Recent developments and applications of invariant integrals Yi-Heng Chen School of Civil Engineering and Mechanics, Xi’an Jiao-Tong University, Xi’an 710049, P.R. China;
[email protected]
Tian Jian Lu Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, UK;
[email protected] Although invariant integrals 共path independent integrals兲 have been used extensively in the 20th century, mainly in the calculation of dominant parameters that govern the initiation and propagation of both linear and nonlinear cracks, new applications are increasingly being identified. This article presents developments and applications of the invariant integrals in recent years, focusing on four major application areas: i) fracture mechanics of functional materials 共eg, piezoelectric ceramics and ferromagnets兲, which exhibit features different from those found in purely mechanical problems due to the coupling of electric, magnetic, thermal, and mechanical quantities; ii) damage mechanics of multiple interacting cracks, and new damage measures; iii) domain integrals, two-state integrals, and their applications in determining the dominant parameters of 3D cracks and in clarifying the role of higher order singular terms in the Williams eigenfunction expansions; and i v ) nano-structures 共eg, stress driven surface evolution in a heteroepitaxial thin film兲. In writing this review article, we have been able to draw upon a large number of published works on invariant integrals over the last three decades, and yet it is impossible to cover the whole subject in the limited space available. Consequently, the main aim of the article is to summarize the major developments and applications in the four important areas mentioned above. Still, 261 references are reviewed in the article. 关DOI: 10.1115/1.1582199兴
1
INTRODUCTION
Over the past decades a variety of methods within the general framework of Fracture Mechanics have been proposed to determine the critical parameters that govern the stability and growth of a crack 共or cracks兲. Among these, invariant integrals 共also known as path-independent integrals, following the fundamental theorem of Eshelby’s energy momentum tensor 关1–3兴 and the finding of the J -integral 关4 – 8兴兲 have proved extremely attractive, for two main reasons. Firstly, several invariant integrals such as J M , and L bear clear physical meaning as the energy release rate (ERR) for a crack-like defect 关4 –11兴. Secondly, such an approach only requires the evaluation of contour integrals along a contour far apart from the crack tip, which is readily available for many crack configurations and loading conditions from numerical solutions 共eg, finite element method, abbreviated as FEM , or boundary element method, BEM ); no special treatment of near-tip singular stress fields or the use of singular elements near a crack tip is necessary 关12–15兴. 共Notice,
however, that fine FEM meshes are still needed in the tip region to capture the asymptotic behavior in order to obtain accurate solutions of stress and displacement fields far away from the tip.兲 As pointed out by Kanninen and Popelar 关16兴, although Eshelby in 1956 关1兴 was the first to derive invariant integrals, Cherepanov 关6,7兴 in 1967 and Rice 关4,5兴 in 1968 were apparently the first to recognize their potential use in Fracture Mechanics. 共It should be mentioned that, although Cherepanov’s work was a little earlier than Rice’s work, his work was first published in Russian and was not widely referenced until his book was translated into English in 1979 关8兴; for this reason the invariant integrals are sometimes called Rice-Cherepanov integrals 关17兴.兲 Subsequently, a large number of researchers have focused on the further development of invariant integrals and their applications. Earlier works include Bueckner 关18兴, Budiansky and Rice 关9兴, Knowles and Stermberg 关11兴, Stern et al 关12兴, Bergez 关13兴, Hellen and Blackburn 关14兴, Freund 关15兴, Blackburn 关19,20兴,
Transmitted by Associate Editor KP Herrmann
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© 2003 American Society of Mechanical Engineers
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Chen and Lu: Developments and applications of invariant integrals
