integration of time correlation functions of fluctuating quantities yields transport coefficients according to the Green-Kubo formalism,
ff -
f0 ~
dt(A(t + r)A(r))
(12)
Equivalently, one can calculate transport coefficients by applying the "Einstein relation"
1 ((A(t + r ) - A(r)) 2) = 2
t
t---~~
(13)
where ff is a transport coefficient, t the time, A(t) a variable that varies for each ff in study, A(t) a time derivative, r an arbitrarily defined time origin, and the brackets ( ) indicate an average over different r. For example, in determination of self-diffusion coefficients, A(t) represents molecule positions/~(t) and its time derivative is ~'(t). Hence diffusion coefficients are expressed as
Dsetf- -~ i f ~ dt(~)i(t -k- r). vi(r))
(14)
or
1 ([/~i(t + r)-/~i(r)[
O s e l f-- 6
t
2) t---~c~
(15)
The factor 51 in Eqs. (14) and (15) comes from the averaging process in all the three directions. Although the time correlation of only one molecule i is sufficient to determine bulk transport coefficients, as expressed in Eqs. (14) and (15), usually an average over N molecules in the system should be taken to improve statistical accuracy. In the second class of NEMD methods, the system is artificially driven away from equilibrium, and system responses are monitored. Two methods are available to implement the nonequilibrium condition. In the first method, Eqs. (1)
Molecular Dynamics Simulation
103
and (2) are modified to include a perturbation term. Although no physical gradient exists in the system, such a perturbation term creates a thermodynamic flux and integral of time correlation of this flux gives the transport coefficient (Allen and Tildesley, 1987; Hoover, 1993). In the other method, boundary conditions are modified to physically induce a gradient. For example, Lukes et al. (2000) imposed different temperatures on two sides of a solid film to induce a temperature gradient, and obtained thin-film thermal conductivities according to the Fourier law.
B. NONCLASSICALMOLECULAR DYNAMICS SIMULATION Any phenomena that involve electrons or photons cannot be treated easily by classical MD simulation, unless empirical equations can be obtained to describe dynamics of these particles. Usually, interactions among electrons and photons have the quantum mechanical nature and must be obtained by solving the Schr6dinger equation. MD simulations involving these interactions are called Quantum Molecular Dynamics (QMD) and are very useful in many areas, such as in the study of radiation-related processes (Kotake and Kuroki, 1993; Shibahara and Kotake, 1997, 1998). However, limited by computational capacity, QMD can simulate a system with only about 100 molecules. Attracted by the fact that classical MD can treat a relatively large number of molecules, researchers attempted to combine QMD and classical MD in one simulation, and such a technique has been applied successfully in simulating chemical reactions (Singh and Kollman, 1986; Field et al., 1990) and gas/vapor interaction with solid substrates (Carmer et al., 1993; Skokov et al., 1994).
III. Liquid-Vapor Interfaces This section studies planar, spherical, and cylindrical liquid-vapor interfaces. Planar interfaces include a liquid film in a bulk vapor and a vapor film in a bulk liquid, with the main goal of determining surface tension, interface stability, and evaporation and condensation coefficients. Spherical interfaces refer to a droplet in vapor and a bubble in liquid, with the main study being the curvature effect on surface tension. For cylindrical interfaces of liquid jets, the main topics are interface stability and the rupture time. Most MD work simulates LJ molecules, but simulations of other molecules are also available (e.g., Dang and Chang, 1997; Daiguji and Hihara, 1999).
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Chang-Lin Tien and Jian-Gang Weng
FIG. 1. System configuration for a thin liquid film in its vapor.
A. PLANAR INTERFACES 1. Liquid Film in Bulk Vapor Several issues are discussed here: the density profile, local pressure components, local stress, surface tension, film stability, and evaporation and condensation coefficients. Figure 1 shows a typical system configuration with periodic boundary conditions applied in each of the three directions. Here, Ls is the initial film thickness, and usually Lx = Ly.
a. Density Profile Figure 2 depicts a typical density profile, where z* = 0.0 corresponds to the center of the film. The local density falls from the liquid density (~0.771) to the vapor density ('~0.007) within about 1 rim, and these values agree with the experimental data of saturated argon densities at that temperature. The density profile can be expressed as (Rowlinson and Widom, 1982) p(z) -- 0.5(pt + p~) - 0.5(pt - p~)tanh[2(z - ze)/d]
(16)
where Pl and p~ refer to the liquid and vapor densities, respectively. Two important parameters are defined in Eq. (16): the position of the equimolar surface Ze and the interface thickness d, both of which can be obtained by numerical fitting, also
105
Molecular Dynamics Simulation '
0.8 0.6 i
i
o
~
" ~
~
,
i
,
|
,
!
O MD Simulation -- - Fitting Curve Pt = 0.77155 • 0.0016 Pv = 0.00726 + 0.00105 z~ = 3.70422 • 0.00623
Q'0.4 0.2 0.0 0
i
i
i
2
i
|
4
~
=
_
_
_- r . . . .
6
8
_'r
......
10
Z
FIG. 2. Densityprofile for a liquid thin film in vapor at T* -- 0.818.
shown in Fig. 2. Usually, the overall film thickness is considered to be 2Ze. The interface thickness d is expressed as d -- -(Pl
-
Pg)[dP(z)/dz]
(17)
-1
Z--Ze
and has two contributions: mass diffusion (evaporation and condensation) and random perturbations at the interfaces (surface waves). In MD simulation, gravity is usually neglected, so the surface wave considered here is the capillary wave.
b. L o c a l Pressure C o m p o n e n t s
In a bulk medium, the pressure is spatially uniform and can be expressed by the virial theorem V
3V
. . dp'(rij)?ij 97ij
-
PK -- PI
(18)
Here, all the N molecules in a system of volume V are considered in the pressure calculation. Equation (18) indicates that the pressure has two terms: PK considers the contribution from the kinetic motion of molecules, and is always isotropic; PI counts for intermolecular forces and is isotropic only in a system with a uniform density. In the interracial region where a large density gradient exists, the pressure is neither uniform nor isotropic. Two components in the local pressure, the normal pressure component PN (normal to the interface) and the tangential pressure component P r (parallel to the interface), are used to describe the anisotropic nature, and both of themhave two contributions: P N = P N , K -- P N , I , and PT = PT, K - PT, I. In calculation of the local pressure, one cannot use Eq. (18) by just changing the total system volume V to the volume of the local region: PN,K and PT, K can be
106
Chang-Lin Tien and Jian-Gang Weng
calculated in this manner, but PN,I and PT, I cannotnintermolecular forces acting across the boundary of the local region also contribute to the pressure components. The conventional approach defines normal and tangential pressure components as follows:
1 P N ( k ) - PN,K(k)- P g , l ( k ) - (n(k))kBT - ~sl Pr(k) = Pr, K ( k ) - Pr,1(k)=
i,j
i,j -'J rij ~'(rij)
>
-~(x~ rij+ y~) r (rij)
(19)
(20)
where n(k) is the number density in slab k, V~t the volume of slab k(Vsl = LxLyLsl), Lsl the slab thickness, and r the intermolecular potential. The summation Y~i~,j runs over all particle pairs (i, j), of which at least one of the particles is situated in slab k. If only one particle is in slab k, half of the intermolecular force contribution is given to slab k whereas the total contribution is given to slab k if both molecules are in that slab. The contribution of pair (i, j) is Z2jqb'(rij)/rij for PN and (x~ + yZ)r for Pr (Nijmeijer et al., 1988). This approach excludes the contribution from any pair of molecules if both of them are not in that slab. It should be noted that this is a simplified approach and adopted mainly for computational efficiency. According to the theory by Kirkwood and Buff (1949), the intermolecular force contribution to the local normal pressure component is given as the sum over the normal components of all the pair forces acting across a surface element divided by the surface area. The same argument holds for the tangential pressure component (Rowlinson and Widom, 1982). In other words, an intermolecular force contributes to local pressures in all imaginary planes between these two molecules. Here, the intermolecular force is assumed to act in a straight line. Based on these, Weng et al. (2000) modified the conventional method and proposed equations to calculate the pressure components as
PN(k) = PN,K(k)- PN,l(k)-- (n(k))kBT- ~sl 1 Pr(k) = Pr, l,:(k)-PT, l(k) = (n(k))kBT- ~
i,j i,j
\ ri--~ ~ (x 2 + y2) r rij
)> (22)
where a new variable fk,ij represents the ratio of the size of the vertical interval in which Fij is effective in the slab k to Zij. For example, consider the force between
M o l e c u l a r D y n a m i c s Simulation
107
FIG. 3. Schematicof pressure contribution of the pair i, j. the pair (i, j ) in Fig. 3. If neither molecule is in a slab but the line connecting them crosses the slab, such as slab 2, f2,ij -- Lst/zij; if one molecule is in a slab, such as slab 1, fl,ij = L1/zij; and if both molecules are in a slab (not shown in Fig. 3), f k , i j - - 1.0. Figure 4 compares two typical profiles of the normal pressure component calculated by these two methods, in which P~v, P~V,K, and P~v,i are reduced pressures in Eqs. (19) and (21), respectively (Weng et al., 2000). The result from the conventional method has significant fluctuations across the interface, whereas fluctuations disappear in the result from the new method. This smooth profile agrees with the condition that the film is under mechanical equilibrium in the normal direction.
c. Local Stress Local stress is defined as the difference between the normal and tangential pressure components (PN -- PT, or Pr, l - PN,I), and is important because the integral of local stress from the center of the film to the bulk vapor region gives surface tension •
-
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-- P r ) d z
(23)
Thermodynamically, local stress can be understood as the excess energy per volume. Excess energy refers to the difference between the local potential energy at the interfacial region due to the existence of a large density gradient and that of a
Chang-Lin Tien and Jian-Gang Weng
108 (a)
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bulk liquid and vapor mixture at the same density. In a bulk medium, because the pressure is isotropic (and the excess energy is zero), local stress is zero, whereas in the interfacial region local stress is not zero and serves as an indicator of the location of an interface. Therefore, local stress profiles can show how the interfacial region evolves while system parameters (such as temperature and film thickness) change. Figure 5 lists local stress profiles for films with different thickness (Weng et al., 2000). One can observe that as the film thickness decreases, the two interface regions start to overlap; consequently, local stress at the film center increases. When film thickness decreases below a critical value, film rupture occurs. As discussed later, the critical thickness depends on computational domain sizes, surface tension, and the Hamaker constant.
FIG. 5 9 Profiles of P*N , I ' PT,I, * and local stress PN* -- Pr" * The left column is from the conventional method and the right column is from the modified method. The film thickness decreases from S 1 to $4. (Reprinted with permission from Weng, J. G., Park, S. H., Lukes, J. R., and Tien, C. L. (2000). Molecular dynamics investigation of thickness effect on liquid films. Journal of Chemistry and Physics 113, 5917-5923. Copyright 9 2000, the American Institute of Physics.)
110
Chang-Lin Tien and Jian-Gang Weng d. Surface Tension
Since the pioneering work on MD simulation of surface tension by Chapela et al. (1977), simulation techniques have been improved and system size increased. Nijmeijer et al. (1988) used a much larger system to obtain the local stress profile and to study the temperature effect on surface tension. Mecke et al. (1997) proposed a new long-range force correction method and used it to simulate effects of the cutoff radius and temperature on surface tension. All of these studies indicate that surface tension values decrease as the temperature increases or as the cutoff radius decreases. Moreover, Chen (1995) applied the Gaussian model of capillary waves to study the effect of simulation domain sizes on surface tension a y ( L ) - y(cx~)= L2 (24) where y(cxz) refers to the value of surface tension at an infinitely large surface, L is the dimension of the computational domain (here L -- Lx = Ly), and a is a constant (about 2.2 in a non-dimensional system). Because both surface tension and the interface thickness can be related to characteristics of the dominant capillary wave, it is possible to determine surface tension values directly from density profiles. By assuming that the surface wave contribution to the interface thickness can be decoupled from the mass diffusion contribution, Sides et al. (1999) were successful in extracting surface tension values from the density profiles, and their results exhibited a striking agreement with results from the conventional method.
e. Film Stability and Rupture A liquid-vapor interface is always subject to random perturbations that can be treated statistically as capillary waves with different wavelengths. Thermodynamic models indicate that a capillary wave changes the system energy in two ways: (1) it increases the liquid-vapor interface area and then increases the surface energy (defined as the surface tension multiplied by the surface area); (2) it also changes (in this case, decreases) the interaction energy between two vapor phases across the liquid film. Waves with different wavelengths have different contributions to the two changes. For short waves, the first change is dominant, and the system is stable; however, for long waves, the second change is dominant, and the system is destabilized because the corresponding perturbation decreases the system energy and is amplified. At the critical wavelength, the two changes cancel each other. For a film to be stable, the largest wavelength in the system should be smaller than the critical wavelength ,~max ~ ~.cr
(25)
Molecular Dynamics Simulation
111
FIG. 6. Capillarywavepropagation in a liquid-vaporinterface.
For simplicity, a one-dimensional surface wave is considered here. Figure 6 shows the most unstable condition schematically: two synchronic waves propagate along the x direction. In this case, the critical wavelength is
)~cr- L2f,/4zr3y`~ /V A
(26)
where Lf is the film thickness, Yt, the value of liquid-vapor surface tension, and A the Hamaker constant that describes both the interaction between two vapor phases across the liquid film and interaction between two liquid phases across a vapor film (Israelachvili, 1992). Refer to Edwards et al. (1991), Majumdar and Mezic (1998), (1999), and Weng et al. (2000) for detailed derivation. MD simulation can determine the critical wavelength from a totally different approach. The largest wavelength in MD simulation equals the computational domain size due to the imposed periodic boundary conditions: ~ m a x "-- Lx. For the same film thickness, film rupture occurs when Lx is large (Weng et al., 2000). By fixing the film thickness and simultaneously varying Lx and Ly, one can obtain the critical value of Lcr. Any domain sizes larger than Lcr lead to unstable films; any domain sizes smaller than Lcr lead to stable films: Lcr ~- ~cr. The quantitative comparison between results from the classical thermodynamics prediction [Eq. (26)] and MD simulation shows that the value from the classical method is about twice as large as the value from MD. The difference might come from the fact that MD simulation considers a two-dimensional wave (in both x and y directions), but the classical prediction is based on a one-dimensional model.
112
Chang-Lin Tien and Jian-Gang Weng
Hwang et al. (1998) applied MD simulation to demonstrate the rupture process for a liquid film in vapor and a liquid film on a solid substrate. Simulation results revealed that a large liquid-liquid interaction and a small liquid-solid interaction help to increase the rupture rate, which is reasonable because strong liquid-solid interaction would keep the liquid film on the substrate. Several other interesting issues on liquid film rupture have not yet been addressed. For example, after establishment of a thermodynamic two-dimensional surface wave model [similar to Eq. (26)], determination of the stability condition in MD simulation can be a new way to calculate the Hamaker constant because values of surface tension, film thickness, and critical wavelength are readily known from simulation. However, because direct simulation of the Hamaker constant by MD is not available (Monte Carlo simulation of the Hamaker constant has been established), no comparison can be made between the direct method and the method from the stability simulation.
f. Evaporation and Condensation Coefficients Yasuoka et al. (1994) and Matsumoto et al. (1994) studied evaporation and condensation at the surface of liquid argon and methanol. A strong correlation between evaporation and condensation, termed the molecular exchange phenomenon, has been found in their work in which vapor molecules colliding on the liquid surface knock liquid molecules out to the vapor phase. They also found that temperature dependence on evaporation and condensation behavior is not significant. For other studies in evaporation and condensation coefficients, refer to review articles by Tanasawa (1994) and Carey (1995).
2. Vapor Film in Bulk Liquid After a thorough treatment of liquid films, at first sight it seems redundant to consider the vapor film any further because simulation techniques are similar and results of surface tension, the density profile, and the stability condition should be exactly the same. A detailed investigation on molecular structures of liquid and vapor films, however, reveals an interesting difference. Known from the previous discussion on local stress of liquid film and interface stability, interface overlapping occurs if the film thickness is small. As a result, the center of a film experiences a higher tensile stress, and decreasing the film thickness further breaks the film because the liquid can sustain only a limited stress. Note that the difference between liquid and vapor phases is that vapor cannot sustain any tensile stress. In the vapor film, therefore, interface overlapping should never
Molecular Dynamics Simulation
113
occur. Before two interfacial regions start to "see" each other, interface rupture should occur. It can be concluded, then, that for the same computational domain size, the critical thickness of a vapor film should be larger than that of a liquid film. This conclusion seems to contradict the classical thermodynamics prediction, because for LJ liquids the stability condition depends only on surface tension and the Hamaker constant, both of which are supposed to be identical in these two film cases. In order to solve this discrepancy, detailed analysis and extensive simulation would be required.
B. SPHERICAL INTERFACES
1. Droplets Simulation of droplets started as early as that of liquid films (Rusanov and Brodskaya, 1977). Formation, dynamics, and surface tension of droplets have been the major focus in MD simulation. The temperature effect on surface tension is similar to that in the planar interface, whereas the curvature effect is unique and has been simulated extensively in the past several decades. Disagreement, however, still exists. Figure 7 shows a typical profile of instantaneous molecule position of a droplet viewed from one direction (e.g., x direction) in the computational domain.
FIG. 7. Molecule position for a droplet in the computational domain.
114
Chang-Lin Tien and Jian-Gang Weng a. Droplet Formation and Dynamics
The discussion of droplet formation here considers multiple droplet formation during homogeneous nucleation in the vapor instead of preparation of the initial condition for the liquid-vapor interface. Classically, the nucleation rate is an exponential function of the free-formation energy AG, and A G is expressed as A G = YA + Ag V
(27)
The first term represents the energy required to create a liquid-vapor interface and is always positive, where Y is surface tension and A is the surface area. The second term is usually negative and represents the free energy change between bulk vapor and liquid phases, where Ag is the free energy difference per unit volume and V is the volume. Further derivation that relates A G to the droplet radius R is based on the assumption that the droplet is perfectly spherical. Yasuoka and Matsumoto (1998a-c) simulated homogeneous nucleation in vapor phases of argon and water. They revealed a significant difference between A G(R) from MD simulation (interpreted from the simulated nucleation rate) and classical prediction (with experimental data of Y and Ag). The possible reasons are: (1) the surface tension value in classical prediction is for planar interfaces and is different from that for small droplets due to the curvature effect; (2) the droplet shape is far from spherical; (3) Ag in nanoscale is different from that in macroscale; and (4) the classical theory is based on the infinitely large bulk phase whereas MD simulates the system with a limited number of molecules. Usually, the impact of errors in surface tension is large, and this induces extensive research efforts to study the curvature effect on surface tension. Droplet evaporation simulation reveals another interesting phenomenon. In the classical theory, the evaporation rate is defined as dr/dt. The large droplet evaporates at a rate so that ~ (x t, where R is the droplet radius and t is time. For small droplets, the evaporation rate is constant: R cx t. Long et al. (1996) used MD simulation to confirm this proportionality. b. Density Profile Similar to the planar interface case, the density profile can be fitted with the expression p(r) = 0.5(pt + P v ) - 0.5(pt - Pv) tanh[2(r - Re)/d]
(28)
as shown in Fig. 8. Here, Re is the radius of the equimolar surface and d is the interface thickness. From the comparison between density profiles of a liquid film
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and a droplet, one may find that the vapor density in droplet simulation is usually lower than the vapor density in film simulation and the liquid density is higher. The reason is that the pressure in a droplet is higher than that in the vapor. One may also notice that the local density (and the local pressure, as discussed later) near the droplet center exhibits large fluctuations, which are caused by statistical calculation over a small volume (or surface). Another interesting issue is that if the simulation starts by melting and vaporizing the same crystalline solid in vacuum (as discussed in Section II.A.3), the final vapor density differs for different computational domain sizes. The vapor phase is formed as molecules evaporate from the crystalline structure to the vacuum, which would take quite some time to reach equilibrium. Criteria to determine whether the system is at equilibrium have not yet been established. c. Pressure Tensor
In a spherical system, the pressure decomposes into two components: the normal pressure PN and the tangential pressure PT. Different from the liquid film case, the two pressure components are not independent because of the mechanical equilibrium in the system r d PN(r) PT(r) = Pu(r) + - ~ 2 dr
(29)
116
Chang-Lin Tien and Jian-Gang Weng
Therefore, MD simulation need consider only the normal component because it is easier to simulate than the tangential one. The knowledge of the normal component profile not only gives the tangential component, but yields surface tension as well, as expressed by the Laplace or the Young-Laplace equation
Pl-- P~ = 2?, Rs
(30)
where Rs is the radius of the surface of tension. The pressure at the droplet center, Pl, and the pressure in the bulk vapor phase, P~, can be obtained from the profile of the normal pressure component: Pl -- PN (r = 0) and P~ = PN (r = c~). Note that planar liquid films are at mechanical equilibrium only in the normal direction because external forces are acting in the tangential direction at the film boundary to keep the film flat. Based on the assumption that the intermolecular force is acting on a straight line, Thompson et al. (1984) provided the formula to simulate the normal pressure component
PN(k) = (n(k))kBT
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where the subscript k indicates the local value of the kth shell surface. Note here that the pressure is taken as the surface average, as compared to the case of planar interface, in which the pressure is a volumetric average [in Eqs. (19) and (21)]. The reason is that in a shell with radius from r to r 4- Ar, the normal component of an intermolecular force has a different direction in each concentric surface. If one force crosses a shell surface twice, its contribution to that plane is doubled. Figure 9 shows pressure components for a droplet and a bubble (discussion on the pressure tensor of a bubble is given later) (Park et al., 2000b). Just as in the planar interface case, the local stress is defined as the difference between the normal and tangential components. Fluctuations of the tangential pressure (and therefore local stress) on the liquid side are very large.
d. Surface Tension Classically, Tolman's equation is applied to express the curvature effect on surface tension of spherical interfaces (Tolman, 1949; Thompson et al., 1984) y
= 1--
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where g ~ refers to the surface tension value for planar interfaces and ~ is called the T o l m a n l e n g t h and is defined as the difference between the radius of the equimolar surface and the radius of the surface of tension
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118
Chang-Lin Tien and Jian-Gang Weng
The Tolman length 6 is usually considered to be positive for droplets and negative for bubbles. Other researchers applied the Density Functional Theory (DFT) and found that 3 varies with the droplet radius and becomes negative when the curvature is small or the temperature is high (e.g., Kalikmonov, 1997; Koga et al., 1998; Baidakov and Boltachev, 1999; Bykov and Zeng, 1999; Koga and Zeng, 1999). Two methods are available to calculate surface tension of droplets, both of which require the knowledge of the normal pressure component profile and the surface tension value for planar interfaces. One is from Eqs. (30), (32), and (33) (three equations for three unknowns: y, 3, and Rs, and the radius for the equimolar surface Re can be derived from the density profile). The other way is to integrate the local stress as ?' -- fo
~r
[PN(r) -- Pr(r)]dr
(34)
and Rs is from Eqs. (32) and (33). Neijimeijer et al. (1992) tried this method and found fluctuation in the liquid phase is too large to perform integration. However, it is still possible to integrate the fitting curve (Thompson et al., 1984; Park et al., 2000b). Usually the Tolman length 3 is very small (< 1.0 or) and varies with temperatures. Haye and Bruin (1994) took a different approach to calculate 3 directly from the liquid film simulation and found that 3 is about 0.2 o. However, because of statistical errors, MD simulation has not clearly revealed the temperature and curvature effects on 6. One may find that in Fig. 7, the droplet is not perfectly spherical because of capillary waves at the spherical interface. Similar to the planar interfaces, it should also be possible to calculate surface tension directly from the density profile. Powels et al. (1983) studied the capillary wave contribution to the overall interface thickness, but this contribution was obtained with a known surface tension instead of directly from the density profile.
2. Bubbles a. Bubble Formation and Initial Conditions
Typically, a bubble forms either as the temperature increases (constant pressure condition) or as the pressure decreases (constant temperature or energy condition). In MD simulation, the latter condition is easier to achieve. Kinjo et al. (1999) obtained a bubble by stretching the bulk liquid (in other words, by decreasing the pressure to negative values). The main objective of their work was to study the
Molecular Dynamics Simulation
119
incipient stage of homogeneous nucleation and bubble dynamics, and therefore it was not necessary for the bubble to be stable. In studies of the surface tension or density profile of a bubble, it becomes important to be able to maintain a stable bubble surface. Park et al. (2000a, 2000b) obtained a bubble by taking a liquid sphere away from an otherwise uniform bulk liquid and allowing the liquid to equilibrate itself. They found: (1) the minimum size of the stable bubble that can be obtained was different from that of the droplet; and (2) the condition of maintaining a stable bubble depended on the computational domain size. For a smaller domain, the average number density (defined as the total molecule number divided by the total domain volume) should be lower to guarantee a phase separation; for a larger domain, the average number density can be higher. Such observations indicate that small domains are more stable than large domains, and the density should be decreased more in order to destabilize the liquid phase, which is physically true because the metastable condition is destroyed by perturbations with long wavelengths and in a large system the allowed wavelength is longer. For detailed information of phase stability, refer to the work by Abraham (1979) and Koch and coworkers (1983). Note that even if a phase separation occurs, the resultant "bubble" may not be spherical because the interaction of the bubble in the computational domain with other "bubbles" in its neighboring domains (i.e., its images) results in a vapor column.
b. Surface Tension
The surface tension simulation scheme is same as that used in droplet cases. Figure 9b plots typical profiles of normal and tangential pressure components. Figure 10 shows simulation results for the curvature effect on surface tension of bubbles and droplets, as compared with Tolman's prediction (Park et al., 2000b). It is found that surface tension of droplets follows Tolman's prediction quite well, but surface tension of bubbles does not. The reason for this discrepancy is still unknown.
C. CYLINDRICALINTERFACES
Currently, simulation of liquid jets in vapor studies only the stability condition and the interface rupture characteristics. Classical treatment of liquid jet breakup relies mainly on Rayleigh's studies on linear stability of an infinitely long, circular, incompressible liquid jet. The axisymmetric perturbation can be expressed as (Bogy, 1979) r = a + ae exp(ott - i k z / a )
(35)
Chang-Lin Tien and Jian-Gang Weng
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where r is the "local" radius in the jet, a the average radius, z the direction along the cylinder axis, e a non-dimensionalized perturbation amplitude (different from the energy parameter in the LJ potential), and c~ the wave growth rate. The wave number k is non-dimensionalized according to the radius: k = 2roaN~L, where N is an integer and L is the jet length. In inviscid fluids, the growth rate is given as ll(k)(1 - kZ)k 1/2
?' -
10(k)
(36)
and in highly viscid fluids, it is given as lY - - ~ V ( 1
6a#1
-k
2)
(37)
where V is surface tension of the liquid jet, p the bulk liquid density, #1 the liquid viscosity, and In the nth-order modified Bessel function of the first kind. Despite increasing importance of the liquid jet in industry applications, its MD studies are very limited. Koplik and Banavar (1993) and Kawano (1998) used MD to study the rupture time and compared their results with Rayleigh's theory in Eqs. (36) and (37). Recently, Moseler and Landman (2000) simulated nano liquid jets emitting from a solid nozzle and found that nozzle exit condition has profound effects on the nanojet stability.
Molecular Dynamics Simulation
121
Several issues regarding liquid jets have not been studied sufficiently and are discussed in the following subsections.
1. Stable Cylindrical Interfaces In order to evaluate thermophysical properties such as density profile or surface tension, a stable cylindrical interface should be maintained because rupture occurs in a time scale that is short enough for the MD simulation method to detect. One possible way to accomplish this is to control the length of the liquid jet so that the wave growth rate ot is less than zero (the wave exponentially decays with time). According to Eqs. (36) and (37), the non-dimensional wave number k should be larger than 1 or L < 2Jra.
2. Density Profile and Surface Tension A typical density profile can be found in Kawano (1998) and is reproduced in Fig. 11. Note the y axis is in the log scale. Similar to planar and spherical liquidvapor interfaces, the interface thickness is related to the surface waves. Comparison of density profiles for liquid cylinders with different lengths may reveal the effect of different capillary waves on the perturbation amplitude in each system.
10.000
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0.100
0.010 L
0.001
0
. . . . . . . . . . . . . . . . 2
4
6
r1
8
FIG. 11. Densityprofile of a liquid cylinder in vapor. (Reprinted with permission from Kawano, S. (1998). Molecular dynamics of rupture phenomena in a liquid tread. Ph~'sicsReviewE 55, 3068-3071. Copyright 9 1998, the American Physical Society.)
122
Chang-Lin Tien and Jian-Gang Weng
Surface tension of cylindrical interfaces, as a result of interface instability, has not been addressed by MD simulation. However, in order to obtain accurate values of the wave growth rate, it is necessary to study the curvature effect on surface tension instead of using the surface tension values for planar interfaces. In a cylindrical system, the force can be decomposed into three components along radial (normal), azimuthal, and longitudinal coordinates: Fr, Fr and Fz, respectively. Two are particularly important: the normal component Fr can determine the surface tension, and the longitudinal component Fz can reveal detailed information on tensile stress that the liquid jet experiences when interface rupture occurs. The simulation algorithm has not yet been established.
3. Interface Rupture The ability to control the length of liquid cylinders provides another opportunity to study the interface rupture phenomenon. By gradually increasing the jet length (increase the computational domain size in z direction), one should be able to observe the trend of destabilizing the nanojet. A plot of rupture times with the nanojet length could provide some insights on how the capillary wavelength can change the stability.
IV. Liquid-Liquid Interfaces Liquid-liquid interfacial studies are of great interest to researchers in various fields because many biological, chemical, and physical processes occur at liquidliquid interfaces accompanied with phase separation or transport of electrons, ions, and molecules. Here, surface tension and diffusion coefficients play critical roles, and sufficient understanding of them is highly desirable. Currently, most studies of these processes are still based on macroscopic models or experiments. Molecularlevel details on liquid-liquid interfaces not yet available. By exploring the molecular structure at interfaces, MD simulation can provide a theoretical tool to interpret experimental data, offer new physical insight, and benchmark applicability of existing phenomenologic models. Basic simulation techniques have been presented in previous sections. However, several difficulties that have not been encountered in the study of liquid-vapor interfaces arise. In simulation of liquid-vapor interfaces, presence of space with a dilute vapor phase eases the equilibration process because the liquid phase has enough room to adjust its size or shape, and molecules can evaporate from or condense to the liquid phase freely. As a result, the system pressure is constant and close to zero
Molecular Dynamics Simulation
123
FIG. 12. Systemconfiguration for liquid-liquid interfaces.
(as shown in Fig. 4). In the simulation of liquid-liquid interfaces, in contrast, no space exists for liquid phases to adjust themselves and, consequently, the normal pressure component becomes difficult to control; It not only shows large fluctuations, but also differs significantly for different initial conditions. Solutions for this problem are either to introduce a vapor phase into the system (shown in Fig. 12) or to adopt different simulation schemes (Allen and Tildesley, 1987). Refer to Zhang et al. (1995) for a more detailed discussion. Moreover, in studies of liquid-liquid interfaces, two or more types of molecules are present, and the intermolecular potential model between unlike molecules is slightly artificial.
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Chang-Lin Tien and Jian-Gang Weng
This section reviews studies of various liquid-liquid interfaces. In addition to planar and spherical interfaces, mixtures of two or more liquids are discussed as a result of their significance in real applications.
A. PLANAR INTERFACES The typical setup is a liquid slab sandwiched by two slabs of the other liquid, as shown in Fig. 12, with periodic boundary conditions applied in all three directions. Most studies adopt Figs. 12a and 12c, and different normal pressures in these studies may influence their simulation results. The two liquids can be either immiscible or partially miscible. Several important issues are discussed here: density profiles, diffusion coefficients, surface tension, and interface stability.
1. Density Profile The simulation method of local densiti,~s is same as that in liquid-vapor interface studies. The only difference is that three profiles are required here: the local density of Phase A, the local density of Phase B, and the total density. A typical density profile is shown in Fig. 13 (Dfaz-Herrera et al., 1999). Many factors affect the shape of density profiles, including normal pressure, temperature, and affinity between unlike molecules. Due to the existence of three curves, a clear definition of the interface thickness is not available. Different from the liquid-vapor interfaces, the density profile of Phase A may not show monotonic decrease from its bulk phase region to the region of the other liquid. An oscillatory structure has been revealed by MD simulation (Toxvaerd and Stecki, 1995; Stecki and Toxvaerd, 1995). Density profiles at an interface of two immiscible liquids at low temperature and high pressure are shown in Fig. 14. MD simulation also shows that such oscillation is prominent only at interfaces of small cross-sectional areas because capillary waves with long wavelengths smear it. The cause of such oscillation is similar to the layering process when a liquid approaches a solid (discussed in Section V).
2. Surface Tension Surface tension simulation, again, follows the same procedure as that in liquidvapor interfaces. Simulation results indicate that surface tension values depend on not only system temperature and cross-sectional area, but also normal pressure and interaction between unlike molecules. D/az-Herrera et al. (1999) found
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1
0.8
i 0.6
~ a
0.4
~
0
~,
4
8
2'"
. . . . . . . . . . .
T'=0.827, p ~
12
16
20
FIG. 13. Densityprofiles at a liquid-liquid interface, p* is the total density. (Reprinted with permission from Dfaz-Herrera, E., Alejamdre, J., Ramffez-Santiago,G., and Forstmann, E (1999). Interfacial tension behavior of binary and ternary mixtures of partially miscible Lennard-Jones fluids: A molecular dynamics simulation. Journal of Chemistry and Physics 110, 8084-8089. Copyright 9 1999, the American Institute of Physics.)
non-monotonic behavior of v(T) with respect to the system temperature T in the interface of two partially miscible liquids (Fig. 15), as compared to monotonically decreasing behavior in the liquid-vapor interface. The reason is that the diffusion coefficient usually increases with temperature, which may change the local stress profile in the interface. The relationship between these two parameters has not yet been addressed.
3. Transport Coefficients As stated in previous sections, an interface is a highly anisotropic region, and this leads to the anisotropic behavior in transport coefficients at the interface. It is also expected that values of transport coefficients at the interface will be different from those in the bulk phase. Such differences are prominent in studies of nanoscale systems. Among all the transport coefficients, the diffusion coefficient might be considered the most important because it is closely related to processes of chemical and physical reactions and the density profile across the interface. Let DAB be the diffusion coefficient of molecules A in a bulk phase B. At equilibrium, DAB = DSA.
0.8 0.7
0.2
0.1 n 4.5
5
5.5
6
6.5
z position
7
7.5
6
0.2
15
15.5
16
16.5
17 17.5 porition
16
18.5
19
19.5
2
FIG.14. Oscillatory profiles for (a) density of phase A and (b) total density at liquid-liquid interfaces. The full line is for the domain size L , = 7.5 u, the dashed line for L , = 15 u , and the dotted line for L , = 22.5 u . (Reprinted with permission from Toxvaerd, S., and Stecki, J. (1995). Density profiles at a planar liquid-liquid interface. Journal of Chernisfry and Physics 102, 7163-7168. Copyright 0 1995, the American Institute of Physics.)
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|
127
9
2.2 2.1
f
1.9 1.8
9N = 1 7 2 8
1.7
9N = 2 5 9 2
1.6 1.5 1.4
0.5
l
I
1
,
I
1.5
,
I
T
.2
i
I
2.5
,
I
3
FIG. 15. Temperature effect on surface tension at liquid-liquid interfaces for systems with two different molecule numbers. (Reprinted with permission from Dfaz-Herrera, E., Alejamdre, J., RamfrezSantiago, G., and Forstmann, E (1999). Interfacial tension behavior of binary and ternary mixtures of partially miscible Lennard-Jones fluids: A molecular dynamics simulation. Journal of Chemistry and Physics 110, 8084-8089. Copyright 9 1999, the American Institute of Physics.)
The method to determine DAB can be found in Allen and Tildesley (1987). As with surface tension, one might find DAB to be a function of temperature, normal pressure, and the miscibility of the two components. Another interesting problem is to consider the diffusion process of a third type of molecules across the interface. Benjamin (1991, 1992) used MD to simulate electron and ion transport across the interface of two immiscible liquids. For different types of liquid (e.g., polar or nonpolar) and difference particles (e.g., changed or uncharged), the diffusion process is completely different, and more extensive simulations are required to solve this problem.
4. Interface Stability When a liquid slab is sandwiched by two slabs of another liquid, the stability issue arises. If the material properties (e.g., the dielectric constants and refractive index) are very close, the Hamaker constant is very small [for calculation of the Hamaker constant, refer to Israelachvili (1992)]. According to Eq. (26), if the surface tension value is larger than or comparable to that at liquid-vapor interfaces, the liquid film can be very thin before rupture occurs.
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Chang-Lin Tien and Jian-Gang Weng
This phenomenon might bring about the question of applicability of concepts of the Hamaker constant (or disjoining pressure) and the surface tension when the liquid film is atomically thin.
B. SPHERICAL INTERFACES The spherical liquid-liquid interface refers to the case of a large molecule or a droplet (i.e., a molecular cluster) in a bulk liquid consisting of another type of molecules, a condition frequently encountered in practical engineering applications. The droplet usually adopts a spherical shape in order to minimize its surface energy. Complete understanding of spherical liquid-liquid interfaces involves studies of three problems: droplet formation, growth, and transport. Droplet formation in liquids is similar to that in vapors: the nucleation rate exponentially depends on the formation energy A G that is expressed in Eq. (27). As in spherical liquid-vapor interfaces, the curvature effect on surface tension is important here. However, due to involvement of different molecules (and therefore different interaction potentials) in this study, such an effect is much more complicated. After formation of droplets, how they grow is another concern in engineering processes. Laradji et al. (1996) used large-scale MD simulation to monitor the dynamic growth process of droplets during the final stage of spinodal decomposition, and their simulation results were in agreement with classical prediction: the droplet size R(t) is proportional to t, where t is time. Many parameters may affect such a proportionality, including temperature, intermolecular potentials, and pressure. It has been known that droplets dispersed within another liquid can change the transport coefficients of the host liquid significantly. An example is that shapedeformation at surfaces of dilute droplets results in a macroscopic viscoelasticity in an otherwise continuous Newtonian fluid (Edwards et al., 1991). Effects of droplets on transport coefficients is different if system parameters such as temperature and pressure change, all of which deserve more detailed investigation.
C. MIXTURES
Mixtures of two or more liquids can be categorized further into isotopic and nonisotopic mixtures. In an isotopic mixture, the intermolecular potential between like and unlike molecules is equivalent; the only difference is the mass of each component. If the gravity effect is neglected or the mass differences are small,
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these components achieve an ideal mixing condition. In contrast, in the case of nonisotopic mixtures, things are quite different. Consider a case in which B molecules disperse in a bulk phase consisting of A molecules. If the attraction force between two B molecules is stronger than that between an A molecule and a B molecule, the B molecules tend to bind together and form a small droplet if attraction between unlike molecules is stronger, many A molecules surround a B molecule, consequently changing the structure of the host liquid. For both mixtures, the main concern is how to express its properties as simple functions. For an isotopic mixture, the effective transport coefficient depends on the mass ratio, mole fractions of two components, and the bulk properties of each component. Kerl and Willeke (1999a) simulated these effects on selfdiffusion coefficients. Other properties such as viscosity and thermal conductivity are still unknown. In a nonisotopic system, simulations are more complicated, as is interpretation of simulation results. Kerl and Willeke (1999b) considered the effect of the mass ratio on diffusion coefficients of an equimolar binary mixture. Additional work must be done in exploration of the effects of molecule sizes, mole fractions, intermolecular potentials, and system parameters such as temperature and pressure. In many engineering applications, properties (especially thermodynamic properties such as enthalpy) of a liquid can be represented as functions of its density and temperature. One of the main goals in simulating mixture properties is to explore whether a mixture could be treated as a "pure" liquid with its properties depending on its "effective" density and temperature. Generally, mixture properties can be obtained by two mixing rules: Kay's (mole fraction-weighted) rule and Enskog's rule (Heyes, 1992). For example, for a mixture of different LJ molecules, Kay's rule is expressed as ~x -- ~
Xi(7i
(38)
XiEi
(39)
ximi
(4O)
i
Sx -- Z i
mi -- Z i
where cri and ei are parameters in the LJ potential for molecules i, mi is the molecular mass, and xi is the mole fraction. The effective densities and temperature for this mixture are given as Px = p(~rx Icrl )3
(41)
Tx = T ( s l / S x )
(42)
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Chang-Lin Tien and Jian-Gang Weng
where p and T are the actual (overall) density and temperature in the mixture, and liquid properties are expressed as a function of Px and Tx. Heyes (1992) demonstrated that for some mixtures (Ar-Kr, Ar-CH4, CH4-N2), different mixing rules lead to similar results. More extensive simulation is necessary to solve this problem.
V. Liquid-Solid Interfaces Subjects in liquid-solid interfacial study are a little different from those in liquid-vapor and liquid-liquid interface cases. Here, the main efforts are on nanoscale boundaries and liquid-solid transition. Nano-boundary problems include determination of viscosity, velocity slips, and temperature slips between solid substrates and the liquid. The study of liquid-solid transition considers mass transfer and surface morphology during melting and solidification. Only a few studies address liquid-solid interfacial tension because large fluctuations in the local stress profile exist at the liquid-solid interface.
A.
PLANAR
INTERFACES
1. Nano-Boundary Problems
Boundary conditions are essential in solving momentum transport problems at a fluid-solid interface. Traditionally, a no-slip condition is enforced, and the underlying assumption is that interactions between fluid and solid molecules are strong enough to keep the interface at thermodynamic equilibrium. If such an assumption is not satisfied, the empirical linear Navier boundary condition must be applied V~ ,,all - L~ y A Vx - Vx,J~,id -- Vx,wa, = L s -~z
(43)
where Ls is the constant slip length, ~, the strain rate, x the flow direction, and z the direction along which the velocity gradient exists. In gas-solid interfaces, the equilibrium assumption requires a high frequency of collisions between gas and solid molecules or the Knudsen number K,, - - lmfp/D < 0.01 (Gad-el-Hak, 1999), where lmfp is the mean free path and D is the length scale in the boundary problem (in some literature, the Knudsen number is defined as D/lmfp). For liquid-solid interfaces, however, the macroscopic criteria to determine whether the
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no-slip condition is valid for a specific problem are not well established, and the nonequilibrium Molecular Dynamic (NEMD) method is often applied to study the liquid-solid boundary condition. Detailed molecular-level simulations reveal that three basic boundary conditions exist at solid-liquid interfaces: slip, no-slip, and multilayer locking. The third case occurs when liquid molecules adjacent to the solid substrate are adsorbed to the solid due to strong attraction between liquid and solid molecules and, consequently, they form highly ordered layers and follow the motion of the solid. Because the thickness of these ordered layers is in the nanometer range, the multilocking condition is always considered to be the no-slip condition in macromicro-scale boundary conditions; in nanoscale studies, however, they become critical, and neglecting them could introduce a significant error. One difficulty encountered in studies of flow boundary condition is that the apparent viscosity deviates from its bulk value when the system dimensions shrink to the nanometer range. This deviation is usually accompanied by a change of the molecular structure in liquid (e.g., multilayer locking). Both experimental study (Gee et al., 1990) and numeric simulation (Thompson et al., 1992) indicate a transition from the liquid to solid-like structure when an ultra-thin liquid film is confined between two solid plates. Although it is known that the critical thickness at which liquid structure transitions occur depends on the strength of interactions between liquid and solid molecules, a general guideline to predict such thickness has not been developed. The other difficulty involves inability to determine, from first principles, the condition under which slip, no-slip, or multilocking occurs, and MD simulation is often applied to provide insight into this problem. An interesting example is the simulation work by Thompson and Troian (1997), who simulated a Couette flow and studied velocity slips as a function of interaction parameters between solid and liquid molecules. Their velocity profiles are shown in Fig. 16, in which a significant velocity slip is observed when the density of the wall (pw) is large or the wall-fluid interaction (characterized by LJ parameters 0 "wf and e ~f) is weak. Thompson and Troian (1997) further obtained a general equation that relates Ls to }~
L--~
( c),J2 1-
(44)
where L ~ $ and ~,~ are the scaling values that depend on the intermolecular potential parameters. Note that although it is obtained from the molecular level simulation, Eq. (44) is a general boundary condition and can be applied to solving macroscale problems. Other studies (e.g., Thompson and Robbins, 1990) also reveal the multilayer locking condition in a Couettee flow due to strong solid-fluid interactions.
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Z
9
I
II
I
I
'
'
i
,
~F
0.5
oooOO~J .
e,~li 9
F-wf(Twf
. * ~y~ J~
0 o~ 0
0 0 . 6 1.0 *0.60.754 9 0.2 0.75
_PlWI I
0.5
z/h
FIG. 16. Velocity profile in Couette flow geometry, p~' is the nondimensionalized solid density and cr wf and e w f are LJ parameters for the wall-fluid interaction. (Reprinted with permission from E A. Thompson and S. M. Troian, N a t u r e . A general boundary condition for liquid flow at solid surfaces. 1997;389"360-362. Copyright 9 1997, Macmillan Magazines Limited.)
Many other questions in flow boundaries remain unsolved. Examples include flow boundaries of liquids consisting of diatomic or polyatomic molecules and Hagen-Poiseuille flow. In the former case, not only the characteristics of interaction potential between solid and liquid are different, but other degrees of freedom (rotational and vibrational) contribute to the slip condition as well, complicating the analysis further. The Hagen-Poiseuille flow, a flow in the pipe of a circular cross section, is another flow dominated by viscosity and, therefore, by molecular interactions across solid-liquid interfaces. In boundary conditions for the energy transport equation, an analogous nojump condition is applied: T f l u i d l w a l l - - Twa11.Similar to the no-slip condition in flow boundaries, the no-jump condition occasionally fails to provide accurate description at solid-liquid interfaces. Generally, a temperature jump is always present between two different materials due to boundary scattering of phonons--for example, a contact resistance between two solids. If one of the materials is a fluid, the temperature jump becomes larger. Maruyama and Kimura (1999) and Ohara and
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Suzuki (2000) proved that the temperature jump would be significant in nanoscale studies or when the temperature gradient across the interface is large. However, a general temperature boundary condition, similar to Eq. (44) in flow boundaries, has not been established. When a chemical potential gradient exists across the interface, mass diffusion occurs. Geysermans et al. (2000) simulated the interface between solid copper and liquid aluminum and found the diffusion coefficient of liquid adjacent to the solid is four times as small as that of the bulk liquid. The dramatic decrease of diffusion coefficient, probably due to the multilayer locking process at the solidliquid interface, would lead to a concentration jump and deserve further studies.
2. Liquid-Solid Transition Studies of melting and solidification are of both fundamental and practical interests, especially in the area of semiconductor material processing, because any defect or impunity introduced during crystallization significantly deteriorates microelectronic devices. Usually, two theoretical methods are available to study crystalline growth: the thermodynamic method and the kinetic rate theory. However, neither can provide detailed dynamic information of the crystal facet or the diffusion process of foreign molecules (e.g., oxygen or hydrogen molecules) across the liquid-solid interface. These problems can be solved easily by MD simulation because MD can monitor motion of molecules at any time. Film growth during Liquid Phase Epitaxy (LPE), which involves the precipitation of a crystalline film from a supersaturated melt onto the parent substrate when the system temperature decreases under a threshold value, is one of the most important topics of MD simulation of solid-melt interfaces. It has been known that temperature gradients across solid-melt interface have significant impacts on film growth rates, facet morphology, and the temperature at which epitaxy is initiated (the initiation temperature can be noticeably lower than the liquid-solid coexisting temperature). Landman and coworkers (1988) and Luedtke et al. (1988) conducted interesting research on such effects. Later, Nishihira et al. (2000) found that stress of the grown film also depends on the temperature gradient. The diffusion process of defects and foreign particles is another important problem. Defects refer to the vacancies and interstitials; foreign particles include hydrogen, oxygen, and other impunities (such as sulfur). If the production processes (e.g., LPE, Czochralski pulling, or floating-zone methods) are not well controlled, densities of defects and impunities increase by orders of magnitude. This study differs from the study in the concentration slip condition in that here a temperature
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gradient always exists across the solid-melt interface, which greatly complicates the analysis. Only a few MD studies have been taken to simulate dynamics of foreign particles (Kakimoto et al., 2000) and transport of defects (Ishimaru et al., 1998) in conditions of different temperature gradients. Moreover, during Czochralski pulling or floating-zone purification, macroscopic motion of melt usually exists, and such motion also influences formation of defects and impurities. No studies have yet addressed such an effect.
3. Density Profiles and Interfacial Tension
Many microscale studies on liquid-solid interfaces have revealed that when a liquid phase approaches a crystalline solid, liquid molecules near the solid surface form ordered layers that are similar to the solid structure due to the strong repulsive force between the liquid and solid molecules. Such a layering process is different from the multilayer locking in that the liquid molecules do not necessarily move with the solid. Because liquid molecules are localized in specific regions and other regions have no molecules, large fluctuations are encountered in density profiles, making evaluation of the interfacial thickness impossible. Similarly, fluctuations in local stress make it difficult to determine interfacial tension (Sikkenk et al., 1988; Nijmeijer et al., 1990) because interfacial tension is just an integral of local stress across the interface, according to Eq. (23). Because sufficient understanding of interfacial tension is critical, more extensive simulation or better simulation techniques should be developed in the future.
B. SPHERICALINTERFACES Spherical liquid-solid interfaces are encountered in colloidal liquids. If the particle size is close to the size of the liquid molecules, analytical treatment based on classical statistical and kinetic theories might be problematic due to the increasing importance of layering liquid molecules near the solid surface. Formation and growth of solid particles are similar to those of liquid droplets discussed previously, but the transport process is quite different because in the latter case interface deformation could occur. Two problems are discussed here: drag force and the diffusion coefficient of solid particles. Classically, for a sphere moving in unbounded fluids the drag force can be expressed as Fd = 6rc# R U
(45)
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where/z is viscosity, R the sphere radius, and U the particle velocity. Note here a "no-slip boundary" is assumed. For the sphere moving adjacent to a solid wall, the expression becomes Fd = 6zr# R2 U / h
(46)
where h is the minimal distance between the wall and the sphere. Vergeles et al. (1996) found that as h is small, a velocity slip occurs and Eq. (46) is not applicable, and as the particle size is small, layering of the liquid molecules adjacent to the solid sphere makes the definition of the "sphere radius" in Eqs. (45) and (46) slightly ambiguous. The diffusion coefficient is related to the drag force by Einstein's expression, and can be expressed as D# =
ksT
6zrR
(47)
Simulation techniques are similar to those discussed previously.
C. CYLINDRICALINTERFACES
The Hagen-Poiseuille flow, a flow in a tube with a circular cross section, is one of the most important flow types in industry applications. Classically, the velocity inside the tube follows a parabolic distribution and vanishes at the inner wall of the tube. As the tube radius decreases, the velocity profile is quite different. For a gas flow, the gas mean free path is on the order of 10 nm at atmospheric pressure and room temperature; if the tube size is on the order of 1 micron or the Knudsen number is on the order of 0.01, the slip boundary condition must be considered (Harley et al., 1995). Furthermore, Wu (2000) proved that in addition to the slip condition, the gas compressibility, nonparabolic velocity profiles, and channel deformation due to high pressure in the channel should be considered in order to predict the mass flow rate in micro channels accurately. For a liquid flow, macroscopic criteria to determine the onset condition of velocity are not available, and molecular level simulation must be used. Only a few MD studies are related to the Poiseuille liquid flow; some considered flows in nano tubes (Heinbuch and Fischer, 1989) whereas others simulated two-dimensional Poiseuille flows confined between two solid plates (Koplik et al., 1988; Travis et al., 1997; Travis and Gubbins, 2000). According to simulation results of these studies, the velocity profile deviates dramatically from the classic parabola when the length scale of the flow shrinks to the same order of the molecular radius because the viscosity is no longer uniform (instead, it shows
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large fluctuations). The boundary conditions, slip, no-slip, and multilayer locking, however, have not received enough attention. For example, in most studies, interaction parameters between solid and liquid molecules are at least equal to the the interation parameters between liquid molecules, which seems to guarantee establishment of the no-slip or multilayer locking condition.
VI. Three-Phase Systems The three-phase system refers to the condition in which solid, liquid, and vapor phases are all present. Because of its complexity, it has been less extensively simulated by MD techniques than other interface cases. The main research efforts have been focused on the spreading wetting and contact line region but, as discussed later, interfacial problems in planar and spherical systems are also important and interesting. A. PLANAR INTERFACES
The planar three-phase system includes (1) a liquid film sandwiched by a bulk vapor and a bulk solid and (2) a vapor film by a bulk liquid and a bulk solid. The first case exists in heterogeneous condensation and the second case exists in heterogeneous boiling. The Lithographically-Induced Self-Assembly (LISA) technique recently developed by Chou and Zhuang (1999) also belongs to the second case. The major problem in this interface case is film stability. Similar to the planar liquid-vapor interface case, the stability conditions for a liquid film and a vapor film are different. Using thermodynamics theories, Majumdar and Mezic (1998, 1999) considered stability of a water film on an atomically smooth and hydrophilic substrate and obtained a stability region map. This stability study is similar to that in the case of a liquid film between two vapors except that more interaction potentials are involved, including hydration, electrostatic force, and elastic strain between solid and liquid. With properly given intermolecular potentials between liquid molecules and the solid wall, MD techniques can certainly simulate this condition and then compare simulation results with classical predictions. Since a liquid film can form on a smooth, solid substrate during condensation, it is also reasonable to assume that a vapor film could form on the solid substrate during boiling, called explosive boiling. Yang and Tsutsui (2000) used microfabricated line resistors to study heterogeneous boiling in a bulk liquid. They found that if the heater surface was very smooth and the heat flux was high (~ 106 W/m2), a vapor layer formed first between the bulk liquid and the heater surface (shown
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(a)
FIG. 17. Schematic of (a) electrical heating in liquid and (b) Lithographically-Induced SelfAssembly (LISA).
schematically in Fig. 17a), and as the vapor layer thickness grew, it became unstable and turned into a bubble. Note that this boiling mechanism is different from classical heterogeneous boiling due to the unavailability of nucleation cavities on the heater substrate (Carey, 1992). It is not known what causes the film to break, and both experimental studies and numerical simulations are required in order to solve this problem. With the exchange of the liquid and solid positions, one obtains the condition for LISA (see Fig. 17b). Chou and Zhuang (1999) found when they put a mask (solid) above the polymer (liquid) surface (the vapor phase was in between) and increased the temperature, the polymer formed pillars that reached the mask. In contrast to the previous case, here gravity helps stabilize the interface, and the interface instability was caused primarily by attraction between the solid and the liquid.
B. SPHERICAL INTERFACES A simple example of the three-phase spherical interface case is that a droplet containing a solid core suspends or moves in its vapor, and this condition occurs when a solid particle induces condensation in the vapor phase and a liquid layer
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Chang-Lin Tien and Jian-Gang Weng
consequently covers the solid surface. In this system, the curvature effect on surface tension for both solid-liquid and liquid-vapor interfaces exists; and, because of the small thickness of the liquid layer, these two effects may be highly correlated. Such a problem has not yet been addressed.
C. SPREADINGWETTING AND CONTACT LINE REGION
MD simulation of spreading wetting of a droplet or a bubble on a solid substrate and the contact line region has revealed some unique phenomena. The concept of a contact angle is different from the classical definition: Because several layers of liquid molecules cover the solid substrate due to absorption, usually no vaporsolid interface exists, and neither does the contact line where three phases meet. Yang et al. (1991, 1992) and Maruyama et al. (1998) demonstrated the dynamic evolvement of liquid-vapor interface shape with time. In addition, Maruyama and Kimura (2000) also conducted simulation of a bubble on a solid substrate. Results from these studies lead to the conclusion that the ordered liquid molecules on the solid substrate influenced the spreading wetting significantly. In all MD simulation of spreading wetting, the system temperature is kept constant and uniform. In real applications, however, a temperature difference, called superheat, usually exists between the solid substrate and liquid. Under this condition, the liquid film and the contact line region may have different behavior as a result of evaporation at the liquid-vapor interface. Currently, many interesting problems regarding this subject are still unsolved; for example, there are two totally different arguments regarding whether a significant heat flow occurs in the nanoscale contact line region. MD simulation can be a useful method to understand these issues. The capillary phenomenon refers to the case in which the liquid level in a small tube is different from the level of outside liquid, and it occurs when the contact line region moves up or down in order to achieve equilibrium. In macro- and microlength scales, the capillary phenomenon has been well understood and used in applications (e.g., heat pipes). When the size of the capillary tube reduces to nanoscale, the scenario changes. Two things can happen: (1) the curvature effect of surface tension becomes important, and (2) the bounded liquid molecules near the tube wall further decreases the area of flow path. Ugarte et al. (1996) observed that open carbon nanotubes with inner diameters larger than 4 nm could be filled with molten silver nitrate, whereas carbon nanotubes with inner diameters about 2 nm could not. The authors concluded that this was due to increased solid-liquid interfacial tension caused by increased stress of the tube wall when the inner
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diameter decreases. However, the curvature effect on surface tension as well as the decreased flow path might also contribute to this discrepancy.
VII. Other Interfacial Phenomena This section discusses the interfacial problems that are not included in previous four sections: sonoluminescences (SL) and surfactants. These two problems are generally more complicated and require more detailed physical understanding, more computational power, and sometimes nonclassical MD simulation techniques. However, as nanoscale interfacial study develops, these problems will receive more attention because of their practical importance.
A. SONOLUMINESCENCE
When a small bubble experiences a large-amplitude volume oscillation driven by a sound field, it emits visible photons in every acoustic cycle. This phenomenon is called sonoluminescence (SL). The first single bubble sonoluminescence (SBSL) was observed around 1990 (Gaitan and Crum, 1990), and one of the earliest serious studies was conducted by Barber and Putterman ( 1991). Since then, an intense amount of research work in this area has been published (e.g., Moss et al., 1997; Vuong et al., 1999; Yasui, 1999; Jensen and Brevik, 2000), but thorough and unanimous understanding of the generation mechanism of SL is still not available. Two different theories exist: One theory explains SL based on a shock-wave model, and predicts that the temperature at the bubble center can reach several million degrees, which causes photon emission (Vuong et al., 1999); the other theory indicates that SL is the first macroscopic demonstration of quantum vacuum radiation, in which photons are emitted due to the interaction between the rapidly decelerating dielectric interface and the quantum vacuum (Yasui, 1999; Jensen and Brevik, 2000). The entire experimental apparatus is relatively simple, but the accurate measurement is extremely difficult. Under this condition, theoretical and numerical approaches might be preferred. In SL the input and expected (calculated) system parameters are as follows: the initial pressure is about 1 bar; the applied acoustic pressure is also about 1 bar, and frequency is on the order of 10 kHz; the initial size of bubbles is on the order of 1 #m; the minimum size of bubble is on the order of 0.1 #m; the temperature at the bubble center is on the order of 10,000 K, at the final stage of bubble collapse the time duration is about 50 ps (Moss et al., 1997; Vuong et al., 1999). The
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Chang-Lin Tien and Jian-Gang Weng
macroscopic numerical approach is based on equations of conservation of mass, momentum, and energy for the system. Many simplifications are made, some of which have not been justified. If SL is indeed induced totally by electron dynamics (the second school of theory), MD simulation cannot be used because although quantum MD (QMD) deals with dynamics of electrons and photons, the length and time scales QMD can handle are far too small compared to the scales in SL. For the shock-wave model, however, classical MD simulation can be applied as a benchmark. In this case, MD need only simulate the final period of acoustic oscillation when SL occurs. Here, the time duration is about 50 ps and the bubble size is about 0.1 #m. Note that the bubble size is one order of magnitude larger than what MD can simulate, which may indicate that MD simulation can yield only qualitative results. Pressure oscillation can be realized by changing the boundary size and at the same time keeping the total molecule number constant. This can be achieved by forcing each molecule to move inward or outward for a short distance simultaneously. When the system reaches the steady state, a temperature distribution can be measured and the temperature at the bubble center indicates the possibility of SL. In order to prevent vapor molecules from moving out of the bubble, different intermolecular potentials can be applied to vapor and liquid molecules.
B. SURFACTANTS
Surfactants refers to surface-active materials that are strongly adsorbed at an interface in the form of an oriented monomolecular layer (monolayer) and can dramatically reduce the surface tension of water and other aqueous solutions. They are usually long-chain hydrocarbon molecules and are relatively insoluble in water. Using surfactants actively and efficiently makes it possible to engineer and control the interface. Limited work in MD simulation of surfactants has been published (e.g., Laradji and Mouritsen, 2000), and more work should be done to provide more insights on this subject.
VIII. Concluding Remarks This chapter presents a review of MD simulation of various nanoscale interfacial phenomena in fluids. The liquid-vapor interface simulation is emphasized because publications in this area outnumber publications concerning all of other interfacial studies combined. Despite the extensive efforts, many fundamental problems in this
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area are still not solved satisfactorily; for example, the curvature effect of surface tension and interfacial stability. For interfacial phenomena involving solid materials, the critical issue seems to be how to model solids (thermally and mechanically) in the most realistic way, because very large fluctuations in local properties occur in "bulk" solid and solid-liquid interfaces. Determination of transport coefficients is also important in studies of interfaces because any interface is highly anisotropic and results in large differences between transport coefficients in different directions, and flow, thermal, and diffusion boundary conditions change dramatically. Sonoluminescence and surfactants are very interesting interfacial phenomena that could be simulated by MD methods after some assumptions are made. As stated previously, the advantage that makes MD simulation unique is its capability to obtain molecular-level structures and capture short-time molecular behavior. It should also be noted, however, that in MD simulation the computational domain size is so small and the simulated period is so short that direct comparison of MD results with macroscopic behavior (i.e., experimental observation or classic analysis) is almost impossible. It is, therefore, crucial to apply statistical or scaling theories to bridge this gap. The developments of these "bridges" will not only make MD a versatile tool to calculate micro- nanoscale properties, but also an economical tool to optimize micro- nanosystems.
Acknowledgments The authors thank Professor Arunava Majumdar in Department of Mechanical Engineering, University of California at Berkeley, and Professor Theodore Wu in Department of Engineering Science, California Institute of Technology for valuable discussions and numerous comments during preparation of this chapter, and Professor Seungho Park in Department of Mechanical Engineering, Hongik University, Seoul, Korea, for extensive collaboration in the course of this study. The financial supports from the U.S. Department of Energy and the National Science Foundation are gratefully acknowledged.
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Fracture of Piezoelectric Ceramics qONG-YI ZHANG, MINGHAO ZHAO, and PIN TONG Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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IV. Piezoelectric Dislocation and G r e e n ' s Function . . . . . . . . . . . . . . A. Interaction of a Piezoelectric Screw Dislocation with an Elliptical Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Interaction of a General Piezoelectric Dislocation with an Elliptical Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Extended Line Force on the Elliptical Cavity Surface . . . . . . . . D. Solutions for Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Force on a Piezoelectric Dislocation . . . . . . . . . . . . . . . . . . . .
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V. Conductive Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Uniform Remote Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Antiplane Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Interaction of a Dislocation with a Conductive Crack . . . . . . . .
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VI. Interface Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. I m p e r m e a b l e Interface Cracks . . . . . . . . . . . . . . . . . . . . . . . . B. Other Relevant Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Three-Dimensional Electroelastic Problems . . . . . . . . . . . . . . . . . A. General Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. An Ellipsoidal Cavity under Remote Loading . . . . . . . . . . . . . C. An I m p e r m e a b l e Planar Crack of Arbitrary Shape under Arbitrary Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. A Permeable Planar Crack of Arbitrary Shape under Arbitrary Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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VIII. Nonlinear Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Electrostriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Domain Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Domain Wall Kinetics Model . . . . . . . . . . . . . . . . . . . . . . . . . D. Polarization Saturation Model . . . . . . . . . . . . . . . . . . . . . . . . IX. Experimental Observations and Failure Criteria . . . . . . . . . . . . . . A. Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Failure Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. Introduction
Piezoelectricity is a phenomenon of interaction between mechanical and electrical responses of materials. When a mechanical load is applied to a piezoelectric body, a voltage is induced. Conversely, when a voltage is applied to such a material, the shape and dimensions of the body change. Thus, piezoelectric materials have the ability to convert mechanical energy into electric energy, and the converse is also true. There are classes of crystalline materials possessing such properties. The most widely used piezoelectric materials are ferroelectric. For a crystal composed of ferroelectric material, at temperatures higher than the Curie point, the local positive center of a unit lattice cell coincides with the negative charge center of the cell. In such instances, the material is said to be in the paraelectric phase. When the temperature is lower than the Curie point, spontaneous polarization occurs along crystalline directions, wherein the local positive charge center of a unit cell deviates from the negative charge center. The material is then said to be in the pyroelectric phase because the value of the spontaneous polarization is dependent on temperature. The deviation generates an electric dipole as well as internal local strains. The direction of the spontaneous polarization is called the polar axis. If the direction of the spontaneous polarization in a pyroelectric crystal can be reversed by application of an electric field, the crystal is called a ferroelectric crystal (Jona and Shirane, 1993). The electric dipoles may align only over a region of the crystal, whereas in another region the direction of the spontaneous polarization may be different or even reversed. The region of uniform polarization is called an electric domain. Figure l a shows a schematic depiction of the domain structure of a piezoelectric ceramic, and Fig. 1b shows the microscopic domain structure as revealed through scanning electron microscopy.
Fracture of Piezoelectric Ceramics 180 ~ d o m a i n wall
/ (a)
\ J
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. . . .
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FIG. 1. Domain structure. (a) Schematic depiction and (b) a scanning electron microscopy photograph of an etched fracture surface showing grain and domain structure of PZT-841 ceramics. Many of piezoelectric materials used in contemporary applications are polycrystalline ceramics. Piezoelectric ceramics have quick response. Their physical, chemical, and piezoelectric characteristics can be tailored to specific applications. Hard and dense bulk ceramics can be manufactured in almost any shape and size by the powder metallurgy technique. Ferroelectric thin films can be grown by deposition, such as metal-organic chemical vapor deposition, sol-gel, sputtering, and so on.
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Piezoelectric ceramics are chemically inert and immune to moisture and other atmospheric conditions. In addition, the mechanical and electrical axes of these ceramics can be oriented precisely in relation to the shape of the ceramic. These axes are set during poling, a process that induces piezoelectric properties in polycrystalline ceramics. Before poling, the electric domains in a polycrystalline ceramic are distributed randomly. When a highly static electric field is applied to the polycrystalline ceramic near its Curie point, all domains align along the direction of the applied electric field, which is called the poling field. Then, decreasing the temperature to room temperature while holding the applied poling field causes the aligned domains to freeze, to a large extent, and thus the material exhibits macroscopically piezoelectric responses. It is the poling that gives polycrystalline ceramics their macroscopically piezoelectric properties. The orientation of the static poling field determines the orientation of the mechanical and electric axes of the material. Different poling processes may cause a polycrystalline ferroelectric ceramic to exhibit macroscopically piezoelectric responses in different directions or in a combination of directions. Figure 2a schematically shows a hysteresis loop of the polarization P versus the electric field strength E, whereas Fig. 2b depicts a butterfly loop between the strain, e, and the electric field strength. The relationship of P - E or e - E is generally nonlinear. However, a linear relationship of P - E or e - E holds approximately in certain narrow regimes. The linear constitutive relationship is described in detail later in this article. In addition to the piezoelectric property, ferroelectric ceramics exhibit the pyroelectric effect, whereby the thermal energy can be converted into the electric energy, and the converse is also true. As a result of these unique properties, ferroelectric ceramics are being more widely used in smart structures, microelectronics, and microelectromechanical systems. Clear understanding of the mechanical and fracture properties of ferroelectric ceramics is becoming increasely important as the applications of piezoelectric materials expand rapidly. This article summarizes current knowledge concerning the fracture of piezoelectric ceramics. Attention is confined to fracture mechanics studies, yet experimental results are also examined for comparison with theoretical predictions. Basic equations, including the thermodynamic functions, kinematic equations, static equilibrium equations, principles of piezoelectric virtual work, linear constitutive equations, energy release rates, and the J integral, are introduced in Section II, where we also list the material matrices of commonly used piezoelectric crystals and introduce thermal stresses. Section III presents Stroh's formalism for two-dimensional electroelastic problems and studies an elliptical cavity. To demonstrate physical insights explicitly, antiplane problems are solved in parallel with using Stroh's
Fractureof PiezoelectricCeramics
151
I P Ps
E ,....-
E~
(a)
l
E
Fro. 2. Schematic plots of (a) a hysteresis loop of electric polarization versus electric field strength and (b) a butterfly loop of mechanical strain and electric field strength.
formalism. We examine closely the electric boundary conditions along the crack faces and their influences on the intensity factors and the energy release rate. Section IV describes Green's function for an elliptical cavity and the interaction of a piezoelectric dislocation with an elliptical cavity. An image method, which converges much faster than the Faber series, is used to construct the solution. In Section IV, we also determine the force acting on a piezoelectric dislocation. The result is an extension of the well-known Peach-Koehler formula. Section V establishes the fracture mechanics and Green's functions for conductive cracks. In Section VI, we consider interfacial cracks in piezoelectric ceramics, with special
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attention given to electrically impermeable cracks due to their mathematical simplicity. The linear solution is oscillatory near the crack tips. Three-dimensional electroelastic problems are discussed in Section VII. The electrical boundary conditions along crack faces play an important role also in the three-dimensional fracture mechanics. Four types of nonlinear approaches (i.e., electrostriction, domain switching, domain wall kinetics, and polarization saturation) are introduced in Section VIII. In Section IX, we review experimental observations and failure criteria. Finally, summary remarks are provided in Section X. There are voluminous theoretical studies on the fracture of piezoelectric materials, not all of which can be discussed in this article. The number of experimental studies, although increasing greatly in recent years, is still limited. The experimental results indicate that fracture behaviors under electrical, mechanical, and combined loading are complex and cannot be explained totally by existing theoretical work. Moreover, the experimental data presented by different researchers sometimes contradict. More theoretical and especially more experimental studies are needed to provide a complete understanding of the fracture behaviors of piezoelectric ceramics.
II. Basic Equations Parton and Kudryavtsev (1988) summarized the thermodynamic functions of piezoelectric materials. The internal energy, free energy, electric enthalpy, mechanical enthalpy and full Gibbs energy are described briefly here. The internal energy per unit volume u can be expressed in the differential form using the index notation as du = (YijdEij -']- E i d D i + T d s ,
(2.1)
where s is the entropy per unit volume, T is the absolute temperature, Ei is the electric field strength, Di is the electric displacement, O'ij is the component of the stress tensor, and 8ij is the component of the strain tensor. Hereafter, the repeated indices denote summation, unless special notice is given. The free energy per unit volume f is defined as f = u - Ts.
(2.2)
Then df = r
-Jr-EidDi - s d T .
(2.3)
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The electric enthalpy per unit volume h is defined as (2.4)
h - u - E i D i - T s,
with dh
=
oijdEij
DidEi - sdT.
-
(2.5)
Let the mechanical enthalpy per unit volume w be (2.6)
113 = U - - O ' i j e i j -- T s .
Then dw = -6ijdaij
+ EidDi
- sdT.
(2.7)
The full Gibbs energy per unit volume g is defined as g = u
-~ijeij
-
Ei Di -
Ts,
(2.8)
with dg = - e i j d c r i j
-
DidEi
-sdT.
(2.9)
There are three types of independent field variablesmmechanical, electrical, and thermal--that determine the six thermodynamic functions. The independent m e c h a n i c a l variables can be strains or stresses, whereas the independent electric variables can be electric field strengths or electric displacements. The thermal variable is usually temperature. For constant temperature (i.e., the isothermal condition), Eqs. (2.3), (2.5), (2.7), and (2.9) reduce, respectively, to df
= c r i j d e i j 2r- E i d D i ,
dh = aijdeij
-- D i d E i ,
(2.10) (2.11)
dw
=
-Eijd(Yij
"1- E i d D i ,
(2.12)
dg
=
-6ijdoij
-
DidEi.
(2.13)
For piezoelectric solids that have zero body forces and are free of electric charges, the kinematic and static equilibrium equations are given by
1 8ij ~. -~(ui,j -Jr-uj,i),
oij,j -- O,
(2.14) Ei = -~,i,
Di.i = 0
Tong-Yi Zhang et al.
154
where ui is the displacements, 4~ is the electric potential, and the subscript ,j denotes differentiation with respect to xj. We have the following principles of piezoelectric virtual work of an isothermal process for the domain rI (Zhang, 1994b; Zhang et al., 1998)
f
(tit~Ui-
ck6nr)dF-
fn6f dl-I -O,
(2.15)
0,
where F denotes the boundary of the domain, t is the traction vector along the boundary with the component ti = o'ijn j, ~ = Dj nj is the boundary value of the electric displacement, and n is the unit vector normal to the boundary and outward from the domain. Let us introduce the following four isothermal potential energies:
PF = fr (tiui -- ~r~)dF
-
fn f dl-I,
(2.19)
where the tilde (~) denotes the prescribed quantities on the boundary, which are tractions and electric potential;
Pu -- fr (~iui + g~ck)dF - fn h dFl,
(2.20)
where the prescribed quantities on the boundary are tractions and the boundary value of electric displacement;
Pw=fr(-aiti-~$)dr'-fnwdIq,
(2.21)
where the prescribed quantities on the boundary are displacements and electric potential;
Pc - fr (-~iti + ffrdp)dF - fn g drI,
(2.22)
where the prescribed quantities on the boundary are displacements and the boundary value of electric displacement. Then, the piezoelectric virtual work can be
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155
rewritten as 8PF = fc([iaui - ~Sttr)dF - s
3 f dII = O,
(2.23)
(~PH = fr([iaui + ffrac/))dr - s
ah dFl = 0 ,
(2.24)
aPw = f r ( - a , Sti - a S ~ ) d F - s
aw dFl = O,
(2.25)
8Pc = f r ( - t i i S t i + O a q ~ ) d F - s
agdFl = 0.
(2.26)
If the generalized force P, the generalized displacement A, the generalized voltage V, and the generalized charge Q are used, then the changes in the total free energy F, the total electric enthalpy H, the total mechanical enthalpy W, and the total full Gibbs energy G for the entire sample are given by gF = PdA + VaQ,
(2.27)
dH = PdA - QdV,
(2.28)
+ VdQ,
(2.29)
d G - - A d P - Qd V.
(2.30)
dW = -AdP
Now, consider a piezoelectric material containing a crack. Adding the energy change associated with the crack extension into each of Eqs. (2.27)-(2.30) leads to Jda,
(2.31)
QdV - Jda,
(2.32)
dF = PdA + VdQd H --- P d A dW = -AdP
Jda,
(2.33)
QdV - JdA,
(2.34)
+ VdQ-
d G --- - A d P -
where A is the area of the crack faces and J is the energy release rate for crack propagation defined as
J=
~
A,o_
a,v
= -
-07
P,o
= -
P,V
.
(2.35)
Alternatively, the energy release rate can be evaluated by following Rice's (1968) treatment of elastic fracture mechanics, and is given by J--
OPF
aPH
3A
3A
-
3Pw 3A
-
3Pa 3A "
(2.36)
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156
For two-dimensional problems, if the crack grows along the crack plane (along the x~ axis), the energy release rate can also be calculated from each of the following four path-independent J integrals as long as the integration path encloses the crack tip:
J
iF (fnl --~Yijtljui,1 - - D1Eini)dF,
(2.37)
J - fr (hnl -crijnjui,1 + DiElni)dF,
(2.38)
J - fr (wnl + crij,lnjui - D1Eini)dF,
(2.39)
J -- fr (gnl + crij,lnjui + DiElni)dF.
(2.40)
--
Cherepanov (1974) derived the J integral [Eq. (2.38)] and Pak and Herrmann (1986) presented a similar path-independent integral, which included the Maxwell stresses, for dielectric materials. It is noted that no constitutive equations are involved in the derivation of the piezoelectric virtual work, the energy release rate for crack propagation, and the J integrals. This means that Eqs. (2.37)-(2.40) can be applied to piezoelectric solids with linear or nonlinear constitutive laws. The isothermal linear constitutive equations define the linear relationships between the field variables O'ij , gij , E i , and D i in the form E
(7ij = Cijkll3kl -- ekij E k ,
(2.41)
O i - - eiklEkl + Kike E k ,
where ci~kl is the isothermal elastic constants, e k i j is the isothermal piezoelectric constants, and tcik e is the isothermal dielectric constants at constant strains. In Eq. (2.41), the stresses and the electric displacements are expressed in terms of the strains and the electric field strength. The constitutive equations can also be written in the form E
p
Eij --" Sijkl(Ykl @ dki j Di --
Ek,
(2.42)
d~takt + K i ~ E k ,
where siejkt is the isothermal elastic compliance constants at constant electric field strengths, xi~ is the isothermal dielectric constants at constant stresses, a n d dkPij
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157
is the isothermal piezoelectric moduli. The relationships between the material constants in the index notation are E E __ r CijklSklmn
'
dPmij m emklSklij, E
(2.43)
Kika ~ Kike -~- eij m d~jm ,
where r i s the Kronecker delta. We can also take thermal stresses into account. Iesan and Scalia (1996) discussed thermoelastostatic deformations in detail for conventional solids. In thermoelastostatics, thermal stresses are treated as "body forces." In the absence of time dependence, the entropy does not change with time. Temperature is controlled by the steady state heat transfer equation, which ignores the temperature change induced by the mechanical and electrical fields. We name this approach the pseudoisothermal approach because the temperature field is calculated from heat transfer only, and not affected by the mechanical and electrical fields. Once the temperature field is available, the mechanical and electrical fields can be calculated in a manner similar to the isothermal process. In the widely used pseudoisothermal approach, the temperature distribution is calculated from the steady state heat transfer equations
hi
hi,i = O,
= --(TlijO),j,
(2.44)
where hi is the ith component of the heat fluxes, l']ij is the heat conduction coefficients, 0 - T - To and To is a reference temperature. For a given system, solving Eq. (2.44) with appropriate boundary conditions yields the temperature field. In this case, we may define the pseudoisothermal stresses and pseudoisothermal electric displacements as O-i*j
- -
O-ij Af_ ~. iEjjO
- -
CijklSkl -- ekij E k , E
(2.45)
D~ -- Di - pEo -- eiktekt + Ki~Ek, which satisfy the equilibrium equations
9
Di,i - - - P i
E
(2.46) O, i,
where ~.E is the temperature stress coefficients and pE is the pyroelectric
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Tong-YiZhanget al.
coefficients at constant strain and constant electric field strength. The virtual work for a crack-free piezoelectric solid in static equilibrium may be expressed as
fF(~
fll()~iEjo'j~tti J-~pE~O'j)dI-IfI-I~f*dI-I-O, (2.47)
fF(O'ijnj'ui-J-D;l'lj'~))dI-'f. (~.Eo, j'Ui--pEO,j,~))d[" I--f, ,,,,,_o.
(2.48)
fF (--Ui'~
f~ (ui)~E'o'J-dpp~,Oj)dFl-frI ,w*dFl--0,
+ Djnj6~)dI-' + fo (ui~ij ,j f.(--ui(~oijnj * *
(2.49)
j
--
f,,...-o, (2.50)
where f*, h*, w*, and g* are the pseudoisothermal thermodynamic functions, which have the same forms as Eqs. (2.10)-(2.13) except that the stresses and electric displacements are replaced by the corresponding pseudoisothermal ones. For a piezoelectric ceramic containing a crack, we may define pseudoisothermal potential energies, similar to the isothermal potential energies given by Eqs. (2.19)(2.22), as follows:
(2.51)
PH: fF (~
L)jnj~))dI"-f~ (~.EO,jUi--pEO, j~)dIl-f
~*.~ (2.52)
P~V--fF(--ui'i;'nJ-~D;nj)d~-JfFl('i)~Eo'J-~)PEO'j)dI-If.w..., (2.53)
P~= fr (--UiO'ijnj-~-[)~njdp)dF+fl-l(Ui)~Eo'Jqt-pyO, jdp)dl-I-fN g*dI-l. (2.54)
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Consequently, the energy release rate for crack propagation is calculated from
J =
OP~ OA
OPh OA
=
=
aP~ OA
=
aPa OA
.
(2.55)
In this case, the path-independent integrals for two-dimensional problems are given by
J=fv
( f * n l --
J =
(h*nl - - ~ i j n j u i ,
f
oijnjui, l -- D~Eini)dF Jr- s
().iEjO,jUi, 1 + qbpEO,jl)dl-[, (2.56)
*
l --]- D i Elni)dF +
*
(~.iEjOjUi,l + pj OjE1
s
~
)dn
,
(2.57)
J = f v (w*n 1 + oij,* lnjui - D~Eini)dF -flq (ui~.EOjl -ckp~O ,jl )dFl , (2.58)
J = fv (g*nl --~-oij,lnjui + D*Elni)dF - fl-I (ui~EO'jl - p~O jEl)dH. (2.59) Note that the surface area F is the surface area of the volume H of a closed domain. Kishimoto et al. (1980) and Prasad (1998) presented a path-independent J integral with thermal stresses for conventional solids that also included an area integral similar to Eqs. (2.56)-(2.59). In the pseudoisothermal approach there is an alternative way to derive the virtual work and the J integral. We still use Eqs. (2.10)-(2.13) as the definitions of the pseudoisothermal thermodynamic functions. The virtual work for a crack-free piezoelectric solid in static equilibrium can be expressed as
l"
fv (6ijnj~ui - ~6Djnj)dF - Jn 6f dFl
O,
(2.60) (2.61)
f
f
-
- O,
(-ai~aijnj + D j n j ~ ) d F -
-
s
~gdFl - 0 ,
(2.62) (2.63)
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160
where the prescribed quantities on the boundaries are correspondingly the same as those for Eqs. (2.19)-(2.21). Accordingly, the pseudoisothermal potential energies are
P~.-fv(6ijnjui-~Djnj)dF-frfdI-I,
(2.64)
P[_l-fv(Sijnjui+~jnj~)dF-filhdrl,
(2.65)
P~v-fv(-~icrijnj-~Djnj)dF-frlwdI-I,
(2.66)
P~ --
(2.67)
(-Si~ijnj + Djnjd~)dF-
gdl-I,
for a piezoelectric ceramic containing a crack. The energy release rate for crack propagation is calculated from Eq. (2.55), and the path-independent integrals for two-dimensional problems are given by J -- fv (f*nl -0-ijnjui.1 -- D 1 E i n i ) d I - ' ,
(2.68)
0-ijnjui,1 + DiElni)dF,
(2.69)
0-ij.lnjui - D1Eini)dF,
(2.70)
+ 0-ij, l n j u i + O i E l n i ) d I " .
(2.71)
J
--
iF (h*nl -
J - fv (w*nl + J --
fF (g'n,
In the second pseudoisothermal approach, the thermal stresses are implicitly involved in the pseudoisothermal thermodynamic energies. Gurtin (1979) derived a J integral similar to Eq. (2.69) for conventional solids. In Eq. (2.41), we can take into account the symmetry of piezoelectric crystals to reduce the number of independent coefficients in the linear constitutive equations. If the contracted notations are introduced, 0-1 ~
fill,
81 - - 8 1 1 ,
0"2 - - 0"22,
0"3 --" 0"33,
0"4 --" 0"23,
82 =
83 - - 8 3 3 ,
84 =
822,
2823,
0"5 =
0"13,
85 - - 2 e 1 3 ,
0"6 - - 0"12, 86 =
(2.72)
2812,
the constitutive equation [i.e., Eq. (2.41)] can be rewritten as
0"• = c•
7 - ek• Ek ,
Di = ei•215 + Kik Ek,
V, 0 = 1,2 . . . . . 6;
i,k=1,2,3,
(2.73)
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161
where the subscripts given in Greek notation are from 1 to 6 and the subscripts given in the Roman alphabet are from 1 to 3, and E
ekij
Cyrl - - C i j k l ,
-
e
ek•
(2.74)
Kik -- Kik.
The following rules 11~-+ 1, 2 2 ~ 2 ,
23 or 3 2 ~ 4 ,
33~3,
13 or 31+-~5,
12 or 2 1 ~ 6 (2.75)
are applied to replace ij by y. Nye (1972) presented a comprehensive review of the physical properties of crystals in tensor and matrix forms 9 Table I lists the nonzero matrix elements for the different symmetrical classes, including quartz (class 32), lithium niobate (LiNbO3, class 3 m), cadmium sulfide (CdS, class 6 mm), poled piezoelectric ceramics (class 6 mm), and gallium arsenide (GaAs, class 3~3 m). TABLE I MATERIALS CONSTANTS
Dielectric constants Piezoelectric constant matrix e~i
Elastic constant matrix c,~
Crystal of trigonal symmetry (classes 32, 3 m)
t
Cl I C12 C14
C12 C13 Cll
C13
C14
9
--C14
Class 3 m
Class 32
9
ell
"
--e22 e22 9 e51
9
-e,,
--C14
9
"
9
C44
9 c66 = (Cll --
9 "
"
C44 C14 C14 C66/
;i
41
c12)/2
--e41
e51
--ell
e22
(Kl ") Kll
K33
:!3,)
Crystal of hexagonal symmetry (class 6 mm) Cll c12 c13 c12 ell c13
e13'~ r / e33J
m a t r i x Kij
9
e13~
ci13 c13. c33. c44"
i!
e |
c44 9
Kll
e51
K3
C6
C66 -- (ell -- C12)/2
Crystal of cubic symmetry (class 3~3 m)
t
Cl 1 C12 C12 C12 e l l C12 C12 C12 Cll 9 9 9 C44 9 . c44 9
9
KI1 9
e l
KII
e41 c
9) KI1
e41
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Tong-Yi Zhang et al.
III. Two-Dimensional Electroelastic Problems and Stroh's Formalism
There are two commonly used coordinate systems in the study of fracture mechanics for poled piezoelectric ceramics. The first is the material coordinate system (~l, ~2, ~3), wherein the poling direction is often in the -~3 axis. The second is the sample coordinate system (Xl, x2, x3), wherein the xl axis is often chosen to be along the crack direction. The nonzero elements listed in Table I are expressed in the material coordinate system ( ~ , ~2, ~3). Note that these material constants may have different nonzero elements and values after they are transformed into the sample coordinate system (x l, x2, x3) following the tensor transformation rules. In two-dimensional deformation, mechanical and electrical fields are functions of Xl and x2 only, whereas the poling axis could be along any direction in the sample coordinate system. Of all formalisms for two-dimensional deformation of anisotropic elastic materials, the Lekhnitskii fornaalism (1950, 1957) and the Stroh formalism (Eshelby et al., 1953; Stroh, 1958; Ting, 1996) are the two most extensively used by the engineering community. As Ting (1996) stated, the Lekhnitskii formalism essentially generalizes the Muskhelishvili (1953) approach for solving two-dimensional deformations of isotropic elastic materials. The Lekhnitskii formalism begins with stresses and assumes that they depend on Xl and x2 only. Therefore, the Lekhnitskii formalism is stated in terms of the reduced elastic compliances. The Stroh formalism starts with displacements, and the resulting formalism is in terms of elastic constants. The general solution form of the Stroh formalism resembles that of the Lekhnitskii formalism. Thus, we introduce only the general solution of the Stroh formalism for piezoelectric solids.
A. GENERAL SOLUTION BASED ON STROH'S FORMALISM
Using the Stroh formalism, Barnett and Lothe (1975), Suo et al. (1992), Pak (1992a, 1992b), Park and Sun (1995b), and Chung and Ting (1996) established the general solution in terms of strains and electric field strengths for linear piezoelectric solids undergoing an isothermal process. The solution, in turn, is expressed in terms of displacements and electric potential. Following Barnett and Lothe's (1975) treatment, we extend the three-dimensional space to four dimensions, as follows: 1. Define a new operator 8 Ox4
-0
(3.1)
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2. Denote the electric potential 4~ as U 4 3. Define new strain components as Sij = Eij , Si4 - - S4i -
1
and
--~ Ei
i,j=
$44=0,
(3.2)
1,2,3
4. Define new stress components as ~aij = O'ij ,
and
~2i4 = ]~4i = Di
E44 = 0,
(3.3)
i, j = 1, 2, 3
5. Define new elastic constant components as E C i j k I m Cijkl,
e Cakl4 - - Cnk4l - - Ck441 = Ck414 - - --Kkl, C 4 j k l - - C j a k l = C k l a j - - Cklj4 = e j k l ,
i,
j, k, 1 = 1, 2,
(3.4) 3
C44kl = Ckl44 - - C j 4 4 4 = C4j44 - - C 4 4 j 4 = C444j = C4444 - ' - 0 .
Note that O/Ox3 = O / O x 4 - - 0 . The extended elastic constant tensor still retains the following symmetric properties i, j, k, l = 1, 2, 3, 4.
Cijkl - - Cklij - - C j i k l - - C i j l k ,
(3.5)
Consequently, the governing equations can be expressed in the four-dimensional system as 1
Sij - - -~(ui, j + u j , i ) ,
(3.6)
]~ij = Cijkl Skl,
E i j , j = O,
i, j, k, l = 1, 2, 3, 4.
Hereafter, the repetition of a subscript in a term denotes a summation with respect to that subscript from 1 to 4, unless otherwise specified. If u i are functions of Xl and x2 only, the general solution can be written in the form !
uj = Aj f (z)
or
u--A
!
f (z),
(3.7)
where Z = Xl + p X 2 .
(3.8)
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164
Combining Eqs. (3.6) and (3.7), we have [Q + (R + Rr)p + Tp2IA ' = 0,
(3.9)
where Oik = C i l k l ,
and
Rik = Cilk2
Tik =
(3.10)
Ci2k2.
A non-zero solution of A' requires that det[Q + (R + Rr)p + Tp 2] = 0.
(3.11)
Equation (3.11) has eight roots that cannot be real because of the positive definiteness of the strain energy and electric energy densities (Barnett and Lothe, 1975; Ting, 1996). The eight roots form four complex conjugate pairs. Without loss of generality, we define the four roots as p,~ for c~ - 1, 2, 3, 4, with Im(p~) > 0. Introducing the vector L' - (R r + pT)A' - _ I ( Q
p
+ pR)A',
(3.12)
we can rearrange Eq. (3.12) in the standard form of eigen-equation N~ = p~',
(3.13)
where
."i)
N1 = _ T - 1 R r ,
(::)
N2-T-
1 - N 2 r,
N3 -
RT-1R r -Q
- N~',
(3.14)
with ~" being an eigenvector and p being an eigenvalue. The eigenvectors are uniquely determined up to an arbitrary multiplicative constant. They are normalized to satisfy AA T + A A r = L L T + L L T = 0 , LA T + L A r = A L T + A L T = I ,
(3.15)
where I is the identity matrix, the overbar signifies a complex conjugate of the corresponding function, and A-(A'~
A~
A~
A~),
L-(L'~
L'2
L'3 L~)
(3.16)
are the eigenvector matrices, in which (Li) -- ~'7 is thejth eigenvector of Eq. (3.13), which is associated with the jth eigenvalue p j, j = 1, 2, 3, 4. J
Fracture of Piezoelectric Ceramics
165
The general solution for two-dimensional problems can be written in the form
where f -
U = Af + Af,
(3.17)
~p = Lf + Lf,
(3.18)
[fl(Zl) f2(z2) f3(z3) fa(Z4)] r, ~ is the extended stress potential, and
Z -- X l -k- pjx2, j = 1, 2, 3, 4. If we express the extended stresses in matrix form E1 "- (0-11
0-21
0-31
Dl) r,
E2 = (0-12 0-22 0-32 D2) r,
(3.19)
then the extended stress matrices, 0-33, and D3 become ]El = - - 0 , 2 ,
0"33 - - ]~33 - -
(3.20)
E 2 - - 1//,1,
1
-~C33ij(ui,j Jr- uj.i),
D3 = E34 - - Z43 - -
(3.21)
1
-~C34ij(ui.j + uj,i)
It can be shown (Lothe and Barnett, 1976; Suo et al., 1992) that the matrix B - i A L -~
(3.22)
is a Hermitian matrix (i.e., B - !~v) and can be partitioned as
The upper left block B1 is a 3 x 3 matrix, and the lower fight element B44 is a scalar. For stable materials, Lothe and Barnett (1976) showed that B1 is positive definite, B44 < 0,
(3.24)
and -T
B2 - B 3 .
(3.25)
For coordinate transformation, an in-plane coordinate rotation matrix may be introduced (Suo et al., 1992)
_
cos0
sin0
0
0
-sin0 0
cos0 0
0 1
00
0
0
0
1
'
(3.26)
166
Tong-Yi Zhang et al.
where 0 is the angle between the x l axis and the rotated x]~ axis, in which the superscript asterisk denotes the quantities associated with the rotated coordinates. Then, we have A* = ~A,
L* = .qtL,
B* : ~ B ~ T.
(3.27)
The pure plane-strain deformation requires Ul = UI(X1, X2),
U2 = U2(X1, X2),
//3 = A3f + A3f = 0,
~b "-" t~(Xl, X2),
(3.28) while the pure antiplane deformation requires Ul
--
A l f + A l f = 0,
U2 = A2f + A2f = 0,
U3 m= U3(Xl, X2),
t~ "-- ~(Xl, X2) (3.29)
where A j, j - 1, 2, 3, is thejth row of the eigenvector matrix A. For purely elastic materials, Ting (1996) expressed the condition for decoupling the antiplane deformation from the in-plane deformation in terms of zero components of elastic constants. Although it is difficult to express the decoupling condition for general piezoelectric ceramics explicitly, for poled ceramics, the in-plane deformation decouples from the antiplane deformation when the poling direction is perpendicular to the (Xl, x2) plane (Suo et al., 1992).
B. GENERAL SOLUTION FOR ANTIPLANE DEFORMATION For antiplane deformation, the electric potential ~b and the displacement U3 can be expressed in terms of two complex potentials (Pak, 1990a, 1990b; Zhang and Hack, 1992; Zhang and Tong, 1996) 4~ = Im[~(z)],
U3 = Im[U(z)],
(3.30)
where U and ~ are analytic functions of z = Xl + ix2 and Im denotes the imaginary part of an analytic function. Then, the electric fields and the shear strains can be written as E2 + iE1 = - ~ ' ( z ) ,
F3e + iy31 = U'(z),
(3.31)
where the prime denotes differentiation with respect to z and ~'3i = 2e3i, i = 1, 2, are the antiplane engineering shear strains. The constitutive equations become O'32 + i0-31 -- c44U'(z) + e l S ~ ' ( Z ) ,
D2 + i D1 = elsU'(z) -- Xll ~'(z).
(3.32)
Fracture o f Piezoelectric Ceramics
167
Gao et al. (1997) developed a strip saturation model using a simplified electroelastic constitutive equation for in-plane deformation. They followed the approach used by Rice et al. (1994) for three-dimensional dynamic crack propagation with a slightly wavy crack front. The simplification allows them to solve in-plane problems in the same manner as antiplane problems.
C. AN ELLIPTICAL CYLINDER CAVITY UNDER REMOTE LOADING
Consider an elliptical cylinder cavity (x2/a 2 + x 2 / b 2 -- 1) in an infinite plane under combined remote mechanical-electric loadings (Fig. 3a), where a and b are, respectively, the major and minor semi-axes of the ellipse. The electric loading is in-plane, whereas the mechanical loading can be in-plane tension and/or shear (mode I and/or mode II) and/or antiplane shear (mode III). The boundary conditions on the surface of the cavity are as follows: crijnj = 0,
i = 1, 2, 3
(traction-free),
Djnj -- D~nj (surface charge-free), q~ = qbc
(continuity of electric potential),
(3.33) (3.34) (3.35)
where the range of summation of the repeated index is from 1 to 3 (i.e., j --- 1, 2, 3) and the superscript c refers to the quantities in the cavity whereas the parameters associated with the piezoelectric medium are without superscript, and n is the unit outward normal vector of the cavity surface.
1. Solution Inside the Cavity
Because only an electric field exists in the cavity, the governing and constitutive equations, respectively, are reduced to (3.36)
vzt~ c = 0,
Ocbc
Djc = K c Ejc = _ x c , Oxj
j-
1,2,
(3.37)
where x c is the dielectric constant in the cavity. The cavity is assumed to be electrically isotropic. The dielectric constant has a value ranging from 8.85 x 10 -12 F/m (Farad/meter) for a vacuum cavity to infinity for an electrically conductive cavity. The general solution of Eq. (3.36) is r
= im[~C(z)],
(3.38)
Tong-YiZhang et
168
al.
o'i2 or e~2
L.
L. L.
(a) z- plane
E7
L. O"il
~
a
(~or
b
D;+
e2
L.
t
T
E 2 or D 2 c,o
L.
O'i2 or e~2
&
g.
(b) w- plane
g.
or -----~~
D~
or oo
.-?
g.
1
T
Eil
T
E~' or D ; FIG. 3. An elliptical cylinder cavity in an infinite piezoelectric medium under combined remote loadings in (a) the z plane and (b) the w plane, where the ellipse is mapped into a unit circle.
169
Fracture of Piezoelectric Ceramics
where ~ C ( z ) is a complex analytic function of z with z = Xl + ix2. The complex electric field E c and electric displacement D c, respectively, are given by Ec
_
~ _ E 2+iEcl-
d~ c dz
,
D c_
c c= D2 + i D l
d~ c
-x
c~. dz
(3.39)
In order to determine ~c, we introduce the transformation z = R(w
(3.40)
+ m),
which maps the ellipse in the z plane to a unit circle in the w plane, where w = vl + iv2, R = (a + b ) / 2 , and m = (a - b ) / ( a + b). The inverse mapping of Eq. (3.40) is w =
Z -'1- ~/Z 2 - - C 2
2R
.
(3.41)
The line segment ( - c , c), where c 2 = a 2 - b 2, o n the Xl axis in the z plane is mapped to a circle of radius ~ in the w plane (Fig. 3b). The electric field must be single-valued along the line segment in the z plane, which requires
~c(~/--mei~ ~c(w/me-i~
(3.42)
where 0 is the polar angle in the w plane. Under remote loading, the complex potential, ~c, takes the following form ~c _ RC
( ml w + -w
-
Cz,
(3.43)
where C is a constant to be determined. In other words, the electric field inside the cavity is constant.
2. S o l u t i o n in the P i e z o e l e c t r i c M e d i u m
The general solution for a piezoelectric material is given by Eqs. (3.17)-(3.18). Consider the four planes defined by z~ - xl + p ~ x 2 , c~ - 1, 2, 3, 4, where p~ is the eigenvalues of Eq. (3.13). In each of the four planes, the elliptical cavity is distorted. The transformation z,~ - R~ w~ +
(a not summed)
(3.44)
maps the distorted ellipse on the z~ plane to a unit circle on the w~ plane, where a - ip~b
R~ = ~
,
2
m~ =
a + ip~b a - ip~b
(a not summed).
(3.45)
Tong- Yi Zhang et al.
170
The inverse mapping function is +
wo" --
,
2Ro"
c~ -- 1, 2, 3, 4,
(c~ not summed)
(3.46)
where 2 -- a 2 + po" 2b2 . co,
(3.47)
To construct the general solution for remote loading, we assume the components of f of Eqs. (3.17) and (3.18) in the form
2Ro'ao'2 fo" -- Ro" ao'l too" nt- ao'2 ) _ ao'l (Zo" -Jr-v/Z 2 -- C2) 2 wo" 2 + zo" + V/z 2 - c 2 c~ - 1, 2, 3, 4
(c~ not summed)
(3.48)
with ao'l and ao'2 being constants to be determined by the boundary conditions. Using Eqs. (3.17)-(3.18), (3.20), and the mapping functions, one can rewrite the boundary conditions (3.33-3.35) on the unit circle in the wo" plane in the form
Ljka~l + Lji, a[,2 - - 0 ,
j = 1, 2, 3
Lnka~l + Lal~a~2 -- --Kc(c + m C ) / 2 Aaka~l + Anka~2 -- (C - m C ) / 2 i where Ajk and Ljk tively, and
,
aj -
(traction-free),
(3.49)
(surface charge-free),
(3.50)
(continuity of electric potential),
(3.51)
are components of the eigenvector matrices A and L, respec(R1
R2 R3 R4) T ---~alj ---~a2j ---~a3j ---~a4j ,
j-
1,2.
We can solve for C and a 2 in terms of I11 C
~2(B 4 +
4B--?)m(1 + x c B44)La~ + (1 - x c B44)La~ m2(1 + x c B44) 2 - (1 - x c B44) 2
a~ - L-ld- L-1La~',
,
(3.52) (3.53)
where
B4 - (B41 B42 B43 B44), d-
(0 0 0 d) r,
d - -xc(c
aj--(alj
+ mC)/2,
a2j a3j a4j
(3.54)
)r ,
j -- 1,2,
Fracture of Piezoelectric Ceramics
171
in which Bij are defined in Eq. (3.22). Note that B44 is a negative real parameter in the unit of the reciprocal dielectric constant (Lothe and Barnett, 1976; Suo et al., 1992). The constant vector a 1 is to be determined from the remote loading conditions. The remote electric loading is normally the electric field strengths Ei or the electric displacements Di, whereas the remote mechanical loadings are the stresses tTij or the strains 6ij, a s shown in Fig. 3a. The remote extended stresses are given by lim E2 = Lal + Lal = E ~ , z--,o~ lim E 1
Z----~ OO
- L (pu)al
(3.55)
L(p--~)a-i-= E ~176
where the superscript infinity (o~) refers to the remotely applied extended loads and ( ) denotes a diagonal matrix. There are only seven independent equations in Eq. (3.55). An additional supplementary equation is needed to determine al, a complex vector of four components. We may take the zero rotation around the outplane axis at infinity as the supplementary condition (Sokolnikoff, 1956), which leads to the equation A2al - (A(p~))lal + A2al - (A(p~))lal -- 0,
(3.56)
with A2 --- (A21 A22 A23 A24) and (A(p,~))l -- (AllPl A12P2 A13P3 A14P4). From the constitutive equation, we have lim $ -- C -l lim E = S ~176 Z---+ O~
(3.57)
Z---~ OO
where S is the extended strain tensor (or matrix) and C -1 is the inverse of the extended elastic constant tensor (or matrix), called the extended compliance tensor (or the extended compliance matrix). We can use Eq. (3.56) and (3.57) to determine al in terms of the applied remote strains. The extended displacements and electric potential in the material can be calculated from Eqs. (3.48) and (3.17), and the extended stresses can be calculated from Eq. (3.20), which can be written in the form 2~2 - -
Lf, l + Lf, l,
where f,j = (fl,j f2,j f3,j f4,j)r,
f~,l =
(
4 R~a~2
a~l - (z~ + v/Z 2 - c~) 2
fu,2 -- Potfa,1
E1 ---- - L f , 2 - Lf,2,
(3.58)
j = 1, 2 and
)
z~ +
-
2v/Z ~ - c 2
c~- 1,2,3,4 (ct not summed) (3.59)
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Tong-Yi Zhang et al.
The extended strains can be calculated from the stress-strain relation [Eq. (3.6)] or from the extended displacements [Eq. (3.17)].
3. Electric Field Inside the Cavity or Crack Equations (3.39) and (3.43) give a uniform electric field inside the cavitymthat is, E c - E~ + i E ~ -
-C.
From Eq. (3.52), the uniform electric field can be written as Ec
-
-
- ( B 4 4- B4) x
(1 - ct)(1 - / 3 ) L ( 1 - i p j u } a l + (1 + c~)(1 4-/3)L{1 - i p j u ) a l 2(c~ + fl)(1 + aft)
(3.60)
where c~ and fl are the dimensionless ratios defined as ot - b / a ,
(3.61)
(Zhang and Tong, 1996), and ,c elf is the effective dielectric constant defined as Keft-
-- 1/B44
(3.62)
(Zhang et al., 1998). Multiplying Eq. (3.60) by the dielectric constant of the cavity yields the electric displacement inside the cavity D c -- D~ + i DCl
__ (B4--]-B44~ (1 - o ~ ) ( 1 - / ~ ) L ( 1 - i p j o t ) a l + ( 1 + c~)(1 + j 6 ) L ( 1 - i p j o t ) a l 2(1 + ct/fi)(1 +cq~)
\
/
(3.63)
For fi --+ 0 (i.e., the dielectric constant of the cavity is much smaller than that of the piezoelectric material), the electric field inside the cavity still has a finite value Ec
~ ( B 4 + B4)
(1 - c~)L(1 - ipjot)al + (1 + c~)L(1 - ipjot)al. 2c~
(3.64)
Equation (3.64) indicates that the dimensionless parameter c~ has a great influence on the electric field inside the cavity. The smaller the or, the larger the electric field will be. The electric field approaches infinity as c~ approaches zero. Another
Fracture of Piezoelectric Ceramics
173
extreme of Eq. (3.60) is that the electric field inside the cavity tends to zero as the dielectric constant of the cavity grows large (i.e.,/3 ~ ~ ) . This is the result for a conductive cavity. When c~ --+ 0, the cavity is reduced to a crack. For simplicity, our present discussions focus on electrically insulating cracks only. Conductive cracks are examined in Section V. Readers may also refer to the work by Zhang et al. (1998), in which three limiting cases of a conductive crack were reported. An electrically insulating cavity or crack has a finite value of fl, which reduces Eq. (3.60) to
E c -- -(B4 -+- B--~)(1 - / ~ ) L a l + (1 + fl)Lal 2/3
(3.65)
Equation (3.65) shows that the smaller the value of/3 is, the larger the electric field inside the crack will be. An explicit result is given later for antiplane problems. As c~ --+ 0, Eq. (3.63) has three limits, which are, respectively,
Dc - (B4 + B4) [E~ + fl(Lal DC=(
D c = 0,
B n) 2+BB444
1 +E~c~//3'
forc~/fl--->0,
(3.66a)
for c~//3 ~ a finite nonzero value,
(3.66b)
for c~//3 --+ cx~.
(3.66c)
In Eq. (3.66b),/3 has the same order as c~, and in Eq. (3.66c),/3 is smaller than c~ by a few orders; therefore,/3 is negligible there in comparison with unity. Equations (3.66a-c) show that the electric displacement inside the electrically insulating crack approaches zero only if the ratio ~/fl approaches infinity. For the other two limits of ~/fl, the electric displacement does not equal zero. Equations (3.66a) and (3.66c) are related to the two most frequently used electric boundary conditions, which are discussed later.
4. Antiplane Solution To demonstrate explicitly the role of each physical property in affecting the fracture behavior of a piezoelectric ceramic, we examine the facture mechanics for mode III cracks. Equations (3.30)-(3.32) give the general solution for antiplane deformation. For simplicity, we consider a remote mechanical load cry2 and a remote electric field E ~ only. From the mapping function Eq. (3.40) and
174
Tong-Yi Zhang et al.
the boundary conditions Eqs. (3.33)-(3.35), we find the two complex potentials (Zhang and Tong, 1996) U'(z)
1 [ ( 0 ~ + e l s E ~ ) ( z + ~/z 2 - c 2) can
+
l
2
((ot--fl)el5 Oe + / ~
Ey
)
q- O'3C~
dp,(Z)___E~[Z'+'~/Z2--C2
2R2
]
1
~
Z + ~/Z 2 --C 2 ~/Z 2 --C 2'
ot--fl
2R 2
]
(3.67)
l
O~+ / ~ Z-~- ~/Z 2 --C 2 ~/Z 2 --C 2
Under the given loading condition, the electric field inside the cavity is uniform and the only nonzero component is in the x2 direction, which means l+c~ E c - E~ + iECl = ~ E~. c~+/~
(3.68)
For this case, the effective dielectric constant, x eff, of the piezoelectric material is given by
Keft --
KII +
e~5/r
(3.69)
For a given c~, if/3 -+ 0, meaning that the dielectric constant of the cavity is much smaller than that of the piezoelectric material, the electric field inside the cavity still has a finite value l+c~ U = ~E~. (3.70) O~
Similar to the general loading case shown by Eq. (3.64), Eq. (3.70) gives the explicit relationship between the electric field inside the cavity and the parameter c~, which indicates that the smaller the c~, the higher the E c is. However, for a conductive cavity,/3 --+ cx~ and the electric field strength in side the cavity becomes zero, no matter how large the finite value of c~ is. For c~ ~ 0, the electric field strength in the cavity is EC
1 x eft -- fi E y = x---7 E y .
(3.71)
For most of the widely used piezoelectric ceramics with vacuum or air-filled cavities, the ratio of xeff/x C is of the order of 1000. Therefore, the electric field strength inside a vacuum or air-filled crack could be three orders of magnitude higher than the applied field strength. The similar result is also true for the general loading conditions (Zhang et al., 1998). Multiplying Eq. (3.68) by the dielectric constant of the cavity yields the electric displacement inside the cavity. Similar to Eqs. (3.66a-c), we have the three limiting
Fracture of Piezoelectric Ceramics
175
cases for antiplane deformation
D c = D~ + i DCl = KeffE~, D c
KegE~ = 1 + oe//3'
D C = 0,
for ~//3 ~ 0,
(3.72a)
for ot/fl ~ a finite nonzero value,
(3.72b)
for c~/fl ~ oo.
(3.72c)
Equations (3.72a-c) explicitly indicate that the electric displacement inside an electrically insulating crack depends on the ratio of oe/fl. The electric displacement approaches zero only if the ratio of c~/fl approaches infinity, resulting in an electrically impermeable crack. When ot/fl -+ O, the electric displacement inside the crack is identical to that in the piezoelectric medium on the crack faces, and the crack becomes electrically permeable. Many researchers (e.g., Sosa, 1991; Sosa and Khutoryansky, 1996; Chung and Ting, 1996; Gao and Fan, 1998a, 1998b, 1999) studied the elliptical cavity problem for piezoelectric ceramics, and McMeeking (1989) discussed the appropriateness of the electrically impermeable boundary conditions for isotropic paraelectric materials by analyzing an elliptical flaw.
D. ELECTRIC BOUNDARY CONDITIONS ON ELECTRICALLY INSULATING CRACK FACES
In the study of the fracture mechanics of piezoelectric materials, a crack is usually treated as a mathematic slit. It is inconvenient to use the exact electric boundary conditions, Eqs. (3.33)-(3.35), because of the difficulty in calculating the electric field inside a slit if one does not treat the crack as the limit of an elliptical cavity. Therefore, simplifications are often used for the electric boundary conditions. Parton (1976) published a fundamental result on the fracture of piezoelectric materials. He assumed that the crack is traction-flee but electrically permeable. An electrically permeable crack requires that the electric potential and the electric displacement normal to the crack surface are continuous across the slit. Mathematically, the electric boundary conditions are O + -- 3 2,
(3.73)
~b+ = 4~-,
(3.74)
where the subscript n denotes the normal component of the crack face and the superscripts plus (+) and minus ( - ) mean the upper and lower crack faces, respectively.
Tong-Yi Zhang et al.
176
We can check the validity of Eq. (3.74) from the electric field inside the cavity. The electric potential drop across the crack faces equals the electric field strength multiplied by the cavity width. We have A~b = q~+ - q~- = ac~ sin 0 ( B 4 + B44) X
(1 - or)(1 - / 3 ) R e [ L ( 1 - ipjot)al] + (1 + or)(1 + fl)Re[L(1 - ipjot)al] (a + fl)(1 + aft) (3.75a)
from Eq. (3.60) for the general solution, where Re denotes the real part of a complex function, and A4~ = ~b+ -- qS- = --2ac~ sin 0
l+c~
c~+/~
E~,
(3.75b)
from Eq. (3.68) for antiplane deformation, where 0 is the elliptical parametric angle in the z plane (the polar angle in the w plane) ranging from 0 to zr. As ot approaches zero, Eqs. (3.75a) and (3.75b), respectively, reduce to AO = q5+ -- ~ - = a sin0(B4 + B 4 4 ) ~
(3.76a)
Aq0 -- 4~+ -- 4~- -- --2a sin0 ot/flE~ 1 + ~/r
(3.76b)
and
which indicate that AO5 -- 0 only if ot/fl --> O. The dielectric constant of a piezoelectric material can be three orders of magnitude higher than that of air or vacuum. Thus, ot/fl can be finite for a physical slit and the electric potentials at the two crack faces may be different. The permeable boundary condition Eq. (3.74) corresponds only to the case c~//3 --+ 0. Deeg (1980) analyzed dislocation, crack, and inclusion problems in piezoelectric solids. To simplify the analysis, Deeg set the normal component of the electric displacement to zero at the upper and lower crack faces D + -- D,~- -- 0.
(3.77)
This approximation is equivalent to treating a crack as an electrically impermeable slit by neglecting the electric field within the crack. Pak (1990a) gave a detailed argument for neglecting the electric displacement within the crack. From Eqs. (3.66c) and (3.72c), we see that the impermeable condition Eq. (3.77) corresponds to the case when oe//3 approaches infinity.
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Fracture o f Piezoelectric Ceramics
Hao and Shen (1994) introduced the following electric boundary conditions D + -- D~-, D+Au,
(3.78)
-- --KcA0,
(3.79)
on the crack surfaces, where AO is the potential drop across the crack and Aun is the crack opening. Equation (3.79) reduces to the permeable condition [Eq. (3.74)] if Au,, = 0, or to the impermeable condition [Eq. (3.77)] if K c -- O. We call the electric boundary conditions of Eqs. (3.78)-(3.79) the semi-permeable electric condition. When the crack lies in the xl axis, Au,, is Au2. This boundary condition is based on the crack profile after deformation and requires that the ratio of AO to Au2 be spatially independent. We may use the solutions of AO and Au2 derived from the exact boundary conditions to demonstrate the spatial independence. From Eqs. (3.17) and (3.83), we have the extended crack opening Au = ia[A(al - a2) -/il~(a-i-1- a22)] sin 0,
0 < 0 < Jr.
(3.80)
Because AU4 - - A ~ , Eq. (3.80) yields A~b Au2
=
[A(al
-
a2) -
A(~-
a22)]4
[A(al -- a2) - A ( a ] ( - a2)]2
,
(3.8~)
which is a constant. Combining Eq. (3.79) with Eq. (3.81) shows that the normal component of the electric displacement along the crack face is also a constant that depends on the material properties and the loading conditions. Because the profile of the crack opening is elliptical, even when the crack lies on the x l axis, D2 may not be exactly perpendicular to the deformed crack surfaces. However, if the crack opening is small, D2 may be treated as approximately D,,. Thus, in practice, one may first use the electric boundary condition D + = O~- -- D o
(3.82)
to solve the boundary problems in terms of a constant D o and then use Eq. (3.81) to determine the value of D ~ It is more convenient to evaluate the electrical boundary conditions on undeformed crack surfaces. However, under combined mechanical and electrical loads, the crack opening is very sensitive to the electric field inside the opened crack, and the electric field, in turn, is affected by the profile of the opened crack. Zhang et al. (1998) conducted a self-consistent calculation of the crack profile using the exact boundary conditions that showed the effects of the boundary condition approximation on the energy release rate. Interested readers may refer to the work presented by Zhang et al. for the self-consistent calculation, which is a geometrically nonlinear
Tong-Yi Zhang et al.
178
electroelastic analysis. Dunn (1994b) and Shindo et al. (1996) also investigated the electric conditions along crack faces. In summary: We examined four sets of electrical boundary conditions on electrically insulating crack faces for studying the electroelastic fracture of piezoelectric ceramics. The four electric boundary conditions are, respectively, the exact [Eqs. (3.34) and (3.35)], the electrically permeable [Eqs. (3.73) and (3.74)], the semipermeable [Eqs. (3.78) and (3.79)], and the impermeable [Eq. (3.77)] conditions. In later sections, we examine the consequences of the different approximations in the electric boundary conditions.
E. INTENSITY FACTORS AND ENERGY RELEASE RATES
1. Electric and Mechanical Fields of a Slit Crack a. General Case When the cavity reduces to a slit, the extended displacements and stresses are given by Eqs. (3.17), (3.18), and (3.58), with f,~, f,~.l, and f~,2 reducing to
f~=
a~, (z,~ + v/Z2 - a 2) 2
+
a2a~2 2(zc~ + v/Z~ - a2) '
a2aa2 f~,l --
ac~l -
) Za + ~//Z2 --a 2
(zot + v/z 2 - a 2 ) 2
2 v / z 2 - a2
(3.83)
c~ = 1, 2, 3, 4 (c~ not summed)
fa,2 -- Pa f~,l An intensity factor vector can be defined as K* -
lim L (v/2Zr(z,~ - a ) ) [ , l
(3.84)
Z~ "--+O
for the fight crack tip. A substitution of Eqs. (3.83) into Eq. (3.84) yields K* = v/-Y-~L(a, - a2) 2
~/-7ra ( E l - d). 2
(3.85)
Recall that d - (0 0 0 d) T in Eq. (3.54). Using Eqs. (3.52) and (3.54), we find
d_(B4+B4)(1-~176176176 2B44
(1 + or)[ 1 + c~2 + c~//3 + or/3]
~ ' (3.86)
Fracture of Piezoelectric Ceramics
179
for the cavity. Again, we consider here only an electrically insulating crack that has a finite value of/3. Then, letting ct ~ 0 leads to three limiting results: m
d = (B4 + B 4 ) E ~ , 2B44 A
d =
(B4 -+- B4) ~ 2B44
1 1 + ct//~'
d - 0,
for ct/fl ~ 0,
(3.87a)
for ct//~ --+ a finite nonzero value,
(3.87b)
for ct//3 ~ c~.
(3.87c)
The three limiting results in d correspond respectively to the three limiting cases, shown by Eqs. (3.66a-c), of the electric displacement inside the crack. Zhang et al. (1998) showed that K* is complex for a conductive crack. For electrically insulating cracks, Eqs. (3.87a-c) indicate that d is real; consequently, the intensity factor vector K* is also real. The mode II, I, and III stress intensity factors and the electric displacement intensity factor are defined as twice the real part of K* K-
K* + K* -
K -
(Ktl
Kt
-d],
Kin
(3.88)
Ko) r.
Because d - (0 0 0 d) r, Eq. (3.88) may be written as ( g ll
g l
glll
g D ) -- x / - ~ ( t7 ~2
t72~
0"3~
O ~Z - d ) 9
(3.89)
Equation (3.89) shows that the stress intensity factors of modes I, II, and III are the same as those in elastic media and independent of the remote electric load, whereas the intensity factor of electric displacement can be a function of the remote mechanical loads through the piezoelectric effect. It can be seen in Eq. (3.87c) that d = 0 only ifct//~ ~ c~. Thus, for electrically impermeable cracks, the electric displacement intensity factor is simply (Suo et al., 1992)
Ko -- x/-~--aD~.
(3.90)
For most PZT ceramics containing a vacuum flaw (which has the smallest dielectric constant among all media), the ratio of the dielectric constant inside the crack to the effective dielectric constant of the material is of the order of 10 -3. To satisfy the condition d -~ 0 requires that ot = b/a > 0.01. Thus, caution is necessary in using fracture mechanics in treating a real physical cavity. Zhang (1994a) discussed the crack width effect. However, for electrically permeable cracks (ot/fl--+ 0) with d given by Eq. (3.87a), the electric displacement intensity factor induced by the piezoelectric
180
Tong-Yi Zhang et al.
effect is If"
KD =
~ / / ~ L(B41 -[- 841)0-12 -[- (842 -[- 842)0"22 -[- (843 -'[- 843)0"32 ]. 2B44
(3.91)
Equation (3.91) shows that the intensity factor of electric displacement is completely induced by the piezoelectric effect rather than by the applied electric loads. If c~/fl is finite, depending on the value of c~/fl, the applied electric displacement contributes to the intensity factor of electric displacement--that is,
KD = - 2B~(1 + ot/fl)
-[- (843 -[" 843)0"3T -
2844(ol/fl)O~].
(3.92)
Moving the origin of the coordinate system to the fight crack tip and introducing new variables z~ = z~ - a, we can express the electrical and mechanical fields near the right crack tip in terms of the complex stress intensity factor:
Z~-L ,/2~z: ~l
--
-L
/ / / / P'~
L-1K*
/-/ / +/
L
-
~
P'~
~_.
,
0.93) L-1K*.