Rice et al 关21兴, Herrmann 关22兴, Fletcher 关23兴, Ishokawa et al 关24兴, King and Herrmann 关25兴, among many others. Kirshimoto et al 关26,27兴 established the Jˆ integral for dynamicproblems. Lorenzi 关28兴, Murakami and Sato 关29兴, and Bakker 关30兴 applied the J -integral to 3D crack problems. Rice 关31兴 summarized conservation integrals and energetic forces in detail, and later established the weight functions theory for 3D crack analysis 关32兴. Yau and Wang 关33兴, Park and Earmme 关34兴, Matos et al 关35兴, Rice 关36兴, and Chen and Hasebe 关37,38兴 made use of invariant integrals to determine dominant parameters for interface cracks. Other applications in traditional materials with a single crack can be found in Sinclair et al 关39兴, Wu et al 关40兴, Tsamasphyros 关41兴, Wu 关42兴, and Bueckner 关43兴, whereas Suhubi 关44兴, Maugin 关45兴, and Ani and Maugin 关46兴 considered conservation laws in nonlinear cases, and Atluri 关47兴 and Epstein and Maugin 关48兴 extended these investigations to account for finite elasticity and inelasticity with body forces and inertia effects. For applications in Damage Mechanics with micro cracks, Hutchinson 关49兴 and Ortiz 关50–52兴 applied the J -integral in microcrack shielding problems, assuming that the global J -integral is identically equal to the local J -integral when the contour enclosing the microcrack zone surrounding a macrocrack tip becomes infinitely small. For applications in functional materials 共eg, piezoelectrics兲, Cherepanov 关8兴 established the path-independent integral, ie, the ⌫-integral, in coupled mechanical-electric fields. Using the Eshelby energy momentum tensor 关2,3兴, Pak and Herrmann 关53,54兴, Pak 关55,56兴, McMeeking 关57兴, Maugin and Epstein 关58兴, and Dascalu and Maugin 关59兴 extended the J -integral 共or the J k -vector兲 concept to piezoelectric materials. Suo et al 关60兴 proposed to use the J -integral as a piezoelectric fracture criterion. Park and Sun 关61,62兴 divided the J -integral representation into two distinct parts: the mechanical part and the electric part, and then by entirely neglecting the electric part introduced the Mechanical Strain Energy Release Rate (M SERR) as the piezoelectric fracture criterion. Gao and coworkers 关63–70兴 extended the Dugdale crack model 关71兴 to describe piezoelectric fracture, and introduced the local-global J -integral concept in an electric saturation strip ahead of an impermeable crack. Their results show that the local J -integral accounting for electric yielding may be used as a fracture criterion. Wang and Shen 关72兴, Sabir and Maugin 关73兴, and Fomethe and Maugin 关74兴 attempted to use invariant integrals to solve direct electromagnetic coupling problems 共eg, electrostriction兲, with both linear and nonlinear constitutive equations considered. For single crack problems, another significant application of invariant integrals is based on Betti’s reciprocal theorem 关75兴 and the work-conjugate integral concept 关18兴. Following the pioneering work of Bueckner 关18兴, Chen 关76兴 in 1985 found that the associated pseudo-orthogonal property of the Williams eigenfunction terms of a plane crack could be used to derive new weight functions. In fact, about ten years earlier, Mazya and Plamenevskij 关77,78兴 共published in Russian兲 had already found this important relation in a general form for elliptical boundary value problems, and used it to develop effective methods for determining crack-tip stress intensity factors. They called this relation the biorthogonal
Appl Mech Rev vol 56, no 5, September 2003
property 关78,79兴, as seen in the two Russian books by Leguilon and Sanchez-Palencia 关80兴 and Mazya et al 关81兴. More recently, Chen and Hasebe 关37,38兴 extended this property to treat interface cracks and established useful weight functions, whereas Ma and Chen 关82,83兴 extended this property to treat cracks in piezoelectric solids and dissimilar piezoelectric solids. Another new development received much attention in recent years. Chen and Shield 关84兴 established three conservation laws for two equilibrium states, and called these the two-state conservation integrals. The two-state integrals in particular have been widely employed 关85– 89兴 to calculate stress intensity factors (SIFs) and elastic T-stresses ahead of a crack tip, and to determine dislocation strength. For example, Choi and Earmme 关89兴 used the two-state L -integrals to evaluate SIFs for a circular arc-shaped interfacial crack, while Im and Kim 关90兴 applied the two-state M -integrals to treat singular fields associated with a generic wedge. Hui and Ruina 关91兴 examined in detail the higher order singular terms (HOSTs) and their physical significance in describing the influence of the nonlinear zone surrounding a crack tip. It is established that, unlike the classical fracture mechanics where the HOSTs are always neglected due to the bounded values of elastic strain energy and displacement at a crack tip, in the presence of a relatively large nonlinear zone the contribution of the HOSTs to the global J -integral cannot be neglected 关91兴. This contribution is induced from the interaction between the HOSTs and higher order non-singular terms 共ie, the higher order regular terms兲. The work of Hui and Ruina 关91兴 provides a useful note as well as a better understanding of the well-known Small Scale Yielding (SSY ) concept; subsequent development on this topic can be found in Chen and Hasebe 关92兴. Jeon and Im 关93兴 applied the two-state J - and M -integrals to clarify the role of the HOSTs in elastic-plastic fracture. Chen and