(3.94)
It can be shown that the extended crack opening near the crack tip is given by
AU = -~-[B
+
I],IK,
(3.95)
where r is the distance from the crack tip. Hereafter, the phrase near a crack tip means that r << a.
b. Mode III
For antiplane deformation, Eq. (3.67) gives the two complex potentials for the elliptical cavity. Consequently, the mechanical and electrical fields in the material
Fracture of Piezoelectric Ceramics
181
are expressed by 1 F (0-~ + e , s E ~ ) ( z + ~/z 2 - 0 2)
V = V32 + iy31 =
+(~/~
c44
/
2
- 1
1 + ot/fl e l S E ~ + E=
Z -'1- ~ / Z 2 - - C 2
~ / Z 2 - - C2
(3.96)
E2+iE1 C2
_ wy[ z + ,/z ~
I
0"
0.32
--- 0"32
-5
+
ot/fl - 1 1 +Ot/j~
l+c~[
Xl
]
Z ' J - 4 Z 2 - - C2
+ i0"31 = C44V -- elsE,
As z approaches a along the
2R 2 D
= D2
1
J 4 Z 2 - - C2 '
-+- i D I
elsv +
=
Kll
E.
axis, the values of the four functions are
~
)"32 =
~
~32 - -
0"32 =
C44V32 - - e l 5 E 2 ,
elsflE~
]
(~ + t3)c44
,
l+c~
E2 -- ~ E ~ ,
c~ +
(3.97)
D 2 = e 1 5 v 3 2 -~- K l l E 2 .
Equation (3.97) shows that the shear strain approaches infinity as c~ approaches zero for finite/3. Singular strain produces singular mechanical stress; it also causes singularity in the electric displacement through piezoelectricity. However, as long as the dielectric constant of the cavity is not zero, and then I3 has a finite non-zero value and the electric field strength remains finite no matter how small c~ is. When the cavity is reduced into a crack, Eq. (3.96) is reduced to V =
Y32 -k- iV31
= ~
1 [
C44
z
~
0"32 ~ / Z 2 -
c~/fl z --elsE~ a2 -t- 1 + ot/fl ~ / Z 2 - - a 2
ot/~ E-
0-
Ez + i E 1 --- 0"32
-~- i0"31
E~ =
z
l + o~/fl ~/z 2 - a 2
C44Y - - elsE,
+
1 + ~ e l 5 1 + ot/fl
E~], (3.98)
, ] 1 + ot/fl
'
D = De + i Dl = elsv +
Kll
E.
As expected, the mechanical and electrical fields are related to the ratio of or~ft. Let us define the real strain, stress, electric field strength and electric displacement intensity factors at the right crack tip as Ko32 III _ lim v/27r(z - a)0-32 z~a E2 KIn -- lim V/27r(z - a)E2, Z-.--> a
KII>'~2 -- lim v/2rr(z - a)v32, z~a D, K i d -- lim v/2n'(z - a)D2. z.--~ a
(3.99)
182
Tong-Yi Zhang et al.
The near crack tip strain, stress and electric displacement can be expressed in terms of the strain, stress, and electric displacement intensity factors as K0-32 0 HI 0-32 - - ~ COS ~ ,
~32 - - ~
E2-
KY32 III
0
D2 KIH
0
~cos~,
2~~
/(,-Y32 "'~ HI
0 COS ~ ,
E2 Ki11
k" ~ " 9IH
0-31 =
}'31 -- -- ~
0
sin ~, (3.100)
E2 Ki11
El =
0
2v/~_7 s i n ~ , D2 KIII
.-.
cos D2 -- ~ - ~ r 2'
0
sin 2'
ul -
~
0
sin - . 2
If c~ is treated as zero first (i.e., ot/fl ~ 0), we find the mechanical and electrical fields in the material for a permeable crack. In this case, we have E -- E ~ from Eq. (3.98), indicating that the permeable crack has no influence on the electric field strength. Then, using Eq. (3.99) results in /,f,"~32
"'III
=
0-32x/~a C44
E2
'
oo / / ~
,
(3.10~) D2
K II1 -- O,
k,. 0-32
""111 n 0-32
Kin =
e150-32 C44
The strain and stress intensity factors are the same as those in elastic solids. There is no intensity factor of electric field strength and the intensity factor of electric displacement is induced by the piezoelectricity. If the dielectric constant of the cavity is treated as zero first (ot/fl --+ e~), the crack becomes electrically impermeable. Then, the various intensity factors take the following form: K•
III
_ ~0-~2
KIII - ~ - a E ~ ~
+ elsE~ C44
o32 ~ ~ 0 - ~ 2 , K I11
Klli -~/-~--d
(3.102) C44
+
1+
C44 K 11
Equation (3.102) shows that the electric field strength intensity factor is no longer zero for electrically impermeable cracks. As a result of the piezoelectric effect, both the applied mechanical and electrical loads induce singularity in the elastic strain and electric displacement.
Practure of Piezoelectric Ceramics
183
However, if the ratio u/fl has a finite value for u ---> 0, the intensity factors are "'~///
=
K cr3z __ 111
Dz
Kill
C44
0"32
VCffa 0-~2
-- ~
'
+ 1 + ~/flelsE~
e2 _ ,r
KIll
,
E~
[ el5 oc ( ~0-32 + gll E~ 1 + c44
ot/ fl 1 + ~ / fl '
c44xll
(3.103)
1 + c~//~
Comparing Eq. (3.103) with Eqs. (3.101) and (3.102) indicates that, for a finite non-zero value of ot/fl, the intensity factors, except for the stress intensity factor, have values between the corresponding values for impermeable and permeable cracks.
2. Energy Release Rates for Crack and Cavity Propagation In facture mechanics, the characteristic of the energy release rate is extremely important because it provides, from a thermodynamic point of view, a clear physical picture of crack growth. Many researchers have specifically studied the problem. For example, McMeeking (1999), Suo et al. (1992), and Park and Sun (1995b) evaluated the energy release rate of piezoelectric cracks. To demonstrate the relationship between the energy release rate and other parameters, particularly the intensity factors, next we look at the general cracks and mode III cracks. Explicit formulas are available for mode III cracks.
a. General Case As described in Section II, the energy release rate can be evaluated from each of the four thermodynamic functions: the free energy, the electric enthalpy, the mechanical enthalpy, and the full Gibbs energy, or from each of the four associated potentials. Using these functions, we can derive the energy release rate for the growth of an elliptical cavity, as shown in Fig. 3a, under the constraint that the ratio of the minor to the major semi-axis (i.e., the dimensionless parameter ~ = b/a) remains fixed. When the elliptical cavity reduces to a crack, the energy release rate automatically becomes that of crack extension. Because only two-dimensional problems are treated here, the crack area A(=I x a = a) and all properties are based on unit thickness. Thus, the energy release rate is given by
2 J - O P t 4 - f r-( O a
ti--~aOUi-Jr-Di Odpni) drOa
0aa
fn harI
(3.104)
184
Tong-Yi Zhang et al.
A factor of 2 is added here due to the presence of two-crack tips in this system (i.e., a is the length of the major semi-axis, or the half-length of the crack). Using the solutions given in the previous sections and after some tedious algebraic manipulation, we find J
~_
7ra T 4 {(E~) (B+I])E~-(1
+u)(E~)r(Bd+l]d)
+ ~ 2 ( E ~ ) r ( B + I~)E~ - i u ( 1 + u ) ( E ~ ) r ( B d - l~d)
(3.105)
- [i(1 + c~)(d - d) - 2orES] r (Aa~ + Aa~) - u [ ( 1 + u)(d + d) - 2 E ~ ] r [ A ( p j ) a l + ,~(p--j-)a]-i]}. Equation (3.105) shows that the energy release rate for an elliptical cavity depends on mechanical and electrical loads, the dimensionless parameter ot - b/a, as well as the value of d. For an electrically insulating crack, the dielectric constant of the crack medium x c has a finite value (for example, x c = 8.85 x lO-12F/m for a vacuum crack) and d is real. When the cavity reduces to a slit (i.e., ot = b / a ~ 0), the energy release rate for crack extension reduces to ara T J -- 4 [ ( E ~ ) (B + I~)(E~ - d)].
(3.106)
In terms of the intensity factors, J can be written as 1
J - ~ [K:r(B + I])K + 7/ad(B4 + ]~4)(X~ - d)].
(3.107)
Substituting the expression of d [Eq. (3.86)] into Eq. (3.107) leads to J --
Kr(B + I~)K 4
+
rca(ul3) [(B4
4
-+-
B4)X~z] 2
2B44(1 +
Ot//~) 2 "
(3.108)
For a finite value of o~/3, the intensity factor for electric displacement KD is given in Eq. (3.92). For both electrically impermeable cracks (o~/3 --+ cx~) and electrically permeable cracks (or~3 --+ 0), Eq. (3.108) becomes J =
K :r (B + I])K 4
,
(3.109)
in which the intensity factor of electric displacement is given by Eq. (3.90) for electrically impermeable cracks and Eq. (3.91) for electrically permeable cracks. For electrically permeable cracks (oe/fl --+ 0), the intensity factor of electric displacement is determined by the remote mechanical loads only and is independent
Fracture of Piezoelectric Ceramics
185
of the remote electric load. Substituting Eq. (3.9 l) into Eq. (3.109), we can further reduce the energy release rate for the electrically permeable cracks to
4J = (Kll K,
_
Bll + Bll
B13 -q- BI3~
Bl2 -k- B12
]~ "- /B21 -k- B21 B22 q- B22 ! \B31 + B31 B32 + B32
(3.110)
B23 -q- B23
_
1
(",4 + B14~
B33 + B33
|B24"+'-ff24|(B41+B41
2B44/ / \ B+ 3B-~34,/ 4
B42 + B42
B43 + B43),
where Bij, i, j = 1, 2, 3, 4, is defined in Eq. (3.22). Equation (3.110) indicates that the energy release rate is independent of the applied electric field. In other words, the applied electric field does not drive or impede the propagation of electrically permeable cracks.
c. Antiplane Deformation The energy release rate for the elliptical cavity propagation under a fixed c~ with antiplane deformation can be written as Jm = -g--(1 zrca +or) I cr3~ ( Y32~ -
(l+~176
9
\l+~/fl
(3.111) When the cavity reduces to a crack (i.e., a --+ 0), Eq. (3.111) becomes
Jill =
7ra [
-~
a3~
(y~ _ 32
e,sE~ ) ( c~/fl )1 ot/fl)c44 -E~Dy 1 + ot/-----fl "
(1 +
(3112)
"
We may express the energy release rate in terms of a 32 ~ and E ~ as 7ra[ (cr~2)2 Jill-- T c44
(1+
e~5 )xll(E~C)2( ot/fl
/~11c44
l _ql..0///~
)].
(3.113)
For electrically permeable cracks (c~/fl ~ 0), Eq. (3.113) is further reduced to Jill --
IH 2c44
""
(3.1.14)
As expected, the electric field contributes nothing to the energy release rate of
186
Tong-Yi Zhang et al.
electrically permeable cracks. If the crack is electrically impermeable (i.e., ct/ fl --~ c~), Eq. (3.113) becomes 1 [ t'k'~r32 2 ( ~,"~,II ) 1+
Jill = ~
C44
e~5 ) KllC44
E22] 9
K,I (KII 1 )
(3.115)
The contribution of applied electrical loads to Jm as given in the second term on the fight side of Eq. (3.115) is always negative. Thus, the electric field impedes the mode III crack propagation for electrically impermeable cracks. If the ratio of ~//~ has a finite value, the energy release rate is given by Eq. (3.113) and the intensity factors are calculated from Eq. (3.103). It should be pointed out that the energy release rate calculated from Eq. (3.104) depends on the value of d b / d a , while the J-integral for a slit crack is uniquely determined because the crack tip fields are obtained after the limiting process of b --, 0. Very recently, Zhang and Zhao (2001) derived the condition for a mode III crack, under which the engergy release rate calculated from Eq. (3.104) is identical to the J-integral. The condition states that the area of the ellipse should be remained unchanged during the calculation of the energy release rate.
IV. Piezoelectric Dislocation and Green's Function
Piezoelectric dislocation, an important crystalline defect that can adversely influence the performance of piezoelectric materials, has attracted a great amount of research interest (Faivre and Saada, 1972; Barnett and Lothe, 1975; Farvacque et al., 1977; Deeg, 1980; Pak, 1990b; Chung and Ting, 1995a, 1995b; Liu et al., 1997b; Meguid and Deng, 1998; Deng and Meguid, 1999). A piezoelectric dislocation is defined as a mechanical displacement jump across a slip plane, identified as the Burgers vector b, and an electric potential jump across an electric dipole layer. The dislocation core can be subjected to a line force and/or a line charge. Clearly, the solution for a piezoelectric dislocation is the same as a Green function. The characteristics of various Green's functions have been studied extensively by many researchers (Lee and Jiang, 1994; Chung and Ting, 1995a, 1995b; Liang and Hwu, 1996; Liu et al., 1997a; Gao and Fan, 1998b; Lu and Williams, 1998; Qin and Mai, 1998). Usually, one uses the Laurent or power series to analyze elliptical inclusions and the interaction of a piezoelectric dislocation with an elliptical cavity (Gao and Fan, 1998a, 1998b; Meguid and Deng, 1998). To demonstrate the physical picture, Eshelby (1979) adopted an image dislocation approach to solve the boundary value problems of dislocation-cavity interaction. We use the Laurent series to examine the interaction of a general piezoelectric dislocation with an elliptical cylinder
Fracture of Piezoelectric Ceramics
187
cavity, and use the image dislocation approach to investigate the interaction involving screw piezoelectric dislocations. The series solution from the image dislocation approach converges much faster than the Laurent or power series.
A. INTERACTION OF A PIEZOELECTRIC SCREW DISLOCATION WITH AN ELLIPTICAL CAVITY
If a piezoelectric screw dislocation with its line being along the poling direction that is chosen to be in the x3 axis, the deformation associated with the screw dislocation is antiplane. The general solution for antiplane deformation can be expressed in terms of two complex potentials [see Eqs. (3.30)-(3.32)]. For a piezoelectric screw dislocation located at Zd in a homogeneous infinite medium, the mechanical and electric complex potentials are simply (Pak, 1990b) U - ,~ ln(z -- l~ ln(z --
Zd), Zd),
(4.1)
where ,~ -- ,41 + i,4e and l~ - / 1 1 + i/3e are named the mechanical and electrical complex constants, respectively, representing the characteristics of the dislocation. The real parts of,~ and l~ are related to the Burgers vector b3 and the jump in electric potential A~b, respectively. The imaginary parts of ,~ and !~ are related to the line force and the line charge. For any Burgers circuit enclosing the dislocation only, we have
frOU3dl=b3'fr O d p d l - A c ] ) ' O31l
(4.2)
The force and charge balance require the following equations"
~ aijnjdl -- -Fi,
~ Djnjdl = q,
(4.3)
where Fi is the line force per unit thickness or per unit dislocation length in the xi-direction and q is the line charge per unit thickness. For anti-plane deformation, F1 -- F2 -- 0. Using Eqs. (4.1)-(4.3), we find b3
lil = 2Zr'
--xll F3 + elsq /~2 --
2rr (C44Kll -}- e25) '
/~1 -/32 --
ar 27r
,
(4.4)
e15F3 -+- c44q
- 2rr (c44x,, + e125) "
(4.5)
Figure 4a shows schematically a piezoelectric screw dislocation near an elliptical hole in the z plane. Figure 4b shows the corresponding screw dislocation near the circular hole in the w plane and the image dislocations used to construct the solution for the material. For antiplane deformation, the boundary conditions
Tong-Yi Zhang et al.
188
X2 (a) Z- plane _1_ z~
b
..~a
h.
x,
V2 1
(b) w- plane
"~'~ N" . ; . ~ 1
./
/ .L,r
/.'w)m Wd
....\.\ :" : m w, i I "'"....! i . . /
. vl
m/w; --";:L.
FIG. 4. A piezoelectric screw dislocation near an elliptical cylinder cavity in (a) the z plane and (b) the w plane with image dislocations for construction of the solution in the material. Eqs. (3.33)-(3.35) at the cavity surface can be written in terms of the three complex potentials as
c44(U + O) + e15(~ + ~ ) -- O, e15(U + U) - x11(~ + ~ ) -- - x c ( ~ c + ~c),
(4.6)
~-~--~c-~c. From Eq. (4.6), we find the following relationships between the complex electric potentials in the piezoelectric material and in the cavity:
xeff(O + ~)) -- xC(OC + Oc)'
(4.7)
189
Fracture of Piezoelectric Ceramics
where x eff, defined in Eq. (3.69), is the effective dielectric constant of the material under antiplane deformation (Zhang and Hack, 1992; Zhang and Tong, 1996). We use the image method to solve the problem. First, consider a circular cavity of unit radius, where a = b -- 1 and m = (a - b ) / ( a + b) = 0. In this case, the single-valued condition [Eq. (3.42)] is satisfied automatically, and the w plane is identical to the z plane. Consider a piezoelectric screw dislocation located at W d ( W d > 1) with a given electrical constant i~ in the w plane. To satisfy Eq. (4.7), we added an image dislocation of electrical constant f21~ at 1/~--d and an image of electrical constant -f21] at the center. It can be shown that a dislocation of electrical constant AI~ located at Wd generates the electric field inside the circular hole, where K c -- K eff
fl -- 1
x ~ + x~ff
fl + 1'
2 K eff
2
K c _Jr_K ef f
1~ + 1
m
A
(4.8)
Thus, the solution to Eq. (4.7) is given by r
1~ln(w
Wd) -{- ~21~In ( 1
~
fl3
9 ~ -- AI~ ln(w -
W d
)
(4.9)
,
(4.10)
Wd).
Consequently, from Eq. (4.6), we calculate the mechanical complex potential as --
--
Wd
-U9
-- ~ ( f 2 + 1)!~ In C44
Wd
9
t/)
(4.11) For a general elliptical cavity, m - r 0 and the single-valued condition of Eq. (3.42) must be satisfied. However, the complex potential given by Eq. (4.10) does not meet the single-valued condition. We use image dislocations to satisfy the single-valued condition Eq. (3.42) and the boundary conditions in Eq. (4.7) alternatively. The approach is thus called the i m a g e d i s l o c a t i o n m e t h o d . To make Eq. (4.10) satisfy the single-valued condition, we add to the complex potential ~c an image dislocation of electric constant AI~ at m / W d and another image of - A l l at the center. The two images generate a potential AI~ l n ( m / w - Wd), which together with the dislocation of Eq. (4.10) meet the single-valued condition inside the cavity. Unfortunately, the new ~c violates the boundary conditions of Eq. (4.7). To enforce the boundary condition again, we add to the complex potential 9 an image at Wdd/m associated with an electric constant of f2B and use the two images ^
Tong-Yi Zhang et al.
190
with each having an electric constant I~ inside the hole and, respectively, at m/wa and at the center, as shown in Fig. 4b. These images produce additional potentials 1~ln(m/w - Wd) + f2[I ln(m w -- Wd) in the material. At this stage, we complete the first round of the image dislocation approach. Now the solution satisfies the boundary conditions and the single-valued condition except generating a new dislocation at ~-~/m associated with ~1~. It is interesting to note that the new dislocation is 1/ m (> 1) times farther from the cavity, and its electric constant is f2 (< 1) times weaker than the original one. To cancel this new dislocation, we introduce a dislocation at ~dd/m associated with -E2B to ~, and thus start the second round of the image dislocation approach. Finally, we have the sequent solution ^
= 1~ln(w -- Wd) + ~'-2]~In ( 1
tV
+
(--1)1~1+11~ In (mw
Wd
)
wa) +"ln( mw Wd) -Jr"
(-- 1)1 ~'21]~In
(m2 1/)
nt-(-- 1)2~22+11~In ( m2w
Wd) +
+ (--1)l~21l~ In
(--1)2f221~ln ( mgw w d ) + ' " ,
(4.12)
[(mw---~-j) (m2 tV
}
+(-1)2~2"ln[(mZw-wcl)(; ~
,413,
The associate mechanical potential is then given by
U ~i'ln(w -- wd) -- ~l'ln ( w c44e15{ (f2 + 1)1]In ( lw + (-1" f2'+'l~ In ( m W
+(--1)2f22+lflln
( m2 W
Wd) We) + I~ In (mw
We)
lV d ) + (--1)l~ll~ln ( m2 lV
)
Wd
+ (--1)2f22]~ln
wd)
(4.14)
( m3 W
Application of the mapping function to Eqs. (4.12)-(4.14) yields the potentials in the z plane.
P'racture of Piezoelectric Ceramics
191
B. INTERACTION OF A G E N E R A L PIEZOELECTRIC DISLOCATION WITH AN ELLIPTICAL CAVITY
For a general piezoelectric dislocation, we use Stroh's formalism with the general solution given in Eqs. (3.17)-(3.20). For a dislocation at w~d in an infinite body in the w plane, the solution takes the form of f0 = (In ( w ~ - w~a))q.
(4.15)
The parameter vector q is related to the extended Burgers vector, b = ( b l b2 b3 A4~)r, of the dislocation and the extended force, F = (F1 F2 F3 q)r, acting at w~a through (Ting, 1996) 1
q -- ~-~-/(ArF + Lrb).
(4.16)
First we consider an electrically impermeable elliptical cavity. The extended traction is zero along the cavity surface, which is satisfied if all the components of the extended stress potential 1/ti = 0,
i = 1, 2, 3, 4, (traction-free)
(4.17)
at w~ = e iO in the w plane. The complex vector f in Eqs. (3.17)-(3.18) can be written in the form f = f0 + re,
(4.18)
where f0 is given by Eq. (4.15) and fe is induced by the cavity and called the image vectorto be determined. Using Eqs. (3.18), (4.15), and (4.16), we can write Eq. (4.17) as
Lfo(e iO) + Lfe(e iO) -+-Lfo(e -iO) -+-Lfe(e -iO) = O.
(4.19)
Equation (4.19) is satisfied if fe(w~) = Zfl,
(4.20)
,
(4.21)
with
zij . .L .l ln(1 .
tO i
i, j -- 1, 2, 3, 4 (i, j not summed)
The solution of Eq. (4.20) is given in the w plane; then, using the mapping function [i.e., Eq. (3.46)], we obtain the solution in the z plane.
Tong- Yi Zhang et al.
192
For a general electrically insulating elliptical cavity, we use the Faber series to express the complex potential in the w plane for the electric field inside the cavity oo
c~C(w) -
a,,~
Z
-Jr- m n w -n
),
(4.22)
n=l
where a no is a constant to be determined by the boundary conditions and m =
(a - b)/(a + b). The Faber series satisfies the single-valued condition Eq. (3.42). Similarly, we construct (x)
( f j)e --
Cj,nIIOj
j = 1, 2, 3, 4 (j not summed)
,
(4.23)
n:O
for fe in the material, which satisfies the remote stress free condition. The boundary conditions [Eqs. (3.33)-(3.35)] on the cavity surface can be expressed (Lf0 + Lfe + Lf0 + Lfe)j = 0,
j = 1, 2, 3 (traction-free),
(Lfe + Lfe + Lfo + Lfo)4 =--x
' [s c
2
( a o. + a -L--d .m
n ) einO
+ ( a . om
n
+ a . o) e -i'~
n=l
]
(4.24)
(surfacecharge-free), (4.25)
(Afe + Afe + Afo + Afo)4 --2i Z
( a~ _ anm n
einO
- (a.m~
n + an)e-inO-2-5
(continuity of electric potential).
n=l
(4.26) Substituting Eq. (4.23) into Eqs. (4.24)-(4.26) and using the series expansion In (e iO - w j )d-
oo
In ( - w e ) - n ~ !
el'~
j = 1, 2, 3, 4, (j not summed) (4.27)
for f0(e/~ we determine the constants
cj,o -- - q j In ( - w J ) ,
j -- l, 2, 3, 4, (j not summed)
c--~--- L - I L (
g,=(O
0
1 ) n(wJ) n q
0
g,)r,
1 c gn-----~X ( a o. + a %-"5 . m ) ,n
(4.28a)
+ L-lgn,
(4.28b)
Fracture of Piezoelectric Ceramics
(B4L o
an--2
193
,)
(1)
iA4) n(~j)~ ftmn(1 -- fl) 4- (B4L 4- iA4)I n(wi),, q(1 +/3) m2n(1 -- fl)2 _ (1 4- fl)2
(4.28c) Equations (4.15), (4.18), (4.22), (4.23), and (4.28) provide the solution for a piezoelectric dislocation near an electrically insulating elliptical cavity in the w plane. Again, the mapping should be applied to have the solution in the z plane.
C. EXTENDED LINE FORCE ON THE ELLIPTICAL CAVITY SURFACE
Consider an extended line force per unit thickness (or per unit dislocation length) applied to the surface of the elliptical cylinder cavity. In this case, the extended Burgers vector becomes zero and Eqs. (4.15) and (4.16) take the following forms f0(wa) -- (ln[w~ - eiC"])qF,
(4.29)
ArF ~. qF -- 2:ri
(4.30)
The solution for the extended line force can be obtained easily by replacing q with qF and w~d with e ir in Eqs. (4.20) and (4.28). For electrically impermeable cavities, the image vector has the same form as that of Eq. (4.20) f e ( w ~ ) - - L - l [ , f o ( w , ~ ) - ZilF,
(4.31)
with
Zkl -- -LkjlLj----71n( ~1- e
--i~d),
wk
k, 1 =, 12,3,4.
(4.32)
For electrically insulating cavities, these constants are
cj,o -- --(qF)j ln(--eiO"), e-in~Jd Cn = ~ L - 1 L q F ?/
g,,--(0
0
0
j -- 1, 2, 3, 4,
(4.33a)
+ L-lgn,
gn) T,
(4.33b)
g, -- --~l Kc (a,~ + a~ '') o_ 2 an
(B4L - iA4)flFmnein~/d(1 -- fl) + (B--44L+ iAa)qFe-inr n(m2n(1 _ fl)2 _ (1 4- fl)2)
fl) (4.33c)
Again, the mapping function must be applied to obtain the solutions in the z plane.
Tong-Yi Zhang et al.
194 D.
SOLUTIONS
FOR CRACKS
As discussed previously, if we first treat the dielectric constant of the electrically insulating cavity as zero and then let the cavity shrink to a slit, we have the solution for an electrically impermeable crack. However, if we reduce the cavity to a crack while keeping the dielectric constant of the cavity unchanged, we obtain the solution for an electrically permeable crack. In the following subsections, we manipulate the general cavity solution in these two ways to derive the solutions for electrically permeable and impermeable cracks.
1. Electrically Permeable Mode III Cracks When the cavity is shrunk to a slit, we have m ---> 1. For electrically permeable mode III cracks, 13 has a finite value, and the solution to antiplane deformation [Eqs. (4.12) and (4.13)] consequently reduces to
= I] ln(w -
w a ) + [I
In ( 1
wd),
(4.34a)
W
(pc
--
(1 + fl)l]ln [( w 2/~
__
wd)( 1 w
//3 d
)1 - ~ f - l B l n x 2f
[ ( w - ~--j) ( 1 ) wd ] w (4.34b)
If a constant 1~ln[--a/(2Wa)] is added to Eq. (4.34a), we have = 1~ln(z - Zd).
(4.34C)
Because a constant in the potential does not affect the solution, Eqs. (4.34c) and (4.34a) are equivalent. Equation (4.34c) indicates that a permeable crack does not affect the electric field of the material. For electrically permeable mode III cracks, m --+ 1 and the mechanical potential is reduced to
Wd) e,5{.ln(1 ) ( 1 )}
U = ,~ ln(w
-
l/3d) --
~ ' l n ( lto
+ I] In
C44
//3
wd
W
.
(4.35)
9
195
Fracture o f Piezoelectric Ceramics
Using the potentials and the mapping function, we calculate the mechanical and electrical fields of the piezoelectric medium, which are
/~(Z "if-~/Z 2 - - a 2)
y=
Z - Zd + ~/Z 2 -- a 2 -- ,u/ Z ] -- a 2
-+-
~
a 2 - (z + ~/z 2 - a2)(Z-d"+ V/~2d 2 - - a 2)
(Be15 / c44 )a 2 a 2 -(z
E
(~k nt- ~el5/C44)a 2
+
/ _
]
+ ~/z--~2 - a--~2i(Z--da; ( z 2 -- a 2)
1
a~/z2 ' - - - --~
'
(4.36)
Z -- Zd
D
cr = c44Y - e l s E ,
=
el5Y -t--t C l l E .
Using the definition for the real strain, stress, electric field strength, and electric displacement intensity factors at the right crack tip [i.e., Eq. (3.99)], we calculate the intensity factors at the right crack tip, which are given by
{ ~, + f3els/cn4
Ky3•
III = -
KltI E2
Zd -- a + ~/Z2 - a 2
--0,
o32
K Ill
~
.
{
Ac44 +
.
.
.
I~e15
+ ~e15/c44
+ Z--d
192
{
a + V/--~d - - a 2
(4.37) ~c44 + l~e15
+ z~-
Ki11 = --el5
] ~/-~a
A + [le15/c44 Zd -- a + ~ Z 2 - a 2
a + V/--~d -- a 2
/ ,ff~a,
I
+ i + 1 /c44 I ,/-YS. Z---d-- a + ~-~d -- a 2
I
As expected, there is no the intensity factor of electric field strength for electrically permeable cracks. In this case, the energy release rate for crack propagation is the same as that given by Eq. (3.114).
2. Electrically P e r m e a b l e General Cracks
For a general piezoelectric dislocation, we have Rj --+ a / 2 andmj --+ 1, j -- 1, 2, 3, 4, when the ellipse is reduced into a crack. For an electrically permeable crack,
Tong-Yi Zhang et al.
196
the solution is still given by Eq. (4.18) with Eqs. (4.15), (4.23), and (4.28), except that the constants gn and an0 are reduced to the following form: 1
g,, = - ~ x C(a ~ + a~
1)
(4.38a)
(1)
n(-~)n (1(1 -- f l ) + (B4L + iA4) n(wd)n q(1 + 13) 0
B
a n
(4.38b)
3. Electrically Impermeable Mode III Cracks For electrically impermeable cracks under antiplane deformation, the two parameters in Eq. (4.8) have the value of f2 = - 1 and A = 2, and the solution is simplified as -- 1~ ln(w -- Wd) -- I] In ( 1
//9
Wd),
(4.39a)
~c = 0, -
(4.39b)
)
-
Wd 9 (4.39c) w As expected, there is no electric field within an impermeable mode III crack. The mechanical and electrical fields in the piezoelectric medium are calculated from the potentials and take the following forms in the z plane a 2)
/~k(Z + x/Z 2 -
V--
/_ Z -- Zd
a t- X / Z 2 - - a
2 --
,/Z~ -- a 2 u ~,
-
]
,~a 2
+
a~ - ( z + ,/z ~ - a ~ ) ( ~ + V / ~ - a~) E -- - [
1 ,/z 2
a 2'
l~(Z + X/Z 2 - - a 2)
(4.40)
z - Zd + x/Z 2 - - a 2 -- v/z 2 - a 2 l~a 2 a~ - (~ + ,&~ - a ~ ( ~ O" =
C44 V - -
el5E,
D = el5?' +
] + V / ~ - a~) K'll
E.
1 "/z~ - a~
197
Fracture o f P i e z o e l e c t r i c C e r a m i c s
Consequently, we have the following real intensity factors at the right crack tip
f K Y32 __ --/
Ill
KIll -
k,.o-32
"~m
__
I Zd --
{
/~ a +
+ z~
--
z. - a + v/z 2 - a 2
-
{
Z---d-- a + V/--~d -- a 2
a 2
} ,/-~a,
+ z~-
Ac44 --I- Bel5
-
/ el5 "ll
a2
_
}
Ac44 + Bels
+
z d -- a + V/ z ~ -- a 2
Km - -
a + V/~-
(4.41) ~/-~a,
z---~_ a + v/--~d _ a 2
+
Z d -- a + V/ Z2 -- a 2
e15
}
~ffa.
z---~_ a q_ v/--~d _ a 2
The energy release rate takes the same form as that presented in Eq. (3.115).
4. E l e c t r i c a l l y I m p e r m e a b l e G e n e r a l C r a c k s
For a general piezoelectric dislocation, the solution is the same as that given by Eqs. (4.15), (4.18), (4.20), and (4.21). In mapping the solution into the z plane, one should take into account that Rj ~ a / 2 and mj --+ 1, j = 1, 2, 3, 4. The extended stresses can be calculated from Eqs. (3.18) and (3.20) with
fO, 1 -~- [e, 1 =
v/z2
Zij, 1 = E
a2
z~ + , / z ~
- 02 -
~
-
V//( z ~ ) ~ - 02
4 1 ( L ~ I Lkj
k=l
q+Z,l(l,
)
02
v/Z 2 - a 2
a 2-
(4.42a)
(Zi q- v / Z f - a 2) (Z--d q'- V/(7) 2_ a 2)
f0,2 -+- re,2
gij,2
,/~2_ a2
4
= ZL~I-~kJv/z2
k=l
zo + , / z 2 - a 2 _ z~ -
,i( --
a 2
V/( z ~ )
2 - a2
02
a2--(Zi-}-
q+Z2~,
-_ v/Z 2 - - a 2 ) ( Z J -}- V/(~fd.) 2 a 2)
(4.42b)
) !
198
Tong-Yi Zhang et al.
The intensity factor vector at the right crack tip induced by the dislocation at z~d is K* -
lim L(v/2rr(z~ - a))(fo,1 + re, l)
Zot----~a
1
-
z d - - a 4- ~(Zd)2 -- a 2
) ( q+L
1
z d - - a 4- ~(~d~d)2 -- a2
)1 fl
9
(4.43) Equation (4.43) shows K* is real. Thus, the intensity factor vector, K, for the mode II, I, and III stress intensity factors and electric displacement intensity factor are twice that of K* (i.e., K = 2K*). Consequently, the energy release rate can be calculated by Eq. (3.109).
E.
FORCE
ON A PIEZOELECTRIC
DISLOCATION
Pak (1990b) calculated the force on a piezoelectric dislocation under external mechanical and electrical loads. The force acting on a piezoelectric dislocation is a configuration force, which relates the change in energy when the dislocation moves an infinitesimal distance. Thus, from a thermodynamics point of view, we may add the changes in isothermal thermodynamic functions associated with the dislocation movement into each of Eqs. (2.27)-(2.30), d E = P d A + V d Q - J . dl,
(4.44a)
= P d A - Q d V - J . dl,
(4.44b)
dn
dW = -AdP
4- V d Q - J . dl,
(4.44c)
dG = -AdP
- Q d V - J . di,
(4.44d)
where l is the dislocation displacement vector and J is the force acting on the dislocation, which is defined as -
_
A,Q
_
445,
p, Q
A,V
-~i
P, V
The force acting on the dislocation can be evaluated from the each of the isothermal potential energies defined in Eqs. (2.19)-(2.22) such that Ji
--
0 PF Oli
--
0 PH 31i
--
0 Pw Oli
=
0 PG 31i
.
(4.46)
199
Fracture of Piezoelectric Ceramics
For a straight dislocation line along the x3-axis, the generalized Peach-Koehler forces are J, = a~2bl + o'2a2b2+ o'~2b3 + D~ Aq~ + u~, 1F, +
J2 - - - ~
2,1F2 +
Ua
b,
a 3,1F3 + Elq
Ua
- - o'~2b 2 - O~l
,
b3 - D? A~b +
(4.47) U a1,2
F,
.+_ua2,2F2 .+_U3,2 a F3 "+- E2q a .
V. Conductive Cracks
Internal electrodes in electronic and electromechanical devices made of piezoelectric ceramics may act as conductive cracks or notches, causing the devices to fail under electric and mechanical loads. It is therefore of practical importance to study the fracture mechanics and failure behavior of conductive cracks in piezoelectric ceramics. Many researchers have worked on this topic (McMeeking, 1987; Furuta and Uchino, 1993; Suo, 1993; Zickgraf et al., 1994; Lynch et al., 1995; Chung and Ting, 1996; Ru and Mao, 1999). Using compact tension samples with conductive notches, Fu, Qian, and Zhang (2000) performed extensive fracture tests on lead zirconate titanate (PZT) ceramics under purely electrical or mechanical loading. The experimental results indicate that both purely electric and mechanical fields can propagate conductive cracks (notches) and fracture the samples. Under purely electric loading, there is a critical energy release rate at fracture called the electric fracture toughness. The electric fracture toughness is about 25 times larger than the mechanical fracture toughness, the critical energy release rate at fracture under purely mechanical loading. Like the mechanical fracture toughness, the electric fracture toughness is a material property, which is defined as the resistance of a material against fracture or as the energy per unit area absorbed by the material as the crack propagates. Following the approach used by Orowan (1952) and Irwin (1956, 1958), Fu, Qian, and Zhang (2000) attributed the high electric fracture toughness to electrical plastic deformation. In this section, we study conductive cracks within the framework of linear fracture mechanics. As in the preceding sections, we establish the solution for an elliptical cylinder cavity first, and then reduce the cavity to a crack. For simplicity, we consider here only the case that no net free charge exists on the conductive cavity or crack (i.e., f DinidF -- 0 for any integral path enclosing only the cavity or crack).
Tong-Yi Zhang et al.
200
A. UNIFORM REMOTE LOADING For a conductive cavity under uniform remote loads, the components of the vector f of Eqs. (3.17)-(3.18) have the same form as those of Eq. (3.48). Using the mapping functions Eqs. (3.44)-(3.46), we write the boundary conditions on the conductive cavity surface in the w plane along the unit circle in the following form:
Ljiail* -Jr-Ljiai2* -- 0 ,
j -- 1 2, 3 (traction-free),
A4iai* 1 -Jr-A4iai*2 -- dpo (continuity of electric potential),
(5.1) (5.2)
where 4~0 is a reference potential. For simplicity, we take 4~0- 0 and rewrite Eqs. (5.1) and (5.2) in the compact form of
Qa~ + Qa~ -
O,
(5.3a)
where
Q
Lll L21 L31 A41
L12 Lee L32 A42
L13 Le3 L33 A43
-7 a2 -
_Q-1Qa~.
L14'~ L24 / L341 A44] |
(5.3b)
o
|
Equation (5.3) gives (5.4)
Recall that
a; -
RI ---~alj
R2 ---~a2j
R3 ---~a3j
R4 --ff a4j
)T
,
j -- 1,2.
From the remote loading conditions,
J -Qa, +Qa,,/ 3vJ
-- -Q
(5.5)
we can determine al. Similar to electrically insulating cracks, there are only seven independent remote input loads in Eq. (5.5). We need a supplementary condition to determine a unique al. As before, we take the null rotation around the x3 axis at infinity as the supplementary condition (Sokolnikoff, 1956) by requiring A2al -- (A(pa))lal + A2al -- (A(p,~))lat -- 0.
(5.6)
Fracture of Piezoelectric Ceramics
201
When the conductive cavity is reduced to a conductive crack, from Eq. (3.48) we have
f,~ =
aoel (zoe + ~/-z~ -- a e)
a2aoee
+
2
2(zoe + v/Z 2 - a 2 ) '
a2aoe2 foe, 1 = foe,2 - -
) zoe + v/z 2 - a 2
aoel - (zoe + v/Z~ Poe
foe,1,
2v/z~ -- a 2
a2) 2
(5.7)
O~ -- 1, 2, 3, 4 (a not summed).
Equation (5.7) together with Eqs. (3.17)-(3.18) and (3.20) give the mechanical and electrical fields for a conductive crack in an infinite piezoelectric medium under remote loads. We can define an intensity factor vector at the right crack tip as K* -
lim Q(v/27r(zoe - a))f,1
(5.8)
Z ct ---+ a
Equation (5.8) gives
x/2~rz~ )Q-IK*
f, 1 - -
for
Iz~l << a
(5.9)
where z~ = zoe - a and Iz~l is the absolute value of the complex variable z~*. For Iz~l << a, we consequently have f--
2
* ~/Q-IK
/
*
f2--( Poe ) 1
v/2n.z * Q - K*
(5.10)
(5.11)
A substitution of Eq. (5.7) into Eq. (5.8) yields
( ~r~2 / a2~
K*- ~ -
2
I ~73~
\-E 7
( KII 1 ] KI
~
~ I Kill
(5.12)
\-KE,
which shows that the intensity factor vector K* is real and its components equal half of the stress intensity factors of modes II, I, and III and the intensity factor of
202
Tong-Yi Zhang et al.
electric field strength. Let
p __
All
A12
AI3
A14
A21 A31 L41
A22 A32 L42
A23 A33 L43
A24 A34 L44
(5.13)
The near crack tip fields can be written as
uUi 1
O-12 ~ / -~ / -El/
l
o-11 0"21 ~ E2
\-Ke,
/
/( K//
2,~ Q~Q-'+0
~
i,,,, '
(5.14)
\-KE, { K//
_
2v/~- Q ~
Q_
p~
~
KI \-KE,
The extended crack opening near the crack tip is ul A
u2 u3
KII
_
lP4
i~2-7 ~ [pQ-l _ p Q - 1 ]
KI glll
(5.15)
--KE,
The energy release rate can be calculated by the methods introduced in Section II. Using the solution for conductive cracks, we find the energy release rate { K// J -- (Kit
KI
Kill
_ K e , ) i [ p Q - 1 _ pQ-1] / KI 4 | KIII \--KE,
(5.16)
B. ANTIPLANEDEFORMATION
To illustrate the dependence of the energy release rate on the material constants, we examine the antiplane deformation of a conductive crack. The general solution
Fracture of Piezoelectric Ceramics
203
of Eqs. (3.30)-(3.32) is valid for the antiplane deformation of conductive cracks. From the traction-free and constant electric potential boundary conditions,
o31nl -+- tY32n2 = Im[(c44U' + elsdP')e iO] = 0, (traction-flee)
(5.17a) (5.17b)
Im[~] = 0, (continuity of electric potential)
we find the two complex potentials for a conductive elliptical cavity
U(z) = H1
~(Z)--
H3
Z "-[- ~/Z 2 - - C 2
2 Z -'l- "~/Z2 --C 2
2
-+-
-+-
2R2H2 z
-+- x / Z 2 -
C2
(5.~8)
2R2H4 z + x/z2 - c 2
where CX)
9
el___55(E~ + i E ~ ) ,
H i __ 0"32 "1"-/O'31 + C44
H2
--
-
C44
cry2 - i o ' ~ C44
el5 + --(E~
(5.19)
-iE~~
C44
Ha -- - ( E ~ + i E T ) ,
Ha -- - ( E ~ a - i E ~ ) .
Then, using Eqs. (3.31) and (3.32) yields the mechanical and electrical fields
Z + ~/Z 2 - - C 2 Y = 1"32 nt- iy31 - -
2
H1
2R2H2
]
1
z + ~/z 2 - c 2 ~/z 2 - c 2
E= E2+iE1 F,(E~a + i E ~ ~ z +
L
I
X/Z 2 - - C 2
2
(7 - - 0"32 -I-- it731 = C44(}/32 -+-
2R2(E~a - i E ~ ) ] Z + ~ / Z 2 - - C2
iv31) - els(E2 +
i E1),
D = D2 -k- iD1 = els(Y32 -+- iv31) + Xll(E2 + iE1).
1 ~ / Z 2 - - C2
(5.20)
204
Tong-Yi Zhang et al.
When the cavity is shrunken to a crack, Eq. (5.20) reduces to e 15
~
C44
C44
Y =Y32+iy31 = ~ E ~ + i
~
+
C44
e 15
+i~E
C44
z N/Z 2
0 2'
iE~z E = E2 + i E1 = E ~ + ~/z 2 - a 2 ,
(5.21)
OG
(7" --" 0"32 + i0-31 - - 10"31 " c~ +
D= "--
Dz+iD1
(
KI1
2
O'32 Z ~ / Z 2 __ a 2
)
+ e----~5 E ~ + iel5 C44
O'31
+
C44
I
e150-~xz 2 C44
+i
x11+~
C44 ,,/
E~
%/CZ2 - - a 2
We define the strain, stress, electric field strength, and electric displacement intensity factors at the right crack tip as K~I = lim V/2Zr(z - a)y,
KIWI1 -- Z----~ lima v/2rr(z - a)0-,
KEI = lim v/2rr(z - aiE,
KIID -- Z---+ lima ~/2rr(z - a)D.
z---~ a
Z-.-+ o
(5.22)
Then, the strain, stress, electric field strength, and electric displacement near the crack tip may be expressed in terms of their corresponding intensity factors as K lall
KYII
0- = x/27r z* ' E =
Kfll
Y -- ~/2rr z* KIID
~/2jrz ~,
D-
~/27r z* '
(5.23) z* - - z - a .
Substituting Eqs. (5.19) and (5.21) into Eq. (5.22) yields K ~I1 =
~0"32 + i ~el5 E ~ C44
KEI_.iE~,-~--d,
x/~ a ,
oc/-~a K lall "-- 0"32 ,,
,
C44
KIID- [e150-~2 ic44 +
(5.24)
( Kll+ e~--~5)E~l~/-~-d'c44
Equation (5.24) indicates that the strain intensity factor and the intensity factor of electric displacement are complex, and the intensity factor of electric field strength is purely imaginary. Both 0-3~ and E ~ have no contribution to the intensity factors. In this case, the energy release rate for the mode III conductive crack is given by 1 Y cr )/ cr E D Jill -- ~ ( KHI KIH ~- KHI Kill - KIll KIll - KEI KID) 9
(5.25)
205
Fracture o f Piezoelectric Ceramics If we define the real intensity factors as • _ lim x/2n'(z - a)Y32, K HI
Ko32 lit = lim ~/2zr(z - a ) 0 3 2
Ki1Elt
Di K m = lim ~/2zr(z - a)D1
Z--~ a
--
lim x/2zr(z - a ) E l ,
z..-+ a
z--~ a
(5.26)
Z---~ a
the energy release rate takes the following form"
1 t' t,,-•
k"O'32
El
Di
(5.27)
Jzn = -~ t " m "tit + Kilt K m )
C . INTERACTION OF A DISLOCATION W I T H A C O N D U C T I V E CRACK
1. General Piezoelectric Dislocation The complex solution vector of a dislocation at w~a in an infinite piezoelectric body is given by Eq. (4.15). In the presence of a conductive crack, the solution has to satisfy the traction-free and the constant electric potential boundary conditions, which take the following form in the w plane: Q(f0 +
fo
fe)
+ Q(f0 + fe) = 0,
(5.28)
= (ln(e i~ - w~))q
The solution that satisfies Eq. (5.28) is
fe -- Yet,
4 Q~ l~-~kj In (1- - - ~/d.)
(5.29)
Yij ------ Z
k=l
//)i
where i, j = 1, 2, 3, 4 and i and j are not summed. The same notations are used in the following equations of Eqs. (5.30)-(5.32). Mapping back to the z plane gives
fo, l
+
fe,1
( ( 2 1
,/z~ - c~ 4
Yij, 1-- ~
z~ + , / z ~
Z~ + ~/Z~-C~
)lq + Y,l el,
~ ~ - z~ - v/ (z~) ~ - c~
1
Q~I Qkj ~/z2 - c 2
x(
1-I=, +
(5.30a)
1
)
Tong-Yi Zhang et al.
206
fo~ + fe~ :
4z~
c~
z~ + 4 z ~ - c~ ~ - z~ - V / (z~) ~ - c~~
4
Yij,2 -- Z Q~l--~kJ k=l
•
Pi
v/Z2 _ c2
(5.30b) l
1--(Zi + ~/Z2 --C2)(Z--~j-'~~/(~jd) 2 - --~j) / 4 Ri-~j )
When the cavity is reduced to a crack, Eqs. (5.30a-b) reduce to
f0,1 + f e , , -
( ~/z21_ ae (
z'~+v/zZ-a2 zo + , / z ~
4
1
~=l
v/z/~-a~
Yij,1 -- ~ Q~I Qkj x(
))q + Y ,(1,
- a~ - z~ - v / ( z ~ ) ~ - a~
(5.31a) a2
)
aZ- (zi + v/zZi -aZ)(-~j + V/(-~j)Z-a2 ) f0,2 + fe,2 --
4Z 2 __ a2
4
V/ z~ + 4 z ~ - a~ - z~ -
q + Y,2(I,
(z~) ~ - a~
Pi
(5.31b)
aZ- (zi + ~z~ -aZ)(z~ + ~(-~j)Z -a2) For a semi-infinite crack with the origin of the coordinate system at the crack tip, we have
f0,1 + fe, l --
( 2~/_~(~_~_ l x/~a) )q + Y,ls (5.32a)
4
1
k:,
2~/~-(~7 + v/z j)
Yij, l -- -- ~ O~l Okj
/-..:_
Fracture of Piezoelectric Ceramics 2~/~(~/~-~-
fo,2 + fe,2 =
~/~d)q + Y , 2 q ,
4
r'ij,2 = - ~
207
(5.32b)
Pi
Q;I Qkj
2~-7(~&-7 + V~jd.)
k=l
near the crack tip. The intensity factor vector at the right crack tip induced by the dislocation at z~a is K*
=
lim Q(v/2zr(z,~
-
a))(fo,1
+
fe, l)
Z ot ----~ a
= Eo(
z~ - a + r
1 -
e- a 2
)q + O (
z--~- a + r
1
-
2- a 2
)1q
(5.33) for a finite conductive crack and K * = lim Q(v/2Zrz~)(f0,1 + fe,1)Za---~0
2
Q ~ z ~d q +
0(5)0] (5.34)
for a semi-infinite conductive crack. Equations (5.33) and (5.34) indicate that the intensity factors induced by a dislocation are real, and therefore Eq. (5.12) holds. As a consequence, the energy release rate is also given by Eq. (5.16).
2. Piezoelectric Screw Dislocation Now consider a piezoelectric screw dislocation near a conductive elliptical cavity. In this case, the two parameters introduced in Eq. (4.8) have the values of g2 -- 1 and A = 0 because/3 --+ cx). We do not need to consider the single-valued condition inside the cavity because of the zero electric field inside the conductive cavity. Consequently, the complex potentials in the w plane are derived in a similar manner as those for deriving Eqs. (4.9) and (4.11) and take the following form: U
,~ln(w
m
Wd) _ ( t ] - + 2 e l S l ] ) l n ( c44 /
-- I~ ln(w - Wd) + !~ In ( lw
1 w
l/3d ) ,
(5.35)
Wd)
where .~ and !~ are complex constants defined in Eqs. (4.4) and (4.5). Using the
Tong-Yi Zhang et al.
208
mapping function, we calculate the mechanical and electric fields in the z plane
~,(z + ~/z 2 - c 2)
y= z -
zd + ,/z 2 -
-t-
c2 -
V/z 2 -
c2
( i + 2~els/C44)4R 2
}
1
4R 2 - ( z + ~/z 2 - c2)(Td + V/~2d 2 - c 2) E = -{
~/z2 - c2' (5.36)
l](z -']- ~/Z 2 --C 2)
z - ~ + ,/~2 - ca - V/Z~ -
~2
l~4R 2
}
1
. .....
4R 2-
(Z q- X/Z 2 -
cr = c44Y - elsE,
-
C2)(Z--d if- V/~2d 2 --C 2)
~/z2
C2'
D = elsy + tcllE
When the cavity is shrunk to a crack, Eq. (5.36) reduces to 2) /_ Z - zd + ~/Z 2 - a 2 - v / z ~ / - a 2 ~k(Z + ~/Z 2 - - a
y=
nt-
(ilk -+- 2__~el__~5/c44)a2
]
a 2 - (z + ,,/z 2 - a2)(z--d" + V/Z-~d- a 2) E =-[
! Z-
I](z+x/z 2-a
1 x / z 2 - a2 (5.37)
2) /
Zd + ",/Z 2 - a 2 - v/z 2 - a 2 l~a 2
a 2 - (z + ~/z 2 - a2)(z--d + V / ~ d - a 2)
}
1
~ / z 2 - a2
D = el5g -t- Kll E
O" --" C44V -- elsE,
Using Eq. (5.22), we obtain the intensity factors at the right crack tip
/r
- -{
,i
+
Zd -- a -+- v/z 2 -- a 2
(~ + 2~e~5/c44) z---d- a + V/--~d - a 2
1~
K,~,- { Zd - a + v/z2 - a 2
K~II_ c44 K~I - e 15KIII, E
D KIII
},r
z~-a ~
+ V/~-
,. }vqa, a2
el5 KI.Y + Ir K~I
(5.38)
Fracture of Piezoelectric Ceramics
209
Moving the origin of the coordinate system to the fight crack tip and letting the crack length approach infinity, we obtain the mechanical and electrical fields of a semi-infinite conductive crack
(~ if- 2~e15/c44) } Y -- 2v/-~ ~/~ - v/T~E--2--- ~ o" =
C44F
+
v/-~_V/-~- + V ~ + v / ~ d --
elsE,
,
'
(5.39)
D = elsF + tell E
The intensity factors at the tip of a semi-infinite conductive crack induced by a piezoelectric screw dislocation at za are /G=-
,
IG-{ K~t -- c44K~III--
(5.40) el5K1EI ,
Kffl -- e,sK~l '
+ KllKIEI
Using the intensity factors and Eq. (5.25), we can calculate the energy release rate induced by a piezoelectric screw dislocation.
VI. Interface Cracks
In engineering applications, piezoelectric ceramics often take the form of multilayered structures that use the accumulative results of stacks to enhance the ceramic's efficiency and sensitivity. The integrity of interfaces between the stacks is of great concern because interfacial fracture is one of the major failure modes in multilayered structures. Interface cracking of piezoelectric ceramics is, therefore, of paramount importance and has drawn much attention (Kuo and Barnett, 1991; Suo et al., 1992; Beom and Atluri, 1996; Chen, Yu, and Karihaloo, 1997; Qin and Yu, 1997; Zhong and Meguid, 1997; Deng and Meguid, 1998; Ru et al., 1998; Shen and Kuang, 1998; Gao and Wang, 2000; Qin and Mai, 1999; Herrmann and Loboda, 2000; Ru, 2000). Because the analysis of an interface crack in piezoelectric media is enormously complex, we study only electrically impermeable piezoelectric interface cracks in detail and discuss the fracture mechanics for other types of piezoelectric interface cracks only briefly.
210
Tong-Yi Zhang et al. X2
Material 1 Xl
Material 2
FIG. 5. A semi-infinite interface crack.
A. I M P E R M E A B L E I N T E R F A C E C R A C K S
1. Semi-Infinite Interface Cracks Figure 5 shows a semi-infinite piezoelectric crack with the X 1 axis along the interface and the coordinate origin at the crack tip. For an electrically impermeable crack, the extended traction-free boundary conditions along the crack faces require Z~
--" ~-]2 - - 0,
Xl
<
0, X 2 --- 0
(6.1)
where the superscripts + and - indicate the limiting values taken from the upper and lower half-planes, respectively. On the bonded interface, the continuity of the extended traction and extended displacement is signified by
~ - = ~2
(6.2)
u + = u-
(6.3)
for Xl > 0, X 2 - - 0. Using Eqs. (3.17)-(3.20), we express the extended displacements and stress functions, respectively, in the form U(k) - A(k)f(k)(Z) + A(~)f(~), Y]2(k) - - lP'(k),l,
r
El(k)-
--1/](k),2,
k - 1, 2(k not summed),
(6.4)
-- L(k)f(k)(Z) + L(k)f(k)
where the subscript (k) denotes the upper half-plane when k - 1 or the lower
Fracture o f P i e z o e l e c t r i c C e r a m i c s
211
half-plane when k = 2. Following the one-complex-variable approach (Suo, 1990; Beom and Atluri, 1996), the vector function, f, is defined as f(z) = [fl(z) f2(z) f3(z) fa(z)] T
(6.5)
in which f represents analytic functions of z = X l + px2 ( I m p > 0). Once the solution of f(z) is obtained, a replacement of z by zi(i -- 1, 2, 3, 4) should be made for the component fi(z) of f(z). By analytic continuation, one finds from Eqs. (6.1) and (6.2) (Suo et al., 1992; Beom and Atluri, 1996) that L1F~l)(Z)- L2~'~2)(z) = L2F~2)(z) - Ll~'~l)(Z) = 2h(z)
(6.6)
where h(z) is analytic in the entire z plane and F
-
A,(2)~'(2)(Z)
--
A~e)F<e)(z) -
2~k(1)~"(1)(Z)
(6.7)
which holds in the entire z plane except on the crack line. Define F~)(xl) = F~l)(xl + 0•
(6.8)
Equation (6.1) can be reduced to a homogeneous Hilbert problem in the following form L
xl < 0.
(6.9)
To determine F
(6.10)
where I is the 4 x 4 unit matrix and
M~ : (D~,)- D,e))(D,I)+ D,2))-', 1
-1
Mt~ = (D~, + D,2))
-l
( W , , ) - W,2)),
D,k) = -[Im(A(k)L(-~l)] -l, W~k) : -[Re(A~k)L~-~I)],
(6.11)
k - 1, 2, (k not summed)
in which the two real matrices M~ and M~ are the generalized Dundurs parameters for piezoelectric bimaterials, D~) and D~2) are real and symmetric, whereas W~) and W~2) are real and asymmetric (Suo et al., 1992). If materials in the upper and lower half-planes are identical, M~ and Mt~ become zero.
Tong- Yi Zhang et al.
212
Substituting Eq. (6.10) into (6.9) and noting that ~"(1)(Xl -qt. 0 - p ) - F~I)(Xl), we obtain (I + iM~)L(l)F~l)(Xl) + (I - iMt~)L(l)F~)(Xl) - 2(1 + M,~)h(Xl),
forxl < 0 (6.12)
The homogeneous solution of Eq. (6.12) may be expressed in the form (Beom and Atluri, 1996) L(1)F(1)(z) - z- 89
(6.13)
where I is a characteristic length and v is a constant vector. The constant vector v is determined by the eigenvalue equation (M~ + iXI)v = 0,
X = tanh(rr ~)
(6.14)
in which X is an eigenvalue. The four eigenvalues of Eq. (6.14) are (Suo et al., 1992), ~l :
17,
~.2 :
--17,
Z3
= -i0),
~4 = i0)
(6.15)
where / / _ {I(~tr(M~) 1 2)
2
__
det(Mt~)]1/2 _ 1~tr (M~)}1/2,
0)--{ [(~tr(M~)) 2- det(M/~)l 1/2-+- ~tr(Mg)}1/2
(6.16) (6.17)
Then, the homogeneous solution of Eq. (6.12) can be further written as L(l)F(l)(Z)- 2 2 ~1~ (i + i M ~ ) y E ( ; ) ie (;)Klg(z),
(6.18)
where g(z) is an arbitrary analytic function over the entire plane, and
1 e -- --1 tanh-l //, x----tan 7/"
2"/"
,2
Y(~' ~') -- 2 //2 ..}_0)2 (~ + ~:-1) + //2 +0)2 (r
1[
-1
(6.19)
0)
1
_+_~.-1) I
i0)2
+ ~ ~(~2 + ~o2)(~ - ~ - l ) - (-0(//2~2 + 092)(r
'[_
+2
1[
,
l
//2 +0)2 (~ -+-~" 1)+ //2 +092 (r + r
i
1
__ ~.-1) 1 M~
_,]
) M~
-1]
2 7(//2 + 092) (~ -- ~-1) + 09(//2 + 0921(~. __ ~. ) M~,
(6.20)
Fracture of Piezoelectric Ceramics
213
in which ~ = ( z / l ) ie and ~" = (z/l) K. The matrix Y(~, ~') satisfies (Beom and Atluri, 1996) Y(1, 1) = I,
Y(s~l~'2,~'1~'2) - -
(6.21)
~'l)Y(s~2, ~'2)
Y(~l,
The general solution of Eq. (6.12) is the sum of the homogeneous solution plus a particular solution
1
-1
F(1)(z) -- 2 ~ ~ z L ~ l ) ( I + iM~)Y
[(ztie ( Z)X ] g(z) + L(~)(I -1 + 7 ' 7
M,~)h(z) (6.22)
A substitution of Eq. (6.22) into Eq. (6.10) gives h(z) = -h(z),
(6.23)
g(z) = g(z)
a. Stress and Electric Displacement Intensity Factors Substituting Eq. (6.22) into Eq. (6.6) yields F(2)(z) - 2~_~zL(2)(I - iMt~)Y
~-
Z ~-
,
- 1) (I - M , ~ ) h ( z ) g(z) + L(2
(6.24) Using Eqs. (6.4), (6.22), and (6.24), we obtain the singular extended stress vector along the bonded interface near the crack tip ~2(xl)-
1
~/2zrxl
Y
,
xl --/-
x l / l << 1
g(xl),
(6.25)
As in elasticity, the solution is oscillatory unless e = 0. From Eqs. (6.16) and (6.19), we know that e - 0 if M~ -- 0, and there is no oscillation. The stress and electric displacement intensity factors can be defined as
K = [Kit Kt
Kilt KD]T -- x,--,olimv/2zrxlY
[(/)-ie ( )-x1 ,
--/-Xl
~2(xl) (6.26)
This definition is an extension of the definition for anisotropic elasticity (Wu, 1990; Qu and Li, 1991) to piezoelectricity. Because both Y[(xl/l) -i~, (Xl/1) -K] and Ez(xl) are real, K is real. Note that the intensity factors defined in Eq. (6.26) have the same physical dimensions as those for homogeneous piezoelectric materials [see Eq. (3.88)] and can be reduced to those intensity factors when the two materials become the same.
214
Tong- Yi Zhang et al.
U s i n g ~]2 -- LF + LF, we find the analytic functions F~k)(z), k -- 1, 2, near the
crack tip as 1 - ' ( I - k - i M / ~ ) Y [ ( ~ ) ie F(l)(z) = 2 2~--~L(1) (6.27) F(z)(z) : 2 2v,,~-~L(2)
}-
For an interface crack, one may define the intensity factors in slightly different ways. Suo et al. (1992) proposed the definitions of one complex intensity factor and two real intensity factors based on the components of the traction vector in the principal (eigenvector) coordinate system. The intensity factors defined by Eq. (6.26) depend on the characteristic length. However, one may define the intensity factors without using a characteristic length, as Beom and Atluri (1996) did. Rice (1988) reexamined the elastic fracture mechanics concept for conventional interface cracks and suggested the use of a characteristic length in the definition for intensity factors. Zhang and Lee (1993) discussed this issue in detail for conventional interface cracks.
b. J Integral Based on the solution to Eq. (2.38). Choosing J and substituting Eqs. intensity factors (Beom
given previously, we can calculate the J integral according a small circle around the crack tip as the integral path for (6.27) into (2.38), we find the J integral in terms of the and Atluri, 1996)
1
J -- - K T U - I K , 4
(6.28)
where, from Eq. (6.11), U -1 = (D~I + D~I)(I + M ~ ) -
-1)(i + iMt~). (I + iMt~) r (D~)l + D(2 )
(6.29)
2. Several Important Cases Using the solutions described previously and the mutual integral, Boem and Atluri (1996) studied several important cases of interface cracks. Here, we briefly present their results of the intensity factors for semi-infinite and finite cracks subjected to distributed loads on the crack faces and the results of the interaction between an interface crack and a piezoelectric dislocation.
Fracture of Piezoelectric Ceramics
215
a. A Semi-Infinite Crack with Crack Face Loading Consider a semi-infinite interface crack subjected to prescribed tractions t + (X l) and t-(xl)(Xl < 0) on the upper and the lower crack faces, respectively. The intensity factor can be calculated from K = ~-
1
CO
Ia - ~ Y[(21 )-it, (.~)-x 1 ](I + i M , ) -
x [(I - M~)t+(xl) - (I + M ~ ) t - ( x l ) ]
1
dx------L-1
(6.30)
where 21 -xleiJr/I. If the tractions are equal but opposite in sign on the upper and lower faces, -
t+(xl) -- - - t - ( x l ) -- t(xl)
(6.31)
Eq. (6.30) reduces to K-
~
fO___ Y[(3;1) -ie, oo
(21)-x](I + i M ~ ) - I t ( x l ) ~ dxl ~/-Xl
(6.32)
For the homogeneous case with material 1 being identical to material 2, we have e -- x = 0, thus Eq. (6.32) is further simplified to K -
~
ffJ~
dxl t(xl) ~_----------~1
(6.33)
b. Interaction of a Semi-Infinite Crack with a Piezoelectric Dislocation If a piezoelectric dislocation with an extended Burgers vector b (bl b2 b3 A4~)r - (bl b2 b3 b4)T and an extended concentrated force F = (F1 F2 F3 q)r _ (Fl F2 F3 F4)r is located at z d in the lower half-plane, the intensity factors are
Km---~Re{~~kZ
Y[(zf)-ie,(Zf)-X]m nH~l1g-1 (2)kl(L(2)jkbj-k- A(z)jk Fj )] ,
k=l
m -- 1, 2, 3, 4
(6.34)
where g l - - gll, K2 _ KI, K3 _ KIll, K4 -- KD, and z^dk _ Zd / l The same notation is used hereafter unless specifically noted, and H -l - (I + i M ~ ) U
(6.35)
Tong-Yi Zhang et al.
216
x2 Material 1
Xl -a Material 2
FIG. 6. A finite interface crack.
For a homogeneous medium, A ( l ) - A ( 2 ) - A, L ( 1 ) - L ( 2 ) - L, Eq. (6.34) reduces to
Im ~k=l ~ k
g m -- -
AjkFj),
m -- 1, 2, 3, 4
(6.36)
c. A Finite Interface Crack under Crack Face Loading
Consider a finite crack located from - a to a on the Xl axis (i.e., Lc = [ - a , a]), as shown in Fig. 6. The crack is subjected to the prescribed tractions t +(xl) and t - ( x l ) ( - a < Xl < a) on the upper and the lower crack faces, respectively. The intensity factor at the right crack tip is given by
K
m
1 f~a Y[(21 )-iE , (21)-~1(I +
2C%-S
-(I
+ M~)t-(xl)]
i M , ) - I [ ( I - M~)t+(xl)
a + Xl dxl - ~ y [ ( f i ) - i E ,
a --Xl
2v/-~
• (I + M~) -1 [M~(I - M ~ ) + (I - M~)S/l)]tc
(fi)-x] (6.37)
where a n
2a l '
tc --
F a
~ ( - - x l e i~r -- a) Ycl - - a ( - X l eizr + a ) '
[t+(xl) + t - ( x l ) ] d x l ,
S(l) -- W(1)D(l)
(6.38)
Fracture of Piezoelectric Ceramics
217
If the materials are homogeneous and the prescribed tractions are uniform and in opposite directions t + ( x l ) -- - t - (x l) = Z ~
(6.39)
then Eq. (6.37) is reduced to the intensity factor as that given by Eq. (3.88). If the characteristic length l is chosen to be 2a, we have Y[(5) -iE, (5) -x ] - I.
d. Interaction of a Finite Crack with a Piezoelectric Dislocation Similar to the case of a semi-infinite interface crack, the piezoelectric dislocation located at z d in the lower half plane generates the following intensity factor
=
-- -
gk -Jf-ay((~d -- a
Re
, (~d))mn
k=l
X H~ll 1L(2)k - l I (L(2)jkbj -+- A(2)jk Fj) ] -+- 1 Y(({t) -ie , ({t)-X)mnUnjbj -1 __Y((cl) -is, (cl)-X)mnUnl(n(-ll S~l , - M~D(,) )tjFj, ~/7r5a
(6.40)
m-1,2,3,4 at the right crack tip, where d
(6.41)
zkd + a For a homogeneous medium, Eq. (6.40) reduces to
Km -" - ~ I m
k--1
Vz~ - a
1
Lmk(Ljkbj + AjkFj) 1
+ 2 r--(Dmjbj~/Tra - Sjm Fj),
m -- 1, 2, 3, 4
(6.42)
Tong (1984) obtained a similar result for purely elastic media. Using the relations of D(k) -- -2iL(k)L(k7"),
S(k) -- i (2A(k)L/~) - I) ,
one can show that Eq. (6.42) is equivalent to 2K* in Eq. (4.43).
(6.43)
Tong- Yi Zhang et al.
218 B.
OTHER
RELEVANT
WORK
1. Curve-Shaped Interface Cracks
Zhong and Meguid (1997) developed a mathematically rigorous method to solve partially debonded circular inhomogeneity problems under antiplane shear and inplane electric fields with impermeable crack boundary conditions. The energy release rate can be derived and given by a closed-form solution. The solution for a straight crack is a special case wherein the radius of the circular inhomogeneity tends to infinity.
2. Permeable Interface Cracks
Gao and Wang (2000) studied the electrically permeable collinear interfacial cracks. They reduced the problem to a Hilbert problem and obtained explicit, closed-form solutions for the electric and mechanical fields in both piezoelectric media.
3. Conductive Cracks and Conductive Inclusions
Thin internal electrodes are widely used in devices and structures made of piezoelectric ceramics. The electrodes act like conductive inclusions. For a slit conductive inclusion in the (xl, x3) plane, the boundary conditions along the inclusion faces, x2 = 0, are the continuity of traction and displacements (Ru, 2000)
cr~j -- cr2j , -
+
-
uj -- u j ,
j -- 1, 2, 3,
E-~ -- E 1 -- O, f
llD2(Xl)lldxl = 47rq,
Xl
~ Lc,
xl ~ Lc, X2 =
X2 =
0,
0,
(6.44) (6.45)
IID2(xl)ll = D ] ( x l , O) -- D 2 ( x l , 0)
(6.46)
where q is the total electric charge of the electrode. If the electrode layer is much more compliant than the surrounding piezoelectric ceramics, the inclusion can be approximately treated as a crack with traction-free faces and Eq. (6.44) can be replaced by O'~j - -
0"2~ - -
O,
j -- 1, 2, 3,
Xl 6 Lc,
X2 - -
0
(6.47)
If the electrode layer is much stiffer than the surrounding piezoelectric ceramics, the electrode can be approximately treated as a rigid and conductive slit (Deng and
Fracture of Piezoelectric Ceramics
219
Meguid, 1998). In such a case, the displacement boundary conditions are
+-uj
Uj
-
--
UjO,
j=
1, 2, 3,
xl 6 Lc,
X2 = 0
(6.48)
w h e r e UjO are rigid body displacements. Note that the net force and net moment on the rigid and conductive slit should both equal zero, which are supplementary conditions to determine the rigid body displacements. Ru (2000) studied conductive interface inclusions subjected to the conditions given by Eqs. (6.44)-(6.46). He discussed the case in detail that the two piezoelectric half-planes are of the same material but have opposite poling directions. The poling directions of the upper and lower half-planes are along the positive and negative x2 directions, respectively. He found that the electroelastic field exhibits the inverse square root singularity without the normal interface oscillatory feature. Ru (2000) expressed the stress and electric fields in the vicinity of the crack tip in the form
B23 ] r 2q (crzl cr22 D2) r, = + 0 :q: B22 1 ~/a 2 - x 2
x E Lc
(6.49)
where B22 and B23 are two elements of the matrix B defined in Eq. (3.22), and + and - denote the upper and the lower edges of the electrodes, respectively. Equation (6.49) shows that the electrical and mechanical fields disappear if q -- 0 and that debonding is likely to occur at the interface between the electrode and piezoelectric ceramic blocks when the electrode carries an electric charge.
4. Contact Zone Model Many of the solutions for interfacial cracks in piezoelasticity have oscillatory singularities, as in the case of elasticity (Williams, 1959; England, 1965; Erdogan, 1965), which causes the overlapping of crack faces, a physically unreasonable phenomenon. To correct this shortcoming, Comninou (1977) developed a crackface contact model for conventional solids. Qin and Mai (1999) developed this model further for electrically impermeable interface cracks in linear thermopiezoelectric media. Assuming the contact zone to be friction-free, Qin and Mai (1999) obtained a set of singular integral equations supplemented by the condition that o"22 equals zero at the ends of the contact zone for the determination of the contact zone size. The stress intensity factors and the size of the contact zone can be calculated numerically by solving the integral equations. Herrmann and Loboda (2000) studied electrically permeable interface cracks by assuming that an artificial contact zone existed. Thus, they were able to reduce the problem for transversely isotropic piezoceramics to a combined Dirichlet-Riemann boundary value problem, which
220
Tong-Yi Zhang et al.
can be solved analytically. Numeric results for electrically permeable interfacial cracks of CTS-19/PZT-4 bimaterials show that the contact zone is four orders smaller than the crack length when the applied loads satisfy the condition
O'1~176 2
< 20,
D~~ = 0.
(6.50)
If the applied shear load is dominant, the contact length becomes significant. For example, when 0"12/0"22 oo oo - - 100 and D~~ = 0, the contact length is about oneseventh of the crack length (Herrmann and Loboda, 2000).
VII. Three-Dimensional Electroelastic Problems
Three-dimensional problems are of great importance in the fracture of piezoelectric ceramics and in practical applications. As early as in 1980, Deeg solved the electro-elastic problem for a three-dimensional piezoelectric inclusion embedded in an infinite piezoelectric medium. In the 1990s, many important threedimensional results were obtained for transversely isotropic piezoelectric media (Sosa and Pak, 1990; Wang, 1992a, 1992b; Dunn, 1994a; Huang and Yu, 1994; Wang, 1994; Wang and Huang, 1995a, 1995b; Wang and Zheng, 1995a; Kogan et al., 1996; Dunn and Wienecke, 1997; Huang, 1997; Zhao et al., 1997a, 1997b, 1999a-c; Chen and Shioya, 1999, 2000; Chen, Shioya and Ding, 1999). In practice, most poled piezoelectric ceramics possess 6 mm symmetry, as discussed in Section II. Thus, we limit the discussion to three-dimensional problems in transversely isotropic media.
A. GENERALSOLUTIONS The governing equations are the same as those given in Section III. Let the (xl, xe) plane of the Cartesian coordinates coincide with the isotropic plane, and let the poling direction be along the x3 axis. By virtue of the constitutive equations and the strain-displacement relations, the governing equations can be written in terms of the displacements, U l, U2, U3, and the electric potential, 4~
OZul 02//1 02Ul 1 02U2 Cl,-~Xl2 + C66 OX 2 -']-C44-~X23 -'1- -~ ( Cl , + C12)OXlOX2
02//3 02~ +(C13 + C44) OXIOX3 -k- (el5 + e31) OXlOX-----~-- O,
Fracture of Piezoelectric Ceramics 02U2
02U2
c66- Xl
+ 44
02U2
1
221
02Ul
+ (Cl, +Cl )0 10
02U3 02(]) + (C13 Jr- C44) OX20X3 nt- (el5 -~- e31) OXzOX3 = O, 02U2
C441k OX~ Jr- OX2 / nt- C33-~X32 + (C13 -~- C44) 0X;~X3 t OX20X3
+el, 0Xl + 02U2
el5\ Ox 2 -Jr- OX2 ~] + eaa--~x32 + (el5 '1- e31) Ox15x3 t OX20X3 --Kll
~
) (7.1)
)
+ OX2 / -- K33-~X32 -- 0,
where cij with C66 -- (Cll --
C 1 2 ) / 2 ; and eij and Kij are the elastic, the piezoelectric, and the dielectric constants, respectively. For three-dimensional deformation, the displacements and the electric potential may be expressed in terms of potentials U and X (Wang and Zheng, 1995b; Ding, Chen and Liang, 1996; Kogan et al., 1996; Zhao et al., 1997a)
OU Ul "-- OXl
OX OX2 ,
OU OX UZ -- ~X2 + -~Xl'
OU U3 -- k,-~x3'
OU 4) -- k20x3
(7.2)
where kl and k2 are constants to be determined. After substituting Eq. (7.2) into Eq. (7.1), we have
02Ui 02Ui 02Ui = 0, x--~l-~O x--~2 O nt- ~,i x---~3 O OZx Ox 2
i = l, 2, 3(i not summed) and
02X
02X
(7.3a) (7.3b)
where ~.4and ~i(i
-
2c44 Cll
(7.4a)
-- C12
l, 2, 3) are the roots of the characteristic equation
C1 ~.3 --~ C2 ~.2 Jr- C3 ~ -+- C4 - 0,
(7.4b)
222
Tong-Yi Zhang et al.
in which (?l, C2, C3, and C4 are materially related constants defined as follows: C;1 -- Cll(e25 + C44Kll), C2 -- 2el5(el5cl3 -- e33c11) -1--2c13(e15e31 + C44Kll)
-c44(e~l + cllK33) + Kll(C~3 -- CllC33),
(7.4c)
C3 -- e33(e33Cll --I- 2e15c44) -- 2e33(e15 -t- e31)(C13 + c44) -- K33(C23 -k- 2C13C44 -- C11C33) -k- K11C33C44 -I- c33(e15 -q- e31
C4 --
--c44(e23 +
)2
,
C33K33) 9
Equation (7.4b) is derived from the elimination of kl and k2 to obtain Eq. (7.3a). In the following discussion, we consider only the case that the first three roots ~i(i = 1, 2, 3) are distinct. In many cases, it is more convenient to use the polar cylindrical coordinates (r, 0, z), where the z axis coincides with the x3 axis. Letting U4 = X, we can express the displacements and the electric potential in a more compact form in terms of the four potentials Ui. Equation (7.3) can then be written as 02Ui 1 OUi 1 oZui -----5Or + -r -~r -~ r2 O0 2
oZui
where zi = si z and si -- 1 / ~ i . Ur-
i -- 1, 2, 3, 4 (i not summed),
~- OZ2 = O,
3 OUi Z O; i=1
(7.5)
The results of Eq. (7.5) are
1 OU4 r OO '
Uo-
3 OU i Uz -- Z k l i ~ , i=1 Oz
~
-
3 1 OUi OU4 Y~ r O0 ~ ~Or , i=l
(7.6a)
3 OUi Z k2i Oz ' i=1
where kli and k2i(i -- 1, 2, 3) are constants related to ~-i by C44 + (C13 + c44)kli + (el5 + e31)k2i
c33kli + e33k2i
Cll
C13 "]- C44 + c44kli -'}- elsk2i
=
e33kli - K33k2i el5 -+- e31 + el5kli - Kllk2i
(7.6b) = ~.i.
Consequently, the stresses are given by 3 F 02Ui 10Ui 1 02Ui - - 002 ~ -~- (Cl3kli d- e31k2i)--~z 2 tTrr -- i")~9 - LCll--~r2 + C12 r 0r + Cl2r2
1 02U4) + (Cll -- C12)
r2 00
r O00r
'
223
Fracture o f Piezoelectric Ceramics
3 E
02Ui
C12---~-F2 + Cll
~176176 -- /El "__
10Ui
1 02Ui -~- C1 l -r2- ~ 002 -Jr-(cl3kli Jr- e31k2i) Oz 2
r 0r
02U4) r OOOr ' 1
-- (Cll -- C12) r 2 O0
3 02Ui tTzz -- Z ~.i(C44 -Jr-Cl44kli + el5kzi) Oz 2 , i=l
3[
tYzO -- i 9~ 1 (C44 "+-c44kli-1t- elsk2i) 102Ui r OzO0
(7.7)
]
0204
OrOz
+c44~
3 [ O2Ui 1 02U4 tTzr -- i ~ (c44 -~- c44kli -1I- elsk2i) OrOz - c44 -
rS-b
3 [ tTr~ ~---/El'= (cll -c12) 1
-~- ~(Cll --C12)
(1 r
02Ui
0rO0
(02U4
r 2 00
10U4 r Or
0r 2
1
02U4)
r 2 002
'
and the electric displacements are given by
3 [ Dr -- E
i=l 3[
Do --/~l'= Dz --
O2Ui 1 02U4 (el5 d- el5kli - gllk2i) OrOz -- el5-
r OOOz
(el5 q- el5kli -
3
~.i(e15-~- el5kli E i=1
1 02Ui]
02U4
Kllk2i)-r aOaz + e 1 5OrOz ~
(7.8)
O2Ui - Kllk2i) Oz 2 .
B. AN ELLIPSOIDAL CAVITY UNDER REMOTE LOADING For a transversely isotropic piezoelectric medium, Fig. 7 shows an ellipsoidal cavity in an infinite body under remote loading. The center of the cavity is located at the origin of the coordinate system, and the ellipsoidal surface is described by
r2
Z2
a2 q- ~-~
1,
(7.9a)
224
Tong-Yi Z h a n g et al.
az~, D7
X3
X2
,,o ~
( ~ rr
Xl
Fro. 7. An ellipsoidal cavity under combined remote loadings.
which can be rewritten as r2
z2
q2----~ + q---~ -- C2'
(7.9b)
where C 2 - b 2 - a 2 and q0 = b/Co. Then,
n
--
{nr,
O, nz}
(7.10a)
is an outwardly normal unit of the ellipsoidal surface with r nr =
a2N'
n, -"
z b 2N
,
N-
/r 2
22
~/ --a-~ + _-7.
(7.10b)
q'b
The boundary conditions on the cavity surface are the same as those given by Eqs. (3.33)-(3.35) and take the following form in the cylindrical coordinate system r - - 4~c
(continuity of electric potential),
~ r r n r q- ~Trzrtz - - O, Crrznr + O'zzn z - - 0
O'Orrlr + croznz -- O,
(traction-free),
D r n r + Dznz -- DCn~ + D~nz,
where the superscript c denotes the cavity.
(surface-charge-free),
(7.11)
225
Fracture of Piezoelectric Ceramics 1. A x i s y m m e t r i c L o a d i n g
Under the remote loads of {rrr and {r=~ and the remote electric displacement D z , the corresponding displacements and electric potential are (X)
{30
OO
Ur -- errr,
{90
{~)OO
u z -- ezzz,
-
OO
- E z z + Cko,
(7.12a)
where 4}o is a reference electric potential, and co
oo
O'rr ~ -- (Cll + Cl2)er~r + Cl3e=z -- e31E z , ~ -- e33 E .,~ , {rzz~ -- 2Cl3Srr + C33e,z. Dz
(7.12b)
~ 9 -- 2el3err~ + e33ezz~ + x 3 3 E =
Under remote loading, the electric field strength E c inside the cavity is uniform with the electric potential ~c _ _ E ~ z + 4}~,
(7.13)
where 4}~)is another reference electric potential and E c is to be determined from the boundary conditions. In this case, the four potentials take the following forms (Kogan et al., 1996): Ui -- A i H ( r ,
U4 - - 0 ,
i = 1, 2, 3,
Zi),
(7.14)
where l[z~qgl- (qi) 4- r 2 q92(qi) - C2i990(qi)], n ( r , zi) -- -~ l(qi-t-1) 990(qi) = ~ In qi - 1
1
991(qi) = ~ In
(qi+l) qi-
'
(7.15)
1 1
qi
1 ln(qi+ 1) 1 qi 992(qi) -- --~ qi -- 1 -~ 2 q 2 - - - ~ ' with constants, A i , i = 1, 2, 3, and qi(r, zi) defined implicitly by r2 z2 - C 2 (i not summed) with C 2 = _b2 _ q 2 _ 1 +- q2 ki
_ 0
2
"
(7.16)
On the ellipsoidal surface, using Eq. (7.9a) and z 2 - z2/)~i yields q2 _ q2i,O ~ b 2/ ( b2 -- ~.ia2) 9
(7.17)
226
T o n g - Y i Z h a n g et al.
A substitution of Eqs. (7.14)-(7.17) into the boundary conditions of Eq. (7.11) gives the following equations for the constants Ai and E c as follows: ~0
r
-
Ez
-
3[
/~l
93i~l ~k2i i g)l (qi ,o)Ai
Ec -
(continuity of electric potential),
-
c44(1 + kli) nu el5k2i ] ).i qgl (qi,o) Ai nt- ff r~r -- 0
(C12 -- Cll)gO2(qi,o) --
".I
(traction-free in the r direction),
3 2 Z[c44(1 -t- kli) "k- elsk2i]q92(qi,o)Ai - Ozz -- 0 i=1 (traction-free in the z direction), 3 2 Z [el5(1 q- eli) - Kllk2i]g)2(qi,o)Ai - D z - trCE c (surface-charge-free), i=1 (7.18) where x c is the electric permeability of the cavity. Note that the traction-free condition associated with ~Or and CrOz is satisfied automatically due to the axial symmetry. Once the coefficients A i , i = 1, 2, 3, and E c are determined from Eq. (7.18), the stress and electric displacement fields can be calculated from Eqs. (7.14), (7.7) and (7.8). As discussed in Section III for a two-dimensional cavity, caution should be used when shrinking the cavity to a crack because the results depend on the sequence of how the limits are reached. Letting b approach zero (i.e., oe = b / a ~ 0) and using Eqs. (7.7), (7.8), and (7.18), we obtain the stress and the electric displacement on the crack plane (z = 0) =
D z ( r , O) =
1 715
+ A r c t a n V/~ 2 -
%/~2 __ 1
tOz
,
Jr
%/~2
1
1
]
+ A r c tan v/? 2 - 1
? -- r / a > 1
] + d*,
~ -
(7.19a)
r/a > 1
(7.19b) where d* = D ~
(X)
~
+ azz M5
for c~/fl --+ O,
(7.20a)
Fracture of Piezoelectric Ceramics
227
d* = D z + CrzzOOM5 for c~/3" -+ a nonzero finite value, 1 + ~/3"
(7.20b)
d* = 0
(7.20c)
for ~ / 3 " --+ oo,
in which
* - xClx3o ,eff
x3oeff= det [M(1)]/det [M (3)],
/~5 - det
[M(5']/det [M(3)]. (7.21a)
In Eq. (7.21a) M (1), M (3), and M (5) are 3 x 3 matrices with the elements
M(S) 2i --
M li(1) = si[c44(1 + kli) + elskzi],
zri -~-[c44(1 + kli) + el5k2i],
M(1) zri 3i = -~--[els(1 + k l i ) - K l l k 2 i ] ,
M(li3) =
/!,4(1) ~'~li '
M(5) li
Aar(1)
--*'*li
/IA(3) ~(1) ~"~2i = "'~2i ' 13'(5)= isik2i,
'
*"2i
(7.21b)
/i//(3) -isikzi, ~'~3i =
~r
*"3i
__ M(~) 3i '
i -- 1 ' 2 ' 3 "
Equation (7.21a) gives the effective dielectric constant for three-dimensional piezoelectric solids. The results indicate that the mechanical and electrical fields are strongly dependent on the ratio of ot/fl*. This is similar to the two-dimensional case, in which the solution depends strongly on the ratio of ot/fl*. The two limiting results given by Eqs. (7.20a) and (7.20c) correspond, respectively, to the electrically permeable and impermeable boundary conditions along the crack faces. Defining Mode I stress intensity factor K I and electric displacement intensity factor Ko as KI
=
lim v/2rr (r - a)~rzz,
r---+ a
K o = lim v/2:r(r - a)Dz, r-+
(7.22)
a
we have
KI
=
2~rzz ~ , a
KD -- 2 ( D ~ - d . ) V ~a"
(7.23)
These results show that, as in the two-dimensional problems discussed in Section III, the Mode I stress intensity factor is the same as that in purely elastic media and is independent of the applied electric displacement. The electric displacement intensity factor relates not only to the applied fields, but also to the material properties in terms of d*. For the two limiting cases, we have
7 #I ~0 ~ KD = -2O'zzO 0 Ms~/~-
(7.24a)
228
Tong-Yi Z h a n g et al.
for electrically permeable cracks and KD -- 2D~//-f-n"
(7.24b)
for electrically impermeable cracks. Using the electrically permeable or impermeable boundary conditions, Kogan et al. (1996) and Huang (1997) obtained the intensity factors for electrically permeable cracks, whereas Zhao et al. ( 1 9 9 7 b ) obtained the intensity factors for electrically impermeable cracks. For a conductive crack, the electric field inside the crack disappears (i.e., E C= 0). Then the traction-free and electric potential continuity boundary conditions require 3
Z
sjk2jAj -- 0
(continuity of electric potential),
j=l 3 Z sj[c44(1 + k l j ) + e l s k 2 j ] A j j=l
7ri
- 0
(traction-free in the r diection),
3
(traction-flee in the z direction).
[c44(1 + klj) -+- el5k2j]Aj = r
2
(7.25)
j=l
Equation (7.25) yields the constants Ai, i - 1, 2, 3. Once the electrical and mechanical fields are calculated from the potentials Ui, i -- 1, 2, 3 of Eq. (7.14) with Eqs. (7.7) and (7.8), then we can calculate the intensity factors, which become or ~//~_ K* -- 2a=z 7r'
KD = 2 F ~ %z~ ~ //rS7'
(7.26)
where F o _ det[FU]/det[Ft],
(7.27a)
in which F u and F t are 3 x 3 matrices with F1u. -- F[i -- s i k z i A i ,
F2]. -- F~i -- si[c44(1 + kli) + elskzi],
F3u -- els(1 + kli) - Kllk2i,
F~i -- c44(1 -+- kli) -+- elskzi,
(7.27b) i -
1, 2, 3.
Equation (7.26) shows that the mechanical load induces the intensity factor of electrical displacement Ko due to the piezoelectric effect.
229
Fracture o f P i e z o e l e c t r i c C e r a m i c s 2. A s y m m e t r i c L o a d i n g
If only 0"~3 and D ~ are applied at infinity, the corresponding non-zero displacement and electric potential at infinity are related by the constitutive equations u ~ = 2e~3r cos0,
4) ~ -
oc 2c44e~3 - e l 5 E ~ -- 0"13,
cos0 + ~b0,
-E~r
(7.28)
2ea4e~ ,3 + X l , E ~ -- D ~ .
The electric potential inside the electrically insulating cavity is cDc -
- E c r cos 0 + q~D.
(7.29)
The potential functions become (Kogan et al., 1996) i = 1, 2, 3,
A i H ( r , Zi)COSO,
Ui -
U4 -
-A4H(r,
Z4)
sin 0,
(7.30)
1
(7.31)
where 2ziC 2
H ( r , Zi) -- rzi[(~l(qi) -]- ~ 2 ( q i ) ] -k- ~ ,
O1 (qi) -- qi
[l+q In (q/1)] -~-
qi-k- 1
rqi ,
02(qi)
qi(q 2 - - 1 )
9
The dependent variables qi(r, zi) and constants Ci are defined by Eq. (7.16). Substituting Eq. (7.30) into Eqs. (7.5) and (7.7) and then into the boundary conditions of Eq. (7.11) yields the equations for the five constants Ai, i - 1, 2, 3, 4, and E c, ~o
-
~;,
3
Z
AiC3 -- A4C3 -- 0
(boundness condition at r - 0),
i=1 3
Z
sikzi[l~l(qi'~ + 3(~2(qi,o)]Ai - E ~ -- - E c
i=1
(continuity of electric potential), 3
Z
Si[Ca4(1 -Jr-kli) nt- elskzi][l~)l(qi.o) nt- 3~2(qi,o)]Ai + sac44[l~)l(q4,o)
(7.32)
i=1 + 3 O 2 ( q a , o ) ] A 4 + o'13
= 0
(traction-free in the r and 0 directions),
3
Z
si[c44(1 + kli) + el5kzi]|
- 0
(traction-free in the z direction),
i=1 3
y ~ si[el5(1 + kli) - Kllk2i][~l(qi,o) -- 3~2(qi.o)]Ai i=1
+ saels[Ol (q4,0) + 3O2(qa,o)]A4 -t- D ~ - x c E c -- 0
(surface-charge-free).
230
Tong-Yi Zhang et al.
The mechanical and electrical fields can be determined from the four potentials Ui as defined by Eq. (7.30) after the five constants Ai, i = 1, 2, 3, 4, and E c are determined. However, the general solution of Eq. (7.32) is complicated. For example, here we consider only the case of c~ --+ 0 and ot/fl* --+ O. In this case, the cavity is shrunk to a crack, and the boundary conditions of Eq. (7.32) are simplified as 3
Z
Ai
-
-
(boundness condition at r - 0),
A4 -- 0
i=1
3 3 -7ri Z s j k z j A j - E ~ - - E c 2 j=l
-7ri 2
j=l
(continuity of electric potential),
sj[c44(1-+-klj) + el5k2j]Aj -k- s4c44A4
(7.33)
-k- o13
(traction-free in the r and 0 directions), 3 ~
[c44(1 -!- kli) + elsk2i]Ai = 0
(traction-free in the z direction),
i=1 3 Z [els(1 -k- kli) i=1
- Kllk2i]Ai
= 0
(surface-charge-free).
The stress at 0 - 0, z = 0 is given by (Kogan et al., 1996) 0-13(F,
0 O) -- 20-~3 F
?2 _+_ 1 + Arc 7/" L 72 ~/72 __ 1
'
6s4 A4i c44 , ?2 ~,/72 _ 1
+
tan
V/? 2 -
1]
? - r / a > 1.
(7.34)
Defining the Mode II and Mode III stress intensity factors as Kll = lim V/27r(r
r-+a
-
a)0-rz,
Kill -- lim V/2zr(r -- a)0-zO,
r--~a
(7.35)
we have
KII--(40-13 ~ Km=
a
Im[A416s4c44~/-~)cosO,
-Im[Aa]6sacan~/na sin 0.
(7.36)
The results show that the electric displacement has no singularity near the crack tip. In other words, the electric loading D ~ at infinity does not contribute to the
Fracture of Piezoelectric Ceramics
231
singularity at the crack tip. Wang (1992a) and Chen and Shioya (2000) obtained the same result for electrically impermeable cracks.
C. AN IMPERMEABLE PLANAR CRACK OF ARBITRARY SHAPE UNDER ARBITRARY LOADING
Consider a planar crack of arbitrary shape in the isotropic (Xl, X2) plane. The upper and lower surfaces of the crack are denoted by s + and s-, respectively. The prescribed tractions, Pl, P2, and P3, along the respective x l, x2, and x3 axes and the prescribed electric displacement boundary value nr(x, y) are all equal but opposite in sign on the upper and lower faces of the crack; that is, PI(X1, x2, 0 +) = -PI(X1, x2, 0 - ) -- Pl(x, y), P2(x1, x2, 0 +) -- - P 2 ( x I , x2, 0 - ) = P2(x1, x2),
(7.37)
e3(xl, x2, 0 +) = - P 3 ( x I , x2, 0 - ) = P3(xI, x2), U)'(X1, X2, 0 +) -- --UY(XI, X2, 0 - ) = UT"(X1,X2).
The boundary integral method can be used to solve this type of crack problems (Cruse, 1996). Zhao et al. (1997a) established the Green functions for the loadings of concentrated electric potential discontinuity and concentrated displacement discontinuity in a three-dimensional piezoelectric medium. Using the fundamental solutions, Zhao et al. (1997b) derived the boundary integral equations for electrically impermeable cracks of arbitrary shapes f
+
[tz~l Ilu311 + tzl IlOll]~ds(~, 1 r/) = P3(xl, x2),
f+ [Lzz211u311 + Lz2llcblll-~ds(~, 1 s+{[Lzr3
O) = nr(xl, x2),
(7.38a) (7.38b)
cos 2 0 -k- Lz03 sin 2 0]llul II
+ [Lzr 3 -
s+{[Lzr3 -
Lz03]
Lzo3]
1
sin0 cosOllu2ll}-~ds(~, rl) - Pl(Xl, x2),
(7.38c)
sin0 cos011Ul II 1
-+- [Lzr 3 cos 2 0 -k- Lzo3 sin 20]llu211}-~ds(~, O) - P2(xl, x2),
(7.38d)
232
Tong-Yi Z h a n g et al.
where r 2 --- ( X l - - ~ ) 2 ..]_ (X2 -- 0) 2, the symbol II II denotes a jump of the quantity across the crack, and the material constants are given by 3
3
Lzzl = ~ - ~ a i m i , i=1
1
Lzr3
-
-
-
-
2
1
Lz~
~
~
1
3 i=1
3
H123 i=1
3
Lzz2--Zaiti,
i=1
- - ~H123
~
3
Lzl--Zbimi,
Lz2 -- ~
i=1
biti,
i=1
si Hi(c44 + c44kli -+-el5k2i) -+- c44s4
(7.39)
,
Si Hi(c44 + c44kli -1- el5k2i) - 2c44s4 ,
in which
ai --
--c13 +
(c33kli -~- e33k2i)s 2,
bi -
- e 3 1 --t- (e33kli
m l - - [k12k23 - k22k13 + k23 -
k22]/(47rSl A),
m2 - - [k13k21 - k l l k 2 3 -k- k21 -
kz3]/(47rszA),
m3 - - [ k l l k 2 2 - k12k21 d- k22 -
k21]/(47rs3A),
t2 - -
[el
~(k21kl3 c44
- k l l k 2 3 ) + kl3 - k l l
]/
- K33k2i)s2, (7.40a)
(47rs2A),
(7.40b)
t3-- Iel---~5(kllk22-k12k21)-t-kll -k12]/(47r,3/~)'c44 A - - k12k23 - kl3k22 + kl3k21 - k l l k 2 3 + k l l k 2 2 - kl2k21, H1 -
1,
H2
-- (a3bl -
H3 - - ( a l b 2 - a 2 b l ) / ( a 2 b 3
alb3)/(a2b3 -a3b2),
a3b2),
(7.40c)
H123 - - H1 d- H2 d- H3.
Equations (7.38a-d) are hypersingular boundary integral equations (HSBIE). Without loss of generality, the coordinate system can be chosen such that its origin is at the crack tip, and the x~ and x2 axes are along the normal and the tangential directions of the crack front, respectively. We define the intensity factors at the crack tip as
KI -- lim
xl-~0
Km -
lim
~/27rXlO'33(Xl, X2, 0),
xl--~0
~/27gXlO'32(Xl, X2, 0),
KII --
lim x/2rrXlCr31(Xl, X2, 0),
xl--+0
KD -- lim ~/27rxl D3(Xl, X2, 0). xl--+0
(7.41)
233
Fracture of Piezoelectric Ceramics
Then the intensity factors can be expressed in terms of the jumps of the displacement and the electric potential across the crack faces K o = - J r x,--.0 lim ~2~/xl [tzl Ilu311 + L~2114~11], KI--Jr
lim 2r x,-~o
[tzzlllu311 + tzz2114~ll],
(7.42) Kit -- -7r lim x l ---~o Kill-
--Tr
lim
xl~O
2~xx~[~L , r 3
+ 31 -~ Lz03
Lzr3 + -s
1 Ilulll,
Ilu211.
We can solve Eqs. (7.38a-d) to obtain the jumps of electric potential and displacements across the crack faces. Solutions for HSBIE problems for conventional materials are available (Ioakomidis, 1982a, 1982b; Tang and Qin, 1993). By analogy, we have the solutions of Eqs. (7.38a-d) for the piezoelectric solids, and then we can determine the intensity factors.
1. Symmetric Problems
Under symmetric loading, P1 = 0 and P2 = 0, the boundary integral equations involve only Eqs. (7.38a, b). Correspondingly, the HSBIE solution takes the form I1~11 4Jr(1# - v) fs+ ---~-ds(~, 11) - -t3(Xl, x2),
(7.43)
for isotropic elasticity, where II~ II is the jump in the normal displacement across the crack surfaces; # and v are, respectively, the shear modulus and Poisson's ratio; and t3 is the prescribed normal traction. One can show that the stress intensity factor is (Tang and Qin, 1993) # lim ~ 27r II~ II K ~ ( t 3 ) - 4(1 - v)x,--,0 x---7 "
(7.44)
The stress intensity factor is related to the loading through the displacement jump of Eq. (7.43). If the equations for piezoelectric solids have the similar form to
234
Tong-Yi Zhang et al.
that given by Eqs. (7.43) and (7.44), we are able to solve the piezoelectric HSBIE problems. To do so, we linearly combine Eqs. (7.38a, b) to yield f + [Lzlllu311 + Lzzllc~ll]-~ds(~, 1 17) = GE M P3(Xl,X2) -] GEEuff(XI,X2),
f+ [Lzzl Ilu311 + Lzz21lckll]-~ds(~, 1 rl) -
(7.45a)
G MMP3(X1, x2) + GME~o(XI, X2),
(7.45b) where GEM
_
L z 2 ( L z l
--
Lzz2)/AL,
2 GMM ._ ( L z z l L z 2 - Lzz2)/AL,
G EE
--
G Me-
(L zz I L z 2 - L2z l ) / A L , Lzzl(Lzz2- Lzl)/AL,
(7.46)
AL -- L z z l L z 2 - LzzzLzl.
It has been proved (Chen and Shioya, 1999; Zhao et al., 1997b) that G eE - G MM = 1,
G TM -- G ME -- O.
(7.47)
Equation (7.47) indicates that the electric and mechanical fields are decoupled in Eq. (7.45). Comparing Eqs. (7.42) and (7.45) with Eqs. (7.44) and (7.43), we can write the stress and the electric displacement intensity factors directly in the form KI -- KM(p3),
Ko-
Kzm(~r).
(7.48)
Equation (7.48) indicates that the stress and electric displacement intensity factors can be obtained from the stress intensity factor of the corresponding elastic problem by simply replacing the load t3 by P3 or ~ , provided that the corresponding solutions for the isotropic elastic materials are available. To demonstrate how to implement this method, we present the solution for a penny-shaped crack of radius a centered at the origin of the coordinate system. The crack is subjected to two point forces in opposite directions and at the same magnitude P at (rp, Op, 0 +) and two point charges Q at (ro, Oa, 0 +) on the upper and lower surfaces of the crack. From the stress intensity factor of the corresponding elastic problem (Cherepanov, 1974), we find the intensity factors at point (a, 0, 0) to be P
KI=
KD=--
(a 2 - r2) 1/2
7r ~'-~--d a 2 + r 2 - 2arp cos(O- Oe) Q (a 2 - r~)1/2 Jr ~ / ~
(7.49)
a 2 + r2a -- 2arQ cos(0 -- Oa) "
Using an extended potential function method (Fabrikant, 1989; Ding, Chen, and
Fracture of Piezoelectric Ceramics
235
Liang, 1997) for piezoelectricity, Chen and Shioya (1999) obtained the same results as those presented in Eq. (7.49). 2. Asymmetric Problems
In this case, P3 = ~ - 0. Introducing two new parameters Y and 8 to replace Lzr 3 and Lzo 3 through the relationship Y 8zr(1 - 8 )
1 - 28 Y 8~(1 - 8 2)
-- -Lzr3,
-- -Lz03,
(7.50)
we can rewrite the boundary integral equation (7.38b, c) as rl) f~+ {[(1 - 26) + 38cos 2 0]llu~ II + 36 sin0 cosOIlu21l}-~ds(~, 1
87r(1 --8 2)
Pl(Xl, x2),
(7.5 la)
rl) f + {38 sin0 cos011Ul II + [(1 - 28)+ 38cos 20]llu211l-~ds(~, 1
8zr(1 --8 2)
Pz(xl, x2). (7.5 lb) Y Equations (7.5 la, b) are in the same form as those of the HSBIEs for purely elastic problems. It can be shown that the stress intensity factors are = -
Y
.
r
KII(P1, P:) = 8(1 -- 82) J,l~m0 Z ]]ul ]l, i
KIll(P1, P:) =
r
(7.52)
Y
lim ./~2zr IIu 2 II. 8(1 + 8) x,--,o V xl
Solutions to Eqs. (7.51) and (7.52) can also be obtained easily from the corresponding elastic solutions (Tang and Qin, 1993) by replacing 2#(1 + v) and v with Y and 6, respectively. In summary, the solutions for impermeable planar cracks located in the isotropic plane of an infinite transversely isotropic piezoelectric medium can be obtained directly from the corresponding isotropic elasticity solutions. Analytic solutions have been established for the elliptical (Zhao et al., 1997b) and penny-shaped cracks (Chen and Shioya, 2000). Zhao et al. (1999a, b) also studied a pennyshaped crack in a half-space. Numeric methods such as the boundary element method or the finite element method are generally required for more complicated crack problems (Chen and Lin, 1993, 1995; Kumar and Singh, 1997a, b; Fang et al., 1998).
236
Tong-I'7 Z h a n g
et al.
D. A PERMEABLE PLANAR CRACK OF ARBITRARY SHAPE UNDER ARBITRARY LOADING
For a permeable planar crack without flee-charge loading on the crack faces, the electric boundary conditions on the crack faces are given by Eqs. (3.73) and (3.74). We repeat them as follows for clarity DJ- -- D 3,
4~+ -- 4~-.
(7.53)
In this case, if the applied mechanical loads on the crack faces are in the form of the first three equations of Eq. (7.37), the boundary integral Eqs. (7.38a, b) become
fs + [Lzzl fs +
Ilu3 II]~ds(~, 7) - P3(Xl, x2), 1
(7.54a)
-O3(x1, x2),
(7.54b)
[ g z z z llu311]-~ds(es , rl) -
whereas Eqs. (7.38c, d) remain the same. Then, the stress and electric displacement intensity factors are given by K I -- Kff(P3),
Lzz2
KD = L-~-z~K I .
(7.55)
In this case, it can be proved that the quantity 11)5 in Eq. (7.21a) equals to ~ I 5 -- - L zz_.___~2.
Lzzl
(7.56)
Equation (7.55) indicates that for electrically permeable cracks, the electric displacement intensity factor is proportional to the Mode I stress intensity factor due to the piezoelectric effect.
VIII. Nonlinear Approaches The linear electroelastic analysis of electrically insulating or conductive cavities and cracks is the first and most fundamental step toward understanding the fracture behavior of piezoelectric ceramics. As mentioned in Section I and shown in Fig. 2, both the polarization and the strain versus the electric field strength are nonlinear. Experimental observations, which are described in Section IX, show that the fracture behavior of piezoelectric ceramics is also nonlinear. Various models have been proposed to explain the nonlinear fracture behavior of piezoelectric ceramics. In this section, we briefly introduce four types of nonlinear approaches--namely,
Fracture of Piezoelectric Ceramics
237
the electrostriction, domain switching, domain wall kinetics, and polarization saturation at a crack tip.
A. ELECTROSTRICTION
If we apply an electric field to an electrically insulating material (dielectric material), we change its shape; however the strain remains unchanged if we reverse the direction of the applied electric field. This effect is called electrostriction, and the strain is proportional to the square of the applied electric field. The electrostrictive effect is induced by the Maxwell stress (Landau and Lifschitz, 1960), which is a tensor whose divergence equals the body force induced by the electric field in a dielectric material. Considering the Maxwell stress, Smith and Warren (1966, 1968) studied an elliptical inclusion in an infinite dielectric material, where the dielectric material has zero piezoelectric constants and its mechanical and electrical properties are isotropic. When the inclusion is a conductive crack, they found that the elastic stress field at the crack tip has a 1/r singularity, where r is the distance from the crack tip. McMeeking (1987) reanalyzed the problem of a dielectric solid containing a conductive elliptical flaw. He distinguished the material stress, which represents the force transmitted per unit area between neighboring elements of material, from the Maxwell stress field. At electric conductors adjacent to the boundary surface of the dielectric material, the normal components of the Maxwell stress cause the tractions, which induces electrostriction and material stresses (McMeeking, 1987). McMeeking (1987) believed that material stresses cause deformation, yielding, and cracking of the material. His results show that the material stresses have a l / x / 7 singularity at the crack tip. Due to the electrostriction, a remote electric field E ~ acting on a conductive crack with a length 2a located at ( - a , a) on the xl-axis generates a Mode I stress intensity factor, Kt ,~ v / [ x # : r a / ( l -
(8.1)
where x, #, and v are, respectively, the dielectric constant, shear modulus, and Poisson's ratio of the dielectric material. Using the relationship between the stress intensity factor and the energy release rate for plane strain, J -- K~(1 - v)/2#, and Eq. (8.1), we have 1 J - -~xrca(EC~) 2-
1 F~'K O ' . -~K
(8.2)
Equation (8.2) is a special case of Eq. (5.27), which indicates that in terms of the energy release rate, McMeeking's approach is equivalent to the linear approach introduced in Section V.
238
Tong-Yi Zhang et al.
Considering the electrostriction effect in an isotropic paraelectric material, Suo (1991) introduced the isothermal mechanical enthalpy per unit volume, which took the following form
w(D,
(Y ) = -- ~
O'j i O'ij -- ~1O+' m m PfYnn
-glammDnDn] -t-
--
0[(1 + 0)O'ji D i D j (8.3)
1 ~f ( A ) d A ,
where ~) is the electrostrictive coefficient, 0 is a dimensionless number, and D -(Dn Dn) 1/2. The function f gives the nonlinear dielectric response in the absence of stress, (8.4)
E = f(D).
The relationship between E and D may be simplified as three piecewise linear functions (Hao et al., 1996)
D-
xE, ]E] < Es, Ds, E > Es, -Ds, E <-Es,
(8.5)
where Es denotes the characteristic electric field at which nonlinearity becomes important, and Ds is the saturated electric displacement associated with Es. Gong (1994) and Hom and Shankar (1994) used the following D-E relationship in finite element calculations: E--f(D)--
ESln[I+D/Ds] -21 - D/Ds
"
(8.6)
Differentiating Eq. (8.3) with respect to (Yij and Di gives the constitutive equations Eij -- [(Yij -- I)(Ymm6ij/(1 -Jr- v)]/2#
+ [(1 + q ) D i D j
Ei -- -2[(1 + gl)gijDj - glammDi] + D i f ( D ) / D ,
- qDnDn6ij]O,
(8.7)
where r is the Kronecker delta. In Eq. (8.7), strains are linear in stresses and quadratic in electric displacements. The electric field is nonlinear in electric displacements, but linear in stresses. Using the constitutive Eqs. (8.5) and (8.7) and following the crack nucleation model of small-scale saturation proposed by Suo (1993) and Yang and Suo (1994), Hao et al. (1996) evaluated the Mode I stress intensity factor at an electrode edge,
Fracture of Piezoelectric Ceramics
239
where the electrode is taken to be a plane of vanishing thickness. The stress intensity factor, which may nucleate cracking from the electrode, is given by KI -- A
2(1 + v ) # y s K E Es
,
(8.8)
where Ys is the strain associated with Es. For a paraelectric material, Es may be selected as the electric field at which the electric displacement is almost saturated, Ys is defined as the saturated electrostrictive strain, and A is a dimensionless parameter. The analytic estimations of Eq. (8.8) have been confirmed by nonlinear finite element analysis (FEA) (Gong and Suo, 1996). The FEA results indicate that the crack nucleation condition deduced from the small-scale saturation is still approximately valid when the applied electric field approaches the saturation value. Various nonlinear constitutive equations have been proposed for ferroelectric ceramics (Chan and Hagood, 1994; Hwang et al., 1998; Huber et al., 1999). For ferroelectric ceramics, one must consider domain switching, which is discussed in detail in the next section. B. DOMAIN SWITCHING
As described in Section I, spontaneous polarization generates electric domains in ferroelectric ceramics. Due to the crystalline anisotropy, the direction of the spontaneous polarization of a domain is along one of the polar axes. Under an applied electric field, a domain can switch its orientation from one polar axis to another to align its orientation as close as is possible to the applied field. When the applied electric field changes its direction, reverse switching occurs, which causes the polarization hysteresis loop and the strain butterfly loop, as shown in Fig. 2. Many researchers developed domain switching based models to describe these nonlinear loops (Hwang et al., 1995; Arlt, 1996; Chen et al., 1997; Chen and Lynch, 1998; Hwang et al., 1998; Kim, 1998; Fan et al., 1999; Huber et al., 1999; Lu et al., 1999). Switching a domain by 180 ~ causes little or no mechanical distortion because the spontaneous strain has about 180 ~ symmetry. The spontaneous strain, whose definition in terms of the lattice constants is presented later, is a measure of the lattice distortion induced by the spontaneous polarization. Thus, 180 ~ domain switching is activated primarily by the electric field. However, 90 ~ domain switching is another matter. It involves changing the orientation of the spontaneous strain by about 90 ~ and inducing a high local strain field. Both applied mechanical and applied electrical fields can cause domains to switch by 90 ~ which in turn produces internal stress fields. The induced internal stress field may help or retard the applied mechanical field to fracture a piezoelectric sample.
Tong-Yi Zhang et al.
240
If the internal stress is high enough, the switching-induced stress can damage the sample by itself. If the 90 ~ switching occurs in the vicinity of a mechanically or electrically loaded crack tip, the switching-induced internal stress field may shield or antishield (depending on the nature of the internal stresses) the crack tip from the applied loads, resulting in switching toughening or switching weakening. Switching toughening may be regarded as a kind of phase-transformation toughening (McMeeking and Evans, 1982) because polarization switching in ferroelectric ceramics is a twinning process. The stress intensity factor criterion and the stress criterion are widely used failure criteria in domain-switching models. Both criteria are purely mechanical. Polarization switching is controlled by kinetics. However, thermodynamic approaches can be used to establish switching criteria and calculate the driving force of the polarization switching (Arlt and Pertsev, 1991). We briefly introduce the switching criteria used by Hwang et al. (1995) and Yang and Zhu (1998a) in the following. A 180 ~ switching, where stresses are irrelevant, is activated if
Ei A Pi >__2Ps Ec,
(8.9)
where Ei and A Pi are the electric field strength and the polarization switch vectors, respectively; Ps is the magnitude of the spontaneous polarization; and Ec is the absolute value of the coercive field. The 90 ~ switching criterion includes the work done by the mechanical stresses,
(Tij AEij -3t- Ei A Pi >_ 2 Ps E~.
(8.10)
We may use the absolute value of the spontaneous strain, es
co - a0
--
ao
,
(8.11)
where co and ao are the lattice constants of a ferroelectric phase, to normalize the changes in strains as Aeij = 8sgij. Then, we can define a stress-type criterion for 90 ~ switching
o'ijei---f
P~Ec ( >
-
o c
~
"
es
2
Ei APi ) E~
P~
'
(8.12)
where a~ is called the stress threshold, which is a function of the electric field (Yang and Zhu, 1998a). Note that 90 ~ switching may occur without the presence of external electric fields. Lynch et al. (1995) proposed a simple domain-switching model for electrically conductive and electrically impermeable cracks on the basis of steplike switching and by ignoring the effect of stress on the electric field. The electric field in
Fracture of Piezoelectric Ceramics
241
front of the crack is high enough to cause domain switching. The switching zone is determined completely by the electric field, and the switching occurs when E > Ec. In the absence of stress, the switching strain is zero when E < Ec, or es when E > Er Thus, an internal stress field is generated due to the switching. Then, the internal stress field contributes to the driving force or resistance in the crack propagation. This model captures the essence of the switching behavior and gives insight into the source of the stress field that drives the crack growth. In the phase transformation-toughening theory, a transformation wake shields the crack tip stress intensity factor. By analogy with the phase-transformation toughening theory, Yang and Zhu (1998a) studied switching toughening. The stress intensity factor induced by 90 ~ switching may be expressed generally as AK = -r/f2,
(8.13)
Yes2 r/= ~ , P~ Ec
(8.14)
where Y is a generalized Young's modulus and f2 is related to the electrical and mechanical loads as well as the domain orientation prior to switching. Yang and Zhu (1998a) studied switching toughening for monodomain and multiple-domain ferroelectrics based on two basic assumptions: the stress-assisted switching zone is confined within the specimen, and the electric field is uniform. They calculated the value of f2 similar to that proposed by McMeeking and Evans (1982). When the stress intensity factor induced by 90 ~ switching is available, the local stress intensity factor is the sum of the applied intensity factor plus the induced one; that is, Kl= K a + AK,
(8.15)
where the superscripts 1 and a denote local and applied, respectively. When the local stress intensity factor exceeds the critical value Kc; that is, K l >__Kc,
(8.16)
the crack grows. Equation (8.16) is the commonly used failure criterion in the domain-switching model. Using the domain-switching model, Yang and Zhu (1998b) and Zhu and Yang (1999) provided a mechanistic explanation for the electric field-induced fatigue crack growth. They considered an electrically impermeable crack, which induced electric singularity at the crack tip. The high electric field drives domains to switch and, consequently, the domain switching generates an internal stress field. A cyclic electric load causes a domain switching sequence that generates a cyclic internal stress field, which mechanically fatigues ferroelectric ceramics. In this case, the
242
Tong-Yi Zhang et al.
term f2 in Eq. (8.13) is induced completely by the applied electric field, and the crack is driven solely by AK. Similarly, microcrack nucleation may also occur during repeated domain switching, which degrades the electric properties of ferroelectric ceramics (Jiang et al., 1994a, 1994b; Zhang and Jiang, 1995). In general, a domain-switching model requires the development of new constitutive equations and extensive numeric calculations. Recently, Huber et al. (1999) developed a nonlinear constitutive model for ferroelectric polycrystals under a combination of mechanical stress and electric field. They used a self-consistent analysis, as an extension of the self-consistent crystal plasticity scheme of Hill (1965a, 1965b) and Hutchinson (1970), to address ferroelectric switching and estimate the macroscopic response of tetragonal crystals under a variety of loading paths. In a qualitative way, this model captures several observed features of ferroelectrics, such as the shapes of the dielectric hysteresis and the butterfly loops, a Bauschinger effect under mechanical and electrical loads, and the depolarization of a polycrystal by compressive stresses (Huber et al., 1999).
C. DOMAIN WALL KINETICS MODEL
Polarization domain switching may be regarded as a result of domain wall motion. Studying domain wall kinetics (DWK) can bring insight to the mechanisms of failure processes in piezoelectric ceramics. In addition, the use of the DWK model can greatly simplify the mathematics involved in domain switching modeling. Under applied electric and mechanical fields, domains with low energy orientation grow and domains with high-energy orientation shrink as a result of domain wall motion. The displacement of 90 ~ domain walls causes a shear deformation. Thus, both domain wall displacement and volume deformation affect the material properties of ferroelectric ceramics (Herbiet et al., 1989). Arlt and Pertsev (1991) evaluated the force constant and effective mass of 90 ~ domain walls in ferroelectric ceramics. Domain size and grain size are involved in their model, which links the microstructures to the macroproperties of ferroelectric ceramics. The domain wall motion model explains the internal friction and the dielectric dispersion of ferroelectric ceramics (Pertsev and Arlt, 1993) quite well. Fu and Zhang (2000b) proposed a DWK model to explain the effects of temperature and electric field on the bending strength of PZT-841 ceramics. Either an applied stress or an electric field can drive domain walls to move. The equation for the motion of a domain wall under the electric and mechanical fields can be written as (Arlt et al., 1987) ~,ti + ~:u -- fM + fE,
(8.17)
Fracture of Piezoelectric Ceramics
243
where y zi and ku are, respectively, friction and clamping forces; u is the displacement; and fM and f e are the configuration forces exerted on the domain wall by mechanical and electrical fields, respectively. The inertial term has been neglected in Eq. (8.17) for static loads, monotonic loading at a low loading rate, or cyclic loading with a low frequency because of the light mass of the domain wall. For a cyclic load f = foe i~ the solution to Eq. (8.17) is
e
f o ~i Wt
u =
^
k + ioJy
,
ti=
. z" .,i wt iwJOe
,,
k + icoy
.
(8.18)
The corresponding damping factor is _ wy
(8.19)
For a static load, after a period of time, At, the displacement and the average velocity of the domain wall become
u=--fo k -
2
, (8.20)
2
li---~tf~
2u ~
A----~'
where 2 - 1 - e x p ( - A t / f ) , f -- V/~:, and u ~ is the equilibrium displacement. It takes only several f for a domain wall to complete its movement from its initial position to its final position under a static load. When the time period is larger than a few times f, 2 ~ 1 and u ,~ u ~.
1. Temperature Dependence of Compliance We shall consider the motion of a 90 ~ domain wall. Because a 90 ~ domain wall is actually a twin boundary, a local shear deformation, etocat = esu/d, is associated with the domain wall displacement u (Arlt, 1990), where d is the distance between domain walls and es is the spontaneous strain. This local deformation is accommodated by the neighboring medium, resulting in a global deformation UCX~
eD -- ~es d '
(8.21)
where/~ is a proportionality factor linking the local deformation to the global strain. By definition, eD -- s Dcr, where SD is the contribution of the domain wall displacement to the compliance. We then have uCX~
~es --~ -- SDCr.
(8.22)
244
Tong-Yi Zhang et al.
Equation (8.22) links the domain wall displacement to the macroscopically mechanical behavior. Under a static mechanical load, combining Eqs. (8.20) and (8.22) leads to the force exerted on a domain wall by the applied load o fM-
~:dso
~es a.
(8.23)
If we know how u and d change with temperature, we can predict the temperature dependence of so from Eq. (8.22). There is an alternative way to determine so. Letting u = ~ao, where ao denotes the representative lattice constant and ~ is a dimensionless factor, we rewrite Eq. (8.22) as ~ e s a o / d - socrc with ac - o/~.