coworkers 关94 –113兴 applied the J -integral to treat macrocrack-microcrack and macrocrack-microvoid interactions in various structural materials, including brittle materials, metal/ceramic bimaterials, unidirectional fiberreinforced composites, and laminated composites. Parallel investigations in functional materials, eg, piezoelectric ceramics, are reported in 关114 –118兴, where the J -integral 共or the J k -vector兲 analysis for a system of interacting cracks in a transversely isotropic piezoelectric material was developed and used as a consistency check for numerical results. Particularly, Chen and Lu 关118兴 considered many interacting cracks arbitrarily oriented and distributed in a piezoelectric ceramic and found that, when the electric loading as well as the poling direction is no longer perpendicular to crack surfaces, not only the J –integral (⬅J 1 ) but also the J 2 –integral should be accounted for. Moreover, the conservation laws of the J k –vector in such functional materials are established, which can be considered as a direct extension from conventional materials 关119–122兴 to piezoelectric materials 关118兴. Note that, with some exceptions, most of the above investigations are not only limited to single crack problems but also restricted to quadratic energy. The most widely accepted invariant integrals are Rice’s J -integral, the M -integral, and the L -integral 关9–11兴, all of which are related to the crack
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Chen and Lu: Developments and applications of invariant integrals
ERR and can be established by applying Eshelby’s energy momentum tensor 关1–3兴. Physically, the J - and M -integrals can be interpreted as the ERR for crack extension and uniform crack expansion, whereas the L -integral corresponds to the ERR for crack rotation 关9,10兴. However, because the evaluation of the J-, M -, or L -integrals cannot distinguish the contribution by crack opening from that induced by shearing, it is disadvantageous to use these integrals in mixed-mode crack problems 共eg, crack kinking or deflection兲. Subsequently, a number of alternative invariant integrals have been proposed 共see, eg, 关12–15兴兲, which can be used to compute the distinct values of opening and shear mode stress intensity factors. In most practical cases, this presents a major advantage over singular finite elements 关123兴 as well as other special techniques such as the boundary collocation method 关124兴, the singular hybrid FEM 关125兴, or the enriched FEM 关126兴. The advantage is particularly significant in 3D crack problems 关127–134兴 where the decomposition method of the J -integral and various weight functions have been proposed. The J -integral concept has also proved useful in treating crack problems at the nano-scale 关135兴. For example, Gao and coworkers applied the J -integral to treat stress singularities along a cycloid rough surface 关136兴 and stress-driven evolution in a heteroepitaxial thin film structure 关137,138兴, where a very thin single-crystal layer of one material is deposited onto a single-crystal substrate of another material having the same crystalline structure but different lattice spacings. Admittedly, because the published works on invariant integrals since 1967 are so rich and illuminating, it is impossible for us to write a review article covering the whole subject within the limited length available. Instead we aim to provide an overview of the more recent progresses in this field, with focus placed on new and novel applications derived in recent years and potential developments in the future. Because there are four apparent tendencies in recent years for the development of invariant integrals as mentioned in the Abstract, we attempt to review the major features of such applications that are worth noting and further studying. We focus not only on the new and attractive features of invariant integrals associated with the four new applications, but also on the major difficulties that further advances in these applications may be facing. The topics covered include: 1兲 Eshelby’s energy momentum tensor and invariant integrals in functional materials with defects; 2兲 Bueckner’s work conjugate integral and the associated pseudo-orthogonal property 共or biorthogonal property兲 in both structural and functional materials; 3兲 the two-state integral and its applications; 4兲 new conservation laws of the J k -vector, and the role of the M - and L -integrals in multiple interacting crack problems or microcrack damage problems; 5兲 applications of invariant integrals in studying damage and fracture of functional materials; 6兲 the role of higher order singular terms and the associated two-state J -integral and M -integral; 7兲 the domain integral concept and its applications in the calculation of 3D cracks; 8兲 applications of invariant integrals in nano-structures.