(8.24)
The variations of the lattice constant ao and the domain spacing d with temperature are about the same, such that the ratio of ao/d is independent of temperature. Equation (8.24) indicates that if the applied stress, as a function of temperature, required the domain wall to move the distance ao is available, we can determine the temperature dependence of so. It is widely accepted that lattice exhibits a frictional resistance, known as the Peierls-Nabarro stress, against dislocation motion (Peierls, 1940; Nabarro, 1947). Thus, we assume that the lattice also exerts a friction force against the domain wall motion, resulting in the friction term ?,ti in Eq. (8.17). When the applied load is small and the domain wall displacement u ~, is less than aomthat is, ~ < l m t h e domain wall moves elastically around its equilibrium position. As the applied load reaches the critical value at which u ~ = ao, or ~ -- 1, the domain wall begins to sweep over the lattice sites and induces damping. Comparing Eq. (8.24) with Eq. (8.22), we conclude that Crc defined in Eq. (8.24) is simply the critical stress for triggering damping. It should be possible to determine Crc from the damping behavior of the material. As observed in the experiment on PZT-841 ceramics (Fu and Zhang, 2000b), for an applied cyclic stress, there is a threshold temperature Ti at which the damping factor starts from zero and increases rapidly with temperature. The three threshold temperatures are around 41 ~ 91 ~ and 139 ~ C, respectively, for the peak stresses of 30, 15, and 7.5 MPa. This result implies that for a given temperature, there is a critical level of applied cyclic stress at which damping rises. The plot of the log of the threshold stresses against the corresponding temperatures shows a linear relationship ~c = ~o e x p ( - T/ To),
(8.25)
where ~o = 53.8 MPa and To = 70.7 ~ C are parameters determined by the linear
245
Fracture of Piezoelectric Ceramics
regression. Equation (8.26) gives the threshold stress for triggering damping as a function of T. Combining Eqs. (8.24) and (8.25) yields (8.26)
SD(T) = 6~es(T) e x p ( T / To),
where c~ = ~ao/(o'od). The domain wall motion and the volume deformation inside the domains contribute to the total elastic compliance--that is, (8.27)
s(T) = sv + SD = Sv + 6tes(T)exp(T/To),
where sv represents the contribution of the volume deformation to the compliance. Furthermore, we assume s v to be independent of temperature within the temperature range of this study. Fu and Zhang (2000b) measured the spontaneous strain as a function of temperature for PZT-841 ceramics and found es(T)=0.062
1-~c c
+ 0.170 1 - Tc
-0.142
1-
- 0.077
1-
,
(8.28)
where Tc(~272 ~ C) is the Curie temperature. Fu and Zhang (2000b) measured the total elastic compliance directly and found that the measured data fit Eq. (8.27) perfectly (Fig. 8a). Equations (8.27)-(8.28) predict the existence of a maximum compliance, which occurs at Tm= 227 ~ C. The directly measured maximums of the elastic compliance for the three applied stress levels all occur at about 225 ~ C, in good agreement with the prediction.
2. Effect of Temperature on Bending Strength
The fracture of ceramics with well-polished surfaces can be considered to be a crack nucleation-control process. This is because ceramics are very brittle, and their fracture is triggered once a crack is initiated; microcracks may be initiated at pores or grain boundaries. Rice and Freiman (1981) defined the criterion for the crack initiation at grain boundaries as ( I f + o.in - -
v/9Y?'8/g - o.8,
(8.29)
where o.f is the critical applied stress, o.in is the internal tensile stress perpendicular to the plane of the initiated microcrack, Y is the Young's modulus, ?'8 is the grain boundary fracture energy, g is the grain size, and o8 is the corresponding grain boundary strength. Equation (8.29) shows that internal stress plays an important role in initiating microcracks. Moving a domain wall embedded in a grain produces
Tong-Yi Zhang et al.
246
18 .~
17
0
0
16 O n
~ 14 O
r,.) 13
DWK model
12 50
100
(a)
150
200
Temperature
250 Tc
(~
100
~
9o
m
8o
~
7o
DWK model .
.
.
.
|
.
.
.
.
50
(b)
i
.
.
100
.
.
|
.
.
.
.
150
i
.
.
200
.
.
i
.
250
.
.
.
300
Temperature (~
FIG. 8. Temperature dependence of (a) the bending strength and (b) the elastic compliance for PZT-841. The solid curves are based on the domain wall kinetics (DWK) model. a serration at the grain boundary. The representative local tensile strain in the serration is given by (Arlt, 1990) e-
Ss u ~ d "
(8.30)
This corresponds to an internal stress of 8s u ~ O'in
svd '
(8.31)
Fracture of Piezoelectric Ceramics
247
where the volume elastic compliance, s v, is used because the serration is simply a localized volume deformation. Combining Eqs. (8.22) and (8.31), we have the following relationship trin = ~ S D t r ,
(8.32)
where X - 1/(~sv). Equation (8.32) indicates that the internal stress due to 90 ~ domain wall motion is proportional to the applied stress. Combining Eqs. (8.29) and (8.32) at the critical condition of tr = try yields
t r f ~"
trB 1 + Xso'
(8.33)
which relates the bending strength to the compliance contributed by 90 ~ domain wall motion. Equation (8.33) clearly shows that the minimum bending strength corresponds exactly to the maximum elastic compliance, indicating that the anomalous minimum of the fracture strength is a result of elastic softening due to domain wall motion. Substituting Eq. (8.26) into Eq. (8.33) yields tr8 try -- 1 + ~es(T)exp(T/To)'
(8.34)
where ~ - 6 t / ( ~ S v ) = ao/(dsvtro). Thus, the bending strength is a function of temperature. Within the temperature range of 30-272 ~ C, trB may be assumed to be independent of temperature. Thus, the bending strength varies with temperature through the term of es(T) exp(T/To). Using Eq. (8.34), Fu and Zhang (2000b) fit experimental data of bending strength for various temperatures by the least square procedure (Fig. 8b), yielding tr8 = 99.3 MPa and t? = 1.94. They estimated the values of sv and & for the three applied stress levels with the parameters/~ and d from the fitting results. Using tro = 80.7 x 106 N / m 2, t~ = 1.94, t~ -~ 23 x 10 -12 m2/N, and sv .~ 13 x 10 -12 mZ/N, together with the relations t~ = ~ao/(dtro) and t~ 6t/(~sv), Fu and Zhang (2000b) obtained/~ ~ 0.9 and ao -~ 2 • 10-3d. Taking ao ~ 4 • 10 -4 /zm gives d ~ 0.2 #m, which is in approximate agreement with the results evaluated from Fig. 1b.
3. Effect of an Electric Field on Bending Strength We determine the influence of an applied electric field on the bending strength. Miller and Savage (1959) demonstrated that the velocity of the 180 ~ domain wall
248
Tong-Yi Zhang et al.
motion in BaTiO3 single crystals depends on the electric field and is given by the empirical equation {~ = Vo exp(1 - Eo/IEI),
(8.35)
where Eo is the activation field strength and Vo is the domain wall velocity under Eo. We assume that Eq. (8.35) also holds for 90 ~ domain walls. We may express Vo by V o - Uo2/ZXt, where Uo is the equilibrium displacement of a domain wall under the activation field. If only an electric field is applied, combining Eqs. (8.20) and (8.35) gives f e -- kuo exp(1 - Eo/IEI),
(8.36)
which is the force exerted on a domain wall by an applied electric field E. Substituting Eq. (8.24) and Eq. (8.36) into Eq. (8.20) with f/ = fM + fE, and )? ~, 1 and u ~ u ~, we have u ~ - SDdo-/(fles) + Uo exp(1 - Eo/IEI).
(8.37)
Substituting Eq. (8.37) into Eq. (8.31) leads to O-in -- S D O - / ( ~ S v )
"[- O'E,, exp(1 - Eo/IEI),
(8.38)
where O-eo -- esUo/(Svd) is the equivalent internal stress induced by the activation field of Eo. Substituting Eq. (8.38) into Eq. (8.29) yields O-f =
O-8 - O-e,,exp(1 - Eo/IEI)
1 +)~SD
.
(8.39)
Equation (8.39) gives the fracture strength as a function of the applied electric field. For simplicity, we assume )~so to be independent of the applied electric field. Taking O-8 = 99.3 MPa, Fu and Zhang (2000b) fit the experimental data (see Fig. 17) and found that O-eo - 35.7 MPa and Eo = 14.2 kV/cm for the positive fields, and O-eo = 22.8 MPa and Eo ---=14.0 kV/cm for the negative fields. It is interesting to note that the activation field under positive electric loading is almost the same as that under negative electric loading. Equations (8.27) and (8.39) give the elastic compliance and the bending strength as a function of temperature and electric field. Equation (8.39) shows that the anomalous minimum of the fracture strength is directly related to the inelastic relaxation of the 90 ~ domain wall motion. In this model, the internal stress induced by the domain wall motion is the dominant mechanism causing the strength degradation observed in the experiments. An internal stress field develops when
Fracture of Piezoelectric Ceramics
249
a 90 ~ domain wall is moved by mechanical and/or electrical loads. This internal stress field, in turn, assists the applied load/electric field to fracture the sample. Because the internal stress field varies with temperature and electric field, the bending strength also depends on temperature and electric field. Using Eqs. (8.27) and (8.39), one can estimate the parameters used in the model by regression of the experimental data. The parameters obtained by Fu and Zhang (2000b) given in the previous paragraph seem to be reasonable. The DWK model is essentially the same as the domain-switching model. Both models consider the internal stress field induced by applied mechanical and electrical loads. Depending on its nature, the internal stress field may assist or resist the applied loads to fracture the samples. For bending smooth samples, the failure initiates at locations of high total tensile stress. Grain boundaries, domain boundaries, and other defects are the potential locations for the initiation of cracks.
D. POLARIZATION SATURATION MODEL Gao et al. (1997) proposed a strip polarization saturation model to examine the electrical yielding effect on the fracture behavior of electrically insulating cracks in piezoelectric ceramics under combined electrical and mechanical loading. In the strip polarization saturation model, piezoelectric ceramics are treated as mechanically brittle and electrically ductile materials. This saturation model is analogous to the classic Dugdale model (1960). The crack tip is completely shielded electrically by a polarization saturation zone in the saturation model. As a result, the local energy release rate is only mechanical in nature (Gao et al., 1997). Applying the Griffith theory yields the fracture criterion, jl > j1_
2y,
(8.40)
where J is the J integral, the superscript l denotes local, and y is the specific surface energy. To emphasize the physical insight, Gao et al. (1997) considered a simplified piezoelectric material to make the derivation process straightforward. Subsequently, electrically impermeable and conductive cracks with the constitutive equations given by Eq. (2.41) have been investigated (Gao and Barnett, 1996; Fulton and Gao, 1997, 1999; Ru, 1999; Ru and Mao, 1999; Mao et al., 2000; Wang, 2000). We briefly discuss the analysis of an electrically impermeable crack. Figure 9 shows a semi-infinite crack lying on the negative xl axis in an infinite piezoelectric medium. The applied stresses and electric displacements are expressed in terms
Tong-Yi Zhang et al.
250
a (~) Km
t
--~ K,]
1' K~~
X2
D2=Ds c
-x, I
Saturation s t r ~
FIG. 9. Schematic depiction of the strip saturation model for a semi-infinite crack.
of the intensity factors K a - (K']I K'] K']II K~) r. For the electrically impermeable crack, the extended traction vanishes on the crack facesmthat is, 2~2 - - O,
for xl < O.
(8.41)
In the strip electric saturation zone 0 < x~ < c, the boundary conditions are D2 -- D s ,
(8.42)
f o r 0 ~ Xl ~ c ,
where Ds is the saturation electric displacement. The Green function for an electrically impermeable crack of a finite length 2a is given by Eqs. (4.42a-b). We rewrite the solution in the form below 1 fl
V/Z2
=
(
Z~ + v/Z2~ - a 2
a 2 Z~+v/Z 2 - a 2 - z ~a - V/ (z~) 2
1 Lik 1Lkj I _ k=l ./Z 2 __ a 2 V"
__
a 2
)>tq+ Z, li[l,
4
Zij,1 = Z
X
(
02
)
m
a2-
(Zi + v / z 2 - a 2 ) ( z J
+ v/(-Z~j)2-a 2)
(8.43a)
f,2 --
(( z2 a2
4
Fracture of Piezoelectric Ceramics
z + z2a2/
z~ + , / z 2 . 02.
~
a2
q + Z,2(I,
Pi
Zij,2 -- Y~ L~ 1Lkj /
.,
(8.43b)
( X
z~ . . V(z~) 2
)>
251
02
)
~
a Z - (zi + V/z~-aZ)(z~ + v/(-~j) 2
m
a 2)
where i, j = 1, 2, 3, 4, and i and j are not summed; and the vector q is given by Eq. (4.16) for a general problem. For the current problem, the extended force, F = 0 , and the extended Burgers vector, b = (0 0 0 A4~)r, we have
q _ ~/A~b(g41
L42 L43 L44) r -
2rriA4~L4r
(8.44)
and L4i, i = l, 2, 3, 4, are the components of the eigenvector matrix L defined by Eq. (3.16). Consequently, we shift the origin of the coordinate system to the right crack tip and then let the crack length approach infinity. The solution for the interaction of a dislocation with a semi-infinite impermeable crack becomes
(
,
>
r~2 - Lfl + E l l - L 2~/g2-~(V~-2~_v/~d) q + L Z l r t -+-L(
Zij,1 -- - Z
4
k=l
1
> (t + LZ~q,
1
L~l Lkj
2v~i(V/~_7 + c-~_~/zj)'
~[~1 -- - - E l 2 - E l 2 -- - L
2~~(~//~-
~d)
~z/~d) ( t - LZzq,
4 Zij,2 -- - Z k=,
(8.45a)
L~I
Lkj
Pi
2v/-~7(v/~-7 + ~-v/z~)'
where i, j = 1, 2, 3, 4 and i and j are not summed.
q -- LZ'21~
(8.45b)
Tong-Yi Zhang et al.
252
If the electric dislocation is located on the xz axis, the electric displacement along the Xl axis calculated from Eq. (8.45a) has the compact form
D2 = ~f~(x, -- x~) (L4q + L4q).
(8.46)
Substituting Eq. (8.44) into Eq. (8.46) yields
~l a A~b (L4L4r _ L4L4r) D2 - ~ ( X l - xla) 2:ri
(8.47)
From Eq. (3.15), we have L4 LT - - L 4 L T. Hence, Eq. (8.47) can be further reduced to D2
--
~la A~bL4LT. ~ - 7 ( X l - Xla) 7ri
(8.48)
The intensity factors produced by the electric dislocation located anywhere are defined as
(K d
K]
Kdl
KdD)T = lim(L(v/2n'z,,)fl+L(v/2n'~--d}~,).
(8.49)
Zu---*0
Substituting Eq. (8.45a) into Eq. (8.49) yields
(K d K d KdI KdD)T = --ff~-~ L ~ d
q-t-L
.
(8.50)
If the dislocation is located on the Xl axis, Eq. (8.50) reduces to
Kd KdI Kd)T = (Kd Kd KdI Kd)T = (K d
a [Lq + L(t].
(8.51)
Substituting Eq. (8.44) into Eq. (8.51) leads to
Kai Kffll Kao)r _ i A4) [LL4r _ LLr] _ i
A ,/5 LL4r.
(8.52)
As given by Eq. (3.93) and discussed in Section III, the mechanical and electrical fields near a crack tip can be expressed in terms of the intensity factor vector. For
Fracture of Piezoelectric Ceramics
253
an electrically impermeable crack, Eq. (3.93) can be simplified as
E2 = 5
L v/2zrz,~ I'-I + 1" v/2ZrT~
(8.53)
Thus, the electric displacement on the xl axis under the applied loads is dependent only on the applied intensity factor of electric displacement and is given by
(8.54)
D2 -- ~ .
~/2zrxl
Let f(x' l) be the distribution function of the electric dislocations such that the number of dislocations located in the interval dx'1 at x'~ is f(x' 1)dx'1. The boundary condition in which the electric displacement D2 equals the saturation value Ds in the polarization zone is described by
K---a--P~~ - f 0 c ~/27rXl ~- B
J
f(xl)~/~l l / dx S(x,
-
x'l)
'1 --
Ds
(8.55)
,
where B -- AqOL4L~/(7ri) is a constant related to the Burgers vector. The uniqueness of a distribution f(x' l ) with zero value at x'l = 0 and c requires that
Kao -- 2 V/-2Ds v/-~/ ~/-~.
(8.56)
Thus, the solution to Eq. (8.55) is
f (x l) -- - ~
In
l
(8.57)
"
The electrical dislocations in the saturation zone produce stress intensity factors and intensity factor of electric displacement at the crack tip, which are given by
(K~
K]
K~t
T fo c f(x'l)dx'= K d o ) T - - i xAq~X~LL4 /~ ~1
LL~ L4L].K~.
(8.58)
The vector of the local intensity factors is the sum of the intensity factor vectors induced by the applied field and the dislocations--that is, K t = K a + K d.
(8.59)
254
Tong-Yi Zhang et al.
Combining Eq. (8.58) with Eq. (8.59) yields LlL4r ( K~t - ~rL4L K ~4
g~
K~III
Klall
L2L4v
a
L4LTKD
(8.60)
L3L4T a L4L T KD 0
2
where Lj is the jth row of L. Equation (8.60) indicates that the electric displacement at the crack tip is completely shielded by the saturation zone. Ru (1999) and Wang (2000) obtained results equivalent to Eq. (8.60) from a different approach. In their results, the ratio L/L4r/(L4L4T) is replaced by a ratio [(B + 1~)-1.]i4/[(B-+- ]~)-1144, i = 1, 2, 3. Using the relation given by Eq. (3.15), we can prove that the two ratios are the same. As stated in Eq. (3.109), the local energy release rate can be expressed in terms of the intensity factors as jl
=
(Kt) r (B + 4
I~)KI.
(8.61)
Substituting Eq. (8.60) into Eq. (8.61) yields
jl
:
K~
-
L1L4r a L4L----~K D
a
g l
L2L~ L4L~ K~9
L3L~ ) K~I -- L4L----~K~9
L4LTK~9 m
(B1 + B1)
g~
L2L4T a
L4LTKD
(8.62)
L3L4y K a
k g ~ l l --
t4t-----~ Dj
where Bl is the 3 x 3 upper left block in B, as introduced in Eq. (3.22). When the local energy release rate is adopted as a failure criterion, it yields a linear relationship between the applied mechanical load and the electric field (Gao and Barnett, 1996). In this case, one may also use the local intensity factor as a failure criterion, which predicts the linear relationship between the applied load and electric field as well (Wang, 2000).
Fracture of Piezoelectric Ceramics
255
IX. Experimental Observations and Failure Criteria In the literature, there are a number of experimental studies concerning the fracture behavior of piezoelectric ceramics under purely mechanical loading. However, experimental reports on the subject of combined electrical and mechanical loading are limited. In this section, we provide an overview of the experimental observations of the fracture behavior of piezoelectric ceramics. Along with the experimental results, we introduce failure criteria. A. EXPERIMENTAL OBSERVATIONS
1. Effects of Microstructure and Temperature In 1988, Pohanka and Smith presented an overview of the fracture and strength of piezoelectric ceramics under purely mechanical loading. Typical values of Ktc for commercially available piezoelectric and dielectric ceramics, such as barium titanate and lead zirconate titanate (PZT), range from 0.8 to 1.7 MPa 4'-~. The fracture strength depends on temperature and the grain size and composition of the material. For barium titanate, the fracture energy at room temperature is a function of grain size and varies from about 3 to 12 J~ m 2, whereas the fracture energy at 150 ~ C (above the Curie temperature of 130 ~ C) is almost independent of the grain size with a value of about 3 J / m 2. The effect of the microstructure on the enhancement of the fracture energy is attributed to twinning toughening and microcracking toughening. The maximum fracture energy occurs when the grain size is around 40 #m, at which point the contribution to toughening due to twinning and microcracking is balanced by a linkup of microcracks (Rice and Freiman, 1981). In PZT, Kzc depends also on the Zr/Ti ratio. The minimum K IC occurs at the morphotropic boundaries between phases of different crystal structures due to the reduction of microcracking toughening, where the piezoelectric coefficients are at the maximum (Freiman et al., 1986). Both the fracture strength and the fracture toughness of piezoelectric ceramics are sensitive to temperature. Cook et al. (1983) carried out controlled flaw tests on unpoled BaTiO3 ceramics of nominal grain size (7 #m). The flaws were introduced at room temperature by indentation with a load of 30 N, which produced cracks well in excess of the grain size. Then, four-point bending tests were conducted in a heated oil bath at temperatures ranging from room temperature to above the Curie point. The results show that the fracture toughness at each temperature during heating is the same as that during cooling. The fracture toughness decreases with
256
Tong-Yi Zhang et al.
increasing temperature between room temperature and the Curie point, indicating the existence of intrinsic thermal effects in the toughness parameter. Mehta and Virkar (1990) reported similar results for unpoled PZT samples, in which the fracture toughness decreased from a maximum 1.4 MPa4Fm at room temperature to about 1.0 MPav/-~ at 500 ~ C, far above the Curie point of 350 ~ C. They examined domain switching under electrical and mechanical loading using x-ray diffraction and attributed the observed temperature dependence to the toughening effect of 90 ~ domain switching. If one measures the fracture toughness at temperatures near the Curie point, one may find the minimum fracture toughness at a temperature just below the Curie point. Zhang et al. (1993) observed that the minimum bending strength and the minimum fracture toughness of both poled and unpoled PZT-4 ceramics are near the Curie point. The similar phenomenon was also observed for PBZT and PZTNV- 1 ceramics (Kramarov and Rez, 1991 ). The reasons that Cook et al. (1983) and Metha and Virkar (1990) did not find this behavior may be a result of the fact that, in their experiments, the increment in temperature near the Curie point was too large. Fu and Zhang (2000b) conducted three-point bending tests on PZT-841 ceramics at 25, 122, 220, and 268 ~ C to measure the temperature dependence of the bending strength. Twenty samples were tested at each temperature. In the tests, the poling direction was parallel with the jig surface, in which configuration the maximum tensile stress induced by bending was perpendicular to the poling direction. The bending strength exhibits valley-shaped floors at a temperature below the Curie point, as shown in Fig. 8b. The bending strength decreases from 97.8 MPa at room temperature to 85.9 MPa at 122 ~ C, and further decreases to 74.8 MPa at 220 ~ C, then increases to 90.5 MPa at 268 ~ C. Figure 8a shows that the elastic softening, which accompanies the bending strength reduction, has a peak at the same temperature, where the bending strength is at its minimum. Note that the smooth curves in Figs. 8a and Figs. 8b are plotted using Eq. (8.18) and (8.25), respectively, indicating that the experimental results were explained well by the domain wall kinetics model.
2. Effects of an Alternating Electric Field
The effects of an alternating electric field on crack initiation and growth have long been a focus of interest in the study of the reliability of piezoelectric ceramics. Four major mechanisms for electric cycling damage have been identified experimentally: aging (Robels and Arlt, 1993; Warren et al., 1996), ferroelectric fatigue (Duiker et al., 1990; Warren et al., 1995), cracking (Winzer et al., 1989; Uchino, 1997) and dielectric breakdown (Desu and Yoo, 1993; Chen et al. 1994). Here,
Fracture of Piezoelectric Ceramics
257
we focus on cracking and the effect of pores, flaws, and cracking on ferroelectric fatigue. Ferroelectric fatigue is characterized by the loss of switchable polarization with repeated polarization reversals. Although no cracking is involved in the definition of ferroelectric fatigue, pores and flaws may play important roles. McHenry and Koepke (1983) observed that in unpoled and poled Navy-III-type PZT ceramics subjected to deadweight loading in a double torsion mode, crack propagation was enhanced by the application of a static or an alternating electric field perpendicular to existing cracks. In the poled PZT, the applied electric field always "turned" the crack in the direction opposite to the poling direction. Furuta and Uchino (1993) observed crack initiation and propagation near the internal electrode tip in multilayer piezoelectric actuators during cyclic loading. Aburatani et al. (1994) studied the failure mechanisms in ceramic multiplayer actuators by three simultaneous observations: (1) visual observation with a charge coupled-device microscope; (2) fieldinduced strain measurement; and (3) acoustic emission measurement. They found that during cyclic electric loading, cracks initiate from the edges of internal electrodes and propagate obliquely to other electrodes in piezoelectric samples. Jiang and Cross (1993) showed that ferroelectric fatigue failure occurred in low-density (93-97%) lanthanum-doped PZT (PLZT) ceramics after 104 switching cycles, whereas the high-density (> 99%) PLZT specimens of the same composition did not fail after 10 9 switching cycles. For low-density ceramics, the ferroelectric fatigue rate, defined as the loss rate of switchable polarization with switching cycles, was also much higher than that for high-density ceramics. This indicates that the porosity is one of the key factors affecting ferroelectric fatigue behavior. Microcracks may be a reason for the reduction of remnant polarization (Carl, 1975; Kim and Jiang, 1996). Jiang et al. (1994b) conducted ferroelectric fatigue tests on hot-pressed finegrain PLZT. All the specimens with conventionally cleaned surfaces showed significant fatigue after 105 switching cycles, but specimens cleaned with a new cleaning procedure did not show fatigue, even after more than 108 switching cycles. This type of fatigue was found to be due to microcracking generated at the ceramicelectrode surfaces. White et al. (1994) also observed microcracks induced by an alternating electric field at the resonant frequency of the material in PZT specimens precracked by indentation. At temperatures higher than 150 ~ C, the microcracks were dispersed in small clusters, whereas at temperatures below 86 ~ C, microcracks were generated in a densely populated region near the indentation site. Jiang et al. (1994a) investigated the effect of composition and temperature on ferroelectric fatigue of PLZT ceramics. Their results show that at temperatures higher than the temperature where the dielectric constant is the maximum, no
258
Tong-Yi Zhang et al.
fatigue effect is detected. They also found that compositions of rhombohedral symmetry exhibit little or no sign of fatigue in comparison to compositions of tetragonal and orthorhombic symmetry. Compositions close to the phase boundaries display significant fatigue behavior. Electric fatigue arises from the pinning of domains or from microcracking. Hill et al. (1996) studied the effect of mechanical cycling (four-point bending) and electrical cycling on the degradation of the mechanical properties of PZT-8 bars. Microcracks were found to originate from second-phase material located at triple junctions. High intergranular microcrack density was observed in the mechanically cycled samples and in samples electrically cycled at temperatures of 80 ~ or lower. Samples electrically cycled at 180 ~ showed much lower microcrack density. Hill et al. (1996) believed that elevated temperatures (,~,180 ~ are necessary to cause depolarization in PZT-8 under resonance-induced stresses. With precracks introduced by indentation, steady crack growth was observed perpendicular to the applied field in both PZT and PLZT ceramics under alternating electric fields larger or smaller than the coercive field (Cao and Evans, 1994; Lynch et al., 1995; Zhu and Yang, 1998; Tajima et al., 2000). Cao and Evans (1994) and Lynch et al. (1995) found that electric fatigue was characterized by step-by-step cleavage. Tobin and Pak (1993) showed that fatigue crack growth took place even at field amplitudes as low as 5% of the poling field (e.g., ~0.83 kV/cm). Jiang and Sun (1999b) investigated the fatigue behavior of PZT-4 ceramics using prenotched compact tension specimens and two types of loading. In the first type of loading, the voltage was kept constant while a tension-tension cycling mechanical load was applied. In the second type, the specimen was under a constant tensile load while a time-varying electric field was applied. The results illustrate that the magnitude and direction of the electric field influence the crack growth rate significantly. Jiang and Sun proposed that the mechanical and electrical loads can be combined into a single parameter, and that the mechanical strain energy release rate alone should be used to characterize fatigue crack growth. They fit the fatigue data with a law similar to the Paris law, wherein the mechanical strain energy release rate replaced the stress intensity factor. Xu et al. (2000) reported an in situ transmission electron microscopy (TEM) study of the effect of cyclic electric field on microcracking in a single crystal 0.66Pb(Mgl/3Nb2/3)O3--0.34PbTiO3 ferroelectric ceramic. They observed microcracks initiated from a fine pore under an applied alternating electric field. The microcracks laid on { 110} planes, which are common domain boundaries in ferroelectric materials. The pore was also distorted by the alternating electric field. Tan et al. (2000) confirmed that the electrically induced crack growth in the <001 >-oriented piezoelectric Pb(Mgl/3Nb2/3)O3--PbTiO3 single crystal was along
Fracture of Piezoelectric Ceramics
259
the 90 ~ domain boundary. Under an alternating field with a peak value of 6.5 kV/cm at a frequency of 0.3 Hz, a crack extension of 2.1/zm was produced after 20 electric cycles, yielding an average crack growth rate of 10 .7 m/cycle. Tan et al. (2000) observed that cracks can grow under a static electric field of 10 kV/cm. It seems that the sample holder in the TEM for in situ observations (Tan et al., 2000; Xu et al., 2000) might have constrained the mechanical displacement of the samples. Thus, in addition to other mechanisms that are still under investigation, the constraint could have induced a stress field through the piezoelectric effect to drive the crack propagation. Nuffer et al. (2000) studied the damage evolution in commercial bulk PZT ceramics induced by bipolar cycling. Polarization, strain hysteresis loops, and acoustic emission were monitored in the experiments. They found that higher cycling fields yield stronger ferroelectric fatigue and higher acoustic emission energy, and that the threshold for the onset of acoustic emission events is lower at high cycle numbers. They suggested that the bipolar cycling led to the agglomeration of point defects, which clamped domain walls, therefore reducing the number of switchable domains. Under electric cycling, fewer and fewer domains can be switched due to the coalescence of point defects; thus, the remnant polarization decreases, resulting in ferroelectric fatigue. Clearly, all the experimental results demonstrate that an alternating electric field can damage piezoelectric ceramics, and the damage mechanisms are still under investigation. 3. Effects of a Static Electric Field
The fracture behaviors of piezoelectric ceramics under combined static mechanical and electrical loads have been studied using indentation induced fracture, three- or four-point bending, and fracture of prenotched compact-tension (CT) specimens. Tobin and Pak (1993) conducted indentation tests on PZT-8 ceramics in a static electric field. Cracks normal and parallel to the electric field (the poling direction) were induced, and crack lengths were measured from the comer of the impression to the crack tip using either optical micrographs or a calibrated eyepiece. Figure 10 shows the average crack length and the associated standard deviation of 10-15 separate indents under a load of 4.9 N. Analysis of the standard deviations of the experimental data suggests that the variation in the crack growth due to the electric field application is statistically significant. For example, the average crack length was 21.50 #m under a negative electric field o f - 4 . 7 kV/cm and the associated standard deviation was 4.82 #m, leading to a relative error of 22.4%. For cracks normal to the applied electric field, Tobin and Pak (1993) found
Tong-Yi Zhang et al.
260 60
9 O
For cracks perpendicular to poling For cracks parallel to poling
50 ::t.
40 30 20 10 |
-6
-4
.
i
-2
.
,
0
.
|
.
2
|
4
6
Applied Field (kV/cm) FIG. 10. Effects of static electric fields on the crack length of PZT-8 under an indentation load of 4.9 N (with the experimental data by Tobin and Pak, 1993). that cracks under a positive electric field (same as the poling direction) were longer than those under a negative field (opposite to the poling direction). This indicates that a positive field assists the applied mechanical load in propagating the crack, whereas the negative field retards the crack propagation. The positive electric field of 4.7 kV/cm produces a crack normal to the poling direction almost 70% longer than the crack for zero applied electric field. In the mean time, the crack under a negative electric field of - 4 . 7 kV/cm is about 30% shorter than the crack under zero applied electric field. The electric field does not appear to have much effect on the propagation of cracks parallel to the poling direction. Wang and Singh (1997) observed the opposite trend in the indentation fracture tests of PZT EC-65 ceramics under load ranging from 4.9 to 11.76 N. They found that for all indentation loads, a positive field of 5 kV/cm generated shorter cracks than the negative field o f - 5 kV/cm in both the perpendicular and parallel directions. For both directions, the difference in crack lengths under positive and negative electric fields is larger at smaller indentation loads. The difference diminishes as the indentation load increases. Other data also show that the fracture behavior of piezoelectric ceramics tested by the indentation fracture technique is load dependent but follows different trends. Figure 11 plots the indentation results on PZT-4 ceramics under the indentation loads of 4.45 and 22.24 N (Sun and Park, 1995). At 4.45 N, the results follow the same trend as those in Fig. 10 (i.e., a positive applied electric field enhances crack propagation normal to the electric field, whereas a negative field retards such crack propagation). However, if the indentation load is 22.24 N, both negative
Fracture of Piezoelectric Ceramics
261
130 (a) 4.45 N Load 120
110 :1. e~o
100 ,
9
,
9
,
.
,
9
,
.
|
.
,
.
,
,
,
.
|
.
360 (b) 22.24 N Load
r,.)
340
320
300 9
-8
,
.
-6
|
-4
.
,
-2
.
!
0
,
2
4
6
8
Applied Field (kV/cm) FIG. 11. Effects of static electric fields on the crack length of PZT-4 under an indentation load of (a) 4.45 N and (b) 22.24 N (with the experimental data by Sun and Park, 1995).
and positive fields facilitate crack propagation. Jiang and Sun (1999a) proposed a wedge model to explain the load-dependent behavior. Without considering the applied electric field, they modeled the tensile stress acting on the indentation crack front as a plastic wedge induced by indentation. As a result of the piezoelectric effect, the wedge elongates at a positive electric field and produces a wedge force. In contrast, if the field is negative, 180 ~ domain switching takes place because the voltage on the crack surface is high and the length of the wedge is small. As a result, the wedge effect takes place under a positive as well as a negative electric field. Jiang and Sun (1999a) introduced a reduction factor to link the wedge force with the applied electric field. Allowing the reduction factor to change with applied mechanical load, the wedge model predicts the load-dependent phenomenon well. Lynch (1998) conducted indentation fracture tests with a 39.2 N load on PLZT ceramics. His results show that a positive electric field enhances the growth of cracks perpendicular to the poling direction in PLZT ceramics. This effect seemed
Tong-Yi Zhang et al.
262
to be saturated at 500 kV/cm, about 1.2 times the coercive field. When the ceramics were indented under high electric field, a series of microcracks developed in the stress field of the indentation at a small distance away from the impression. The microcracks were not extensions of the radial or lateral crack systems. Lynch did not report the effect of a negative field. Fu and Zhang (2000a) conducted indentation fracture tests on PZT-841 ceramics with a load of 49.0 N. Under each of the electric fields of 4-4 kV/cm, about 10 indentations were made. The fracture toughness was determined from the following equation (Anstis et al., 1981):
(Y33) 1/2P K1c - 0.016 - ~ c3/2,
(9.1)
where P and c are, respectively, the applied mechanical load and the crack length perpendicular to the poling direction; H is the hardness; and 1133is Young's modulus along the poling direction. Figure 12 shows the variation of Kzc with the applied electric field. Under pure mechanical loading, the averaged K ic was 1.01 4-0.06 MPaVr~. The mean K1c was reduced by both a positive and a negative applied electric field. The reductions were 0.21 MPav/-m and 0.10 MPaWrm, respectively, for the negative and the positive field of 4 kV/cm. This result seems to indicate that a negative field has a stronger influence than a positive field on the averaged KI c. Using the indentation fracture technique with a 40 N load, Schneider and Heyer (1999) studied the effect of a static electric field on the fracture behavior of
1.1 -9
0 0 0
0
1.0
0 0
0
0.9 0
;
0
0.8
o 0
0.7
!
-6
-4
o
0
o ,
,
-2
mean i
0
,
|
i
2
4
6
Applied Field (kV/cm) FIG. 12. Effects of static electric fields on the fracture toughness of PZT-841 under an indentation load of 49 N.
263
Fracture of Piezoelectric Ceramics
ferroelectric barium titanate. Their results indicate that the measured crack length versus the applied electric field shows hysteresis similar to the strain hysteresis (i.e., the butterfly curve introduced in Section I). Curves for cracks parallel and perpendicular to the electric field direction are symmetric to each other. The lengths of cracks perpendicular to the poling direction under either positive or negative electric field are longer than the length of cracks without the applied electric field. This last result is consistent with that of Fu and Zhang (2000a). Park and Sun (1995a) and Fu and Zhang (2000a) conducted fracture tests on prenotched CT samples of PZT-4 and PZT-841 ceramics, respectively. About 10 samples of PZT-841 ceramics were tested at each level of the electric fields, except that 33 samples were tested at the electric field of 15 kV/cm to study the scattering of fracture toughness. The apparent fracture toughness Ktc was calculated based on the critical mechanical load at fracture only. Figure 13 shows the variation of the normalized K ic with the applied electric field for PZT-4 and PZT-841 ceramics.
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The results for PZT-4 ceramics reveal a nearly linear effect of the electric field on the fracture load; the results for PZT-841 ceramics show that the applied electric field increases the scattering of the measured apparent fracture toughness. The applied electric field, either positive or negative, reduces the mean of the apparent fracture toughness. Under purely mechanical loading, the average K IC for PZT841 is 1.12 -4- 0.05 MPa~/~, which is almost the same as that obtained from the indentation fracture tests. A negative field of 7.5 kV/cm reduces the averaged KIC by 0.25 MPav/-~, whereas a positive field of the same strength reduces the average KIC by 0.10 MPa~/-m. Applying a positive electric field of 15 kV/cm reduces further the averaged KIC to 0.92 + 0.14 MPax/-~, a relative reduction of about 18%. Applying an electric field generally increases the scattering in the measured fracture toughness considerably. The largest scattering in fracture toughness is at the electric field of 15 kV/cm. The distribution of K1c for 33 samples is shown in Fig. 14. The ratio of the standard deviation of 0.14 MPax/~ to the associated mean of 0.92 MPa~/-~ leads to a relative error of about 15%. As shown below, the relative error for the bending strength at an electric field of 10 kV/cm is about 33% (Fu and Zhang, 1998), which is more than double that observed in the CT tests. Using CT specimens, Kolleck et al. (2000) measured the fracture resistance curves (R curves) of BaTiO3 and commercial PZT-PIC 151 ceramics under an applied electric field parallel to the crack front. They found an increase in the fracture toughness with growing the electric field, and proposed a domain-switching model to explain the observed phenomenon.
FIG. 14. Distribution density of the fracture toughness under a static electric field of 15 kV/cm for CT tests on prenotched PZT-841 samples.
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4. Bending Tests The bending strength of PZT ceramics also varies with applied electric field. Zhoga and Shpeizman (1992) performed the axisymmetric bending of ferroelectric disks of a diameter of 20 mm and a thickness of 1 mm. The poling direction was along the thickness direction. Figure 15 gives their experimental results of PZT-19, with each point representing the average of 15-20 measurements. As can be seen in Fig. 15, the distribution of the electrical strength was similar to that of the mechanical strength, implying that local sites of failure under both electrical and mechanical fields were likely the same and governed by the microstructure of the ceramics. A weak electric field, ~ 10 kV/cm, either positive or negative, slightly strengthens the ceramics, whereas a strong field weakens them (Zhoga and Shpeizman, 1992). Makino and Kamiya (1994) conducted three-point bending tests on PZT samples under positive and negative electric fields with the poling direction toward the jig surface, in which configuration the maximum applied tensile stress was perpendicular to the poling direction. The results show that either a positive or a negative electric field, even at values less than 10 kV/cm, reduces the bending strength of PZT ceramics. Fu and Zhang (1997, 1998, 2000b) conducted a comprehensive study of the effect of static electric field on the bending strength of PZT-841 by three-point bending tests. The poling direction was perpendicular to the jig surface so that the maximum applied tensile stress was parallel to the
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FIG. 16. Probability distribution of the bending strength of PZT-841 ceramics under (a) positive electric fields and (b) negative electric fields. More than 50 samples were tested under each static electric field level. poling direction. A static electric field, either positive or negative, up to 20 kV/cm was applied across the sample, and more than 50 samples were tested under each electric field. Figure 16 illustrates the probability distribution of bending strength for both the positive (Fig. 16a) and negative (Fig. 16b) electric fields. Figure 17 plots the mean bending strength together with its error bar versus the applied electric field. An electric field generally causes large data scattering. The maximum scattering was observed under the electric field of 10 kV/cm. Without any applied
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FIG. 16. (Continued)
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electric field, the bending strength along the poling direction is 88 MPa, which is comparatively smaller than the bending strength of 97.8 MPa perpendicular to the poling direction. Under the electric fields of +3.33 kV/cm, the bending strengths along the poling direction are, respectively, 89.9 and 89.8 MPa, slightly higher than 88.0 MPa. However, the statistical u test analysis at a 95% significance level shows that the electric fields of +3.33 kV/cm do not change the bending strength of 88 MPa. The bending strengths under the applied fields of -6.7 and +6.7 kV/cm are, respectively, 81.4 and 77.7 MPa, significantly lower than the bending strength without any electric field. The bending strength is further reduced under a higher electric field, either positively or negatively applied. Using Eq. (8.30) derived from the DWK model, Fu and Zhang (2000b) fit the experiment data by the solid curve in Fig. 17. The fit curve agrees well with the experimental data over the entire range of the applied electric field. 5. Conductive Cracks There are only few reports on experimental observations of conductive cracks in piezoelectric ceramics. Lynch et al. (1995) carried out indentation tests on electroded surfaces submerged in distilled water and in electrically conducting NaC1 solution. In both cases, treelike damage grows from the indented electrode under the cyclic electric field. Heyeret al. (1998) studied the electro-mechanical fracture toughness of conductive cracks in PZT-PIC ceramics. They conducted four-point bending tests on prenotched bars, in which the poling direction was toward the jig surface and the notch was filled with NaC1 solution to make the crack conducting. Wide scattering was found under a large applied electric field (i.e., the applied intensity factor of electric field strength JKE] > 50 kV/ml/2). It seems that the critical stress intensity factor increases as the applied intensity factor of electric field strength changes from 30 kV/m 1/2 to - 9 0 kV/m 1/2. When the electrical intensity factor is in the range of - 1 5 kV/m ~/2 to 15 kV/m ~/2, the experimental data can be described by a model based on domain switching near the crack tip (Heyer et al., 1998). Fu, Qian, and Zhang (2000) performed fracture tests on conductive cracks. The CT samples used in their study were made of poled PZT-4 piezoelectric ceramics with the poling direction parallel to the prenotch. A 0.25-mm wide prenotch was cut in every sample, and the notch tip was sharpened further by a wire saw with a radius of 0.1 mm. Silver paint was then filled into the notch to make it conduct electricity. In the test, the applied static voltage increased gradually until the sample failed. In comparison, fracture tests under purely mechanical loading were also performed on CT samples of the same material. All tests were carried
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out at room temperature, and 30 samples were tested for each loading type. Electrical breakdown was usually accompanied by the fracture under purely electric loading. The fracture surfaces were flat for samples fractured under purely mechanical loads. However, for fracture under electrical loading, the critical voltage caused dielectric breakdown and roughened the fracture surfaces. Finite element analysis was conducted to calculate the energy release rate for all samples. The critical energy release rate under purely mechanical loading was calculated based on the fracture load and the ligament size, as shown in Fig. 18a. The mean of the mechanical critical energy release rate was found to be G/% - 8.7 4-0.4 N/m.
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Clearly, the linear regression is very close to a horizontal line, indicating that the critical energy release rate is a material constant independent of the sample ligament. Similarly, the critical energy release rate under purely electric loading was calculated from the ligament size and the critical voltage at failure or fracture. Figure 18b presents the electrical critical energy release rate versus the sample ligament. The linear regression is also almost a horizontal line, and gives a value of G/ec - 223.7 4- 17.0 N / m for PZT-4 ceramics. This fact, analogous to the mechanical loading situation, means that the electrical critical energy release rate is also a material property. The significance of the existence of G~c is that it enables the concepts of fracture mechanics to be applied to studying dielectric failure, and that G ~c is a useful material property for designers of electronic and electromechanical devices to control the reliability of the design. From this description, we see that fracture failure of piezoelectric ceramics is highly complex as it involves piezoelectricity, spontaneous polarization, and internal strain. The common features of the experimental observations are as follows. First, applied electric field, particularly if it is comparable to the mechanical field in magnitude, causes scattering in measured data. This means that a large number of tests are needed to provide a reliable database and allow the statistical treatment of the data. Second, an internal stress field is produced when the material goes through the transition from the paraelectric phase to the ferroelectric phase. The internal stress field changes its magnitude and distribution during poling as well as under mechanical and electrical loading. The internal stress field may inhibit or assist the applied loads in causing the piezoelectric ceramics to fail. Third, electric charges can be trapped at defects, and thus electric domains are clamped during electric cycling, leading to ferroelectric fatigue. The trapped charges may also cause local electric partial discharge. The electric discharge behavior may make an insulating crack electrically conducting, thus changing the failure behavior of piezoelectric ceramics. Fourth, the electric fracture toughness is a material property and differs from the mechanical fracture toughness. Therefore, the fracture toughness under combined electrical and mechanical loading may depend on the proportion of mechanical and electric loads. This is most likely to be case for conductive cracks.
B. FAILURE CRITERIA
Because the fracture behavior of piezoelectric ceramics is complex and the available experimental data are scattered and limited, there is no single failure or fracture criterion accepted by the scientific community. Models have been proposed
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to explain the specific observed failure behavior. The following failure criteria are widely used in various models and practices.
1. Applied Energy Release Rate Criterion Based on thermodynamics, the applied energy release rate for crack propagation may be a proper physical property to characterize fracture. For linear piezoelectric ceramics, the applied energy release rate equals to the J integral (see Section II for details). Thus, energy conservation requires the critical applied energy release rate in the form of J~ = 2(y +
yp),
(9.2)
where ~, is the specific surface energy and 2yp is the plastic energy per unit area, the superscript a denotes applied and the subscript C stands for critical. Equation (9.2) is the well-known Orowan (1952) and Irwin (1948) formula for ductile fracture. Using numeric calculations, Fu, Qian, and Zhang (2000) obtained the applied J integral for conductive cracks in terms of the applied electric voltage and the sample ligament size. Comparing the experimental data of 30 PZT-4 samples of different ligament sizes, they found the existence of a critical applied electric energy release rate beyond which a conductive crack propagates. This critical applied electric energy release rate is a material property, and can be used as the failure criterion for conducting cracks under purely electric loading. If no plastic deformation occurs during crack propagation, Eq. (9.2) reduces to the Griffith ( 1921) formula J~ = 2y.
(9.3)
This is the case for PZT-4 ceramics fractured under purely mechanical loading at room temperature, wherein no plastic deformation is involved during the fracture. For this case, there is also a critical applied mechanic energy release rate beyond which a crack propagates. This critical applied mechanical energy release rate is also a material property for predicting crack failures in PZT ceramics under purely mechanical loading. For PZT-4 ceramics, the critical applied energy release rate under purely electric load is 25 times higher than the release rate under purely mechanical load, indicating that "electrically plastic deformation" occurs during fracture. As discussed previously, the critical value of the applied energy release rate depends on the field of loading, electrical or mechanical. This is because the degree of plastic deformation is related to the loading field. Nevertheless, as long as the loading configuration is fixed, the applied energy release rate can serve as a failure criterion.
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Tong-Yi Zhang et al. 2. Local Energy Release Rate Criterion
If electrical or mechanical deformation occurs near a crack tip and if there is a local energy release rate, it is convenient to use the local energy release rate as a failure criterion because the critical value of the local energy release rate can be balanced by the surface energy. This concept is expressed by J~ - 2y,
(9.4)
where the superscript I denotes local. The electric strip saturation model proposed by Gao et al. (1997), as described in Section VIII, uses the local energy release rate criterion, which explains successfully the roughly linear experimental results between the applied electric field and the fracture load. The electric strip saturation model is based on the hypothesis that piezoelectric ceramics are mechanically brittle and electrically ductile such that electric saturation (yielding) may take place in the front of an electrically insulating crack. The polarization saturation zone shields the crack tip completely from the applied electric field, leading to a zero intensity factor for the electric displacement, as shown by Eq. (8.60). As a consequence, the local energy release rate, as shown by Eq. (8.62), is purely mechanical, and gives the linear relationship between the apparent critical stress intensity factor and the applied electric field. To explain the roughly linear experimental results, Park and Sun (1995b) proposed to use the applied mechanical energy release rate as the fracture criterion for insulating cracks in piezoelectric materials under combined mechanical and electrical loads. They argued that fracture is a mechanical process in nature. The applied mechanical energy release rate predicts almost the same linear relationship between the apparent critical stress intensity factor and the applied electric field as that predicted by the electric strip saturation model. Because the applied energy release rate consists of the mechanical and electrical energy release rates, the applied mechanical energy release rate model ignores the applied electrical energy release rate. We may say that the electric strip saturation model provides a rational explanation at the microscale level for the model of applied mechanical energy release rate, because the crack tip is completely and electrically shielded from the applied electric loading in the electric strip saturation model. Because there is a relationship between the local energy release rate and the local stress intensity factors, the criterion of the local energy release rate is equivalent to the criterion of the local stress intensity factor only if a Mode I, II, or III stress intensity factor is applied. Wang (2000) used a local stress intensity factor as the fracture criterion in his study of the electric strip saturation model.
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It should be pointed out that the characteristics of the local energy release rate in a model depend on the physical nature of the model. Different models lead to different local energy release rates. In general, the local energy release rate can serve as a failure criterion because it meets the law of energy conservation.
3. Stress Intensity Factor Criterion and Stress Criterion For conventional materials, the fracture strength can be defined as a critical stress at fracture during tensile tests, and the fracture toughness can be defined as a critical Mode I stress intensity factor for crack propagation under the plane strain condition. The two failure criteria have been extended to piezoelectric ceramics under combined mechanical and electrical loading, especially in domain-switching models. As introduced in Section VIII, 90 ~ domain switching produces an internal stress field. If the switching occurs at a crack tip the internal stress field can shield or antishield the crack tip from the applied loads. In this case, the local stress intensity factor may be used as the fracture criterion to calculate the critical applied mechanical and electrical loads. Generally, shielding leads to toughening and antishielding assists applied mechanical loads in driving crack propagation, leading to weakening. Even without any applied mechanical loads, the internal stress field produced by domain switching can damage the piezoelectric material. The domain switch toughening is equivalent to the twinning toughening (Pohanka et al., 1983; Mehta and Virkar, 1990). Using the stress intensity factor criterion, researchers developed various models to explain failures in piezoelectric ceramics. Lynch et al. (1995) proposed a domain-switching model to explain the cyclic nature of the electric crack propagation. Gong and Suo (1996) and Hao et al. (1996) developed a nonlinear finite element simulation for the reliability assessment of ceramic multiplayer actuators based on the domain-switching concept. Zhu and Yang (1999) further developed the mechanics of the domain-switching model to predict the electric fatigue behavior. In Section VIII, we introduced the DWK model, which is a type of domainswitching model. Usually, only a rather simple mathematical treatment is needed in the DWK model (Fu and Zhang, 2000b). The model is able to explain the effects of temperature and electric field on the bending strength of PZT-841 ceramics. Because the DWK model is developed to explain the results of bending tests, the critical stress criterion is actually used. Both the domain switching and DWK models account for the internal stress field induced by applied mechanical and/or electrical loads. Depending on its nature, the internal stress field may assist or resist the applied mechanical load to fracture the tested sample. In bending tests
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of smooth samples, failure may start from the point where the result of the applied stress and the internal stress is at its maximum or where the material resistance against fracture is at its minimum. The potential locations for stress concentration and crack nucleation are grain boundaries, domain boundaries, and other defects. In addition to the internal stress mechanism, partial discharge may be another mechanism affecting the failure of ferroelectric materials. Fracture mechanics analysis (Zhang et al., 1998) for linear piezoelectric media under combined electrical and mechanical loads indicates that the electric field inside an electrically insulating crack can be 1000 times higher than the applied electric field. This is due to the fact that the permittivity of the piezoelectric ceramics is 1000 times higher than that of the air or vacuum in the crack cavity. As a result, local discharge can occur between the insulating crack faces, making the crack electrically conducting and thus altering the fracture behavior of the material. The conductive crack propagates when the external electric field provides sufficient energy. Under combined electrical-mechanical loads, the applied energy release rate for a conductive crack has its maximum when the electric loading direction is inclined at some angles to the crack. The partial discharging model, which is still under development, may also predict the reduction in the bending strength. Shen and Nishioka (2000) and Zuo and Sih (2000) proposed to use the energy density criterion (Sih, 1973) for failure of piezoelectric ceramics under combined electrical and mechanical loading. When a critical value of the energy density factor serves as the failure criterion for cracks perpendicular to the poling direction, along which remote electric and mechanical loads are applied, the local minimum of the energy density predicts that the crack propagates along the original crack plane; and the positive electric field aids the crack propagation, whereas the negative electric field impedes the crack propagation. The prediction agrees qualitatively with the roughly linear experimental result.
X. Concluding Remarks A live system distinguishes itself from an inanimate one by its ability to adapt to changes in environment. Smart materials have the ability of sensing and actuating, and thus are capable of performing the rudimentary aspects of life. Piezoelectric ceramics can sense and actuate by converting mechanical and thermal signals rapidly into electrical ones, and the reverse is also true. The piezoelectric properties and the quick response characteristics have made piezoelectric ceramics among the most commonly used smart materials. However, the intrinsic brittleness of
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piezoelectric ceramics and the damageability of the materials under electric field, making the materials prone to fracture, are of major concern for product reliability. These drawbacks have prevented the materials from being used even broader applications. Thus, the fracture of piezoelectric ceramics under combined electrical and mechanical loading has been among the most prevalent research topics. This article summarizes the current state of the knowledge in this area. Section I briefly describes piezoelectricity, ferroelectrics, spontaneous polarization, and electric domains. We also discuss the poling process, the hysteresis loop of polarization versus the electric field strength, and the butterfly loop of strain versus the electric field. Such basic knowledge of piezoelectric ceramics is helpful in understanding the fracture behavior of the materials. As thermodynamics provides the basis for fracture mechanics, we introduce the basic equations commonly used in the study of the fracture behavior of piezoelectric ceramics within the thermodynamics framework in Section II. These equations include the kinematic equations, the constitutive equations, the static equilibrium equations, the principle of virtual work, the energy release rate for cavity and crack growth, and the J integral. Section II discusses the fracture mechanics under thermal loading only briefly. We have not treated this topic in more detail for two reasons: (1) the limited scope of this article and (2) the lack of experimental results on the subject. For the same reasons, the dynamic fracture of piezoelectric ceramics has not been discussed. Limited theoretic publications in these two areas are available in the literature. Interested readers may refer to Lu et al. (1998), Qin et al. (1999), Ding et al. (2000), and Qin (2000) for fracture under thermal loading, and to Dascalu and Maugin (1995), Li and Mataga (1996a, 1996b, and Wang et al. (2000) for dynamic fracture. In Section III, we present the general solution based on Stroh's formalism, a powerful tool for solving two-dimensional electroelastic problems. In parallel, because of its simplicity, the solution for antiplane deformation is provided to help give a physical insight into the fracture behavior. Electrically insulating elliptical cavities and cracks, also called conventional cavities and cracks, in a homogeneous infinite medium are studied in detail. Special attention is placed on the effects of the electric boundary conditions along the crack faces. The exact electrical boundary conditions on the cavity surface are applied in the present study. The solution depends on two dimensionless parameters c~ and/3, which are, respectively, the ratio of the minor to the major semi-axis of the ellipse and the ratio of the dielectric constant of the cavity to the effective dielectric constant of the material. Reducing the cavity to a slit crack (i.e., c~ --+ 0), we find the mechanical and electrical fields at a crack tip and the associated intensity factors and the energy release rate for crack
276
Tong-Yi Zhang et al.
growth in terms of the ratio of ct/fl. For ct/fl --~ 0, the crack is said to be electrically permeable, and the applied electric field, parallel or perpendicular to the crack, has no contribution to the energy release rate. However, if a i r --~ c~, the crack is said to be electrically impermeable. The energy release rate is independent of the applied electric field parallel to the crack, but it decreases (through the change in the intensity factor of electric displacement) with the applied electric field perpendicular to the crack [Eqs. (3.109) and (3.115)], indicating the applied electric field retards crack growth. As a result, the intensity factors and the energy release rate depend strongly on the width of a cavity (crack) and the sample width, which thus play an important role in the facture of piezoelectric ceramics (Zhang, 1994a). Section IV includes an analysis of Green's functions for insulating elliptical cavities and cracks; in other words, it is a study of the interaction between a piezoelectric dislocation and these cavities or cracks. For generality, we allow the piezoelectric dislocation to have an extended Burgers vector and an extended concentration force vector. Special applications correspond to different non-zero components of the extended Burgers and force vectors, with others being zero. For instance, we use Green's functions to solve for the polarization saturation in Section VIII, in which the solution corresponds to an extended Burgers vector with one non-zero component and a null extended force vector. Green's functions can be used to construct solutions for distributed loading on the cavity or crack surfaces. Green's functions are very useful in the boundary element method (BEM). In the BEM, discretization is needed only on the boundaries including the cavity and crack faces. If there are special Green's functions that already account for the boundary conditions on the cavity and crack faces, no discretization is needed on those faces. Thus, the BEM can be more effective in analyzing crack and cavity problems. Cruse (1996) summarized the B EM with Green's functions for purely elastic problems. We study conductive elliptical cavities and cracks in Section V. For simplicity, we consider only the cavities and cracks without any net charges. For a conductive crack under combined mechanical and electrical loading, the mechanical and electrical fields in the vicinity of the crack tip can be expressed in terms of a real intensity factor vector if the new material matrix Q defined by Eq. (5.3b) is used. Otherwise, complex intensity factors have to be defined to describe the crack tip fields (Zhang et al., 1998). The complex intensity factors are demonstrated for Mode III cracks [Eq. (5.24)]. In general, the applied electric field perpendicular to the crack has no effect on the energy release rate, whereas a positive or negative electric field parallel to the crack has a positive contribution to the energy release rate [Eqs. (5.16) and (5.27)]. When the electric energy release rate exceeds a critical value, the conductive crack grows, which has been verified experimentally
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(Fu, Qian, and Zhang, 2000). We also present Green's functions for a conductive cavity and a conductive crack. Section VI studies piezoelectric interface cracks. For simplicity, we focus mainly on the electrically impermeable interface cracks. The linear approach yields oscillatory solutions at crack tips. Although the oscillatory fields result in the physically unreasonable overlapping of the crack faces, the linear solution can still provide insight into the crack behavior if the overlapping zone is small in comparison to the crack length (Rice, 1988). Caution must be used here because the overlapping zone could be very large under certain loading conditions (e.g., for large ratios of shear loading to normal loading; Herrmann and Loboda, 2000). We investigate three-dimensional electroelastic problems in Section VII. As with two-dimensional problems, the electrical boundary conditions along the crack faces play a dominant role in determining the crack tip fields, the intensity factors, and the energy release rate. Although the effective dielectric constant in the three-dimensional problems has a different form from that for two-dimensional problems, we also find the results for the same limiting cases of the ratio ~/fl as those in two-dimensional problems [Eqs. (7.19)-(7.20)]. In Section VIII, we consider four types of nonlinear approaches: electrostriction, domain switching, domain wall kinetics, and polarization saturation at a crack tip. Electrostriction is induced by Maxwell stresses, and is particularly important for conductive cracks. It seems that the energy release rate calculated from electrostriction (McMeeking, 1987) in terms of the stress intensity factor is essentially the same as that calculated by the linear approach in terms of the intensity factor of electric field strength [Eq. (8.2)]. The domain-switching and domain wall kinetics (DWK) models are basically the same in terms of their underlying physics. Both models are concerned with the internal stress field induced by domain-switching or domain wall displacement. Because the electrical and mechanical fields are high in the vicinity of a crack tip, domain switching is likely to occur there, and consequently induces internal stresses. The internal stress field can assist or resist the applied mechanical load to fracture the material. The internal stress field in a specimen is at self-equilibrium, which means that the internal stress is tensile in some regions while compressive in others. For smooth bending specimens, cracks always initiate at sites with high tensile resultant stresses, which include bending and internal stresses. That is why the internal stress always assists the applied load in fracturing a bending sample [Eq. (8.39)]. This has been verified experimentally (Fu and Zhang, 2000b). The merit of the domain-switching model lies in the soundness of the mechanics treatment, whereas the merit of the DWK model is in the simplicity of the mathematics involved. Based on the DWK model, we established the effects of temperature
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on the elastic compliance [Eq. (8.27)] and the bending strength [Eq. (8.34)]. The model predicts a maximum elastic compliance and a minimum bending strength, both of which occur around 225 ~ C (near the Curie point) for PZT-84. The polarization saturation model discussed in Section VIII.D treats piezoelectric ceramics as mechanically brittle and electrically ductile materials. When polarization saturation takes place at an electrically impermeable crack tip, the saturation zone shields the crack tip from applied electric fields. As a result, the local energy release rate is purely mechanical. This model can explain the roughly linear relation between the critical applied mechanical load and the critical electrical field in fracturing PZT ceramics. In Section IX, we provide an overview of experimental observations. The experimental results show that the microstructure and temperature have a profound influence on the fracture behaviors of piezoelectric ceramics under purely mechanical loads. The effect of temperature may relate to the internal stress field induced by the spontaneous polarization. Cyclic electric fields can cause electric fatigue (microcrack initiation and nucleation) and ferroelectric fatigue (loss of polarization switchabililty). In situ TEM observations show that the fatigue cracks grow along domain boundaries. Experimental results also indicate that cracking may be one of the causes for losing polarization switchablilty. Thus, it also may be necessary to consider cracking in ferroelectric fatigue. Indentation, compact-tension, and bending tests have often been used to study the effects of static electric fields on fracture behavior. For PZT-4 ceramics, Fu, Qian, and Zhang (2000) show the existence experimentally of an electrical fracture toughness in terms of the energy release rate, which is about 25 times that of the mechanical fracture toughness of the same material. Generally speaking, the experimental results indicate that the fracture behavior of piezoelectric ceramics is highly complex. Data are still quite limited, and the results from different researchers are not always in agreement with one another. One clear trend is that the applied electric field always increases the scattering of the measured data. In Section IX, we also introduce the commonly used failure criteria. The critical applied energy release rate, at which cracks grow, is defined as the surface energy plus the plastic work per unit cavity or crack extension for ductile piezoelectric materials, or only as the surface energy for brittle piezoelectric materials. The concept of ductility has been extended to include the electrical plastic deformation. Because the degree of electrical plastic deformation depends on loading conditions, the applied energy release rate criterion is appropriate if the loading type is fixed. The electric saturation model introduced in Section IX treats a piezoelectric ceramic as mechanically brittle and electrically ductile. In this case, the electric
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field at the crack tip is completely shielded, and the energy release rate depends only on the mechanical stress intensity factors. Shielding is a local phenomenon, and thus the resulting energy release rate is called the local energy release rate, which is balanced only with the surface energy at the critical condition. Such a local energy release rate criterion is purely mechanical in nature, and has the potential to predict fracture without regard to loading type. Section IX also discusses the stress intensity factor criterion and the stress criterion. Both criteria are purely mechanical in nature. In the near future, more experimental measurements are needed. Because applying an electric field increases the scattering of the measured data, more carefully controlled experiments are required to provide a reliable database. Statistical assessment of the fracture data will provide a better understanding of the fracture behavior of piezoelectric ceramics. Clearly, the discrepancy between theoretical predictions and experimental measurements is still ubiquitous. A comprehensive model(s) is needed that can account for the various mechanisms such as electrostriction, domain switching, domain wall motion, polarization saturation, and partial electric discharge near the crack tip. All these mechanisms affect the internal stresses under applied mechanical and electrical loads, and in turn affect the local energy release rate. It is a challenge to develop such a comprehensive model(s), calculate and measure the induced internal stress field, determine the local energy release rate, and assess the electric and ferroelectric fatigue damage in order to explain the wide range of experimental results. Along with the accumulation of experimental results, a more versatile theory should be established for the design of reliable and durable piezoelectric products.
Acknowledgments The authors are grateful to Dr. Ran Fu for his help in plotting the figures. This work is supported by a grant (HKUST6051/97E) from the Research Grant Council of the Hong Kong Special Administrative Region, China, and a grant (DAG99/00.EG35) from the School of Engineering, HKUST. MHZ thanks HKUST for the Post-Doctoral Fellowship Matching Fund.
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A D V A N C E S IN APPLIED M E C H A N I C S , V O L U M E 38
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion THEODORE YAOTSU WU Division of Engineering and Applied Science, California Institute of Technology Pasadena, California 91125
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Subdivisions of Hydrodynamic Theories for Aquatic and Aerial Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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III. Resistive Theory of Aquatic Locomotion . . . . . . . . . . . . . . . . . . .
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IV. Classical Slender-Body Theory of Fish Locomotion . . . . . . . . . . . A. Slender Body with Only Longitudinal Flexibility . . . . . . . . . . . B. A Generalized Linear Slender-Body Theory of Fish Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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V. A Unified Approach to Nonlinear Theory of Flexible Lifting-Surface Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. A Unified Nonlinear Theory of Two-Dimensional Flexible Lifting-Surface Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Method I--Computational Time-Marching Method . . . . . . . . . B. Method IImGeneralized Wagner-von K~irm~.n-Sears M e t h o d . . C. Method I I I m A Hybrid Analytical-Numerical Method . . . . . . . VII. On Experimental Differentiation between Thrust and Drag in Fish Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Scale Effects in Energetics of Aquatic Locomotion . . . . . . . . . . . . A. Metabolic Rate and Scale Effects . . . . . . . . . . . . . . . . . . . . . . B. Scaling of Swimming Velocity and Energy Cost . . . . . . . . . . . . C. Scaling of Viscous Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ADVANCES IN APPLIED MECHANICS, VOL. 38 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISSN 0065-2165/01 $35.00
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I. Introduction
Locomotion of aquatic and aerial animals in fluid media, or more plainly, fish swimming in water and bird and insects flying through air, must be fascinating phenomena to humans throughout the ages. They attract attention to observe, appreciate, and wonder how some of the highly impressive performances can be attained, and they arouse curiosity strong enough in some observers that they even attempt emulation in some particular aspects of the animals' superb ability. Just imagine where our present-day air travel convenience would be had there been no curiosity or attempts of emulation. Progressive improvement of solutions by animals to particular problems posed by environment and natural hazards has been found repeatedly by research investigations to have led to new engineering possibilities and occasionally to technological breakthroughs. In regard to swimming and flying in nature, there are several salient features worthy of reflection. First, it is the vast scope covered by the phenomena of locomotion of aquatic and aerial animals. By size, it ranges over more than 18 orders of magnitude in body mass, from the Blue Whale (Balaneoptera muscula, the largest of all known animals, with maximum length to 30 m) of 105 kg down to small bacteria (having size of microns and mass of picograms) of approximately 10-14 kg. In terms of the Reynolds number, it extends to as high as 108 for Blue Whale and to as low as 10 -6 for some bacteria. In physiology, the metabolic processes vary from the better-known muscular work in higher animals to the not-so-fully understood processes in prokaryotic flagella of bacteria. Concerning the propulsive movement, there are myriads of modes of locomotion of varied interest to fluid mechanics, mathematics, physiology, and molecular biology. After all, what we are now able to observe in this field is the result of ages old evolution in an ever-competitive world. Another salient feature of the various propulsive modes is the very large amplitude and great flexibility adopted in maneuvering of the lifting-surface part of animal body when desired, such as in the hummingbird's hovering, the eagle's powerful wingbeat in take off, the fish's wavering caudal fin in dashing forward, and so on. In contrast, however, these animals exhibit their seemingly effortless control in maintaining sleek performances at their own command. These remarkable features still pose challenges that our well-advanced theories and technology, which has been successful in bringing up the modern transportation industry to cutting edge, are found still inadequate to comprehend animal locomotion to full satisfaction. Still another feature of complexity concerns experimental differentiation between the thrust and drag interplay in fish locomotion. It seems to be exceedingly difficult to measure the two parts separately and show that at constant cruising, the parts are equal and opposite with desired precision and consistency, especially
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for the high-speed species of fish, such as tuna, and cetaceans, such as dolphin and porpoise. Based on a series of careful investigation, Sir James Gray, the head of Cambridge University's Zoology Department from 1937 to 1961, dean in the field of biology of animal locomotion, enunciated his concrete finding (1936) that either the fish measured could have muscle to deliver specific power (per unit weight of muscle) several times that well attested for warm-blooded animals, or they would experience frictional drag several times less than the experimental data obtained with great care. This conjecture, known as Gray's paradox, has stimulated extensive studies and discussions, yet still remains to be fully resolved. Historically, Gray, talented at promoting interdisciplinary collaboration, has been said to have stimulated Sir G. I. Taylor in making two pioneering investigations, one on the swimming of snakes and eels (1952b) and one that has initiated hydrodynamic studies of flagellar propulsion (1952a). These publications are said to have stimulated imagination of Sir James Lighthill to foster a close collaboration between his student, Geoffrey Hancock, and Gray to produce a valuable analysis of flagellar hydrodynamics (Gray and Hancock, 1955) that was to be used by almost all workers in the field for the next two decades. These creative works also played a leading role in having Lighthill lay a foundation (1960) concerning the swimming of slender fish, and having Wu (1961) make an extension of classical oscillating airfoil aerodynamics to a linear theory of flexible lifting-surface locomotion to examine the performance of bending wings of birds in flapping flight and the lunate tails of fast-swimming percomorph and scombroid fishes and rays in swimming. Academic interest in this field soon spread extensively into both high and low ranges of Reynolds numbers. In his John von Neumann Lecture, Lighthill (1976) delivered his newly accomplished theory on Flagellar Hydrodynamics based on thorough mathematical analysis with a considerable deepening understanding to offer it as an improvement of Gray-Hancock's resistive theory. Separately and at the same time, Chwang and Wu (1971), stimulated by Taylor's works and attracted to the long-standing spirochete paradox, investigated the self-propulsion of a microorganism in a viscous fluid by sending a helical wave down its flagellated tail, thus providing an explanation resolving the paradoxical phenomenon that a flagellate can roll about its own axis without passing bending waves along its tail (Gray, 1962). This study was soon applied to explain the dynamics of spirilla locomotion (a fixed helical cell body attached at both ends with a bundle of flagella) by Chwang et al. (1972) and to determine the underlying mechanism of the helical locomotion of the spirochete (a helically waving flagellum lacking any attached cell body) by Chwang et al. (1974). Related studies on ciliary locomotion were carried out by Blake and Sleigh (1974), by Keller et al. (1975) on a new theoretical
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model (based on representing the ciliary layer action by a distributed force field) for ciliary propulsion, and by groups of scientists at other centers around the world. With these exciting activities progressing in the field, the time soon became opportune to hold the Symposium on Swimming and Flying in Nature (Proc. eds Wu, Brokaw and Brennen, 1975). Various topics of ongoing interest were presented and discussed enthusiastically. The scope included bacterial locomotion, flagellar and ciliary propulsion, swimming of larger animals, flight of birds and insects, and other new problems. One of the exciting discoveries brought to the Symposium was the phenomenon of the hovering of the chalcid wasp, Encarsia formosa, discovered by Torkel Weis-Fogh (1973, 1975), who proposed a new mechanism of lift generation that is of fundamental interest. By the so-called clap-fling operation of the wasp's two wings, a pair of equal and opposite circulations is generated around the wings as the wings swing open (from the tightly closed clap position on top of its back) about their trailing edges, which are held together for a certain duration, until flinging apart horizontally to retain their fully grown circulation to attain a high hovering lift. What is surprising is that throughout this rapidly varying clap-fling operation, no vortices are found to be shed from the wings, making the performance markedly superior to a single wing in similar operation. For a single wing making a step increase in incidence angle, it instantly begins to shed a vortex sheet, generating a sudden lift; however, the lift reaches only half the final stationary value, whereas the final lift is reached after a time delay [known as the Wagner effect; see dicussion with Eq. (6.62)]. For the Weis-Fogh mechanism, there is no time delay in lift generation because no vortices are shed, thus causing no violation at all of Helmholtz's doctrine on vorticity conservation and Kelvin's circulation theorem, as lucidly expounded by Lighthill (1973). The prolific advances in the field led to another very important Symposium on Scale Effects in Animal Locomotion (held at Cambridge University, September 1975; Pedley, 1977) on a theme spanning mechanics, biophysics, and zoology. Topics of strong interest and active pursuance were addressed, including the scaling of terrestrial locomotion by McNeill Alexander (1977), the scaling of aquatic locomotion by Wu (1977), the scaling of aerial locomotion by Lighthill (1977), and the hovering flight by Weis-Fogh (1977). The forum for in-depth interaction with exchanges of views, comments, and arguments between the collaborating disciplines yielded highly fruitful conclusions and comprehensive accounts of what was then known about scale effects, as recorded in the Proceedings. In particular, considerable clarification was achieved on the intricate question about the thrust-and-drag issue and energetics in animal locomotion. This remarkable gain in comprehensive knowledge would not have been possible without such intimate
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collaboration across the relevant fields of specialties. In general consensus, we appreciate the philosophy that to harvest the best fruit in our multidisciplinary field, we must talk to each other and keep talking to each other. This article is not so highly intended to be a general review or survey. It is rather strongly devoted to address, in particular, expository examinations and new developments in three topics of primary interest and importance: the first is different physical concepts required for evaluating various modes of locomotion at high and low Reynolds numbers; the second is a new approach to large-amplitude lifting-surface theory for modeling aquatic and aerial locomotion; and the third is experimental differentiation between thrust and drag in aquatic locomotion aimed for further clarification and resolution. The overall plan is as follows. First, in Section II, the subdivision of hydrodynamic theories for aquatic and aerial locomotion is expressed in terms of two key parameters, one being the Reynolds number characterizing the propulsion dynamics and the other a geometric shape factor called the aspect-ratio of the lifting surface (ratio of wing span squared to the wing's plan area), which is responsible for generating propulsion. The resistive theory is revisited in Section III for its role as an approximation for application to the category of slender aquatic animal locomotion in which the inertial effects are weak and negligible. In Section IV, the swimming motion of typical carangiform, thunniform, amiiform, and their variations characterized by high Reynolds numbers are first analyzed by applying classical slender-body theory for determination of thrust and hydromechanical power required for sustaining the motion. For the case in which vortex sheets are shed from the elongated dorsal and ventral fins and in turn are convected by the stream to pass alongside the caudal fin, theoretical results based on a generalized slender-body theory are presented to elucidate the basic mechanism of interaction between the vortex sheets and the caudal fin. To facilitate further development of modeling aquatic and aerial locomotion encountered in practice, a new approach to establishing a large-amplitude lifting-surface theory is introduced in Section V for modeling three-dimensional flexible lifting surfaces with formulation expressed in terms of the Eulerian description. Further new development is presented in Section VI for two-dimensional flexible lifting surface with the Eulerian formulation of the flow field assisted by a Lagrangian description of the lifting-surface motion, which is free to move along arbitrary trajectory and performing motions of arbitrary amplitude. The intricacies in handling impulsive movements of the lifting surface is thoroughly explored numerically and analytically in the two-dimensional case to obtain a generalized Wagner integral equation for practical application. This theory is applicable asymptotically to flexible lifting surfaces of large aspect-ratio, such as for the lunate-tail mode of
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swimming found in fast swimming percomorph and scombroid fish, the flapping locomotion of skates and rays, and the flight of birds and insects. In Section VII, the problem of experimental differentiation between thrust and drag, especially pertaining to fish swimming (or Gray's paradox), is first argued on the basis that the total force system must be considered as a whole to distinguish between the physically closed system of a fish in uniform self-propulsion and the open system of specimens not in self-propulsion, but with interchanges of momentum and energy with exterior agencies. In parallel to the mechanical approach, the problem of the thrust-and-drag issue is investigated in Section VIII within a joint mechanical and physiological framework. With this multidisciplinary study bringing together biology and mechanics, new results and understanding are attained concerning the necessary balance between the thrust a self-propelling fish or cetacean must produce and the viscous drag it must overcome in steady swimming. As regards expository surveys on this subject, we leave them to the literature (e.g., Lighthill, 1975, 1993; Wu, 1971d, 2001; Azuma, 2000).
II. Subdivisions of Hydrodynamic Theories for Aquatic and Aerial Locomotion The primary distinction that exists between the different types of hydrodynamic theory relevant to aquatic and aerial locomotion is based on the value of the Reynolds number, which is the ratio of the inertial to the viscous force, defined as
R e - Ug./v,
(2.1)
where U is a typical speed attained by animal of length g swimming in water of kinematic viscosity v (typically 1 mmZ/s). The locomotion of animals about 1 mm long (or less) moving at speed of several body lengths per second is a phenomenon of low Reynolds numbers (of order _<1), in which the inertial effects are negligible. Theories of their analysis [involving (3 + 1) dimensions, 3 of space and 1 of time] is based on the Stokes equations, which are linear. They provide a description of linear equilibrium between pressure and viscous stresses, leading to a harmonic equation for pressure and a biharmonic equation for velocities. For bacterial and protozoal locomotion, Reynolds numbers are invariably small--as small as 10 -3 or even less. However, the motions of animals of larger sizes moving with velocity of several body lengths per second or higher are phenomena of high Reynolds numbers, in which inertial effects are important, leaving the viscous effects everywhere
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negligible except for their important role jointly with the inertial effect in generating very thin boundary layers adjacent to body surface and dictating how smooth the flow leaves trailing edges of animal's appended lifting surfaces and tail fins. The viscous boundary layers dragged along by the swiftly forward-moving body surface are generally so thin (with flow separation here regarded as being absent or negligible in well-evolved swimming modes) that they are totally transparent to pressure across them, rendering the inviscid irrotational flow theory (also called potentialflow theory) fully valid to determine the thrust, moment of force, and energetics involved in aquatic and aerial locomotion. Knowing thrust, it is implied by Newton's law that we know the viscous drag of equal precision. Compared with this indirect approach, the task of direct calculation of the viscous drag using unsteady boundary-layer theory seems formidable, primarily due to the lack of knowledge of the laminar-to-turbulent flow transition for such strongly transient basic flows. In the regime of high Reynolds numbers, a further subdivision of hydrodynamic theories arises with an additional parameter pertaining to the geometric configuration of body shape, indicating whether the aspect-ratio of the lifting surfaces used for locomotion is large or small. For the case of small aspect-ratio, there are a great variety of slender animals performing undulatory motions that require appropriate slender-body theory (also called elongated-body theory) capable of handling strongly nonlinear effects, especially for analyzing motions of large amplitude, and with high acceleration arising in maneuvering and stability control. However, there are also animals using lifting surfaces of large aspect-ratio to generate lift, such as the lunate tails used by several families of percomorph and scombroid fishes, highly flexural wings used by skates and rays, and the wings of nearly all birds and insects. For these modes of propulsion, use of two-dimensional liftingsurface theory can provide an appropriate asymptotic accounts of wings of large aspect-ratio, especially for flexible wings moving with large amplitude, as we discuss in Sections V and VI. These added features require the theory to be strongly nonlinear, in sharp contrast with the low-Reynolds-number regime characterized by linear theory. In regard to using undulatory motion of a lifting surface of large aspect-ratio common to bird flight and fish swimming (wing for bird and lunate-tail fin for fish), yet another important factor arises concerning the composition of the heaving and pitching modes--in particular, their difference in phase. Earlier advances can be referred to K~irm~inand Burgers (1934), who showed that high propulsive efficiency cannot be achieved from either the pitching or the heaving mode alone, unless the two modes are superposed with an optimal phase difference. In this respect, it is ingenious of Lighthill (1970) to introduce a single parameter he called the
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proportional feathering (defined as the product of the maximum pitching angle of a flapping wing and the ratio of the flight velocity to the maximum tranverse velocity of the wing) as an indicator of the incidence of the wing section open to the local wind vector and of a limit to not having so large a contribution from leading-edge suction at the verge of leading-edge separation of the boundary layer. With this parameter, it becomes clear how the thrust making and its rate of working in the total power expenditure can be visualized to vary in a competing trend, thus indicating it is an interesting control problem. For optimum efficiency, Lighthill concludes for the feathering parameter to take values of around 0.6 to 0.8. This is a model illustration of Lighthill's mastery in combining geometry, physical principles, and simple yet powerful mathematical methods to reach valuable results and draw lucid conclusions. His methods generally bring out new basic concepts that benefit other researchers working in the field, such as on the comparative studies by Wu (1971b) on theoretical prediction of the feathering of a porpoise fluke with experiments. This feathering parameter has been used effectively by Chopra (1976) to extend the lunate-tail theory to oscillations of large amplitude (for wings of infinite aspect-ratio), and by Chopra and Kambe (1977) to predict the hydrodynamic behavior of a variety of lunate-tail planforms of large aspect-ratio. Their works cast light on the optimization processes that can contribute to reducing the rate of work required by the lunate tail to generate a given thrust. They also discussed in some detail how that thrust is balanced by the hydrodynamic resistance due to body friction. We further remark that this line of approach (using the added mass for lateral motions and an estimated frictional drag) is also applicable to motions with a sudden acceleration or change in direction, as illustrated by Weihs (1972, 1973), who used the unsteady aerodynamic theory of delta wings for the caudal-fin hydrodynamics; for the forces on the anterior part of the body, he used the slender-body theory along with estimates of viscous drag. From the hydrodynamical and morphological viewpoint, the crescent-moon shaped lunate tails have been thought as a product of convergent evolution. They are exhibited not only by many high-speed species of percomorph and scombroid fishes (including mackerel, tuna, marlin, sailfish, and swordfish), but also by some of the fastest sharks and the ever-flying swallows and swifts (Apodidae). It is of basic interest to discover whether these gently sweptback wings and lifting surfaces of large aspect-ratio can achieve benefits from using certain optimum feathering characteristics during flapping flight that may contribute to thrust production at extremely low energy cost. The mathematical problem of analyzing the lunate-tail dynamics using curved lifting lines in unsteady flow, long recognized as a great challenge, has been solved successfully by Cheng and Murillo (1984) and Karpouzian et al. (1990) using an asymptotic perturbation method
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
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for calculating the strip lift distribution, yielding results supporting the previous numerical doublet-lattice calculation by Albano and Rodden (1969). In connection with the flapping locomotion of birds and some aquatic animals slightly denser than surrounding water when both thrust and lift are needed for locomotion, the dynamic cycle may be divided into two parts: the downward pronation power stroke and the upward supination recovery stroke. During the downward-forward stroke, the enhanced incidence angle over the mean is clearly in favor of generating both thrust and lift; therefore, it must be heavily loaded (per unit area of the lifting surfaces). In contrast, the incidence angle required for sustaining lift is a cost to the wing's total incidence that makes thrust production and lift sustaining superficially a matter of conflict of interest. The necessary load changes that occur between downstroke and upstroke may be achieved, in principle, by making variations in the distributed and total circulation, wing span, stroke durations, or in combinations. In this respect, two types of operations are commonly observed: (i) shortening of wing span by chordwise and spanwise bending and flexural twisting, and (ii) shortening the upward stroke duration plus prolonging the downward stroke duration. It can be argued, quite convincingly, that both seem to tend toward maintaining the circulation more than otherwise; accordingly, they would enhance the efficiency of producing thrust under weightload. In regard to these control factors, a new parameter may be singled out and called the proportional flexurality to cover all these configurational factors. Clearly, these factors contain a veriety of nonlinear effects that require more research to clarify the underlying mechanisms. Some of these will be explored in the sequel, but only very selectively. For aerial locomotion, subdivisions of types of theories, other than the topics already expounded, are somewhat simpler since the great majority of flying animals use wings of large aspect-ratio to take advantage of their large lift-to-drag ratio, and they nearly all operate in the high-Reynolds-number regime, aside from a great variety of small insects, such as those in several families of thrips having bodies 0.5-12 mm long and using fringed bristles for wings to fly at Reynolds number of about 10 -2. A theory of their flight mechanism has been proposed by Kuethe (1975). In closing this section, we note that in addition, there are other types of propulsion, such as using inertial reaction of high-velocity jets of water and using variations of these modes. Thus we have taken a brief but central view of the physical features, conceptual outlook, and effective mathematical approaches toward the mechanical problems about the diverse animal kindom. Next, we present an expository investigation, some in historical steps, on the advances in the areas relevant to the main themes stated earlier.
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III. Resistive Theory of Aquatic Locomotion In Taylor's (1952b) study on swimming of elongated animals like snakes and eels at somewhat high Reynolds numbers, he proposed the use of simple resistive forces with tangential or normal component depending on the tangential or normal component of relative velocity, respectively. Like Taylor, Hancock ( 1953) also took a finite step departing from the small-perturbation approach (used in linear theory) by having the flow field around an arbitrarily moving flagellum represented by a line distribution of flow singularities consisting of the stokeslet (first used by Stokes, 1851, which corresponds to a point force acting on fluid) and ordinary source doublet (or simply dipole). The rather elaborate calculations of this theory were subsequently simplified by Gray and Hancock (1955), who, like Taylor, adopted resistive force components related to the corresponding components of the relative velocity, the only difference being that the force-velocity dependence is linear at low Reynolds numbers but quadratic at high Reynolds numbers. This simplification makes results much easier to obtain but at the cost of a cruder approximation. The overall features of resistive theory can be delineated (cf. Gray and Hancock, 1955; Lighthill, 1975) as follows. Consider the self-propulsion of a slender animal swimming in a stream of velocity U along the x axis by passing an undular wave along its flexible slender body, causing the body axis displaced to z = h(x - V t ) from the x axis, with wave velocity V and wave slope h~ = - h t / V, where the subindices x and t denote differentiation. By the kinematic condition, we have for the flow lateral velocity w (along z) at body boundary given by w -- ht + Uhx = ht(1 - U / V ) -- W(1 - U/V),
(3.1)
where by notation, ht =- W, which is the lateral velocity of the body axis. Since the body wave has slope hx = - h t / V = - W / V due to the wave motion of a small body segment at station Sx, the fluid lateral velocity w induces a resistive force - C n w per unit length on the segment, which contributes to a thrust equal to - C n w h x -- C n w W / V per unit length, where Cn is the normal force coefficient. Another contribution to thrust comes from the tangential component, W h x , of the body lateral velocity h t - W; hence, we have the thrust T per unit length as T =(C,,w-CtW)W/V,
(3.2)
where Cr is the tangential force coefficient. The balance between thrust T and frictional drag D = CtU (per unit length) due to the relative stream velocity U
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yields for the swimming velocity U in still water U2 --
W2( 1 - X)(X - r ) / r
(X =- w / W ,
r
-- Ct/Cn),
(3.3)
X being the ratio of fluid lateral velocity w to the body lateral velocity W. Maximum swimming speed (relative to W) is achieved when X - (1 + r)/2,
giving
U/V = (1 - r)/2,
U/W - (1 - r)/2v/-7.
(3.4)
Regarding the energetics, we have the useful rate of working by the thrust (per unit length) given by TU. Its ratio to the power expended against the resistive normal force Cnw (per unit length) by the lateral body motion at speed W gives the Froude hydromechanical efficiency as rl-- C n w W = ( l - x )
1-~-
.
(3.5)
Maximum efficiency is achieved at X = ~,
giving
rlma~--(1 -- ~/~')2, U/V -- 1 -- ~ ,
U / W - - ( 1 - ~/~)/r 1/4.
(3.6) We see that both the maximum Froude efficiency 7/and the corresponding velocity ratio U/W become quite large for 0 < r << 1. As summarized by Lighthill (1975, Chapter 6), the resistive theory holds quantitatively rather satisfactorily for this category of animal locomotion when the inertial effects due to time rate of changes in velocities remain unimportant. Under this premise, the predictions agree with observations for a good number of experimental studies, as reported, for example, by Gray and Hancock (1955) with r _~ 0.54, U/V ~_ 0.3 at Re ~_ 1; by Taylor (1952b) for a polychaete worm with r ~_ 1.5, U/V ~_ -0.2; and by Gray (1939) for a grass snake with r _~0.1, U/V "~ 0.7 at Reynolds numbers as high as 106. At high Reynolds numbers, the resistive theory generally becomes poor for analyzing the swimming of slender animals when the inertial effects become dominant. Next, we discuss this new category of problems.
IV. Classical Slender-Body Theory of Fish Locomotion First, we present an expository review on slender-body theory developed by Lighthill (1960, 1970), Wu (1971 c, 1971d), Wu and Newman (1972), Newman (1973), Newman and Wu (1973), and other contributors for evaluating the
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mathematical biofluiddynamics 1of fish locomotion. It is applicable to a quite general class of body geometry and movement, encompassing, for example, the propulsive motions in carangiform and thunniform on the one hand, and the propulsive forces in amiiform and gymnotiform on the other. In the carangiform mode, body undulation, in form of waves passing distally to the tail, becomes increasingly more significant only in the posterior half, or in an even shorter distal part of the fish's length. The mode of amiiform and gymnotiform describes a variety of species of fishes that propel themselves by passing a wave along an elongated dorsal and ventral fins while the body is held nearly straight, or with only a slight transverse waving movement. Typically, the body may be regarded as slender, characterized by a small slenderness parameter, 6, defined as the ratio of maximum body depth to body length, ~ = 2b/I. The theory is based on the assumption that the amplitude of the undulatory wave motion passed distally along the body is small compared to body length, so that its higher nonlinear effects may be neglected. The Reynolds number in cases of practical interest is generally high, typically ranging from 104 to 107. Insofar as the mechanism of thrust generation is concerned, the body thickness in the direction of bodily displacement (in the z direction) from the stretchedstraight position (in the xy plane) is secondary in effect (Wu and Newman, 1972), which can be neglected or amended if desired, as shown by Newman (1973) and Newman and Wu (1973), who used conformal mapping to map the cross-sectional body contour into a slit so as to make use of the solution for the zero thickness case. Further discussion on this effect has been given by Yates (1977, 1983). With the effect of body thickness neglected, the transverse bodily motion can thus be prescribed, as depicted in Fig. 1, to vary with time t as z = h(x, y, t)
(x, y ~ SblO < x < ~, lYl < b(x)),
(4.1)
where the coordinate system (x, y, z) is fixed in the mean body frame, with a uniform free stream of velocity U in the x direction (equaling the fish's steady swimming velocity in still water), Sb denotes the stretched-straight body planform lying in the xy plane, 2b is the body depth, and IOh/Otl << U and IOh/Oxl << 1 being understood for the motion as assumed. Explicit dependence of h on y is needed for describing undulatory dorsal and ventral fins such as that found in the amiiform and gymnotiform modes analyzed by Lighthill and Blake (1990) and Wu (1983), as well as for analyzing the caudal fin propulsion with vortex sheets shed from upstream side fins (with strength generally varying spanwise). For generalized slender-body theory of fish locomotion, we consider the bodyfin movement to be small, but the movement (in the z direction) may generally IBiofluiddynamics is a word coined by Lighthill (1975).
On Theoretical Modeling o f Aquatic and Aerial Animal Locomotion
(so) [
.._ [ Do=0
re=0
v
~,~ t
I
,L_
r
!
x--O
U
303
I
X--Xm
X--Xc
I X--~
Z
z = h(x, y, t)
FIG. 1. The various sections of flow regions for analyzing fish propulsion: (A) anterior leading-edge section (0 < x < Xm); (13) trailing side-edge section (Xm < x < Xc); and (C) caudal fin section (Xc < x < ~).
depend on both x and y and vary with time t as described by Eq. (4.1). Limited to small amplitudes, the vortices shed from all the fins are assumed to remain in the z = 0 plane for the purpose of applying boundary conditions. With the fluid assumed incompressible and the flow irrotational, we have for the velocity field a scalar potential Co(X, y, z, t) such that r
= Ux + r
y, z, t),
(4.2)
where the perturbation velocity potential r gives the perturbation velocity (relative to the mean body frame of reference) with components Ig -- (lg, U, 11.3) - - V o ~
( V o - - (Ox, Oy, Oz) , Ox - - O/Ox,
etc.),
(4.3)
and satisfies the Laplace equation Cxx + ~yy -Jr- ~)zz = 0 in the flow field. By classical slender-body theory, the slenderness parameter 6 and the displacement function h are both assumed to be of order O(~S) << 1 relative to body length l (which is taken as the length scale). Hence, accordingly, r = O(h6, h2). With
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T h e o d o r e Yaotsu Wu
this order estimate, the three-dimensional Laplace equation for r is reduced, at the leading order, to that for the local cross-flow problem governed by the twodimensional Laplace equation Cyy + Cz., = O,
(4.4)
in which the neglected term, Cxx, is of order O ( ~ 2 , h6) relative to the terms retained. In addition, the Bernoulli equation can be linearized to give. (4.5)
( D - O/Ot + UO/Ox),
(4.6)
= (p~ - p)/p,
where 9 is Prandtl's acceleration potential defined in terms of the pressure p and fluid density p. From Eqs. (4.4) and (4.5), it follows that 9 also satisfies the same two-dimensional Laplace equation [Eq. (4.4)] as for r These two equations then imply that our analysis can be effected in terms of the complex variables (e.g., Wu, 1971c, 1983), ( = y nt- iz,
f = 4) + iO,
F = 9 + iqJ,
v = v - iw = df/d(.
(4.7)
The complex coordinate ~', complex velocity potential f , complex acceleration potential F, and complex velocity v are analytic functions of one another and are related by Df = F
(D =_ O/Ot + U O / O x ) ,
(4.8) (4.9)
Dv = dF/d~.
The boundary conditions for this problem can be expressed in terms of either r or ~, each of which has its own special utilities; for this purpose they are given separately in parallel (with (x, t) serving as parameters in the cross-flow problem) as follows. I. Boundary conditions in terms of 05: II. Boundary conditions in terms of ~: (i) (dpz)+ = D h -- W ( y ; x, t)
(i)
( ~ : ) + -= D W ( y ; x ,
t) ((x, y) e Sb),
(ii)
Ddp+ = O,
(ii)
~ + = O,
((x, y ) e Sw),
(iii)
4)+ = O,
(iii)
~ + = O,
((x, y) e Sc),
(iv)
Dr])+ = O,
(iv)
~ + = O,
((x, y ) o n T.E.),
(v)
f = O(~'-l),
F = 0(~-1),
v = 0(~ "-2) ([~[--~ oo)
(4.10)
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(T. E., trailing edge). Here, the kinematic conditions I(i) and II(i) over the body planform (Sb " lyl < b(x), 0 < x < 1) at z - 0 are clear. 2 In (ii), Sw denotes the wake planform (see Fig. 1) resulting from projecting the vortex sheet shed from trailing edges of side fins onto the z = 0 plane (S,~ : b(x) < lYl < bm = b(xm), b'(x) < 0, with the widest fin tip at x = Xm). In (iii), Sc denotes the region in the z = 0 plane which is complementary to Sb + Sw; this condition is by virtue of the symmetry given by (i). The Kutta condition I(iv) or II(iv) requires the pressure to remain continuous at the trailing edge (T.E.) of side fins and caudal fin [with the pressure p gauged by Eq. (4.6) as with (ii) and (iii)], namely
p(y,O+;x,t)= p(y,O-;x,t)
( y - - b ( x ) , b ' ( x ) < O , xm <X <Xc),
(4.11)
where xc locates the caudal peduncle, from which the caudal fin follows. Also, the asymptotic behavior I(v) and II(v) required of the solution stems from Kelvin's circulation theorem so that the solution excluding the effect of body thickness must be source- and vortex-free at ~" - cx~. In the next section, we first consider the simpler case of motions by slender body with only longitudinal flexibility--that is, with h = h(x, t), independent of y.
A. SLENDER BODY WITH ONLY LONGITUDINAL FLEXIBILITY
Historically, the theory for the special case of locomotive dynamics for bodies with only longitudinal flexibility--that is, with h = h(x, t) independent of y - - i s relatively simple and was first developed by Lighthill (1960, 1970), Wu (1971 c), and others. We present their expositions for different body sections (with or without shedding vortex sheet alongside the body), as shown in the following.
1. The Anterior Leading-Edge Section (0 < x < Xm, d b / d x = b'(x) > O) In the leading portion anterior to the widest section (at x = Xm) of a fish body (0 < x < Xm), its side edges are assumed to be well rounded and its sectional displacement to be straight across (i.e., h = h(x, t) independent ofy), as is quite commonly the case. For this body section, the problem formulated here is a RiemannHilbert problem in its simplest form [namely, with w i = - I m v+ = (r = Dh = W(x, t) for lYl < b, u+ = Re v+ = (r = 0 for lYl > b by Eq. (4.10iii)]; its e The z component velocity written here as W is denoted by V in Lighthill (1960) and Wu (1971b, 1971c).
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Theodore Yaotsu Wu
solution under these conditions and condition (4.10v) is readily verified to be v ( ( , t) = i W ( x , t ) ( ( / H ( ( )
- 1)
(0 < x <Xm) ,
(4.12)
where H((;b)=v/( 2-b
2
(~ = y + i z ,
b=b(x)),
which is defined with a branch cut from ~ = - b to b along the real ~" axis so that H ( ( ) --+ ~ as I~'1 ~ 0o for all arg (, and that on z = +0, H+(y; b) --- v / y 2 - b i for y > b, H+ = - I v / y 2 - b21 for y < - b , and H i = + i v / b e - y2 for lYl < b. Evidentally, this solution is an analytic function of ( satisfying all the conditions I(i), I(iii), and I(v) at all x within this body section. The square root singularity in Eq. (4.12) of the velocity field is typical of linear theory with the body thickness effect neglected. From the corresponding complex velocity potential, f(f;x,t)=qb+i~=iW(x,t)(v/fe-b2-()
(O<X<Xm) ,
(4.13)
we find that the local two-dimensional flow at fight angle to the longitudinal axis has an effective z-component momentum per unit length (in x) given by the line integral of the fluid impluse around the body section, .M - p
(~p_ - ~ + ) d y - m ( x ) W ( x , t) - p A ( x ) W ( x ,
t),
(4.14)
b
where m ( x ) = p A ( x ) -- pzrbZ(x) represents the effective added mass (or virtual mass) per unit length in x (added to the mass of body itself) associated with the fluid of density p in response to the transverse body action. Although this result is based on the simple body shape of vanishing thickness as an approximation, it can be readily extended to the general body shape by assigning A ( x ) the general form A = ~7rb2(x) at a body section of width 2b(x), where/3 is a dimensionless shape factor; in this case, fi = 1 for flat plate of width 2b; also,/3 = 1 for a circular cylinder of radius b, as well as for an elliptical cylinder of major axis b broadwise in motion, and/3 is only slightly less than 1 for a wide variety of geometric shapes (see, e.g., Lighthill, 1970, 1975). As the fluid is sweeping distally past the body with velocity U, the lateral momentum varies at the rate D.M per unit length. By the principle of action and reaction, this gives rise to a differential lift on the plate per unit length in x, equal and opposite to DAd, as L ( x , t) = - D A d
-- -
-~ + U
[ m ( x ) W ( x , t)]
(0 < x < Xm).
(4.15)
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307
This differential lift, valid for the anterior part up to the body section of maximum span at x = Xm, provides a basis for deducing the hydromechanical thrust, energy costs, and so on. Its derivation, first argued by Lighthill (1960, 1970), is recited here with its original simplicity, clarity, and elegance in concept forming.
2. Trailing Side-Edge Section (X m < X < Xc, d b / d x = b'(x) < O)
In addressing the effects of vortex shedding from trailing edge of appended fins on swimming performance, Lighthill (1970) was the first to consider a special class of morphological configuration with the main dorsal fin terminated abruptly with a straight transverse trailing edge at x = xf(Xm ~ Xf < Xc), from which a vortex sheet is shed to fill the region between the dorsal and caudal fins (such as in sp. Cyprinnidae and catfish Siluroidea). Lighthill's theory shows that in addition to the vortex-free added mass re(x) of a section Sx at x ( x f < x < Xc), there is also the added mass fit(x) associated with the vortex sheet in the presence of a frozen body segment, with their conjoint contribution to the lift per unit length given by L ( x , t) = - D { m ( x ) W ( x ,
t) + f i t ( x ) w ( x f , rf)}
( x f < x < xc),
(4.16)
where W(Xf, "t'f) denotes the value of the z-component velocity w evaluated at the fin's trailing edge, which is a t x f , and at the "retarded time" r f = t - (x - x f ) / U , which is the time when the water slice was leaving the fin's trailing edge at x f so as to reach the section Sx (located at x) at time t. It is noted that across the straight transverse trailing edge at x = x f (where the vortex sheet starts), fit(x) has a step jump from fit(Xf - - 0 ) - - 0 t o fit(xf + 0 ) --" pyr(b 2 - b2), where 2b and 2b are the local depths of cross section with and without the vortex sheet included. However, there is no discontinuity between m ( x ) = pyrb 2 for x < x f and the total added mass m ( x ) + fit(x) for x > x f . Nevertheless, it should be pointed out that the pressure has a jump across the trailing edge at x -- x f, and hence does not satisfy the Kutta condition. As a result, the sectional lift has a discontinuity across x f between L ( x = x f - O, t) of Eq. (4.15) and L ( x = x f + O, t) of Eq. (4.16). This, as dicussed previously, is due to a shortcoming of classic slender body theory. However, there is also a broader class of fin configurations exhibiting that posterior to the body section of maximum span (x > Xm), various species of fish (such as dolphin fish Coryphaena hippurus, yellowtail Seriola quinqueradiata, mackerel Pleurogrammus azonas, mullet Mugil cephalus) have dorsal and ventral fins extended out with their edges slanted rearward to form trailing edges from which the flow leaves smoothly to shed vortex sheets trailing along the body side; such a smooth flow at the fin trailing edge is then said to satisfy the Kutta condition, Eq. (4.11). Under this condition, we note that there is a complete analogy
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Theodore Yaotsu Wu
between the boundary conditions I(i) on 4~z - R e ( d f / d ~ ) - W and II(i) on ~z = R e ( d F / d ~ ) - D W, other conditions being equal, including, in analogy, the Kutta condition II(iv) for the trailing-edge flow. This analogy affords the advantage of constructing the solution first in terms of ~, as shown by Wu (1971 c, 1971 d), in analogy with (4.13), to yield
F(r
(X m < X
<Xc)
(4.17)
,
where Xc denotes the longitudinal location of fish peduncle from which the caudal fin ensues with new leading edges. This solution gives on the body surface Sb the acceleration potential 9 + ( y ; x , t) -- q z ( D W ) v / b 2 - y2
(lYl < b(x),
X m < X < Xc),
which clearly satisfies the Kutta condition and gives rise to the differential lift as L ( x , t) =
(p_ - p + ) d y - p b
(~+ - ~ _ ) d y - - m ( x ) D W ( x ,
t),
(4.18)
b
with m ( x ) = prrbZ(x) (same as for the anterior section except here with b'(x) < 0). This result therefore shows that the effect of vortex sheets shed from side fins to trail alongside a body section is to have the fluid momentum transported as if m ( x ) is a constant, although m ' ( x ) < 0. Thus, if this body section is held fixed as a flat plate inclined at small incidence, then W = const., and hence by Eq. (4.18), L = 0, which is a well-known result of classic slender-body theory. Physically, this is explained by the fact that the vortex sheets fill in the gap left by the reduced width of the body plate, making the new flow boundary a flat plate uniform in span (of width bm, x > Xm) on which m W = const., hence L = 0 by Eq. (4.15) as well as by Eq. (4.18). Now, suppose the trailing section is cut off along x = Xm, the anterior delta-shaped plate retains its original total lift (because the cut-off section carries no lift). In this case, however, the Kutta condition is no longer satisfied at the trailing edge along x = Xm, nor is the lift L ( x ) continuous across x = Xm unless the span is stationary there, namely with d b ( x m ) / d x = 0. This is a known shortcoming of the classic slender-body theory. To the same degree of approximation in the present case of unsteady flow, there is no discontinuity between the lift L(xm - 0, t) of Eq. (4.15) and the lift L(xm + 0, t) of Eq. (4.18), provided b(x) and W ( x , t) are both C2(x = X m ) and b ' ( x m ) - - O. To obtain the vortex-induced upwash convected into the caudal fin section to bear influence on the kinematic boundary condition on caudal fin, we must derive the f and v entering the caudal fin section. To derive f ( ( ; x , t) for (Xm < x < Xc),
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
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we first write Eqs. (4.8) and (4.17) jointly as
D{U - i W ( v / (
2 - b 2 - ()} -- - i W D v / (
2 _ b 2 -- - i U W O x r
- b 2.
Integrating this equation along the characteristic line x - Ut - ~ - const, yields, for Xm < x < Xc, f ( ( ; x , t) -- i W ( x , t)(v/( 2 - b 2 - ( ) - i
fx W(Xl, Z'l) OXl /(2
b 2 dXl,
m
Z"1 -- "t'(X, t; Xl) =~ t - (x - X l ) / U,
bl -
(4.19)
b(Xl),
where "c1 is the retarded time with respect to x 1. Here, the integration constant (a function of x - Ut only) in Eq. (4.19) is zero since this f already satisfies condition I(v). Hence, by integration by parts,
f ( ( ; x , t) -- - - i W ( x , t)( + iW(xm, r m ) r "2 -- b 2 + i "Cm = r ( x , t ; X m ) ,
fx r
3
~2 __ b~ -~xtW(x1,
(4.20)
rl)dXl,
m
O
U~W(Xl, Oxl
Z'l) = D W ( x ,
The complex velocity is simply v = d f / d ( ,
t)lx=x,,t=r,,
bm = b ( x m ) .
or
r v = v - iw = - i W ( x , t) + iW(xm, rm) v/(2 _ b2 + i
( m
~ 0 W ( x l , Z'l ) d x l
On Sb + S~, ( -- y + iO, for the trailing-edge section (Xm < x < xr have the result
~ i = zFW(xm, rm)
(4.21)
9
~2 __ b 2 3 x l
b 2 - y2 ztz
we therefore
b~ - y2-~xl W ( x l , t l ) d x l ,
(4.22)
y ~0 W ( x l , r l ) d x l , , / b 2 _ y2 OXl
(4.23)
m
u+
+ W ( x m , rm)
y + , / b 2 _ y2
u
x, -- x
=xf w+ -- W ( x , t) = W ( x , t) - f x / r
(on &,
lYl < b(x))
(on S~,
lYl > b(x), x > x f = b - l ( y ) ) ,
(on Sb,
lY[ < b(x))
y 3 W(Xl, Z'l)dxl , _ b 2 3xl
(b < lyl < bm),
(4.24)
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Theodore Yaotsu Wu
wherexf = b - l (y) is the inverse function of y = b ( x f ) ~ b f forgiven y ~ (b(x) < lYl < bm) so that ( x f , y = b y ) locates the point on the trailing edge from which the vortex line starts to shed to reach a point of given (x, y)(b(x) < lYl < bm) on the vortex sheet. In the tail section (Xc < x < e), the trailing vortex sheets continue to have the z-component velocity transported along the characteristic line x - U t - - ~ , yielding Wv(y; x, t) -- -
fx xc
Y
~
O
i v/y 2 - b ~ o x l
W ( x l , rl)dXl ,
(bc < ]y]
<
bm, x > Xc),
(4.25) y
= --W(Xm' Tm) v/Y 2 -
b2m
_ f x x' ~ y 3 (Xl, ~W m v/Y2 -- b~ OX1
"t'l ) d x l
(on So, JYl > bm),
(4.26)
This result shows that the upwash being transported with a retarded time "t"1 has a spanwise distribution dependent on y. Consequently, this y-variant correction to be imposed on the velocity field in the caudal fin section accordingly requires a generalization of the present simpler theory to a more general theory, which we address in Section IV.B.
3. The Caudal Fin Section (xc < x < g.)
When the vortex sheets shed from side fins enter into the caudal fin section, the interaction between the vortex sheet and caudal fin has been argued as being capable of augmenting the lift and thrust generation handsomely, especially when they become opposite in phase. However, the problem of evaluating this dynamic interaction is complicated as a result of several new issues, a primary one being that the upwash induced by the vortex sheets transported into this caudal fin section bears with it a conspicuous spanwise variation given by Eqs. (4.25) and (4.26). An earlier attempt by Wu (1971 c) contains a deficiency that was removed by Wu and Newman (1972), who took the body thickness effect into account, yielding a solution to the caudal fin problem in terms of an integral equation of the Abel type that can be evaluated numerically. A consistent slender-body theory was subsequently developed by Newman and Wu (1973) and Newman (1973) by taking into account both the kinematic and dynamic interactions and the effect of body thickness on trailing vortices. This problem is discussed in Section IV. B for the generalized case for a thin body-fin system moving with displacement h(x, y, t) varying with both x and y.
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In closing this section, we note that for fishes known for their high performance using the so-called lunate, or crescent-moon shaped tails for propulsion, it calls for abandoning slender caudal fin analysis in favor of adopting high aspect-ratio oscillating wing theory, as commented on in Section II, and is discussed in more detail in Sections V and VI.
B. A GENERALIZEDLINEAR SLENDER-BODY THEORY OF FISH LOCOMOTION The case of body-fin movement that varies with both x and y, as that found in amiiform, gymnotiform, and other similar modes, has been investigated by Wu (1983). With this generalization, the simple and elegant physical concepts and simple calculations presented previously for the simpler case with rigid transverse body sections are no longer sufficient for attaining solutions. However, we may resort to more powerful mathematical methods for resolving such challenging biofluiddynamic problems to attain new results of interest. Next, we present a synopsis of the mathematical construction of the solution before we turn to consider the fully nonlinear, large-amplitude theory of aquatic and aerial animal locomotion. The general problem formulated at the beginning of Section IV.A is in a form of the Riemann-Hilbert problem in its simplest version. Our analysis to derive the required solution to more general problems can be greatly facilitated by using complex function analysis and application of Plemelj's formula (see, e.g., Muskhelishvili, 1953) G(~ ) -- ~
1
g(y~) dy~ Yl -
(4.27)
(~ q~ s
1 1 f c g(Y~) dy~ G+(y) - +-~g(y) + ~ yl - y
(y ~ s
(4.28)
hence g(y) = G+(y) - G_(y), and G(~') = ~
1 fz: G + ( y l ) - G - ( Y l ) d y ~ (~ q~ s yl - ~"
1 f~ G + ( y l ) - G_(Yl)dyl G+(y) + G_(y) = rci yl - y
(y ~ s
(4.29) (4.30)
where G(~') is a complex-valued function of ~ = y + i z, analytic in the entire ~" plane except for ~" 6 s s is an arbitrary smooth line (here taken to be the entire real axis of ~'), G+(y) and G_(y) are the limiting values of G(~') as ~" approaches from the left (upper) or right (lower) side of s as viewed along s respectively, to a point y on s these limits being assumed to be Hrlder continuous, and further, the integral in Eqs. (4.28) and (4.30) assumes Cauchy's principal value.
Theodore Yaotsu Wu
312
By classical analysis and using Plemelj's formula, the required solution can be obtained for different longitudinal sections Sx of the body as follows.
1. The Anterior Leading-Edge Section (0 < x < Xm, d b / d x = b'(x) > O) In this body section, the side edges along y = +b(x) are assumed to be either well rounded or moving well feathered to flow to avoid local flow separation as before, except that the body transverse motion can now have variations in both x and y. To find the solution to v, we first note that its boundary conditions have the symmetry v+ + v_ = - 2 i W for lyl < b; v+ - v_ = 0 for lyl > b; and we next seek its homogeneous counterpart [say, H(()], such that H+ + H_ = 0 for lyl < b; H+ - H_ = 0 for lyl > b. From this, it follows that G(() = v ( ( ) H ( ( ) can be determined immediately by applying Plemelj's formula (4.29) and (4.30), provided that the H ( ( ) so found makes v(() satisfy all the remaining boundary conditions. Thus we find the unique solution as
df ~--p--v-iw--
1 f ~ H+(yl ; b) W(yl" x, t) Yl - ( dyl
d(
7r ~_b H ( ( ; b ) H((;b)=v/(
2 - b2
( 0 < x <Xm),
(4.31)
(( = y + iz).
Here, the function H ( ( ; b) is again as defined in Eq. (4.12). From Eq. (4.31), the corresponding f a n d F can be readily deduced by integration. In particular, if W is independent of y, Eq. (4.31) reduces to the simple solution (4.12).
2. Trailing Side-Edge Section (Xm < X < Xc, d b / d x = b'(x) < O) Within this section, the elongated dorsal and ventral fins have edges slanted to trail behind the flow (with b' (x) < 0), and can move to vary with y, at which edge the Kutta condition is invoked. Under this condition, we note again the complete analogy between the two sets of boundary conditions I(i) on 4~z = R e ( d f / d ( ) = W and II(i) on ~z = R e ( d F / d ( ) = D W, other conditions being equal. Whence, by analogy with Eq. (4.31), we have (Wu, 1983)
d F _--
1 J_f~ H+(yl; b) DW(yl; x , t ) d y I
d(
Jr
b H((;b)
YI-(
(Xm < X
<Xc).
(4.32)
On integration, we obtain F as F ( ( ; x, t) --
l f_~ 7r
d(
cr H ( ( ; b )
b
H+(yl.b) DW(y , 9x,t)dy ' Yl --
which can be seen to satisfy also the Kutta condition and condition (v).
(4.33)
On Theoretical Modeling o f Aquatic and Aerial Animal Locomotion
313
This F field provides the pressure distribution within this trailing-side edge body section. The corresponding velocity potential field can be obtained, straightforwardly, by integrating Eq. (4.8) (i.e., D f = ft + Ufx = F), along the mathematical characteristic lines x - Ut = ~ = constant, starting from x = Xm [at which f is known from Eq. (4.31)], with F given by Eq. (4.33). Then the velocity field can be deduced from v = d f / d ( , thus yielding the ( + )-side limit of v = v - i w over Sb and Sw (for details, see Wu, 1983). The result of w+ = w(y, 4-0; x, t) =-Wv(y; x, t) gives the "sidewash" induced by the entire vortex system, which consists of the bound vorticity lying in the planar body surface Sb and the free vortex sheet over the wake surface Sw : (b(x) < [y[ < bm, Xm < X < Xc). This induced velocity field, when convected with the free vortex sheet into the caudal fin section, is capable of assuming an important role in interacting with the caudal fin for the possibility of enhancing its thrust production and propulsive efficiency.
3. The Caudal Fin Section (xc < x < ~)
In this body section, the caudal fin has new leading edges in the presence of the vortex sheets incident from upstream that in turn interact with the fin motion. In the framework of linear theory, the velocity field can be constructed by superposition of two components, one being vv, which is that induced by the entire vortex sheet convected (invariant along the characteristics) downstream into the caudal fin section (with velocity U, as if without caudal fin), and the other, vc, which is the velocity due to the motion of the caudal fin itself. This gives within the caudal fin section the velocity field as v ( ( ; x , t) = v c ( ( ; x , t) + v~,((; x, t)
(xc < x < ~),
(4.34)
where x = ~ marks the caudal fin trailing edge. Based on the known sidewash velocity, Wv, induced by the vortex sheets on the caudal fin surface (which is assumed to be slender like the entire body configuration), we therefore obtain for v~ the solution in integral form (Wu, 1983) as Vc(~;x, t) =
Jr
H + ( y l ; b ) W ( y l ; x , _..t)- W v ( y , x , t) dyl , b H(~'; b) Y l - - ~"
(4.35)
where W~(y; x, t) is the sidewash induced by the vortex sheets as described in Section IV.B.2. This solution exhibits several important features. First, this v~ reduces to Eq. (4.31) for the anterior section when W~ vanishes. When W~ --/: 0, the interaction between the caudal fin movement and the vortex sheet sidewash is seen as being possible to enhance greatly the thrust generation and propulsive efficiency,
314
Theodore Yaotsu Wu
especially when their own induced velocity fields maintain opposite in phase because the two W terms in Eq. (4.35) then totally augment each other. Typical cases of numerical examples have been investigated by Su and Yates (1983) and by Yates (1983), yielding results that lend quantitative support to our qualitative discussion of Eq. (4.35).
V. A Unified Approach to Nonlinear Theory of Flexible Lifting-Surface Locomotion In pursuing further improvement of slender-body theory for investigating fish swimming, Lighthill (1971) developed a large-amplitude elongated-body theory of eel-like fish locomotion with arbitrary amplitude. By taking the trajectory of a swimming slender fish lying in a fixed horizontal level and using a Lagrangian coordinate a to identify a point x(a, t), z(a, t) on the spinal column, Lighthill generalized his reactive force theory by distributing along the moving spinal column (or backbone) flow singularites that are relevant: here, the momentum carrying dipoles (source doublet), with which he evaluated the water momentum given by the virtual mass of the surrounding fluid in the transverse section perpendicular to the column per unit length times the velocity component of fish in that direction. Lighthill further proposes that propulsive thrust can be obtained by considering the rate of change of momentum within a volume V, which includes the fish and excludes the wake by introducing a geometric plane FI perpendicular to the caudal fin through its posterior end, which separates the wake from flow volume V. The momentum calculation then takes into account both the transfer of momentum across I-I and the action of pressure at the plane FI resulting from the swimming motion. The geometric FI plane, introduced to exclude the wake momentum calculation, is supposed to be thought of as swinging around with the caudal fin throughout the fin's waving motion. With respect to this new concept and the details of calculation, we refer the reader to the original paper (Lighthill, 1971). In the spirit of Lighthill, this present study pursues further improvement in several new aspects, including one attempt to make the trajectory and amplitude of fish body movement entirely arbitrary, another to relax the restriction to bodies being slender, and still another to admit more versatile compositions of the added mass in order to account for the effects due to the free vortex sheets shed from trailing edges of appended fins of arbitrary configuration. Consideration of these issues is believed necessary to determine the nonlinear effects generally involved in fish locomotion studies. With these extensions, the nonlinear unsteady theory of lifting-surface locomotion appears to be applicable in a
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
315
unified manner not only to aquatic locomotion of fish and cetaceans, but also to the aerial locomotion of birds and insects. These objectives actually have had a strong appeal to Wu (2001) ever since his preparation for a lecture he was invited to deliver on 5 January 2000 at a special International Conference on Biofluiddynamics in Memory of Sir James Lighthill held at Technion and University of Haifa. That work, a forerunner of the present study, presented a three-dimensional, large-amplitude, flexible lifitingsurface theory for modeling aquatic and aerial animal locomotion at high Reynolds numbers, with the nonlinear effects fully taken into account, and is for the simulation of the propulsive movement of fish, bird, or insect with flexible lifting surface and with appended fins of negligible thickness, but otherwise of quite arbitrary configuration, moving with arbitrary amplitude and trajectory. For this general case, it is necessary to designate the displaced flow boundary of not only the lifting surface of the body [say, Sb(t)], which is prescribed, but also the vortex sheets shed from the lifting surface to trail as a free wake, denoted by S~(t), with its location unknown a priori (aside from the condition on the vortex sheets remaining attached to the trailing edges of the fins). The theoretical formulation is based on the assumption that the fluid is incompressible and inviscid, and the flow is irrotational, obeying the following principles: (i) The inherently nonlinear kinematic and dynamic boundary conditions are to be expressed in terms of the surface variables on the flow boundary and then applied exactly. (ii) Closure of the system of model equations is accomplished by expressing the solution for the velocity potential in terms of a surface distribution of dipoles as the relevant flow singularities for facilitating comp,Jtational work as well as for analytical studies. (iii) Hydrodynamic thrust and power expenditure in propulsion are to be derived from the pressure obtained from the solution of the exact model equations based on the momentum balance involving the entire flow field in reaction to the propulsive movement of the boundary. Following these principles, the solution for the velocity potential is expressed in terms of a distribution of surface dipoles over Sb(t) + Sw(t); the dipole strength is determined by the prescribed flow velocity normal to Sb(t) together with satisfying the dynamic condition of zero pressure jump across the trailing vortex sheet at Sw(t), including the trailing edge of Sb(t) for the Kutta condition. For the initial value formalism, all the flow disturbances are required to vanish at infinity. Along
316
Theodore Yaotsu Wu
this approach, the analysis leads to a set of differential-integral equations for calculation of the surface dipole distribution over the lifting surface and its vortex wake. In summary, Wu (2001) presented the basic principles and a method of solution to the general three-dimensional problem of flexible lifting-surface locomotion at high Reynolds numbers with the motion of body surface Sb(t) expressed in terms of z = h(x, y, t) for (x, y) lying in the projection of Sb(t) on the (x, y) plane. The final integral equation can be directly applied for computational purposes. Nevertheless, it is noted that the explicit expression of the lifting surface movement in terms of the class of displacement function z = h(x, y, t) is not versatile enough to describe body movement in its more general forms, such as with sharp sideor U-turns in fast maneuvering and impulsive large-amplitude control operations, such as figure-eight wing beats in hovering hummingbirds and highly curved turns in lunate-tail swimming. Such shortcomings have subsequently been overcome by adopting Lagrangian variables for describing body motion for the general purpose. In the following section, we shall consider the two-dimensional theory, and show that considerably more analysis can be carried out further in this manner to expose the intricacies of the challenging tasks involved in computation, and to streamline the complete procedure of calculating the final solution.