517
This review article is divided into seven main sections. Section 2 provides background information for the most famous invariant integrals established in the earlier years, and summarizes their major features and conclusions derived by previous researchers. Only after doing so, the extension of invariant integrals to functional materials, microcrack damage problems, and other complicated cases could become natural and straightforward in subsequent sections. In Section 3, the application of invariant integrals in functional materials is discussed, with focus placed on the novel and distinct features that are different from the basic understandings associated with traditional structural materials. The application of invariant integrals in microcrack damage problems is discussed in Section 4: new conservation laws in multiple crack problems will be reviewed. Section 5 is devoted to recent applications of the two-state integral concept originally established by Chen and Shield 关84兴; in particular, the role of the HOSTs in the Williams eigenfunction expansion form is clarified 关93兴. The mode decomposition of 3D mixed-mode cracks and the application of the domain integral and two-state integrals to 3D crack problems are reviewed in Section 6. Finally, in Section 7, potential applications of invariant integrals in nano-structures are discussed. 2 TRADITIONAL MATERIALS WITH SINGLE CRACKS The essential features of invariant integrals in traditional materials with a single crack are reviewed in this section. Famous integrals established in the early years, their major applications, and the significant results derived by previous researchers are summarized. Particularly, invariant integrals pertinent to functional materials and many interacting cracks, and domain integrals and two-state integrals are discussed in detail. 2.1 Energy momentum tensor Invariant integrals can be constructed in various ways, the most popular being the energy momentum tensor developed by Eshelby 关1–3兴. The original description of the energy momentum tensor starts from the definition of the Lagrangian density L(u i ,u i, j ,X m ) and the Euler equation:
L L ⫺ ⫽0 X j u i, j u i
(2.1)
where u i are displacements and X j are Cartesian coordinates. Distinguishing the explicit partial derivative of L with respect to X i when its other arguments u i ,u i, j and the remaining X m are held constant, one gets:
冉 冊 L Xi
⫽ exp
L 共 u i ,u i, j ,X m 兲 Xi
冏
u i ,u i, j ⫽cont.,X m ⫽cons.,m⫽i
(2.2) Thus,
冉 冊
L L L u i, j L ⫽ u i,l ⫹ ⫹ Xl ui u i, j X l Xl
(2.3) exp
In view of 共2.1兲, Eq. 共2.3兲 can be simplified to
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Chen and Lu: Developments and applications of invariant integrals
Pl j⫽
L u ⫺L ␦ l j u i, j i,l
(2.4)
冉 冊
(2.5)
where
Plj L ⫽⫺ Xi Xl
exp
Here, P l j is defined as the energy momentum tensor, and its static form associated with elastic media is P l j ⫽w ␦ l j ⫺ i j u i,l
(2.6)
where ␦ l j is the Kronecker delta function, w is strain energy density in linear elastostatics, and i j and u i are stresses and displacements, respectively, with w⫽
冕
i j
0
i j d i j
i, j⫽1,2,3
(2.7)
2.2
The J k -vector and the M - and L -integrals In plane elastic media, Eshelby’s energy momentum tensor 关2,3兴 provides a fundamental method for establishing invariant integrals. For example, the most famous invariant integrals, the J k -vector, the M -integral, and the L -integral, can be easily deduced from Eshelby’s formulation 共2.6兲, as 关9–11兴: J k⫽ M⫽
冕 冕
C
共 wn k ⫺u i,k T i 兲 ds,
C
L⫽e 3i j
共 k⫽1,2兲
(2.8)
共 wx i n i ⫺T k u k,i x i 兲 ds
冕
C
(2.9)
共 wx j n i ⫺T i u j ⫺T l u l,i x j 兲 ds
(2.10)
where T k is the traction acting on the outside of a closed contour C, x j ( j⫽1,2) represents a rectangular plane coordinate system, n i refers to the outside normal of the contour C, and u i denotes the displacements, while e 3i j is the alternating tensor depending on the arrangement of the integer numbers i and j: e 3i j ⫽
再
⫺1
when
i⫽2 and
0
when
i⫽ j
1
when
i⫽1 and
Appl Mech Rev vol 56, no 5, September 2003