VI. A Unified Nonlinear Theory of Two-Dimensional Flexible Lifting-Surface Locomotion The theory of two-dimensional lifting-surface locomotion is of fundamental importance in several aspects. First, it provides a valuable limiting case for asymptotic evaluation of lifting surfaces of large aspect-ratio as found in various modes of aquatic and aerial animal locomotion. Furthermore, limiting to two-dimensional configurations may afford the simplicity needed to develop efficient and even elegant methods of solution that can enable generalizations to three-dimensional and more general cases more effectively with improved experience and understanding. Thus we consider the irrotational flow of an incompressible and inviscid fluid generated by a two-dimensional lifting surface Sb(t) of negligible thickness, moving through the fluid in arbitrary manner, even with a good degree of flexibility, as found among various animals in nature. Its motion can be described parametrically by using a Lagrangian coordinate ~ to identify a point X(~, t), Y(~, t) on Sb(t) varying with time t as
Sb(t):x=X(~,t):(X(~,t),Y(~,t))
(-l<~
>0).
(6.1)
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
z = z(~, t)
y, 77
s
\
~
\ \\
-1
3 17
, {
r.,E. {'
+1
(m
x,~
FIG. 2. The Lagrangian coordinate system (s~, 77) used to describe arbitrary motion of a two-dimensional flexible lifting surface moving along arbitrary trajectory through water in a reference frame (x, y) fixed with fluid at infinity.
Here, all the lengths are scaled by the half-chord of Sb. A simple choice of ~ is the initial material position of Sb(t = 0), which is taken to be in its stretched-straight shape such that X(~, 0 ) = ~, Y(~, 0 ) = 0 ( - 1 < ~ < 1), lying in an unbounded fluid initially at rest in an inertial frame of reference (Fig. 2). The position function X(~, t) in self-propulsion is regarded as being the result of active propulsive motion, and the motion of Sb(t) resulting from the fluid momentum then prevailing, including any lateral reactive recoil motions involuntarily produced. The body surface is so thin that we may neglect its thickness distribution, except it is understood that its leading edge (at ~ = - 1 ) is sufficiently well rounded to avoid local flow separation, and to realize the leading-edge suction, and its trailing edge (at ~ = 1) is cusped to satisfy the Kutta condition requiring continuity of local pressure across the trailing edge. The possibly flexible body surface Sb(t) is tacitly assumed to be inextensible, such that X~ + y 2 _ 1,
(6.2)
as it is quite general the case (but this may be somewhat relaxed if needed, such as for describing membrance-like lifting surfaces like bat's wings). The surface has tengential (unit) vector s and normal vector n given by
,(~, t) = OX/O~ = (X~, Y~),
n(~, t) = (-Y~, X~).
(6.3)
The point ~ on Sb(t) moves with prescribed velocity
U(~,t)=OX/Ot=(Xt,
Y,)
(-l<s ~
>0),
(6.4)
318
Theodore Yaotsu Wu
which has tangential and normal components Us(~,t)-U.s:-(X~Xt
-4- Y~Yt), U ~ ( ~ , t ) - U . n = ( X q Y t -
YqXt).
(6.5)
For the flow field, we first express it in the Eulerian field form to begin with, so the velocity u -- (u, v) = (u(x, y, t), v(x, y, t)) of the assumed incompressible (with Ux + Uy "-- 0 ) and irrotational (with Uy - Vx = 0) flow has a velocity potential, 4~(x, y, t), and stream function, ~p(x, y, t), such that u = 4~x = ~y, v = ~ y = -~Px. It follows that the complex coordinate z, complex potential f, and complex velocity w, z -- x + iy,
f -- 4) + i 0 ,
w = u - iv -- d f / d z
(6.6)
are analytic functions of one another. We now take on Principle (i) to express all the boundary conditions in terms of these variables evaluated at the flow boundary. To accomplish this, it is convenient to adopt a set of intrinsic coordinates (~, r/), with the r/axis pointing in the direction of the normal vector n, and 77 - 0 coinciding with Sb(t) for - 1 < ~ < 1, t > 0. During the course of unsteady motion of Sb(t), it is generally the case that a vortex sheet is continuously shed from the trailing edge (T.E., at ~ = 1) to form a free wake surface, Sw(t), which we identify with the extended material coordinate ~ for ~ > 1, Sw(t)"
where
x - X(~, t) = (X(~, t), Y(~, t))
~m(t) identifies the trajectory [(X(~m,
reaching
~m -
(1 < ~ < ~m, t > 0),
(6.7)
t), Y(~m, t)] of the starting vortex
~m(t) at time t after being shed at t - 0 when motion began, whereas
X(~, t) of Sw(t) is to be determined for (1 < ~ < ~m, t > 0) as a part of the solution. To describe the flow in a neighborhood of Sw(t), we adopt the same intrinsic coordinates (~, r/) such that 0 - 0 coincides with S~(t) for 1 < ~ < ~m, t > 0. Physically, it is of theoretical and practical interest to note that the wake S~(t) is continually being opened just beyond the trailing edge to create a new stretch 3X(1, t) - U(1, t ) r t in a small time interval 6t tangentially to the trajectory traversed by the trailing edge at the rate U (1, t). So, in time interval 0 < t' < t since the motion started, the past trajectory of the trailing edge (X(1, t'), 0 < t' < t) is in fact the "birthplace" of the wake vortex sheet, from which the free vortices would have been convected by their local induced velocity field that may be weak for wellfeathered motions of Sb(t) but can be very significant for impulsive body motions. By analytic continuation, the intrinsic coordinates (~, 0) form a complex reference plane ( = ~ + i ~ such that the ( plane and the z -- x + i y plane are related by a conformal transformation, which generally exists Vt > 0 and is denoted by z - z((, t) and has the following general properties. First, for sufficiently smooth bodily movement [X(~, t), Y(~, t)], d z / d ( # 0 or c~ on Sb(t), even including its
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319
end points because Sb(t) is smoothly curved and both Sb(t) and its image plate have zero thickness. In fact, d z / d ( at an end point of Sb has its modulus equal to the distance of the end point to the origin and has its argument equal to the slope angle of Sb(t) at that end, in accordance to the general theory. Furthermore, the point of z = c~ is generally not mapped onto ( = cx~ (unless Sb remains to be a flat plate) but to some point (say, (0) in the finite ( plane, which tends to ( = c~ as Sb reduces its curvature everywhere to 0. Otherwise, z -- z((, t) is analytic in the entire ( plane except at (0 and also possibly at ( = c~. It is unnecessary to determine the actual time-dependent conformal mapping z -- z((, t), a time-consuming and very tedious task, because we need use only the analytic relationship of z -- z((, t) in a small neighborhood of Sb(t) + S~(t). For further analysis, it is convenient to introduce a transformation between the Eulerian field description with (x, y, t) and the Lagrangian material description with (~, 17, t') in terms of: x--s
Y=Y(e,O,t'),
and
t=t',
(6.8)
where the overhead-symbol indicates a functional relation. As understood, the time derivatives with respect to t and t' mean that for a differentiable function
(6.9)
F(x(~, O, t'), y(~, O, t'), t) - P(~, 17, t'), we have, by the chain rule,
OF aP I _ OF OF OF dF at' = Ot7 ~,~ -at + U-~x + V-oy - -dr
(6.10)
signifying that these equivalent operators all mean material time differentiation by following instantaneously the fluid particle identified by (~, 0) at time instant t' - t, and the subindices (~, 0) indicating that they are held fixed in differentiation. In the absence of viscous effect, Sb(t) and Sw(t) are singular surfaces across which certain flow variables may have jump discontinuities. For an arbitrary flow variable F(~, r/, t'), we denote its value on the 4- side of Sb(t) and Sw(t) (located at r / = 4-0) by P(~, r/ -- -+-0, t ' ) - - P + ( ~ , t ' )
(-l<~<~m,t
>0),
(6.1 l)
a relation that holds for z+ - x• + iy+, f+ - r177+ i~+, w+ = u+ - iv+, and pressure p+, all with their functional argument (~, t') omitted as understood. The variables with subindices • are called the boundry surface variables
320
Theodore Yaotsu Wu
[i.e., variables pertaining to the top or bottom boundary surfaces of Sb(t) and S~(t)]. For their derivatives, Eq. (6.10) gives :
-37 •
--
37
+ u+ 9( V F ) i
(U+ --- (hi4-, 11)+), V = (Ox, Oy)),
(6.12) and similarly, = s 9( V F ) i ,
+
-- n . (VF)+.
(6.13)
Here, the operator %, on F+ gives the material rate of change of F, ( d F / d t ) + , along the two sides of Sb(t) and S~(t), and the operator 0~ on F+ gives the surface gradient s 9(VF)+ of F and (0oP) the normal gradient of E For the kinematics, we note that the fluid particles on the two sides of Sb(t) + Sw(t) at X(~, t) are moving with velocities
u+(~, t') = Ox+(~, t')/Ot'
(--1 <~ < ~ m , t ' > 0),
(6.14)
with u+ # u_ -r U in general, while the two fluid particles on the -1- sides are both identified withx+(~, t') = x_(~, t') = X(~, t') on Sb(t) + S~(t) at the instant of interest. In the sequel, the - of P may be omitted for brevity as understood, if obvious. In complex form, Eq. (6.14) becomes 0z+ (~, t') - ~+(~ t') Ot'
'
'
(6.15)
where the overhead -denotes the complex conjugate. In terms of the tangential velocity us(~, t) and normal velocity un(~, t) components, we also have u~s - iu n+ = (u+ - iv+)(X~ + iY~) -- w+(~, t')OZ(~, t')/O~,
(6.16)
where Z = X + i Y. The kinematic condition, asserting that any fluid particle once setting on the boundary surface Sb(t) + S~(t) remains moving along it, can be expressed as on Sb(t)"
Un+(~, t') = Un( ~, t') -- Un(~, t')
( - 1 < ~ < l, t > 0),
(6.17)
on Sw(t)"
Un+(~,t') -----Un(~, t t) -- Un(~, t')
(l < ~ < ~m, t' > 0).
(6.18)
Physically, this signifies that the flow velocity normal to the surface is continuous at the surface and equal to the normal velocity of the (moving) body surface, but the tangential velocity at the boundary surface generally has a jump in magnitude across the surface. With such discontinuities, the surface Sb(t) becomes a bound vortex sheet, whereas Sw(t) forms a free vortex sheet being convected away with the fluid. In viscous fluid, they form a thin boundary layer in which the vorticity is
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distributed within the layer, whose integral across the layer gives the velocity jump in the inviscid region just outside the thin layer (which reduces to zero thickness with the viscous diffusion effects neglected). In this sense, we have the distinction that Un(~, t') is prescribed on Sb(t), but is unknown a priori on S~(t). The latter is to be determined by also imposing a dynamic condition invoking that the pressure be continuous across S~(t), namely, p+(~, t') = p_(~, t')
(1 < ~ < ~m, t > 0),
(6.19)
signifying that free vortex sheet cannot sustain any pressure jumps, including the trailing edge at ~ = 1 for the Kutta condition, where the pressure p is given by the Bernoulli equation P + aq~ + 1 (u 2
p
--~
-p -q a ~
2
-~ s + U , ) - - P
at'
1 (u 2 + u,) 2 -- 0,
2 s
(6.20)
in which the latter form follows from using Eq. (6.10), so that, under conditions (6.17) and (6.18), ~(p-
- p+) -
-g
+
~
_ + ~ ( ~ s + + , s ~ ( , s + - Us)
(z 6 s = Sb + Sw) a
= ~,(,+ -,_~-
(6.21)
1
~(u~ + ~;)(U+s - Us) (z 6 s = Sb + S w ) .
(6.22)
To construct the unique solution to this problem, we now follow Principle (ii) to adopt the integral representation of a surface distribution of mass dipole (or mass doublet) of strength # per unit length streamwise (and unit depth transversewise) over the boundary surface Sb(t) + S~(t), giving the complex potential f = 4~ + i~p as
t)dz, f (z, t) -- ~ 1 fL #(~" z'-------~z As z --+ Z(~, t) 6 E from the r / = •
f+(z, t) - • 1
( z ' = z(~', t'), z ~/2 = Sb + S~).
(6.23)
side, we have, by Plemelj's formula (4.28),
t) + ~il fc #(~"z'----~zt)dz'
(z = Z(~ t) ~ s
(6.24)
with the integral assuming Cauchy's principal value. We therefore have the dipole strength related to the jump in 4~ as /z(~, t) = ~b+(~, t) - 4~-(~, t)
(dipole strength/length)
(6.25)
Theodore Yaotsu Wu
322
while the normal derivative of 4~is continuous across/2 - Sb + S~, [see, Eq. (6.29)]. From this, we derive the complex velocity from w(z, t) - d f / d z to obtain 1
w(z)-
~i
d
#(~', t)
_
1
-
2re i
1
dz'=
dz' z ' - z
~/(~', t ) d ~ ,
y -
2rri ~ / z ( ~ ' , t ) [
~
d~
Ose ' z ' - z
z r
z' - z
-~'
by integration by parts, where t) -
0# o~
t) -
0 o~
(ep+
-
ep_) -
(U+s -
Us)
(6.26)
is the vorticity per unit length along 12 -- Sb + Sw or along the ~ axis, which is equal to the jump in tangential velocity across the boundary/2. By applying Plemelj's formula once again, we find that
1 /dz 1 ~y(~',t) ~z-7 d ~2-' z w+(z, t) = 4--~ y(~, t) -d~ + ~
(z = Z(~,t) 6 12);
(6.27)
and hence, from Eqs. (6.16) and (6.27), there results
1 1 dz ~ y(~', t)d~,. Us~ - i Un -- -+--~)/ ( ~ , t) + 2 rr----~d--~ z '----~-Z
(6.28)
From this equation (6.17) and (6.18) are automatically recovered and further, for both z, z' ~ s
s zdfg b+&. Y(~" 't) d~' z } u+(~, t) - u-~(~, t) - -~1 Re { d-:-7
U sm ~
l(u++u_)_
-2
s
1 2re Im
{dzf -~
Z t
(6.29)
--
y(~',t) } b+&, z ' - z d~'
(6.30)
of which the first equation shows the continuity of normal velocity u,, = Och/On across 12 and the latter gives the algebraic mean of tangential velocity Us across the boundary/2. Moreover, the fact that the us and u,, velocity components arise in Eq. (6.28) so naturally lends simplicity to the subsequent analysis as well as to related computational work. Summing up, we have obtained Eqs. (6.29) and (6.30) that characterize the kinematic boundary conditions as follows: on Sb
:
on Sw :
Un(~, t) = U,(~, t) -- is prescribed,
y(~, t) - is unknown.
y(~, t) - is material invariant,
un(~, t) - is unknown.
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323
To this end, we point out that by the material invariance of free vorticity, it is meant that according to Helmholtz's theorem, vorticity in irrotational flow of incompressible inviscid fluid cannot be generated in any interior bulk of the fluid, but only at a boundary surface; and once generated, it leaves the surface as a conserved property of the local fluid material. In viscous fluid, the boundary layer generated at such high Reynolds numbers is very thin, but distributed with all the vorticities of large values on both sides of Sb(t) that finally merge to be convected smoothly downstream from the trailing edge. With the viscous effect neglected in the ideal fluid (without diffusion), this layer reduces to a singular surface of vortex sheet Sw(t), with the vorticity frozen with the local fluid as a conserved property. It therefore implies that Oyw at
+Usm
Oy~
a~
=0
(1 < ~ <~m, t > O ) , -
(6.31)
-
where Usm is given in Eq. (6.30). Under either condition (6.19) or (6.31) serving as the Kutta condition, the trajectory X(~, t') of the free vortex sheet Sw(t) can be determined either analytically, by means of an integral equation, or numerically, by a time-marching procedure, to be delineated later. To account for the total net vorticity, we evaluate Kelvin's circulation, F(t), given by the line integral of the velocity component along the contour (taken clockwise) from point P_(~, r / = 0 - ) to point P+(~, r / = 0+) without cutting the boundary surface/2 = Sb(t) + S~(t), r ( ~ , t) -
f
u. dx =
d ~ - 4~+(~, t) - 4~-(~, t) -
.(~,
(-l<~<~m,t>0),
t)
(6.32)
#(~, t) being the local doublet strength. Circulation F(~, t) is also related to the vorticity distribution through Eq. (6.26) to give r(~, t) -
u . dx =
y(~, t)d~
( - 1 < ~ < ~m, t > 0).
(6.33)
This also results from Stokes's theorem, which states that the line integral of F is equal to the surface integral of all the vorticity V x u (pointed into the surface) spanning the integral contour. Thus, the circulation around any contour E completely circumventing s must be conserved, Fz(t) --
f0
y(~, t)d~ = F(0),
(6.34)
a constant equal to the circulation of a stationary initial state, which is zero for a rest state. This is known as Kelvin's circulation theorem. Of particular interest
324
T h e o d o r e Yaotsu Wu
is that by Eq. (6.34), the circulation Fb(t) around lifting surface Sb(t) and the circulation around wake S ~ ( t ) are related as Fb(t) =
u . dx -
?,(~, t)des - -
?,(es , t)des -
-Fw(t).
(6.35)
B
Now consider the variation 31-'b = Fb(t + 6t) - Fb(t) during a smooth motion at a small time 6t apart, this 6Fb, by Kelvin's circulation theorem, must be equal and opposite to the new vorticity being shed from the trailing edge into the new gap 6~ created by the forward moving trailing edge with velocity Us plus the fluid motion with tangential mean velocity Usm; that is, 6Fb(t) =
6r~(t) = - y ( 1 , t)6~ = - y ( 1 ,
t)[Us(1, t) + Usm(1, t)]6t,
(6.36)
where U s ( l , t) is the tangential velocity of the forward moving trailing edge. This vorticity balance does not involve the vorticity sheet that has previously been shed because the latter moves on invariant while being convected with the local fluid bulk. With the vortex shedding rate determined, we can set up a primarily numerical method for computation of the initial-value problem as follows.
A. METHOD I--COMPUTATIONAL TIME-MARCHING METHOD
For computation of solution by discrete mathematics, we take a time sequence to = 0, tl, t2 . . . . . (tk+l -- tk = At, k = 0, 1,2 . . . . ), with At sufficiently small. At to, the body surface Sb(0) is taken at the stretched-straight position in an unbounded flow field, which is taken at rest (or at a stationary state). At tl, Sb(tl) assumes its new position given by X(~, tl) of Eq. (6.1) for ( - 1 < ~ < 1), while S w ( t l ) has its first onset grid opened to length A~ as a smooth extension of Sb(t~) beyond the trailing edge to receive the first element of a starting vortex that is being shed to form the new wake, with A~ and the shed vortex strength y(1, tl) given by Eq. (6.36). The unknown vorticity y(~, tl) over ( - 1 < ~ < 1) on Sb(tl) is determined from the integral Eq. (6.29) under condition (6.17) by applying some efficient numeric method of high precision. This way of evaluating the discretized unknown y(~, t) for ~ only over the Sb(tl) stretch of s is considered to be necessary and sufficient because the new vortex sheet Fw(tl ) = --Fb(tl ) involves neither any additional new unknowns because of Eq. (6.36), nor are there any additional kinematic conditions required on S~(tl). After y(~, tl) is so determined over ( - 1 < ~ < 1 + A~), we can deduce u~ from Eq. (6.28) and un on A~ of S ~ ( t l ) from Eq. (6.29), with which we can revise the data on S~(t~) by iteration for any improvement on the length and direction of A~. We can next update all the boundary variables by applying
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
325
the evolution Eq. (6.15) for the displacement of s and Eq. (6.31) for the wake vorticity, which are needed for the next time step at t2. At t2, Sb(t2) assumes its assigned position from X(~, t2) of Eq. (6.1) whereas the starting vortex 6F1 -- 3Fw(1, tl) has moved to its new position by convection with the local known flow velocity while maintaining its magnitude invariant henceforth, leaving a new gap A~(t2) to receive new shed vortex sheet of strength y(1, t2) given by Eq. (6.36), so the same calculation can be performed for y(~, t2) as done for y(~, tl) to obtain the solution at t2. Thus, with the solution accomplished for time step at tk, the computation can be carried forward to obtain the solution at tk+l. In this manner, this time-marching procedure can, in principle, be extended forward indefinitely. The problem is thus reduced to computation of the unknown vortex element just leaving the trailing edge and entering the wake at each time step. This is a point of paramount importance to forming a physical concept, conducting mathematical analysis, and making numeric algorithms and codes for the computation of a solution. Concerning the algorithm, quadratures, and numerical filters to be used for resolving the integral Eq. (6.29) under condition (6.17), reference may be made to the existing numerical methods for evaluating unsteady airfoil dynamics (e.g., Smith and Hess 1967; Giesing et al., 1971; Ashley and Rodden, 1972; Katz and Weihs, 1978; McCune et al., 1990; some being purely numerically aimed). Relative to these existing numerical methods, the present method is a further extension to the general case of a flexible lifting surface moving with arbitrary form along arbitrary trajectory, taking into account the nonlinear effects involved. However, it should be noted that even in the simpler case of rigid airfoils, satisfaction of the Kutta condition would generally involve some sort of artifices to various degrees of discrete mathematics, making each numerical method seem to carry a "fingerprint" characteristic of its own, rendering critical comparisons between theories and between theory and experiment not entirely straightforward. This is particularly true at the beginning of a sudden start or right after a sudden stop, whenever the airfoil undergoes an impulsive or discontinuous movement. To pursue further development and improvement, we explore ways of reducing some of the shortcomings of Method I by investigating another method.
B. METHOD II - - GENERALIZED WAGNER--VON KARMa, N-SEARS METHOD
In the history of pioneering development of linear unsteady airfoil theory, Herbert Wagner (1925) was the first to have generated an integral equation for calculating the wake vorticity shed from the wing. This approach was subsequently further developed by Theodore von K~irm~in and William R. Sears (1938), who
Theodore Yaotsu Wu
326
made a lasting contribution by providing an alternative and innovative formulation of the basic theory, and elucidating the underlying physical significances of the mathematical analysis advanced, and the results they accomplished. The improved derivation is based on an ingenious decomposition of the bound vorticity Fb on the wing into two parts, one for its "quasi-stationary wakeless" flow, F0, and the other, F~, due to wing's reaction to the trailing vortex sheet, with both F0 and F~ lucidly analyzed using linear approximations. In addition, there are other pioneers who have made valuable contributions. Theodorsen (1935) found a fundamental solution for the simple harmonic oscillatory wing motion in heaving and pitching. Ktissner (1936) obtained excellent experimental results and his theoretical explanations for these results. Robert Jones (1940) converted the unsteady-lift functions for wings of infinite span to that for elliptic wings of large aspect-ratio. Nearly all these latter theories are based on using orthogonal function expansions, integral transforms, or other operational methods that are basically for linear analysis, and hence not practical for extension to nonlinear problems. However, of these linear theories, the Wagner-von K~irm~in-Sears method is unique in affording a direct generalization to fully nonlinear theory, as has been accomplished by McCune and co-workers (1990, 1993) for investigating unsteady two-dimensional airfoil problems. Here, we make further generalization to two-dimensional flexible lifting surface performing arbitrary movements for modeling aquatic and aerial animal locomotion. In the spirit of Wagner, von K~irm~in,and Sears, we adopt for t > 0 the following vorticity distribution: on Sb(t) :
T(~, t) = Yo(~, t) + Yl(~, t)
(--1 < ~ < 1),
on Sw(t) :
T(~',t) = Yw(~', t)
(1
< ~ <
~m),
where Y0(~, t) is the bound vortex distributed over Sb for representing the "quasisteady wakeless" flow past Sb that moves with the original prescribed Un(~, t) (here with t fixed to serve merely as a parameter), and ?'l(~, t) is the additional bound vortex in reaction to the trailing wake vortices y~(~, t), which is required to reinstate the original time-varying normal velocity Un(~, t) on Sb(t). To determine these vortex distributions, we adopt the following procedure, composed of: Step 1: Determine Y0(~, t) using the prescribed U,,(~, t), with t fixed and no unsteady wake; Step 2: Determine Yl(~, t) due to the wake vortices Yw(O, t) to reinstate unsteady U.(~, t); Step 3: Determine yw = yw(~, t) by applying the Kutta condition.
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
327
We note that this procedure takes the actual displacemnt of/2 = Sb(t)+ Sw(t) (as well as its nonlinear effects) into account, which can have significant effects on the solution; otherwise, it is the same as for the original linear theory. We also remark that the flow with Y0 in step 1 does have a (uniform) wake connecting the trailing edge of Sb to its own starting vortex at downstream infinity, across which 4~ has a uniform jump to give a constant circulation F0 around Sb. In Step 1, we dismiss the variable wake and regard t as a fixed parameter, reducing Eq. (6.29) to
1Re[dzf_l
Un(~, t) - 2re
-~
Y~
t)d~' ]
1 z' - z
(1<~<1)
(6.37)
as an integral equation for Y0(~, t). To facilitate our analysis, we rewrite Eq. (6.37) to single out the Cauchy kernel in this key step of analytical operation as
gn(~, t) -- ~
lfl l 1
]
~' - ~ + g(~'' ~) yo(~', t)d~' - (Go + al)yO,
g ( ~ ' , ~ ) = Re
{dzl} d~ z ' - z
(6.38)
1
~'-
where (~' - ~)-1 is the Cauchy kernel, g(~', ~:) is a smooth "good" kernel, and the integral operators Go and G1 are defined in the order as shown. In fact, g(~', ~) vanishes for flat-plate airfoil with both z and z' lying on the plate (because then dz/d~ = contant) and likewise, Ig(~', ~)1 is small for slightly curved body movement. This situation makes it convenient to solve Eq. (6.38) by iteration, or by perturbation analysis with a series expansion of Y0(~, t) in terms of the small bound of g(~', ~) Y0 - - Y00 +
Y01 -+- Y02 +
(6.39)
9 9" ,
in increasing orders. Substituting (6.39) in (6.38), we find the reduced equations as
1 fl 1 ~' -1 ~ Yoo(~', t)d~'
GoYoo = ~ GoYok =
lfl
2n"
- gn(~, t),
(6.40)
1 g(~', ~)Yo,k-l(~', t ) d ~ ' - - G l y o , k _ l
(k = 1, 2,--.)
(6.41)
The leading-order reduced equation, as is known in linear theory, has under the Kutta condition a unique solution in closed form as Y00(~,t)=
2~l-~f_l~
zr
1+~
1
l+~'Un(~''t) 1-~'~d~'~'-~
=GoIU,
(-1<~<1), (6.42)
328
Theodore Yaotsu Wu
where the integral operator Go 1 is the inverse of Go. This result can be obtained by solving the underlying Riemann-Hilbert problem as illustrated for Eq. (4.31). Alternatively, it can be directly verified by substituting Eq. (6.42) in Eq. (6.40) and using the Poincare-Bertrand formula (e.g., Muskhelishvili, 1953)
~--~
~;, 2 ~-;
= -Jr
(~, ~) +
d~"
(~' _ ~)(~" _ ~ , ) d ~ ,
where L is a smooth arc or contour, h(~', ~") is H61der continuous on L. Hence, by inversion of the reduced equations of all orders, we find the series solution as N
X 0 -- ~ - ~ ( - 1 ) m ( G i G o l ) m U n ~ G N U n
Yo(~, t ) = G o l ( V n -+- No),
(N <_%00). (6.43)
m=l
where GN denotes the N-term series of integral operators as defined. Thus, given the body motion Z(~, t) of Sb(t) [furnishing U,, by Eq. (6.5)], Vo is completely determined. This solution is virtually the same as the result by iteration. The terms with n > 1 involve z(~, t) in operator G1, which is a function of Sb'S arbitrary movement in general and requires numerical integration, which should be straightforward. These terms represent the contributions from the nonlinear effects due to such factors as curved trajectories and off-course convection of wake vortices. Having found the exact solution (6.43) for V0, we have the corresponding circulation around Sb as Co(t) --
f
1
Yo(~, t)d~ - - 2
i-~
1
[U.(~, t) + No(~, t)]d~.
(6.44)
In Step 2, Eq. (6.29) provides the normal velocity induced on Sb by the trailing vortex Yw as Un l ( ~ , t) --
1Re{dzfl~~215 }
2---~
~
z---~-~-~-z d ~ '
= (Ko + K1)yw = K?%
(-1<~<1)
(6.45)
1 f~'" y~(~', t)d~, ~'- ~
Koym - 2re
1 f~'"g(~', ~)g~(~', t)d~',
K1gw -- 27r
(6.46)
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
329
where g(~', ~) is again given by Eq. (6.38) as first used in Step 1, but they have different ranges, namely, - 1 < (~', ~) < 1 for g(~', ~) in Step 1, but 1 < ~' < ~m and - 1 < ~ < 1 for g(~', ~) here. We next introduce a new bound vortex Yl (~, t) distributed over Sb(t) tO annihilate the normal velocity Unl induced on Sb(t) by the wake vortex ?,~ so as to reinstate the original unsteady boundary condition on Sb(t). This is done to have Yl satisfy Eq. (6.37) just as for Y0, but now with Un replaced by -Unl. Then, in complete analogy with Eq. (6.43) in Step 1, we have for yl the exact solution as t)
Yl(~,
=
-GoI[Unl + Nl], N
Ul -- Z ( - 1 ) m ( G 1 G o i ) m u ~ I - GNUs,
(U < cx~),
(6.47)
m=l
Substituting Eq. (6.45) in Eq. (6.47), we find, after some algebra, Yl(~, t)
=
YlN =
Yl0(~,
t) +
-GoI(K1
Yl0 - - - -
Jr
-+-
(6.48)
GNK)yw
-
l+~,
(-1
~'-~
'-I
1+ ~
YlN -- - ~
Vl0 = - G o 1KoVw,
?'IN(~, t),
Vw(~', t)d~'
< se < 1)
~"
(6.49)
d~" (6.50)
where G(~', ~) is defined by (6.46) with the operator GNK replacing K1 and G(~', ~) replacing g(~', ~). The result of V10 follows from integration with the order of integrations interchanged. This result for Yl(~, t) yields around Sb a new bound vortex given by Fl(t)
--
FIO --
1-'lN - -
f
l
i_' ,/,. 1 YlO(~:,
--yr
(6.51)
Yl(~, t)d~ -- rio(t) + FIN(t), 1
t)d~
--
V~(~, t)d~
s,"(7'+' ) s, 7, ~-
1
i
1 y~(~, t)d~,
1 - ~' [g(~' ~') -t- ~(~, ~')]d~'.
(6.52)
(6.53)
Theodore Yaotsu Wu
330
The total circulation around the lifting surface Sb is 1-'b -- 1-'0 d-- I"l, and the total circulation around the trailing vortex sheet S~ is r~(t)
=
f
~m
•
t)d~.
(6.54)
Finally, we apply Kelvin's theorem of conservation of total circulation, (6.55)
Fb q- 1-'w ---- 1-'o q- 1-'1 q- l"w -- 0, from which we obtain
F0 +
~ _ 1 Yw(~, t)d~ + F1N
--
0.
(6.56)
This is the generalized nonlinear Wagner's integral equation for the trailing vortex distribution Yw. Clearly, this integral equation satisfies the Kutta condition because both Y0 and Yl are constructed on this basis, and yw(1, t) is determined with the same onset conditions for yw. In this equation, the term with No in Eq. (6.44) for 1-'0 and the term FlU in Eq. (6.56) are both due to the nonlinear effects. With these terms neglected in the linear limit, this nonlinear equation reduces precisely to Wagner's famed integral equation. In this connection, it is of interest to note that for a streamlined uncambered airfoil with negligible thickness, the No term drops out [because g(~', ~ ) = 0] and F0 reduces to the linear theory result. However, the FlU term associated with a flat plate airfoil can become significant for zigzag foil motions or impulsive starting and stopping maneuvers. In the latter case of a brisk start-stop process, the equal and opposite starting and stopping vortices would dash off together as a vortex pair riding on their own induced current in a direction generally oblique to the body trajectory while the free body could coast motionless in another direction as a result of action and reaction. In an analogous three-dimensional case of fish swimming, McCutchen (1977) illustrated this flow phenomenon resulting from intermittent impulsive tail flippings of a small fish with a vivid movie recording. In short, the nonlinear effects in both two- and three-dimensional cases associated with wake vortices are motion dependent. In calculating the solution, the integral Eq. (6.56) is used to determine the wake vorticity Yu,, given the quasi-stationary bound vortex F0(t) around Sb. In this calculation, it is important to note that y~(~, t) depends on only one variable because of its own invariance along the mathematical characteristics imposed by condition (6.31), and this variable can be written as ?'~ = Yw(~l, "gl), where ~l ~" 1
On Theoretical M o d e l i n g o f A q u a t i c a n d A e r i a l A n i m a l L o c o m o t i o n
331
refers to the trailing edge and r~ = "rl (~, t;~l) is the retarded time for the present nonlinear case, which satisfies, on account of Eq. (6.31), the following equation drl dt
0rl
0rl
Ot + Usm 0~ = 0.
(6.57)
In linear limit, we have 721 =
t - (~ - 1 ) / U = t - (x - 1 ) / U
so that
( 0 t -1t. U O x ) y w = O.
(6.58)
We recall the similar characteristic variable that arises in Eqs. (4.19) and (4.20) for slender-body theory. For the general nonlinear case of arbitrary motion, it is convenient to have the retarded time function determined numerically. In order to combine the best merits of both the analytical and numerical methods so as to achieve a method of high precision and to reduce computational effort to a mininum, we propose a hybrid method as follows.
C. METHOD III--A HYBRID ANALYTICAL-NUMERICAL METHOD
The basic idea of this method, attempted to combine the best merits of methods I and II, is to curtail as much as possible the need for calculating numerically the mathematical characteristics to obtain the convection pattern of the wake vortex Yw while retaining the high precision offered by the analytical method. Such optimal conjunction of the two methods is desired because this part of the solution can be important when the wake effects are significant, such as in highly curved turning and wake-crossing maneuvers by Sb. To achieve this objective, we propose to adopt the analytical approach of Method II to have the starting vortex ),w determined using the nonlinear integral Eq. (6.56), so that the complete vortex distribution becomes known, yb = ?'0 + yl on Sb and ),~ on S~, and hence also the boundary velocities Usi and u,, by Eq. (6.28). At this point, we shift the computation to Method I so that the new solution can be used together with specified X(~, t) of Sb to move Sb and S~ to their new positions at the next time instant and to have the wake convected by the prevailing local flow velocity to make the wake integral in Eq. (6.28) known over the stretch beyond the very first wake grid being opened (just behind the trailing edge) in order to receive the new wake vortex entering this grid. Thus well prepared, the computation is shifted back to Method II for the next time-step to compute the new wake vortex having entered the first wake grid by using again the nonlinear integral Eq. (6.56). In this manner, the interlaced use of Method I and Method II can be continued to evaluate the solution on forward time marching.
Theodore Yaotsu Wu
332
The merit of this hybrid method can be illustrated by Wagner's (1925) classic problem for a flat plate airfoil making a time-step increase in incidence angle so that F0(t) = H(t), the unit Heavyside step function. With the nonlinear terms dismissed for possible iteration later, Eq. (6.56) then reduces in the linear limit to
f,+utyw(~,t)~
~( +tl d >~ - -01 ) ~ - I
,
(6.59)
with ~,~ properly scaled to have the unit change of the integral as a standard. Following von K~irm~in and Sears (1938), we adopt the new variable s - 1 + Ut - ~, so that yw(~) = y~(1 + Ut - s) - ~,(s), s being the distance measured along S~(t) from the tip of the starting vortex (at ~ = ~m) backward toward the trailing edge, by which Eq. (6.59) is tranformed into
fo U' f,(s, t ) ~ 2 + Ut_ -s s d s - - 1
(t >
(6.60)
If only the starting motion is of interest, 0 < U t - s << 1, the leading order of the small-time asymptotic representation of Eq. (6.60) is
fo Ut
f/(s, t)
~
2 ds--1 Ut - s
(O
This is Abel's integral equation, which has the solution
f,(s,t) = - ( 7 r ~ s )
-l
(Ut << 1).
(6.61)
The corresponding Wagner's lift-deficiency function ~ ( U t ) is found [see von K~irm~in and Sears (1938)] to yield
f
9 (Ut) = - 1
l+u,
Y(~) -d~ - - fo Ut 9(s) ds = -1 v/~ 2 - 1 ~/2(Ut - s) 2
(as t ---->0), (6.62)
which is Wagner's famed result, stating that an impulsively started flat-plate airfoil generates instantly half the final lift, which it eventually achieves, 1 - ~ ( U t ) ~ 1 as t --> o~ (i.e., after a time delay), which is known as the Wagner effect. In addition, von K~irm~in and Sears (1938) showed that the function 9 can be used on linear theory to calculate the lift acting on an airfoil that is subjected to an arbitrary transient variations of quasi-steady circulation F0. Here, the point is that in application of this fundamental solution for constructing solutions to general problems, it is vital to notice that the small-time behavior of its flow field exhibits various singular features, including the bound vortex Yl having square-root singularities at both the leading and trailing edges of the airfoil
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
333
(with the starting vortex sitting right at the trailing edge) at t = 0 [as is clearly indicated by Eq. (6.61) adjacent to the trailing edge], together with the pressure possessing a Dirac delta function 3(Ut) over the whole plate (Nakamura, 1992). Involvement of these generalized functions may be ascribed at least in part to the difficulties in attaining consistent results between various primarily numerical methods when assessed on the basis of equal accuracy. For this purpose, Wagner's fundamental solution stands as a critical test. Having obtained the solution to the entire vortex distribution, we can proceed, heeding Principle (iii), to calculate the net force and moment of force acting on the lifting surface Sb(t) by integrating the pressure distribution of Eq. (6.22) over Sb(t). With respect to the solution at hand, a few remarks are in order. First, we note that to have body Sb(t) possess flexibility and move with time-dependent velocity U(t) is necessary for evaluating such maneuvering operations that are marked with strong nonlinear effects as starting and turning, bending and twisting of the lifting surface, stretching and shortening of wing span, operating duration and loading variations in the pronation-supination strokes, and other operations that are commonly displayed in flapping flight of birds and some insects as well as skates and rays in water. We further note that determination of the wake vortex y~ by using the generalized Wagner equation depends only on wing's quasi-stationary circulation F0 rather than the detailed bound vorticity distribution Y0(~, t), an interesting result that points to the brilliant decomposition of Fb = F0 + F~ introduced by von Kfirm~in and Sears (1938). In addition, the leading term of the solution to the bound vorticity [Y00(~, t) given by Eq. (6.42)] shows that the contribution of the local normal velocity U, of Sb to the differential lift has a weighting function proportional to {~/1 + ~/~/1 - ~ }, which clearly favors greater Un input near the trailing edge than near the leading edge. The feature of this relationship ought to help explain why birds in free flight have their wings more bent down near the trailing edge during the downward power stroke than near the leading edge.
VII. On Experimental Differentiation between Thrust and Drag in Fish Locomotion In Gray's (1936) pioneering study on aquatic animal locomotion referred to earlier, he found evidences of incompatibility between the mechanically implied frictional drag on high-performing aquatic animals and the biologically expected muscle power required of the observed high performance as compared with the known specific muscle power (per unit muscle weight) of warm-blooded animals,
334
Theodore Yaotsu Wu
with the discrepancy found beyond reconciliation by a factor of severalfold. This perplexing conclusion stimulated extensive investigations over several decades with interests from both biological and mechanical fields. There are a great variety of viewpoints held for hypothetic resolution, including such ideas as seeking for low-drag possibilities (e.g., streamlined body shape; compliant skin; mucus secretion; active boundary-layer control, such as expelling water jets from fish gills) on the one hand, and on the other hand, seeking for improved ways of measuring the drag on animal (e.g., in accelerating mode or decelerating in quiet coasting of life specimens). However, the collected data on drag coefficient Co of fish measured under various conditions show such a wide scatter (by a few to tens of times the Co of a fiat plate at comparable high Reynolds numbers; see, e.g., Gray, 1968, and Webb, 1975, for survey) that no sound conclusion seems feasible. In addition, there is also the enlightening assumption that the drag on fish during swimming is different from the drag on the same specimen (or its ideal scale model) that is not self-swimming, such as being held with fixation, being towed, or even in quiet coasting. We refer the reader to the extensive references in the literature (e.g., Lang, 1975; Nagai, 2000). On this issue of differentiating between the thrust and drag involved in animal swimming, we propose that the foremost criterion is to determine whether the net force (including all the forces) acting on the specimen in question is either zero (said to be in State LMO) or nonzero (in State LM1), and, when body rotation is involved, whether the net torque acting on the specimen is zero (in State AMO) or nonzero (in State AM1). It is then obvious that is State LM0 of the linear momentum, the thrust and drag are equal and opposite to each other, but not in State LM1; similarly we can conclude in regard to the propulsive and resistive torques concerning the angular momentum when body rotation is present. Based on this criterion, there is of course no validity in taking the drag measured in State LM1 (of a specimen that physically is an open system) to imply the same drag in State LM0 (of a self-propelling specimen that momentumwise is a closed system). The reason for this criterion is simple: It is because the pressure distributions over the body surface in these two states are fundamentally different, so therefore are all the boundary-layer properties. Furthermore, it is essential to realize that the validity of this criterion is based on Newton's first law (for which this is a sophisticated applicationmnamely, to animated bodies) and is true regardless of what the Reynolds number of the swimming motion in question may be. Let us explore the physical significance of this issue for the case of low Reynolds number regime because the distinction between the two dynamic states of microorganism locomotion is even more dramatic. Here, the two photographs in Fig. 3, taken by Keller and Wu (1977) for comparative studies by visualization, vividly exhibit this distinction between two drastically different types of streamlines, one
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
335
FIG. 3. Streak photograph of streamlines produced by (A) a freely swimming Paramecium caudatum, at left, and (B) a dead specimen of Paramecium caudatum sedimenting under gravity at right. (Adapted from Keller and Wu, 1977.)
for a freely swimming microorganism, Paramecium caudatum (on the left) and the other for an inert impermeable specimen (immobilized, slightly denser than water) falling in water under the influence of gravity, which is an external force (on the fight). On theoretic argument, the inert specimen sedimenting in fluid exerts on the fluid a total force equal to its weight in the fluid. Applying the low-Reynolds number (in this case, Re ~ 10 -3) fluid mechanics, this body action on the fluid can be represented by a distribution of stokeslet (standing for a point force on fluid) and its higher-order poles over the body, with the net stokeslet strength equal to the total force. As the stokeslet is a long-range singularity, with its induced velocity falling off inversely proportional to the distance from it, this singular force moves the fluid from far behind to far ahead, as clearly displayed by the streamlines streaking past the body. In contrast, the freely swimming specimen would still need a (different) stokeslet distribution to represent the ciliary forcing strokes, but of zero net strength, as required by Newton's first law. Accordingly, the effects of the stokeslets appear to stop right at the enveloping edge of the ciliary layer, leaving the exterior flow invariably irrotational at all times, a flow field that can always be represented by a distribution of mass dipoles and its higher-order poles. Nevertheless, the stokeslets, even with zero net, are playing a vital role in generating at the edge of the ciliary layer a differential velocity distribution that is required just to match exactly the exterior irrotational flow so that the flow velocity is continuous everywhere (in and out the ciliary layer) and can be said to satisfy the
336
Theodore Yaotsu Wu
(. FIG. 4. The streamlines in the laboratory frame past a prolate-spheroid of eccentricity 0.9, with the body in (A) self-propelling motion at left, and (B) inert translating under gravity, at right. (Adapted from Keller and Wu, 1977.)
no-slip condition for the exterior irrotational flow at the beating ciliary envelope. It is, therefore, remarkable to observe irrotational flows to manifest at such a small scale (about 50-150/zm in length for microorganisms) with such a marvelous fit with the strongly viscous flow inside the ciliary layer. It is no less important to note that although irrotational flows are also solutions of the Navier-Stokes equations for viscous fluids, and therefore possess viscous stresses, but these stresses acting over a (closed) body surface of arbitrary shape always result in zero net force, as can be readily shown to hold for incompressible viscous fluid with constant viscosity. With the preceding observation, Keller and Wu (1977) pursued in parallel a theoretical study using a prolate ellipsoid to simulate the paramecium body shape. The corresponding results are shown in Fig. 4, one for the streamlines around the body in incompressible irrotational flow for modeling a self-propelling body in the laboratory frame (on the left), and the other for the streamlines around the body in Stokes flow (on the right) for comparison with an inert body translating under an external force (viz. gravity). The comparison between theory and experiment is strikingly similar, notwithstanding the discrepancies that the boundary conditions used in the theory are at variance with the experiment due to the proximity of glass plates and sealing gel used for making microscopic observations (for more details of the experiments, see Keller and Wu, 1977). In regard to energetics, a topic we explore next in Section VIII, we note that despite the remarkable achievement by ciliates in manufacturing a perpetual zero net force environment while sustaining impressive swimming speed (easily from tens
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
337
to scores of body lengths per second), their cilia do have to expend nonvanishing work on the viscous fluid outside at the rate of the local flow velocity at the ciliary layer edge (in addition to the work on the fluid within the layer). For instance, for a spherical ciliate of radius R swimming at velocity U in water of viscosity #, its rate of working can be found to be 127r# U2R. With reference to the intrinsic force fo -- # v (pertaining to the fluid, # = p v being the dynamic viscosity coefficient of the fluid of density p) working at the same rate, their ratio gives a specific energy cost for the ciliate specific energy cost-- 12zrlzUZR/foU = 12zrUR/v = 12toRe,
(7.1)
which is very small since the Reynolds number Re << 1 for ciliates. Summing up, we see that ciliates can always succeed in achieving zero net force to swim at impressive speed at extremely low energy cost while living in an environment abundant with their energy supply. This, then, provides the necessary and sufficient condition to have microorganisms swimming freely and to make the microscopic world a very populous one. Returning to fish locomotion, we note that alternatively, we can also expound the thrust and drag composition from the corresponding momentum theorem applied to the momentum balance between two transverse (geometric) planes, one being put far ahead and the other behind the body, which is either in constant free swimming motion or translating uniformly under tow. Because the fluid is assumed to be at rest far from the body in the fluid frame, the total momentum flux vanishes far upstream, and we need consider only the net momentum flux at the downstream plane. Here, let us first set a familiar reference state with von K~irm~in's vortex street left behind a blunt body translating forward under tow (assuming in an appropriate range of Reynolds numbers giving rise to vortex street). The two rows of zigzag vortices appearing at a specific distance apart are of such opposite senses as to produce a momentum loss within the wake, inducing the fluid there to be somewhat dragged along with the body; this momentum loss is transported across the wake transverse plane at a rate equal to the drag on the body. For free-swimming fish, theoretical studies with the thrust evaluated on potential flow theory without considering viscous drag would sometime present a figure (e.g., Wu 1971 c) depicting the wake vortices much like von K~irm~in's vortex street, but with each vortex reversed, in a sense that produces a net gain of wake momentum to explain the thrust on the body resulting from the wake momentum gain by the action-reaction principle. However, it must be noted that this is an account of the thrust alone. In uniform cruising in viscous fluid, the thrust is balanced by the drag due to the viscous effect, so that the net result to expect in reality is to see a momentumless wake (with the momentum flux over the cross plane integrated
338
Theodore Yaotsu Wu
to zero). The scenario most likely to occur in experimental visualization would be for the vortices of alternating sense (shed from undulating fins) to be aligned all exactly on the centerline so as to provide a zero mean momentum wake. This is a point that can be useful in examining and interpreting experimental results. In concluding on the issue of differentiating between thrust and drag for swimming of larger aquatic animals (fish and cetaceans), we see several plans of good feasibility for complete resolution. One is a theoretical method, by using the result of an accurate potential flow theory, such as along the present approach to carry out calculation of both laminar and turbulent boundary layers for comparative studies versus the theoretical value of thrust. Another plan is a joint theoretical and experimental method based on using a set of microsensors to measure pressure distribution on free-swimming models and on towed models for appraising the accuracy of the theoretical results on thrust for specific flow field, on which basis the drag in free swimming and drag in tow can then be definitively verified and distinguished. Still another method is based on a mechanical and biological interdisciplinary approach by examining the physiological functions of fishes and cetaceans over several orders of magnitude in physical size. Jointly with studies on their dynamical similarity, we attempt to draw concrete inference on the boundary layer flow condition of being laminar or turbulent over the swimmer's surface that would most likely be the case in aquatic locomotion. In this regard, studies on pressure measurement and flow visualization in swimming locomotion can be found in the literature. For instance, the experimental study by Bannasch (2000), on live penguins and measurements with life-sized models with results revealing extremely low drag coefficient, is of great interest. Along the third approach, we present next some findings from the mechanophysiological studies on the energetics of aquatic locomotion.
VIII. Scale Effects in Energetics of Aquatic Locomotion In contrast with the approach of making direct measurement of fish drag, physiological studies of fish energetics are different and highly valuable because they are based primarily on tests with fish specimens in continuous swimming at nearly their natural state. Studies involving such vital factors as oxygen consumption of fish swimming at different levels of activities, their metabolic rate, and muscle efficiency have provided valuable data on energy cost and biochemical-tohydromechanical energy conversion. These results, gathered from various species of very different sizes and examined under dynamic similarity for identifying the scale effects, have brought forth new light concerning the interplay between thrust
On Theoretical Modeling o f Aquatic and Aerial Animal Locomotion
339
and drag on fish in steady swimming to such a degree and extent that it would be infeasible to achieve otherwise. In this section, we present some highlights on joint mechanophysiological studies by Wu (1977), Wu and Yates (1978), and Yates (1983, 1986, 2000) on the energetics of aquatic locomotion. For the steady state of fish locomotion with constant velocity, the mean drag D sustained by a fish must be balanced by the mean thrust T it generates, T = O.
(8.1)
From the biofluiddynamic approach, the viscous boundary layer in which the fluid is dragged along by swimming fish is so thin at high Reynolds numbers that the pressure remains constant across the thin layer, thus rendering the inviscid flow theory valid for evaluating the thrust T, as we discussed in foregoing sections. This approach is useful for predicting, by implication of Eq. (8.1), the viscous drag required for fish locomotion because this data input is also needed for related physiological studies. In order to avoid the uncertainties associated with direct drag measurement or calculation, a useful alternative is to take the physiological energetics approach by measuring the animal's oxygen consumption, taking the advantage of energy balance involving standard energy-conversion factors that are well established to the biologists. By the principle of energy conservation, D V -- P - Ew = qP,
oP = OhPh = OhOmPm,
(8.2)
signifying that the animal's mean rate of working against resistance D at the rate of swimming speed V (denoted before by U) is provided originally by the power P expended by the fish in performing the work with a certain efficiency 0 while casting off, irreversibly, wasted energy (such as the vortical kinetic energy in the wake) to the fluid at the rate Ew. If the power is referred to the hydrodynamic level (i.e., Ph for performing the bodily motion), then 0 = Oh, the hydrodynamic efficiency of propulsion. At the physiological level (i.e., with P referred to the measured net biochemical metabolic power Pm consumed all for swimming, after subtracting the basal metabolic rate spent for sustaining life cycles other than swimming movement), we have 0 = 0h 0m, where 0m is the "muscle efficiency" with which biochemical energy is converted to muscle power during active swimming. On velocity dependence, Ph is seen to be proportional to V 3 in view of the known properties of D (being primarily proportional to V2). Concerning the metabolic power Pm, it can be reasoned that the efficiency with which the muscle system produces power increases in proportion to the frequency of muscle contraction. Hence in both carangiform and lunate-tail swimming, the muscle efficiency 0m
340
Theodore Yaotsu Wu
should be proportional to the swimming velocity V. It then follows that the fish consumes energy at a metabolic rate (8.3)
Pm = Pb + ol e 2,
where Pb is its basal metabolic rate (a constant rate for sustaining swimless life cycle) and a is a constant. The related dimensionless parameter ,f is defined for fish of weight mg as
mgg
= mg
--V + ~
,
(8.4)
known as the specific energy cost, it provides a useful measure of the relative merit of the propulsive system, signifying the energy expenditure in transporting unit weight over unit distance. It was introduced and used by Kfirmfin and Gabrielli (1950) in evaluating the comparative merits of 14 classes of transportation vehicles and animal locomotion, and by Schmidt-Nielsen (1972), Tucker (1975), and others for studies of comparative physiology. For fish locomotion, we see that g has a minimum at the critical speed Vcrit = (Pb/ot) 1/2. This speed gcrit w a s found by Brett (1965b) for salmon (see, e.g., the C p curve in Fig. 7 and was also noted by Weihs (1973) in his theoretical study.
A. METABOLIC RATE AND SCALE EFFECTS
It is important to note that the metabolic rate in fish locomotion depends on several key factors, including: (i) level of activity, (ii) temperature of the water (affecting the concentration of dissolved oxygen), and (iii) such factors as preconditioning (a period of fasting and exercise prior to the test to separate energy spent in digesting and interior physiological processes), as well as the state of maturity and gender of the specimen. The level of activity is generally hard to characterize. For salmon, however, Brett (1965a, 1965b) found three distinct levels of performance: sustained level (speeds that can be maintained almost indefinitely), prolonged level (speeds maintained for 1 to 2 hr with a steady effort, but leading to fatigue), and burst level (speeds achieved at maximum effort lasting for only 30 sec). The physiological basis of each of these levels is different and depends further on the scale of body size, which assumes the general form Pm = a m b,
(8.5)
where m is body mass and a, b are constants. In particular, for the salmon Oncorhynchus nerka, Brett (1965a, 1965b) reported the results with b = 0.775, 0.846, 0.890, 0.926, all with an error bar of 4-0.145 for the standard, J-max,
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
341
3 1-max, and ~-max speed, respectively, and b - 0.970 -t- 0.053 for the maximum speed Vmax. Here, Vmax denotes the speed sustainable for 60 min in freshwater at 15 ~ and the standard state is obtained by extrapolating the 02 censumption versus velocity curve to zero swimming speed. Here, we point out that these b values are all greater than the "surface law" (b = 2/3) and increases with activity level to approach direct proportionality (b = 1). Brett's endeavor rendered studies on scaling of fish locomotion more systematic, as we discuss next.
B. SCALING OF SWIMMING VELOCITY AND ENERGY COST Expressing the drag D in terms of the drag coefficient C D, 1
D = -~pV2SCD,
CD = CD(Re),
Re-
Vg./v
(8.6)
for fish with surface area S (proportional to its length g squared) in water of density p, with CD dependent on the Reynolds number Re and body configuration, where v is the kinematic viscosity of water (v = 0.0114 cmZ/s at 15 ~ From the energy balance in Eq. (8.2), metabolic scaling in Eq. (8.5), and Eq. (8.6), it follows that (8.7)
g 3 ~ e3b-Z/CD
For the carangiform and lunate-tail modes of swimming, both modes being well streamlined, Wu (1977) assumes that Co may be approximated by its relationship with the Reynolds number for steady flow over a flat plate at equal Reynolds number, so that Co is proportional to Re -1/2 or Re -1/5 according to whether the boundary layer is laminar or turbulent, thus yielding the scaling law V = const.~ ~,
(8.8)
with 3 1 3 - g ( 2 b - 1),
3 -i-~(5b-3),
b
2 3'
according to whether the boundary layer is laminar, turbulent, or with CD = const., respectively. The last case (of quite large but constant Co) is for the hypothetical case when flow separation occurs in the cross flow past an undulating fish body in a general braking-type maneuver. A collection of experimental data that relate the swimming speed versus body length for various species of fishes and cetaceans has been assembled by Wu (1977) and Wu and Yates (1978), as shown in Fig. 5, together with the Reynolds numbers and specific speed (in body length per second) given in reference lines, and with the details given in the figure caption. Figure 5 shows the mean line for the maximum
342
Theodore Yaotsu Wu
'\ ..... I,,s "-,,. \ /,-,\
I0
........
Vo\ -t~\ #/~,,'\
y~.,,~, '\
I.)6" \ , - , o ,. ~ t . ' t
\
~1
//
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., ,x
s W w
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/
i
I
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/"
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/
//
/
-
i/
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,'\-
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,,~x
I
/
/
/
B U R S T -%,. /
,/
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/
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FIG. 5. Variation of swimming speed V with body length e at specific levels of activity. Burst speed data:., dace; A, goldfish; O, trout; B+, barracuda; +Ps, porpoise, and +D, dolphin (Bainbridge, 1961). Full activity data: ~, dace; A, goldfish; and O, trout (Bainbridge, 1961); | sockeye salmon (Brett, 1965a); ~, bass; and @, coho salmon (Dahlberg et al., 1968); ]II: C, cod; R, redfish; F, flounder; S, sculpin; and P, pout (Beamish, 1966); >--<, goldfish (Smit et al., 1971); E), larval anchovy (Hunter, 1972); x - - x , herring (Jones, 1963); O, salmon; G, trout (Paulick and Delacy, 1957). 1/4-max data; (~, sockeye salmon (Brett, 1965a). The lines of constant Reynolds number, (Re), and lines of constant specific speed (in e / s ) are for reference. (For the literature cited here, see Wu and Yates, 1978.)
sustained speed (for 60 min at b = 0.97) gives/3 = 0.5 (the slope of the log-log 1 plot), so does the ~-max line (at b - 0.85). In contrast, the burst speed line gives I3 = 0.88; and interestingly, the rule of burst speed = 10 e/s holds for the entire data set, regardless of body lengths [corresponding to b = 1.33 (or even higher values) for laminar (or turbulent) boundary layers]. The results obtained for the scaling of metabolic rate and swimming velocity may now be used to evaluate the scaling of the specific energy cost. From Eqs. (8.4), (8.5), and (8.8), we find that E = P/mg
V --
const.m -•
y -- 1 - b + f l / 3 .
(8.9)
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion o . 6 ~
r
,q--r------ --- -- --T
i
I
..~-,.~----I
0.5 1 1
343
/3,0.5
(observed)
I
0.4 ,B,y 0.3
0.2
O.I I
o -0.86
0.88
l-mox 0.90
t
t
89 0.92
t
-~-mox b
0.94
0.96
mox
0.98
1.00
FIG. 6. Comparison of the observed values of/3 (Brett, 1965a, 1965b) and y, with the similarity prediction based on the three distinct reference states characterized by laminar boundary layer, turbulent boundary layer, and Co = const. (for the case of separated cross flows). The b values shown here exclude the basal rates (Wu and Yates 1978.)
By using the experimental data shown in Fig. 6 and the values of b [with Eq. (8.5)] for the various activity levels, we deduce results showing that the specific energy cost s varies with body mass with y = 0.28, 0.25, 0.22, 0.18,
(8.]0)
1 1 3 at ~-max, ~-max, ~-max, and full level of activity, respectively. These values of y and the observed mean of/~ = 0.5 are plotted versus b (as a measure of the level of activity) in Fig. 6. Also shown are the similarity predictions of ,6 and y evaluated previously for the three distinct reference states of laminar, turbulent, and separated flows. Here, a comparison between experimental data and the similitude calculations brings forth far-reaching implications. First, the observed/~ and y lines both intersect only with their corresponding laminar reference lines of the three flow reference states. This would imply that the boundary layers of these swimming fishes would have to be largely laminar at these high Reynolds numbers
344
Theodore Yaotsu Wu
(8 x 103 < Re < 107), with only a very small (presumably a distal) part turbulent at most. Also, there is no evidence to suggest flow separation. Furthermore, the 1 fact that the two intersection points for the 13 and g lines are about on the g-max activity level implies that the specimens tested would be swimming at somewhere about one-half of their maximum sustainable activity.
C. SCALINGOFVISCOUSDRAG The conclusions with implications just discussed have been further pursued quantitatively by Wu and Yates (1978) using the detailed data obtained by Brett (1963) on the swimming energetics of the sockeye salmon (Oncorhynchus nerka) and measured fish drag. Figure 7 shows the data points of the measured metabolic power coefficient C p and the dead-drag coefficient Cod measured with an anesthetically sacrificed fish in a closed water tunnel over a range of Reynolds numbers Re, where Cp - P /
(1-~p V3S ) ,
CDd-- D /
(1-~pV 2S) .
(8.11)
Here, P is based on the measured metabolic rate spent only for swimming by fish (of a single size group with mean length ~ = 0.178 m) at various levels of activity. A least-square-error fit to the drag data yields C D d ---
15.4 Re -0"4 (2.5 x 104 < Re < 2 x 105).
(8.12)
However, the metabolic power coefficient C p has two branches meeting near its minimum value of 0.07, which is reached at the "critical swimming speed" Wcrit, which divides the sustained-speed and burst-speed ranges. Based on Webb's (1975) estimate of 0m (lying in 0.2-0.3 forthe velocity 0.2 < V//Wcrit < 1 forarange of variations of physical factors, such as size, species, temperature) and using 0h as high as 0.9 (Wu, 1971 a), a reasonable estimate of 0 for this representative case is taken to be 0 = 0.25. With this value of 0, a curve of oCp is shown in Fig. 7, and we should expect from Eqs. (8.2) and (8.11 ) that the actual drag during swimming must be Co -~ 0Cp for 0.2 < V/Wcrit < 1. The result of this analysis thus appears to place CD considerably below the dead-drag coefficient Cod and bring the actual Co closer to the two reference lines near the bottom in Fig. 7 for the laminar and turbulent friction coefficient of a smooth, fiat plate. In addition to this detailed investigation of metabolic rate of a single size group, a comparative physiomechanical study has been made possible by Brett's data
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
I
t'k, ........
I
345
.........
C~~ s-'
~
i" ~
i
~
.
. Cod . ( de~ fish) r Slope =-0.40
\ k
17Cp
I
i
-
SUSTAINED BURST
A
t
~
(~:0.25)
-
Vcri!
_ ~
/-Turbulenl CD - Flol Plole
-
~
~
-
-
1(53
I0 '4
~
~
~ . . ~
(Slope: -0.50) I
I
I
,
I I,,I
105
l
,
,
,
, ,,i
06
Re
FIG. 7. Variation of the measured metabolic power coefficient C p, with the Reynolds number and the variation of the measured dead-drag coefficient CDd of freshly killed salmon (Brett, 1963). The corresponding values of the specific speed are indicated in f/s along the C p curve. An estimate of the swimming drag coefficient CD is provided by the values of r/Cp, shown with a typical value of the overall efficiency r/-- 0.25. The drag coefficient of a smooth flat plate for both laminar and turbulent boundary layers is shown for reference. (1965a, 1965b) on P and swimming speed V for sockeye salmon of five different sizes. The scale effects are clearly exhibited in the Cp ~ Re relationship, as illustrated in Fig. 8, in which there are five size groups (shown in dashed lines), each covering the four activity levels. The least-square-error fit for each activity level (shown in solid lines) of the five size groups bears out the scale effect. In this log-log
346
Theodore Yaotsu Wu ,~_
,
,
,
,
, , , , I
,
,ov
3
,
l
IIII~
q
oo,,o
\
V
,
io 4
,
Re
9
ao 5
9
9
\,,-
iO~
FIG. 8. Scale effects in the variation of the measured metabolic power coefficient (for swimming) with Reynolds numbers for five different size groups of sockeye salmon (adapted from Brett, ]965a). Solid lines are least-square-error fit to C p at specific activity levels; dashed lines show variations of Cp with various activity levels for each size group; dash-dot lines illustrate theoretic thrust coefficient CT computed for the data from Pyatetskiy (| 970) and Webb (] 975).
plot of the Cp Re lines, the slope reads -0.61, -0.54, -0.50, -0.41 for 1-max, 3 1-max, ~-max, and full levels of activity, respectively, of fish over a range of length scales. The result of this analysis seems interesting in comparison with the slopes of the two reference Co ~ Re lines for laminar (with slope -0.50) and turbulent Co (with slope -0.20) of a smooth flat plate shown in the lower part of Fig. 8. ~
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
347
Also shown in Fig. 8 are values of the theoretically predicted thrust coefficient,
Cr-
1
T/-~pUZS,
(8.13)
which is estimated by applying the classic slender-body theory for the body motion
h(x, t) : hi cos(kx - cot),
(8.14)
where hi is constant and k = 2rr/~. is the wave number of the body undulation (parameters taken from the observed data of Pyatetskiy, 1970, and Webb, 1975, in deriving the numerical results). The slopes of these Cr lines are seen to be qualitatively similar to those of the estimated Cp curves for fish of each scale group in spite of the mismatch (in the CT analysis) between species, lack of information on activity levels, and so on. It is hoped that more definitive predictions and comparisons can be made possible with numerical implementation of the extended large-amplitude planar-body theory, as presented here.
IX. Conclusion and Outlook
In this article devoted to an expository discussion on theoretical modeling of aquatic and aerial animal locomotion, several objectives, as brought forth in the Introduction, are focused on exploring how, why, and under what premises such high efficiency and low-energy cost can be achieved in these diverse modes of locomotion, as a result of a long history of (evidently) convergent evolution. The approach attempted here is to take an integral viewpoint from the foundation built by the pioneering leaders in the field, like Herbert Wagner, Theodore von Kzirm~in, William R. Sears, and Sir James Lighthill, followed by other researchers through developing various theoretical and experimental methods used in studies on the subject. In subdividing the various classes of hydrodynamic theories and the underlying physical conceptions, we see that the generalized slender-body theory, required only in aquatic locomotion, is readily capable of expounding the complex interaction between the swimming body and the vortex sheets shed from the appended fins (such as the dorsal, ventral, and pectoral fins), caudal fins, or lunate tails. On this topic, we take note of some important nonlinear effects that remain to be determined. In connection with the flapping locomotion of birds as well as some aquatic animals denser than water when both thrust and lift are needed for locomotion, several physical factors involving maneuvering control of wings and lifting surfaces require further investigation. These maneuvering modes
348
Theodore Yaotsu Wu
appear to be composed of several operations, such as chord- and spanwise bending and twisting of the wings, exerting heavily loaded pronation stroke and lightly loaded supination stroke, and using shortened upward stroke whereas prolonging downward stroke. It is of fundamental interest to examine whether these physical factors are actually making contributions to achieving high efficiency and very low energy cost. Valid answers to these questions seem to rely critically on having a strongly or even fully nonlinear theory to be available at hand. In the direction of developing large-amplitude nonlinear theory of animal locomotion, we have the pioneering contribution from Lighthill (1971), followed by Wu (2001), who formulated a nonlinear theory for three-dimensional flexible lifting surface of arbitrary shape performing arbitrary movement, except for taking a special kind of displacement function that still has limitations. These works can be compared with Pedley and Hill (1999), Wolfgang et al. (1999), and Sandberg et al. (2000) on aquatic locomotion based primarily on computational methods, an approach beyond the scope of this article. Here, new development is presented for a two-dimensional flexible lifting surface free to move along an arbitrary trajectory and performing motions of arbitrary amplitude. With a hybrid scheme using a Lagrangian description of the body motion jointly with the Eulerian description of the flow field, analysis is carried through to obtain a generalized Wagner integral equation for evaluating the wake vortices shed from the wing. In regard to workload involved in using this theory, the adoption of the surface variables pertaining to the boundary of the lifting surface and its vortex wake reduces the original problem of unsteady flow in (2 + 1) dimensions to a problem in (1 + 1) dimension. In generality, this theory can be applied asymptotically to locomotion of aquatic and aerial animals using wings and lifting surfaces of large aspect-ratio. The issue of separate determination of thrust and drag on aquatic animals in free swimming, first enunciated by Sir James Gray (1936) and subsequently pursued by many scientists in related disciplines, is of paramount importance. It applies not only to fish and cetaceans, but also to all aquatic and aerial animals of all sizes. This point is made in Section VII by using a microorganism (paramecium) moving at very low Reynolds numbers in free swimming versus another microorganism moving under gravity. The distinction between the two flow fields is found to be so sharp and dramatic that a first principle becomes self-evident. This principle is now founded with a basic criterion, proposed here: That for differentiating (by separate measurement) between the thrust and drag involved in animal locomotion (whether they are equal and opposite each other), the criterion is specified as distinguishing between the two dynamic states whether the specimen under study has zero net force (being composed of all forces) acting on it, and also has zero net torque acting on it if body rotation should prevail. Concerning body rotation, which is
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
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quite common in the microorganism kingdom, the state of zero net torque is in fact the whole issue about the spirochetes that was addressed by Chwang et al. (1971, 1974). Invoking this criterion implies that it is groundless to apply measurements made of specimens in one dynamic state to interpret or predict the behavior of specimens in another state. The argument is all based on Newton's First Law, for which it is a sophisticated application (i.e., to animated bodies). In parallel, we have made a separate resort to making mechanophysiological studies on energetics and hydromechanics of fish propulsion involving the biochemical and mechanical conversions of energy. The course of study, as delineated in Section VIII, is based on physiological data processed jointly with dynamic similarity analysis on specimens of different size groups covering various items for studies, including (i) the metabolic rate and scale effects, (ii) scaling of swimming velocity and energy cost, and (iii) scaling of viscous drag. The results obtained on these items imply consistently, uniformly, and conclusively that the boundary layers of the specimens tested in free swimming would have to be largely laminar at the high Reynolds number range covered (8 x 103 < Re < 107), with at most only a very small (presumably a distal) part turbulent. This is therefore concrete and compelling evidence for resolving Gray's paradox because it is based firmly on a very comprehensive scope and scale. With what we have achieved in the field of our profession to date, it is more encouraging to have an optimistic outlook for the future. Further hydrodynamic studies of the propulsive mechanisms by a new (3 + 1) dimensional large-amplitude nonlinear theory and experiment at high precision, jointly with well-coordinated observational endeavor concerning specific details of their operations under natural conditions, are extremely desirable. More specifically, a higher-order slender-body theory, with the added-mass effect more accurately resolved under interactions between body and vortex sheets shed from the body taken into account and viscous resistance to the cross-flow included, would be particularly useful. With these further advances, it would be of great significance to achieve resolution of the outstanding nonlinear effects arising in the flapping locomotion of birds, insects, skates, and rays from their using such operations as bending and twisting of flexible wings, stretching and shortening of wing span, adapting different durations for the power and recovery strokes, as well as starting and turning mechanisms in maneuvering control. Moreover, it should be of fundamental interest to integrate these separate studies in a system approach to examine how much they each contribute to the eventual optimal efficiency at very low energy cost as observed under the condition of zero net force in self-propulsion. No doubt other promising lines of research will occur to other workers in the field to continue promoting the advancement of this stimulating multidisciplinary field into the future.
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Acknowledgments M y deep appreciation and gratitude for the benefits of having e n l i g h t e n i n g discussions with Professors W i l l i a m Sears, F r a n k Marble, Y u a n - C h e n g Fung, N i c k N e w m a n , Allen C h w a n g , John Blake, and G e o r g e Yates; and Drs. Stuart Keller, W o o y o u n g Choi, W e n d o n g Qu, and Tao W a n g over various aspects and p h a s e s of this general study and related topics. This w o r k was partially s p o n s o r e d by N A S A u n d e r Grant N A G 5 - 5 2 0 7 .
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Author Index
Numbers in italics refer to pages on which the complete references are cited.
A
Banavar, J. R., 120, 135, 138, 143, 145, 146 Bannasch, R., 338, 350 Barber, B. P., 139, 142 Barnett, D. M., 162, 164, 165, 166, 171,179, 183, 186, 209, 211,212, 214, 249, 254,
Abeyaratne, R., 38, 53, 88 Abraham, F. F., 100, 119, 141, 143 Aburatani, H., 257, 279 Aero, E. L., 7, 88 Albano, E., 299, 350 Aldinger, E, 199, 289 Alejandre, J., 98, 124, 125, 127, 142 Alexander, R. M., 294, 350 Allen, M. P., 97, 99, 103, 123, 142 AI-Shreef, H., 239, 280 Anderson, D. M., 8, 88 Angenent, S., 53, 88 Anstis, G. R., 262, 279 Antman, S. S., 7, 23, 78, 88 Aoki, S., 68, 88, 159, 283 Arlt, G., 239, 240, 242, 243,246, 256, 280,
280, 282, 284, 287
Barnett, R. N., 133, 144 Bash, P. A., 103, 142 Batra, R. C., 32, 88 Bau, H. H., 135, 143 Beale, P. D., 256, 281 Bellur, K. R., 239, 280 Benjamin, I., 127, 142 Ben-Zion, Y., 167, 286 Beom, H. G., 209, 211,212, 213, 214, 280 Berglund, K., 32, 88 Bernardini, D., 8, 75, 92 B inz, E., 8, 88 Biscari, P., 7, 78, 81, 89 Blake, J. R., 293,350 Blake, R., 302, 351 Bofill, 79, 88 Bogy, D. B., 119, 142 Boltachev, G. Sh., 118, 142 Brennen, C., 293, 294, 296, 351, 352 Brenner, H., 111, 128, 142 Brett, J. R., 340, 343,344, 345,346, 350 Brevik, I., 139, 143 Brodskaya, E. N., 113, 145 Brokaw, C. J., 294, 296, 352 Brooks, B. R., 123, 146 Bruin, C., 106, 110, 118, 134, 143, 144
282, 285, 286
Ashley, H., 325,350 Atkinson, C., 68, 88 Atluri, S. N., 209, 211,212, 213, 214, 280 Augusti, G., 8, 87, 88, 92 Azuma, A., 296, 350
B
Bahr, H. A., 268,282 Baidakov, V. G., 118, 142 Bakker, A. F., 106, 110, 134, 144, 145 Balker, H., 268, 282 355
356
Author Index
Buff, E E, 106, 143 Burgers, J. M., 297, 350 Bykov, T. V., 118, 142
Cao, H., 258, 280 Capriz, G., 7, 8, 9, 10, 14, 23, 24, 25, 26, 38, 42, 68, 72, 73, 78, 79, 81, 88, 89, 92 Carey, V. P., 112, 137, 142 Carl, K., 258, 280 Carmer, C. S., 103, 142 Chan, K. H., 239, 280 Chang, K. H., 112, 143 Chang, T. M., 103, 142 Chantikul, E, 262, 279 Chantry, R. A. R., 116, 118, 145 Chapela, G. A., 110, 142 Ch~telain, A., 138, 145 Chen, B., 221,234, 235,281 Chen, L. J., 110, 142 Chen, T., 235,280 Chen, W. Q., 220, 231,234, 235, 239, 28O Chen, X., 239, 256, 280 Chen, Z. T., 209, 280 Cheng, H. K., 298, 350, 351 Cherepanov, G. E, 156, 234, 280 Chopra, M. G., 298, 350 Chou, E C., 96, 142 Chou, S. Y., 136, 137, 142 Chuck, L., 255,281 Chung, M. Y., 162, 175, 186, 199, 280 Chwang, A. T., 293,349, 350 Clarke, D. B., 139, 144 Cleveland, C. L., 133, 144 Coleman, B. D., 25, 42, 89 Colli, E, 8, 76, 89 Collier, L., 199, 240, 258, 267, 273,284 Comninou, M., 219, 280 Cook, R. E, 255,256, 280 Cosserat, E., 6, 7, 89 Cosserat, E, 6, 7, 89 Cowin, S. C., 7, 75, 89, 91, 92 Cross, L. E., 242, 257, 283 Crum, L. A., 139, 142 Cruse, T. A., 231,276, 281 Cuchiaro, J. D., 256, 281 Curtin, W. A., 68, 89
Daiguji, H., 103, 142 Dang, L. X., 103, 142 Dascalu, C., 275,281 Davi, E, 23, 83, 89 Davini, C., 7, 89 Dederichs, H., 242, 280, 282 Deeg, W. E J., 176, 186, 281 Degiovanni, M., 14, 89 de Heer, W. A., 138, 145 de Leon, M., 7, 8, 88, 90 Del Piero, G., 8, 9, 89 Deng, W., 186, 209, 218,219, 281, 285 Desai, R. C., 119, 143 DeSilva, C. N., 7, 89, 90 Desu, S. B., 256, 281 Dhaliwal, R. S., 7, 8, 75, 90, 93 Diaconita, V., 8, 75, 90 Dfaz-Herrera, E., 98, 124, 125, 127, 142 Di Carlo, A., 25, 90 Dimos, D., 256, 288 Ding, H. J., 220, 221,234, 235,275,280, 281 Doukhan, J. C., 186, 281 Drescher, J., 268,282 Du, S. Y., 186, 275,284, 287 Dugdale, D. S., 249, 281 Duiker, H. M., 256, 281 Dunn, J. E., 73, 90 Dunn, M. L., 178, 220, 281
Edwards, D. A., 111,128, 142 England, A. H., 219, 281 Epstein, M., 7, 90, 209, 286 Erdogan, E, 219, 281 Ericksen, J. L., 6, 7, 9, 19, 42, 53, 78, 90 Eringen, A. C., 7, 79, 90 Eshelby, J. D., 30, 53, 68,88, 90, 162, 186,281 Evans, A. G., 240, 241,258, 280, 285 Evans, D. J., 135,145 Evans, W. A. B., 118, 145
Fabrikant, V. I., 234, 281 Faivre, G., 186, 281 Fan, J., 239, 281
Author Index
Fan, W. X., 175, 186, 282 Fang, D. N., 235,239, 280, 281, 284 Farvaque, J. L., 186, 281 Feller, S. E., 123, 146 Field, M. J., 103, 142 Fife, P. C., 76, 93 Fischer, J., 98, 110, 135, 143, 144 Fleck, N. A., 79, 90, 239, 242, 283 Fomin, V. M., 25, 91 Forstmann, F., 98, 124, 125, 127, 142 Fosdick, R. L., 14, 90 Fowler, R. E, 118, 145 Fox, D. D., 7, 90, 93 Freiman, S. W., 245,255,256, 257, 280, 281, 285,285, 286, 288 Fr6mond, M., 8, 75, 76, 89, 90 Frenklach, M., 103, 142, 145 Freund, L. B., 65, 67, 68, 90, 92 Fried, E., 8, 53, 76, 91 Fu, R., 199, 242, 244, 245,247, 248, 249, 256, 262, 263,265, 267, 268, 271,273,277, 278,282
Fulton, C. C., 249, 282 Furuta, A., 199, 257, 279, 282 Futamura, K., 68, 89
Gabrielli, G., 340, 350 Gad-el-Hak, M., 130, 142 Gaitan, D. E, 139, 142 Gao, C. E, 167, 175, 186, 209, 218, 249, 254, 272,282
Gao, H., 249, 282 Gee, M. L., 131,142 Germain, P., 7, 25, 38, 91 Geyersmans, P., 133,142 Giesing, J. P., 325,350 Giovine, P., 8, 23, 25, 26, 68, 72, 73, 89 Gmelin, E., 186, 281 Gong, X., 238, 239, 273, 282 Goodman, M. A., 7, 91 Gorse, D., 133, 142 Grach, G., 8, 53, 76, 91 Gray, J., 293,300, 301,333,334, 348, 350 Green, A. E., 7, 91 Grest, G. S., 99, 110, 131,144, 145 Griffith, A. A., 271,282 Grioli, G., 7, 79, 91
357
Gubbins, K. E., 116, 118, 135,145 Guo, F. L., 275, 281 Gurtin, M. E., 8, 38, 53, 65, 67, 73, 76, 88, 91, 260, 282
Hack, J. E., 166, 189, 288 Hagood, N. W., 239, 280 Haile, J. M., 97, 99, 143 Han, J. C., 186, 275,284, 287 Han, P., 258, 259, 287, 288 Han, X. L., 249, 284 Hancock, G. J., 293,300, 301,350 Hao, T. H., 177, 238, 273,282 Harada, S., 257, 279 Harley, J. C., 135, 143 Haye, M. J., 118, 143 Heinbuch, U., 135, 143 Herbiet, R., 242, 280, 282 Herrmann, G., 156, 285 Herrmann, H. J., 53, 91 Herrmann, K. P., 209, 219, 220, 277,282 Hess, J. L., 325,352 Heyer, V., 262, 268, 282, 286 Heyes, D. M., 129, 130, 143 Hihara, E., 103, 142 Hill, M. D., 258, 282 Hill, R., 242, 283 Hill, S. J., 348, 352 Hom, C. L., 238, 283 Homola, A. M., 131,142 Hoover, W. G., 103, 143 Hou, P. E, 275,281 Hsieh, J. Y., 112, 143 Huang, J. H., 220, 228, 283 Huang, S. H., 220, 287 Huang, Y., 135,143 Huber, J. E., 239, 242, 283 Hughes, T. J. R., 7, 92, 93 Hui, C. Y., 220, 221,225,228, 229, 230, 283 Hutchinson, J. W., 68, 79, 90, 91, 242, 283 Hwang, C. C., 112, 143 Hwang, C. S., 258, 282 Hwang, H. J., 258, 287 Hwang, J. W., 83, 91 Hwang, K. C., 239, 280, 284 Hwang, S. C., 239, 240, 283 Hwu, C., 186, 284
358
Author Index
Iesan, D., 157, 283 Indekeu, J. O., 134, 145 Ioakimidis, N. I., 257,283 Irwin, G. R., 199, 271,283 Ishimaru, M., 134, 143 Israelachvili, J. N., 111, 127, 131,142, 143
Jame, R. D., 38, 91 Jensen, B., 139, 143 Jiang, L. Z., 186, 258, 261,283, 284 Jiang, Q. Y., 83, 93, 242, 257,283, 288 Jona, F., 83, 91, 283 Jones, R. T., 326, 350
Kakimoto, K., 134, 143 Kalikmanov, V. I., 118, 143 K~ilm~in,T. P., 325, 350 Kambe, T., 298,350 Kamiya, N., 265,284 Karihaloo, B. L., 209, 280 Karplus, M., 103, 142 Karpouzian, G., 298, 351 Kataoka, Y., 112, 144, 146 Katz, J., 325, 351 Kawano, S., 120, 121,143 Keblinski, P., 135, 145 Keller, S. R., 293, 334, 335,336, 351 Kenneth, L., 8, 90 Kerl, K., 129, 143 Khutoryansky, N., 175, 286 Kies, J. A., 199, 283 Kim, K. S., 167, 286 Kim, S. J., 239, 257, 283 Kimura, T., 132, 138, 144 Kingon, A. I., 239, 280 Kinjo, T., 118, 143 Kirkwood, J. G., 106, 143 Kiselev, S. P., 25, 91 Kishimoto, K., 68, 88, 159, 283 Knops, R. J., 32, 91 Knowles, J. K., 38, 53, 88 Koch, S. W., 119, 143 Koepke, B. G., 257, 284
Koga, K., 118, 143 Kogan, L., 220, 221,225, 228, 229, 230, 283 Kohn, R. V., 8, 91, 92 Kolleck, A., 264, 284 Kollman, P. A., 103, 145 Koplik, J., 120, 135, 138, 143, 145, 146 Kotake, S., 103, 143, 145 Kramarov, S. O., 256, 284 Kremer, K., 99, 144 Krishnaprasad, P. S., 7, 93 Kuang, Z. B., 209, 286 Kudryavtsev, B. A., 152, 175, 285 Kuethe, A. M., 299, 351 Kumar, S., 235,284 Kuo, C.-M., 162, 165, 166, 171,179, 183,209, 211,212, 214, 284, 287 Kurashige, T., 138, 144 Kuroki, M., 103, 143 Ktissner, H. G., 326, 351 Kuvshinskii, E. V., 7, 88
L Lacasse, M., 110, 145 Lam, C. G., 325,326, 351 Lam, P. S., 68, 92 Landau, L. D., 76, 92, 237, 284 Landis, C. M., 239, 242, 283 Landman, U., 120, 133, 144 Lang, T. G., 334, 351 Laradji, M., 128, 140, 144 Lawn, B. R., 255,256, 262, 279, 280 Laws, N., 7, 91 Lee, J. S., 186, 284 Lee, S., 214, 288 Lekhnitskii, S. G., 162, 284 Li, C. Q., 239, 284 Li, D., 103, 144 Li, Q., 213,286 Li, S. E, 275,284 Li, X. E, 249, 284 Liang, J., 186, 221,234, 235, 281, 284 Liang, X. G., 96, 103, 142, 144 Liang, Y. C., 186, 284 Liao, J. J., 112, 143 Liew, K. M., 275,284 Lifschitz, E. M., 237, 284 Lighthill, M. J., 293, 294, 296, 297, 300, 301, 302, 305,306, 307, 314, 348, 351
359
Author Index
Lin, F. Z., 235,280 Little, E. A., 83, 92 Liu, G. N., 220, 221,228, 231,234, 235,288 Liu, J. X., 186, 284 Liu, Y. J., 220, 221,228, 231,234, 235,288 Lloyd, I. S., 258, 282 Loboda, V. V., 209, 219, 220, 277,282 Long, L. N., 114, 144 Lothe, J., 162, 164, 165, 171,186, 280 Lu, P., 186, 275,284 Lu, W., 239, 284 Luedtke, W. D., 133, 144 Lukes, J. R., 96, 101,103, 106, 107, 108, 109, 111,142, 144, 146 Lupascu, D. C., 259, 285 Lusk, M. T., 8, 91 Lynch, C. S., 83, 91, 199, 239, 240, 258, 261, 267, 273, 280, 281, 283, 284
Meguid, S. A., 186, 209, 218, 219, 281, 285, 289
Mehta, K., 256, 273,285 Melnick, B. M., 256, 281 Meschke, E A., 264, 284 Mezic, I., 111, 136, 144 Micci, M. M., 114, 144 Miller, R. C., 247, 285 Mindlin, G, 7, 79, 92 Molkov, V., 220, 221,225, 228, 229, 230, 283 Moriguchi, K., 134, 143 Moseler, M., 120, 144 Moss, W. C., 139, 144 Motooka, T., 133, 134, 143, 144 Mouritsen, O. G., 128, 140, 144 Munetoh, S., 133, 134, 143, 144 Murillo, L. E., 298, 350 Musesti, A., 14, 89 Muskhelishivili, N. I., 162, 285, 311,328, 351
M
Mai, Y. W., 186, 209, 219, 275,286 Majumdar, A., 111, 136, 144 Makino, H., 265, 284 Mao, S. X., 249, 284 Mao, X., 199, 209, 249, 286 Mariano, P. M., 8, 9, 23, 25, 29, 32, 38, 42, 53, 67, 75, 83, 87, 88, 89, 92 Markov, K. Z., 8, 92 Marlow, R. S., 7, 88 Marsden, J. E., 7, 92, 93 Marshall, D. B., 262, 279 Maruyama, S., 96, 98, 101,132, 138, 144 Marzocchi, A., 14, 89 Mataga, P. A., 275,284 Matsumoto, M., 112, 114, 118, 143, 144, 146 Matsumoto, S., 101, 138, 144 Maugin, G. A., 25, 53, 65, 92, 275,281 McCune, J. E., 325, 326, 351 McCutchen, C. W., 330, 351 McFadden, G. B., 8, 88 McGuiggan, P. M., 131,142 McHenry, K. D., 257, 284 McMeeking, R. M., 83, 91, 175, 183, 199, 239, 240, 241,242, 258, 267, 273,277, 283, 284, 285
McMillan, L. D., 256, 281 Mecholsky, J. J., 255,281 Mecke, M., 98, 110, 144
Nabarro, E R. N., 7, 79, 92, 244, 285 Nagai, M., 334, 351 Naghdi, P. M., 7, 91, 92 Nakamura, Y., 333,351 Narita, E, 178, 286 Nedjar, B., 8, 90 Newman, J. N., 301,302, 304, 305,310, 351, 353
Nicolas, P., 8, 75, 90 Nijmeijer, M. J. P., 106, 110, 134, 144 Nishihira, K., 133, 144 Nishioka, T., 274, 286 Noll, W., 9, 25, 92 Nuffer, J., 259, 285 Nunziato, J. W., 8, 75, 89, 92 Nye, J. E, 161,285
Ogita, A., 101,144 Ohara, T., 132, 144 Ohguchi, K., 118, 143 Okazaki, K., 285,285 Orowan, E., 199, 271,285 Owen, D. R., 8, 9, 42, 89 Ozoe, H., 134, 143
360
Author Index
Pak, Y. E., 156, 162, 166, 176, 186, 187, 198, 220, 258, 259, 260, 285, 287 Park, S. B., 162, 183, 260, 261,263,272, 285, 287
Park, S. H., 101,106, 107, 108, 109, 111,116, 118, 119, 120, 145, 146 Parry, G., 7, 89 Parton, V. Z., 152, 175,285 Pastor, R. W., 123, 146 Paz de Araujo, C. A., 256, 281 Pedley, T. J., 294, 348, 352 Peierls, R., 242, 285 Pence, T. J., 38, 93 Penrose, O., 76, 93 Pertsev, N. A., 240, 242, 280, 285 Ping, T., 154, 172, 173, 174, 177, 179, 274, 276, 288 Podio-Guidugli, E, 7, 8, 25, 79, 89, 91 Pohanka, R. C., 255,256, 273, 280, 285 Pontikis, V., 133, 142 Povstenko, Y. Z., 7, 93 Powles, J. G., 118, 145 Prasad, N. N. V., 159, 286 Putterman, S. J., 139, 142 Pyatetskiy, V. Y., 346, 347, 352
Rice, R. W., 245,255,286 Rifai, M. S., 7, 93 Ritter, A. P., 256, 288 Rivlin, R. S., 7, 91 Robbins, M. O., 131,145 Robels, U., 242, 256, 282, 286 Rodden, W. E, 299, 325,350 Rodel, J., 259, 285 Rogers, R. C., 28, 93 Rosakis, P., 83, 93 Roux, S., 53, 91 Rowlinson, J. S., 104, 106, 110, 116, 118, 142, 145
Ru, C. Q., 199, 209, 218, 219, 249, 254, 286 Rusanov, A. I., 113,145
Saada, G., 186, 281 Sakara, M., 159, 283 Sakata, M., 68, 88 Sandberg, W. C., 348, 352 Sando, M., 258, 287 Savage, A., 247, 285 Saville, G., 11 O, 142 Scalia, A., 257, 283 Schmidt-Nielsen, K., 340, 352 Schneider, G. A., 199, 262, 264, 268, 282, 284, 286, 289
Qi, H., 235,281 Qian, C.-E, 154, 172, 173, 174, 177, 179, 199, 268, 271,274, 276, 277,278, 282, 288 Qin, Q. H., 186, 209, 219, 275,286 Qin, T. Y., 233,235,287 Qu, J., 213,286 Quintanilla, 79, 88
Ramamurti, R., 348, 352 Ramfrez-Santiago, G., 98, 124, 125, 127, 142 Raynes, A. S., 257, 288 Read, W. T., 162, 281 Renardy, M., 28, 93 Rez, J. S., 256, 284 Ribarsky, M. W., 133, 144 Rice, J. R., 29, 68, 93, 155, 167, 214, 277, 286
Scott, J. E, 256, 281 Scott, M. T., 325,326, 351 Sears, W. R., 325,332, 350 Segev, R., 8, 14, 25, 93 Serrin, J., 73, 90 Shang, J. K., 258, 259, 287, 288 Shankar, N., 238, 256, 283, 288 Shchekin, A. K., 118, 143 Shelleman, D. L., 255,281 Shen. S., 274, 286 Shen, S. P., 209, 286 Shen, Y. E, 220, 221,228, 231,234, 235,288 Shen, Z. Y., 177, 282 Shibahara, M., 103, 145 Shillor, M., 8, 90 Shindo, Y., 178, 286 Shintani, A., 134, 143 Shioya, T., 220, 231,234, 235, 280 Shirane, G., 83, 91, 283
361
Author Index
Shockley, W., 162, 281 Shpeizman, V. V., 265,289 Sides, S. W., 110, 145 Sih, G. C., 274, 286, 289 Sikkenk, J. H., 106, 110, 134, 144, 145 Silhavy, M., 42, 93 Simo, J. C., 7, 90, 93 Singh, R. G., 260, 287 Singh, R. N., 235,284 Singh, U. C., 103, 145 Skokov, S., 103, 145 Sleigh, M. A., 293,350 Smith, A. M. O., 325,352 Smith, H. L., 199, 283 Smith, P. L., 255,285 Smith, T. E., 237, 286 Socolescu, D., 8, 88 Sokolnikoff, I. S., 171,200, 286 Sosa, H. A., 175, 220, 286, 287 Spedding, G. R., 298, 351 Stazi, F. L., 87, 92 Stecki, J., 124, 126, 145 Steinmann, P., 78, 93 Stoll, W. A., 239, 281 Storz, L., 255,281 Strang, G., 8, 91, 92 Stroh, A. N., 162, 287 Struthers, A., 53, 91 Su, Y., 314, 352 Subbarao, E. C., 242, 257, 283 Sun, C. T., 162, 183,258, 260, 261,263,272, 283, 285, 287
Suo, Z., 162, 165, 166, 171,179, 183, 199, 209, 210, 211,212, 214, 238, 239, 240, 258,267,273,282, 284, 287, 288 Suzuki, D., 132, 144 Szeri, A. J., 139, 145
Taylor, G. I., 293,300, 301,352 Theordorsen, T., 326, 352 Thompson, P. A., 131,132, 145 Thompson, S. M., 110, 116, 118, 142, 145 Tien, C. L., 96, 101,103, 106, 107, 108, 109, 111,116, 118, 119, 120, 142, 144, 145, 146
Tiersten, H. E, 7, 79, 92 Tildesley, D. J., 97, 99, 103, 123, 142 Ting, T. C. T., 162, 164, 166, 175, 186, 191, 199, 280, 287 Tobin, A. G., 258, 259, 260, 287 Todd, B. D., 135, 145 Tolman, R. C., 116, 145 Tong, P., 166, 167, 172, 174, 189, 249, 272, 282, 287, 288
Toupin, R. A., 7, 32, 93 Toxvaerd, S., 124, 126, 128, 144, 145 Travis, K. P., 135,145 Tricomi, E G., 7, 19, 32, 93 Troian, S. M., 131,132, 145 Truesdell, C. A., 6, 7, 78, 90, 93 Tsutsui, K., 136, 146 Tucker, V. A., 340, 352 Tully, J. C., 101,145 Tuttle, B. A., 256, 288
Uchino, K., 199, 256, 257, 279, 282, 287 Ugarte, D., 138, 145 Umehara, T., 134, 143
van Leeuwen, J. M. J., 106, 110, 134, 144, 145
T Tajima, K., 258,287 Takahashi, K., 96, 142 Tan, M. J., 275,284 Tan, X., 258, 259, 287, 288 Tanaka, K., 178, 286 Tanasawa, I., 112, 145 Tang, R. J., 233,235,287 Tashiro, S., 285,285 Tavares, T. S., 325,326, 351
van Woerkom, A. B., 110, 144 Vaudin, M. D., 257, 288 Vergeles, M., 135, 145 Villaggio, P., 7, 32, 78, 91, 93 Virga, E. G., 7, 8, 9, 10, 14, 25, 38, 89, 90, 92, 93
Virkar, A. V., 256, 273,285 Visintin, A., 8, 76, 89 Voight, W., 6, 93 von Alpen, U., 186, 281 von K~irm~in,T., 297, 325,332, 340, 350
362
Author Index
Vorozhtsov, E. V., 25, 91 Vossnack, E. O., 134, 145 Vuong, V. Q., 139, 145 Vu-Quoc, L., 7, 93
Wu, T. Y., 293, 294, 296, 298, 301,302, 304, 305,308, 310, 311,312, 313,315,316, 334, 335,336, 337, 339, 341,342, 344, 348, 349, 350, 351, 352, 353
W
Wagner, H., 325,332, 352 Wainwright, W. L., 7, 91 Walker, J. A., 348, 352 Walsh, E. K., 8, 75, 92 Walton, J. P. R. B., 116, 118, 145 Wang, B., 186, 220, 231,284, 287 Wang, B. L., 275,287 Wang, H. Y., 260, 287 Wang, J., 7, 8, 75, 90, 93 Wang, M. Z., 209, 218, 282 Wang, T. C., 249, 254, 272, 287 Wang, Z. K., 220, 287 Warren, W. E., 237, 286 Warren, W. L., 256, 288 Wasan, D. T., 111,128, 142 Waser, R. M., 256, 288 Webb, P. W., 334, 344, 346, 347, 352 Weihs, D., 298, 325, 340, 351, 352 Weiner, B., 103, 142, 145 Weis-Fogh, T., 294, 352 Weng, J. G., 101,106, 107, 108, 109, 111, 116, 118, 119, 120, 145, 146 Westneat, M. M., 348,352 Wheeler, A. A., 8, 88 White, G. S., 257, 258, 282, 288 Whitman, A. B., 7, 89, 90 Widom, B., 104, 106, 145 Wienecke, H. A., 220, 281 Willeke, M., 129, 143 Willemsen, J. E, 135, 143 Williams, E W., 186, 284 Williams, M. L., 219, 288 Williams, W., 7, 89 Willis, J. R., 162, 165, 166, 171,179, 183, 209, 211,212, 214, 287 Winet, H., 293,349, 350 Winkelmann, J., 98, 11O, 144 Winzer, S. R., 256, 288 Wolfgang, M. J., 348, 352 Wong, B. C., 114, 144 Wu, K. C., 213,288 Wu, S., 135,146
Xu, Z., 258, 259, 287, 288
Yamaguchi, Y., 138, 144 Yang, J. X., 138, 146 Yang, W. J., 136, 146, 199, 238, 240, 241,258, 267, 273,284, 288, 289 Yao, Z., 235,281 Yasui, K., 139, 146 Yasuoka, K., 112, 114, 118, 143, 144, 146 Yates, G. T., 302, 314, 339, 341,342, 344, 352, 353
Yin, X. L., 256, 288 Yoo, I. K., 256, 281 Young, D. A., 139, 144, 145 Yu, J. S., 220, 283 Yu, S. W., 209, 275, 280, 286
Zemel, J. N., 135, 143 Zeng, X. C., 118, 142, 143 Zhang, L. B., 256, 288 Zhang, Q. C., 256, 288 Zhang, T.-Y., 154, 166, 167, 172, 173, 174, 177, 179, 186, 189, 199, 214, 242, 244, 245,247, 248, 249, 256, 262, 263,265, 267, 268, 271,272, 273, 274, 276, 277, 278, 282, 288 Zhang, Y., 123, 146 Zhao, M. H., 186, 220, 221,228, 231,234, 235,288 Zheng, B. L., 220, 221,287 Zhoga, L. V., 265,289 Zhong, Z., 209, 218, 289 Zhu, T., 240, 241,258, 273,288, 289 Zhuang, L., 136, 137, 142 Zickgraf, B., 199, 289 Zuo, J. Z., 274, 289
Subject Index
Abel's integral equation, 332 Aerial locomotion, hydrodynamic theories, 296-299 Airfoil dynamics brisk start-stop process, 330 nonlinear lifting-surface theory, 314-332 unsteady, 325 Amiiform locomotion, 302 Antiplane deformation conductive cracks, 202-205 elliptical cavity propagation under, 185-186 general solution for, 166-167 Antiplane solution, fracture mechanics for mode III cracks, 173-175 Aquatic locomotion classical slender-body theory, 301-311 energetics, scale effects in, 338-347 generalized slender-body theory, 311-314 hydrodynamic theories, 296-299 nonlinear lifting-surface theory, 314-332 resistive theory, 300-301 Arbitrary loading, planar cracks under, 231-236 Aspect-ratio lifting surface, 295 small and large, 297 Asymmetric loading, ellipsoidal cavity under, 229-231 Axisymmetric loading, ellipsoidal cavity under, 225-228
Balance of interactions at crack tip, 56-57 from invariance of outer power, 16-21 in presence of discontinuity surfaces, 33-38 Bending strength electric field effect, 247-249 PZT ceramics, 265-267 temperature effect, 245-247 Bernoulli equation, 304, 321 Biofluiddynamics International Conference, 315 viscous drag and, 339 Body-fin movement, in slender-body theory, 302-305, 311-314 Boiling, explosive, 136-137 Boundary conditions body-fin movement, 304-305 classic MD simulation, 100-101 electric, on electrically insulating crack faces, 175-178 electric potential, 203 kinematic, 322 Boundary element method, 276 Boundary layer leading-edge separation, 298 microcracked solids, 14 viscous, 297 Boundary surface variables, 319 Bubbles MD simulation, 118-119 sonoluminescence, 139-140 spreading wetting, 138
363
364
Subject Index
Bulk configurational forces in, 43-46 constitutive restrictions in, 39-40 Bulk liquid, vapor film in, 112-113 Bulk vapor, liquid film in, 104-112
Capillary phenomenon, nanoscale tube, 138-139 Cauchy kernel, 327 Cauchy's theorem, 14 Cauchy stress, 16, 37-38, 69, 80 Caudal fin section interaction with vortex sheet, 310-311 leading edges, 313-314 Cavity electric field within, 172-173 ellipsoidal, under remote loading, 223-231 elliptical cylinder, see Elliptical cylinder cavity propagation, energy release rates for, 183-186 shrunken to crack, 204, 208, 226 Ciliary locomotion, 293-294 stokeslets and, 335-336 zero net force, 336-337 Clap-fling operation, wasp wings, 294 Compact tension specimens fracture resistance curves and, 264 poled PZT-4 ceramics, 268-269 prenotched, 258-259 Compliance dependence on temperature, 243-245 volume elastic, 247-248 Computation of nonlinear lifting-surface theory generalized Wagner-von K~irm~in-Sears method, 325-331 hybrid analytical-numerical method, 331-333 time-marching method, 324-325 Condensation, and evaporation: coefficients, 112 Conductive cracks antiplane deformation, 202-205 and conductive inclusions, 218-219 electric fracture toughness, 199 experimental observations, 267-270
interaction with piezoelectric dislocation, 205-209 J integral, 271 uniform remote loading, 200-202 Configurational forces in the bulk, 43-46 on discontinuity surface, 46-53 tip balance of, 59-61, 64 Configurations, bodies lacking in discontinuity surfaces, 9-11 Connection, recognizability, 4-5 Constitutive equations, isothermal linear, 156-157 Constitutive restrictions, multifield theories, 38-42 Contact line region, 138-139 Contact zone model, interface cracks, 219-220 Cosserat and Cosserat's theory, 6-7 Cosserat continua, 76-78 Crack face electrically insulating, 175-178 loading finite interface crack under, 216-217 semi-infinite crack with, 215 Crack propagation energy release rates for, 159-160, 183-186 in materials with substructure, 53-68 Cracks conductive, see Conductive cracks general electrically impermeable, 197-198 electrically permeable, 195-196 impermeable planar, unarbitrary loading, 231-235 interface, see Interface cracks mode III electrically impermeable, 196-197 electrically permeable, 194-195 fracture mechanics, 173-175 mechanical and electrical fields, 180-183 permeable planar, unarbitrary loading, 236 piezoelectric materials containing, 155-156, 158-160 planar moving, kinematics, 54-56 Crack tip balance of interactions at, 56-57 configurational forces, balance, 59-61 driving force at, 63-65 electrically insulating, 249-254 intensity factors at, 232-233
Subject Index near, 180 intensity factors, 182 velocity fields, 55-56 Critical film thickness, MD simulation, 111 Curie temperature, 81 Cylindrical interfaces density profile and surface tension, 121-122 liquid-solid interfaces, 135-136 MD simulation, 119-122 stable, 121
D
Damping, triggering of, 244-245 Decay, elastic energy, substructure effect, 30-32 Defects crystalline: piezoelectric dislocation,
365
bending strength electric field effect, 247-249 temperature effect, 245-247 piezoelectric ceramics, 242-249 temperature-dependent compliance, 243-245 use of critical stress criterion, 273-274 Drag and thrust, in fish locomotion, 333-338 viscous, scaling of, 344-347 Driving force, at crack tip, 63-65 Droplets containing solid core, 137-138 density profile, 114-115 formation and dynamics, 114 formation in liquids, 128 MD simulation, 113-118 pressure tensor, 115-116 surface tension, 116-118
186-199
diffusion process of, 133-134 evolution in materials with substructure, 42-53 Deformation antiplane conductive cracks, 202-205 elliptical cavity propagation under, 185-186 general solution for, 166-167 microcracks, 83-87 Degree of orientation, nematic liquid crystals, 80-81 Density profile droplets, 114-115 liquid film in bulk vapor, 104-105 liquid jet, 121 planar interfaces liquid-liquid, 124 liquid-solid, 134 Diffeomorphism, 19 Discontinuity surfaces, see also Interfaces balance in presence of, 33-38 configurational forces on, 46-53 constitutive restrictions at, 40-42 without and with own structure, 35-38 Domain switching mechanics treatment, 277 piezoelectric ceramics, 239-242 Domain wall kinetics model
E
Elastic materials linear, with voids, 74-75 with substructure, 26-32 Electrical enthalpy, piezoelectric materials, 153 Electric displacement intensity factor, 213-214, 233-234, 252-253 crack tip, 181-182 pseudoisothermal, 157-158 at upper and lower crack faces, 176-177 Electric field effect on bending strength, 247-249 fracture of piezoelectric ceramics, 256-264 inside cavity, 172-173 Electric strip saturation model, 272 Electrostatic problems 2-D, piezoelectric ceramics, 162-186 3-D ellipsoidal cavity under remote loading, 223-231 general solutions, 220-223 planar cracks under arbitrary loading, 231-236
366
Subject Index
Electrostriction, 237-239 Elliptical cylinder cavity interaction with general piezoelectric dislocation, 191-193 piezoelectric screw dislocation, 187-190 under remote loading, 167-175 surface, extended line force on, 193 Elongated-body theory, see Slender-body theory Energetics aquatic locomotion, scale effects in, 338-347 ciliates, 336-337 Energy cost, scaling of, 341-344 dissipated in crack tip process zone, 66-67 excess, 107-108 free, 152 full Gibbs, piezoelectric materials, 153 Energy release rate applied and local, failure criteria, 271-273 for crack propagation, 159-160, 183-186 critical applied, 278 local, as failure criteria, 254 piezoelectric materials, 155-156 Enskog's rule, 129 Equilibrium MD methods, 101 Eshelby tensor, 29, 46 Euler-Lagrange equations, 28 Evaporation, and condensation: coefficients, 112 Evolution damage in PZT ceramics, 259 defects and interfaces in materials with substructure, 42-53 Evolution equation, for interface, 49, 52 Explosive boiling, 136-137 Extended compliance tensor, 171 Extended line force, on elliptical cavity surface, 193 F
Fading memory theorems, 20, 25 Failure criteria fracture behavior of piezoelectric ceramics, 270-274 local energy release rate as, 254 Feathering, proportional, 298
Ferroelectric ceramics domain switching, 241-242 microcracking, 258-259 Ferroelectric solids, rnultifield theories, 81-83 Film stability and rupture, 110-112 thin, ferroelectric, 149 vapor, in bulk liquid, 112-113 Finite difference method, classic MD simulation, 99 Finite element analysis, 269 nonlinear, 239 Fish locomotion classical slender-body theory, 301-314 flexible lifting-surface: nonlinear theory, 314-333 thrust and drag in, 333-338 Flagellar hydrodynamics spirilla locomotion, 293 spirockete locomotion, 293 Flexibility, longitudinal, slender body with, 305-311 Flow boundary condition fish locomotion, 315 questions regarding, 131-132 Force, see also Self-force configurational, 43-61 driving, at crack tip, 63-65 on piezoelectric dislocation, 198-199 Foreign particles, diffusion process of, 133-134 Fracture behavior alternating electric field effect, 256-259 bending tests, 265-267 conductive cracks, 267-270 microstructure and temperature effects, 255-256 piezoelectric ceramics mode III cracks, 173-175 nonlinear approaches, 236-254 static electric field effect, 259-264 Fracture toughness conductive cracks, 268 electrical, 270, 278 electrical and mechanical, 199 piezoelectric ceramics, 255-256 scattering of, 263-264 Free energy, piezoelectric materials, 152 Free energy density, 39-40, 72 along crack, 61
Subject Index
Froude hydromechanical efficiency, 301 Full Gibbs energy, piezoelectric materials, 153, 155
Gauss theorem, 15, 17, 20, 29, 37, 44, 51 Geometry, interfaces, 33-35 Grain boundary, microcracks initiated at, 245-246 Gray's paradox, 293, 349 Green's function in boundary element method, 276 piezoelectric dislocation and, 186-199
Hagen-Poiseuille flow, 132, 135 Hall-Petch effect, 79 Hovering, chalcid wasp, 294 Hybrid analytical-numerical method, 331-333 Hydrodynamic theories, aquatic and aerial locomotion, 296-299 Hypersingular boundary integral equations, 232-233
Image dislocation method, 189-190 Indentation fracture technique, PZT ceramics, 260-263 Inertia effects pertaining to macroscopic motion, 21-26 on standard and substructural balances, 57-59 Initial conditions bubble formation and, 118-119 for liquid and vapor phases, 100 In-plane coordinate rotation matrix, 165-166 Intensity factors at crack tip, 1~ 1-182, 232-233 electric displacement, 184, 213-214, 233-234, 252-253 slit crack electrical and mechanical fields, 178-183 strain, 204 stress, 213-214, 230-231,273-274
367
Interactions balance at crack tip, 56-57 from invariance of outer power, 16-21 measures of, 12-16, 41 surface configurational, 47 Interface cracks contact zone model, 219-220 curve-shaped, 218 electrically impermeable, 277 finite, under crack face loading, 216-217 impermeable, semi-infinite, 210-215 permeable, 218 piezoelectric ceramics, 209 Interfaces, see also Discontinuity surfaces cylindrical, see Cylindrical interfaces evolution in materials with substructure, 42-53 geometry, 33-35 MD simulations liquid-liquid, 122-130 liquid-solid, 130-136 liquid-vapor, 103-122 planar, see Planar interfaces spherical, see Spherical interfaces Interfacial tension, planar interfaces: liquid-solid, 134 Intermolecular potentials, classic MD simulation, 97-99 Internal dissipation inequality, 64 Internal energy, piezoelectric materials, 152 Inviscid irrotational flow theory, 297, 316 Isothermal potential energies, piezoelectric, 154 Isotopic mixtures, liquid, 128-130
J integral for conductive cracks, 271 modified expression of, 65-66 path-independent, 156 in terms of intensity factors, 214
Kay's rule, 129 Kelvin's circulation theorem, 323-324, 330
368
Subject Index
Kinematics fluid particles, 320 planar moving cracks, 54-56 Kinetic energy density, 21-25 Knops-Villaggio's effect, 32 Kronecker delta, 157, 238 Kutta condition, 305,307-308, 312, 315,317, 321,325-327, 330
L Lagrangian densities, elastic materials with substructure, 27-30 Lanthanum, PZT ceramics doped with, 257-258, 261-262 Laplace equation, 303-304 Latent microstructures, 7 Latent substructures, 68-73 Lattices, representative volume element, 85-86 Leading-edge section, anterior: fish body, 305-307, 312 Lead zirconate titanate, see PZT ceramics Lifting surface aspect-ratio, 295 locomotion, flexible, 314-333 undulatory motion, 297 Lighthill's theory, 307 Liquid jets, MD simulation, 119-122 Liquid-liquid interfaces MD simulation, 122-130 mixtures, 128-130 planar, 124-128 spherical, 128 Liquid phase epitaxy, 133 Liquid-solid interfaces cylindrical, 135-136 planar, 130-134 spherical, 134-135 Liquid-vapor interfaces cylindrical, 119-122 MD simulation, 103-122 planar, 104-113 spherical, 113-119 Lithographical Induced Self-Assembly, 136-137 Localization phenomena, of deformation, 87 Local pressure components, liquid film in bulk vapor, 105-107
Locomotion aquatic resistive theory, 300-301 scale effects in energetics of, 338-347 aquatic and aerial, hydrodynamic theories, 296-299 fish classical slender-body theory, 301-314 thrust and drag in, 333-338 flexible lifting-surface, unified nonlinear theory, 314-333 maneuvering modes, 347-348 Longitudinal flexibility, slender body with, 305-311 Lunate-tail theory, 298
M
Maxwell stress, 237 MD, see Molecular dynamics simulation Mechanical dissipation inequality, 39, 45, 5O-52, 61-63 Mechanical enthalpy, piezoelectric materials, 153 Mechanical fracture, toughness, 199 Metabolic rate, and scale effects, 340-341 Microcracked bodies, models, 18 Microcracked materials, multifield theories, 83-87 Microcracks deformation, 83-87 distributions, 3 in PZT specimens, 257-258 as virtual substructure, 14 Micromorphic continuum, 7 Micromorphic materials, multifield theories, 78-79 Microstructure, fracture of piezoelectric ceramics and, 255-256 Mixtures, liquids, MD simulation, 128-130 Molecular dynamics simulation classic, 97-103 liquid-liquid interfaces, 122-130 liquid-solid interfaces, 130-136 liquid-vapor interfaces, 103-122 nonclassic, 103 sonoluminescence, 139-140 surfactants, 140 three-phase systems, 136-139
Subject Index Momentum substructural and standard, 58 wake, 337-338 Motions active propulsive, 317 swimming, Reynolds number, 334-337 velocity fields, 11-12 Multifield theories balance in presence of discontinuity surfaces, 33-38 configurations and balance of interactions, 9-26 constitutive restrictions, 38-42 Cosserat continua, 76-78 elastic materials with substructure, 26-32 ferroelectric solids, 81-83 latent substructures, 68-73 materials with substructure crack propagation in, 53-68 evolution of defects and interfaces, 42-53 material with voids, 74-75 microcracked materials, 83-87 micromorphic materials, 78-79 nematic liquid crystals, 80-81 two-phase materials, 75-76 Muscular power specific, 393
Nanoboundary problems, planar interfaces: liquid-solid, 130-133 Nanoscale tube, capillary phenomenon, 138-139 Nematic liquid crystals, multifield theories, 80-81 Noether's theorem, 6 Nonequilibrium MD methods, 102-103 Nonisotopic mixtures, liquid, 128-130 No-slip condition, at nanoboundary, 131-132
Observer changes of, 16, 19, 24-25, 78 external spatial, 4 fixed and moving, 44, 47-48
369
Oscillation, bubble, and sonoluminescence, 139-140 Outer power, invariance of, 16-21
Patch displacement, 83 in multifield theories, 2 placement, 9-10 second-order tensor for, 3 Phantom molecules model, 101 Phase transformation-toughening theory, 241 Piezoelectric ceramics conductive cracks, 199-209 domain switching, 239-242 domain wall kinetics model, 242-249 electrostriction, 237-239 fracture behavior experimental observations, 255-270 failure criteria, 270-274 mode III cracks, 173-175 nonlinear approaches, 236-254 interface cracks, 209-220 polarization saturation model, 249-254 poling field, 150 2-D electroelastic problems and Stroh's formalism, 162-186 3-D electroelastic problems, 220-236 Piezoelectric dislocation force on, 198-199 interaction with conductive crack, 205-209 elliptical cavity, 191-193 finite crack, 217 semi-infinte crack, 215-216 screw type, interaction with elliptical cavity, 187-190 use of Laurent series, 186-187 Piezoelectric materials basic equations for thermodynamic functions, 152-161 electric domain, 148 elliptical cavity solution in, 169-172 Piola-Kirchhoff stress, 42, 84 Planar interfaces liquid film in bulk vapor, 104-112 liquid-liquid interfaces, 124-128 liquid-solid interfaces, 130-134
370
Subject Index
Planar interfaces (Contd.) three-phase systems, 136-137 vapor film in bulk liquid, 112-113 Planar moving cracks, kinematics, 54-56 Plasticity Cosserat, 78 strain gradient, 79 Plemelj's formula, 311-312, 321-322 Poincare-Bertrand formula, 328 Polarization crystalline materials, 81 ferroelectric domain wall evolution during, 83 switching, piezoelectric ceramics, 240 Polarization saturation model, 249-254 Polarization saturation zone, 272 Poling piezoelectric ceramics, 150 PZT ceramics, 265-267 Pore, as virtual substructure, 14 Potential flow theory, 338 Prandtl's acceleration potential, 304 Pressure tensor, droplets, 115-116 Process zone, crack tip, energy dissipated in, 66-67 Pronation power stroke, 299 Proportional feathering, 298 Proportional flexurality, 299 Pseudoisothermal approach, piezoelectric materials, 157-160 Pyroelectric phase, 148 PZT ceramics bending strength, 265-267 containing vacuum flaw, 179 crack failure, 271 damage evolution in, 259 ferroelectric fatigue failure, 257-258 fracture toughness, 256, 263-264 indentation fracture technique, 260-263
Quantum molecular dynamics, 103 Quasi-stationary wakeless flow, 326
Radial decay, 32 Remote loading
ellipsoidal cavity under, 223-231 elliptical cylinder cavity under, 167-175 uniform, conductive cracks, 200-202 Representative volume element, lattice, 85-86 Resistive theory, aquatic locomotion, 300-301 Reynolds number aquatic and aerial locomotion, 296-297 high Reynolds number, 296, 314-332 low Reynolds number, 296, 334-337 resistive theory and, 301 Riomann-Hilbert problem, 311,328 Rupture cylindrical interface, 122 film, 112
Saint-Venant's effect, 30, 32 Salmon metabolic rate and scale effects, 340-341 scaling of viscous drag, 344-347 Scale effects, metabolic rate and, 340-341 swimming velocity and energy cost, 341-344 viscous drag, 344-347 Scattering, fracture toughness, 263-264 Second-gradient theories, latent substructures, 70-73 Self-force decomposition, 69 elimination of, 77 surface, 37 Self-propulsion microorganism in viscous fluid, 293 slender swimming animal, 300 Semi-infinite piezoelectric crack, 210-214 Shrinking cavity to crack, 204, 208, 226 disc to crack tip, 57-58, 60, 62 Side-edge section, trailing: fish body, 307-310, 312-313 Slender-body theory, 297-298 applied to body motion, 347 fish locomotion, 301-314 Slit crack, electric and mechanical fields, 178-183 Solid-melt interfaces, 133-134
Subject Index Sonoluminescence, MD simulation, 139-140 Spherical interfaces liquid-liquid interfaces, 128 liquid-solid interfaces, 134-135 MD simulation, 113-119 three-phase systems, 137-138 Spirochete locomotion, 293 Spreading wetting, MD simulation, 138 Stability film, 110-112 planar interface: liquit ~-liquid, 127-128 Static equilibrium equation, 153-154 Stokeslet aquatic locomotion and, 300 distribution, 335-336 Stress domain switching-induced, 240 intensity factor, 213-214, 233-234, 273-274 local, liquid film in bulk vapor, 107-108 Maxwell, 237 Peierls-Nabarro, 244 Piola-Kirchhoff, 42, 84 pseudoisothermal, 157-158 Stroh's formalism, solutions to 2-D electrostatic problems, 162-166 Stroke, upward and downward, 299 Structure, own, discontinuity surfaces, 35-38 Substructural interactions balance, 77 at crack tip, 56-57 representational problem, 4 Substructural kinetic coenergy density, 22-23 Substructure elastic materials with, 26-32 latent, 68-73 materials with crack propagation in, 53-68 evolution of defects and interfaces in, 42-53 virtual, pore as, 14 Supination recovery stroke, 299 Surface tension bubbles, 119 droplets, 116-118 liquid film in bulk vapor, 110 liquid jet, 121-122 planar interfaces: liquid-liquid, 124-125 Surfactants, MD simulation, 140 Swimming
371
motion, Reynolds number, 334-337 velocity, scaling of, 341-344
T Taylor expansion, 31 Temperature compliance dependent on, 243-245 effects bending strength, 245-247 fracture of piezoelectric ceramics, 255-256 jump, at flow boundary, 132-133 Thermodynamic functions, piezoelectric materials, 152-161 Thermophysical properties, MD simulation, 101-102 Three-phase systems planar interfaces, 136-137 spherical interfaces, 137-138 spreading wetting and contact line region, 138-139 Thrust, and drag, in fish locomotion, 333-338 Time-marching method, computational, 324-325 Tolman length, 117-118 Tolman's equation, 116 Toughness, see Fracture toughness Traction, crack tip, 65 Transition layers, two-phase materials, 75-76 Transport coefficients interfacial, 141 MD simulation, 102-103 planar interfaces: liquid-liquid, 125-127 Two-phase materials, multifield theories, 75-76
U Unified nonlinear theory, flexible lifting-surface locomotion, 314-333
V Vapor film, in bulk liquid, 112-113 Velocity crack tip, 55-56, 61
Subject Index
372 Velocity (Contd.) swimming, scaling of, 341-344 Verlet algorithm, 99 Virtual work crack-free piezoelectric solid, 158-159 piezoelectric, 154-156 Voids, materials with: multifield theories, 74-75 von K~irm~in's vortex street, 337 Vortex sheets free and bound, 320-321 interaction with body, 349 caudal fin, 310-311 shed from side fins, 308 wake, 318
9
ISBN
0-12-002038-6
W
Wagner effect, 293,296, 332 Wagner integral equation classical, 330 generalized, 330 Wagner-von K~irm~in-Sears method, 325-331 Wake momentumless, 337-338 vortex sheets, 318 Walls, solid, classic MD simulation, 100
Wasp, chalcid, hovering, 294 Weis-Fogh mechanism, 294