of the crack along the x 1 and x 2 axes, respectively 关10兴; whereas the M - and L -integrals represent the ERR induced from the uniform expansion and rotation of the crack, respectively 关9,10兴. Alternatively, when the closed contour C only encloses one tip of the crack, the J -integral represents the ERR associated with the unit advance of the crack tip 关4,5兴. Under uniform remote loading, the two components of the J k –vector calculated along a closed contour C surrounding one single crack completely vanish, since no energy is released when the crack moves along the x 1 or x 2 axis. However, it should emphasized that, as Herrmann and Herrmann found out 关10兴, the traction-free crack surfaces have nontrivial contributions to the second component, J 2 , owing to the outside normal of crack surfaces being n 2 ⫽⫾1. This may lead to the path-dependence of J 2 when the closed contour encloses only one tip of the crack. Herrmann and Herrmann 关10兴 added that the integrals, M and L, provide a more natural description of the ERR associated with plane cracks than the integrals, J 1 and J 2 , due to different contour selections and apparently different physical meanings. If we consider a contour completely enclosing a whole crack, then the trivial results of J 1 ⫽J 2 ⫽0 follow. On the other hand, if we consider a contour enclosing only one crack tip, then the magnitude of J 2 is dependent upon how the starting and ending points of the contour are selected on the upper and lower surfaces of the crack, ie, J 2 is pathdependent. In addition, the M –integral is directly related to the SIFs induced from both tips of a crack as shown by Freund 关15兴, while the L –integral is related not only to the SIFs but also to the contribution induced from the tractionfree surfaces of the crack 关10兴. Recently, following Herrmann and Herrmann 关10兴, Chen 关119兴 found that the M -integral for a plane crack is twice the change of the total potential energy (CT PE) due to cracking 关139兴: M ⫽2U where U is the CT PE defined by U⫽
1 2
j⫽1 (2.11) j⫽2
The Eshelby conservation laws 关2,3兴 reveal that the three invariant integrals of 共2.8兲, 共2.9兲, and 共2.10兲 all vanish when the closed contour C encloses no singularities. Here, as pointed out by Herrmann and Herrmann 关10兴, the closed contour C introduced to define the J k -vector is, generally speaking, different from that used to define the M -integral and the L -integral. For the J k -integral vector, C surrounds either one or both tips of the crack, but for M - and L -integrals, it should be chosen such that the whole crack with both tips is completely enclosed. Physically, when the remote uniform loading remains unchanged and the closed contour C encloses a single crack completely, the two components of the J k -vector represent the ERR induced from the movements
(2.12a)
冕
a
⫺a
⬁ i2 ⌬u i 共 x 兲 dx,
共 i⫽1,2兲
(2.12b)
with ⌬u i representing the displacement jump across the upper and lower surfaces of the crack. Chen found 关119兴 that the M -integral and the L -integral are not independent, related by a very simple formulation owing to Eqs. 共2.12a,b兲 and the definition of the rotational ERR, as: L⫽⫺
1 M 2
(2.12c)
where denotes the crack rotation angle, provided that the remote loading remains unchanged. The important relation 共2.12c兲 has yet been widely recognized. In Section 4 the physical significance of Eqs. 共2.12a,b,c兲 will be discussed in detail when dealing with multiple interacting cracks or microcrack damage problems.
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Chen and Lu: Developments and applications of invariant integrals
2.3 Bueckner’s work conjugate integral The use of Eshelby’s energy momentum tensor, although convenient and representing a general law of continuum physics for any material behavior and any form of energy 共ie, not restricted to quadratic energy兲 关45,81,140兴, is not the only method with which invariant integrals could be established. For example, when the energy in deformable bodies is quadratic, Betti’s reciprocal theorem 关75兴 provides an alternative, from which Bueckner 关18兴 proposed the work conjugate integral: (I) (II) B 共 u (I) , (II) i , i j ,u i i j 兲⫽
冕
␥
(II) (II) (I) 关 u (I) i i j ⫺u i i j 兴 n j ds
(2.13) where the superscripts 共I兲 and 共II兲 represent two different equilibrium stress-displacement fields, and ␥ denotes a closed contour surrounding a single tip of a crack in plane elasticity. Generally speaking, state 共I兲 represents a real stress-displacement field for which the stresses and displacements far away from the crack tip are known 共eg, from FEM ) but the crack tip parameters are unknown, whereas state 共II兲 represents a complementary field for which the stresses and displacements are known analytically and can be used to establish certain weight functions to calculate crack tip parameters associated with state 共I兲. Both states satisfy the traction-free conditions of plane cracks, but state 共II兲 needs not to satisfy external boundary conditions. From Betti’s reciprocal theorem 关75兴, it can be established that Bueckner’s integral is path-independent, irrespective of whether the closed contour ␥ encloses one single tip or both tips of a crack. It should be emphasized that, although Bueckner’s integral based on Betti’s reciprocity 关75兴 can indeed be used to establish weight functions for calculating crack dominant parameters, it is limited to quadratic energy 关45兴. It is Eshelby’s energy momentum tensor 关1–3兴 rather than Betti’s reciprocity 关75兴 that constitutes a general law of continuum physics 关45兴. However, when restricted to cases with quadratic energy 共valid for most engineering applications兲, Bueckner’s integral method is in general more convenient than the energy momentum tensor to develop weight functions for determining the SIFs and subsequently the T -stress 关31,32,37,38,43,141–143兴. Moreover, by selecting two special kinds of the complementary stress-displacement field 共II兲 in a homogeneous plane elastic material, dissimilar isotropic elastic material, anisotropic elastic material, and dissimilar anisotropic material, respectively, Chen and coworkers 关37,38,76,144 –146兴 have shown that the J -integral and the M -integral are merely two special cases of Bueckner’s integral. For example, when the closed contour encloses one single tip of a crack, by taking the complementary state 共II兲 to be the differentiation of the real state 共I兲
(II) ij ⫽
(I) ij , x1
u (II) i ⫽
u (I) i x1
(2.14a)
and substituting states 共I兲 and 共II兲 into 共2.13兲, one gets: B⫽2J
(2.14b)
519
Similarly, when the closed contour encloses both tips of a crack, by taking the complementary state 共II兲 to be (I) (II) i j ⫽ i j ⫹x l
(I) ij , xl
u (II) i ⫽x l
u (I) i xl
(2.14c)
and substituting states 共I兲 and 共II兲 into 共2.13兲, one gets: B⫽2M
(2.14d)
Similar to the L-M relation expressed in 共2.12c兲, relations 共2.14b,d兲 have yet to be widely recognized. The most significant shortcoming for the direct use of the J-, M -, or L -integrals is that none can distinguish the contribution by crack opening from that induced by shearing. It is therefore disadvantageous to use these integrals in mixedmode problems, eg, mixed-mode fracture, crack kinking, and/or deflection. A number of alternative invariant integrals have subsequently been proposed 共see, eg, 关12–15,18 – 22,147–158兴兲, which can be used to calculate directly the distinct values of opening mode and shear mode SIFs. Of great significance in this area may be the finding of the pseudo-orthogonal property of the Williams eigenfunction expansion form (EEF) for a semi-infinite crack in brittle solids or in dissimilar materials 关37,38,76,145,146兴, together with the concepts of two-state integrals 关84兴 and domain integrals 关87,128 –130兴. The pseudo-orthogonal property of EEF can be derived from Bueckner’s integral for a semi-infinite crack in an infinite plane elastic body 关76 – 81兴. The property reveals that: (k) (l) (l) B 共 u (k) i , i j ,u i , i j 兲
⫽ ⫽
冕
再
␥
(l) (l) (k) 关 u (k) i i j ⫺u i i j 兴 n j ds
共 ⫹1 兲共 ⫺1 兲 k⫹l Re关 A k ¯A l 兴 / , 0,
共 k⫹l⫽0 兲
共 k⫹l⫽0 兲
(2.15) where Re refers to the real part of a complex, over bar denotes complex conjugate, A k and A l are complex coefficients, and superscripts (k) and (l) represent two stressdisplacement states corresponding to eigenvalues n⫽k/2 and n⫽l/2 for each pair of terms in the Williams EEF formulated by Muskhelishvili’s complex potentials (z) and (z) 关159兴: ⬁
兺
共 z 兲⫽
n⫽⫺⬁
⬁
(n)
共 z 兲⫽
⬁
共 z 兲⫽
兺
n⫽⫺⬁
兺
n⫽⫺⬁
A n z n/2
⬁
(n)
共 z 兲⫽
兺
n⫽⫺⬁
D n z n/2
(2.16a)
Here, z⫽x 1 ⫹ix 2 , i⫽ 冑⫺1, and the complex coefficients A n and D n satisfy the following dependent relation to ensure the traction-free conditions of crack surfaces: D n ⫽⫺
冋
n A ⫹ 共 ⫺1 兲 n ¯A n 2 n
册
(2.16b)
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Chen and Lu: Developments and applications of invariant integrals
In 共2.16兲, the terms for n⬎0 are physically permissible in the whole cracked plane, n⫽0 represents rigid motion of the body, whereas the terms for n⬍0 are physically permissible only in the region outside of the near-tip nonlinear zone 共see, eg, Rice 关36兴, Hui and Ruina 关91兴, Chen and Hasebe 关92兴, and Jeon and Im 关93兴兲. According to the classical theory of Linear Elastic Fracture Mechanics (LEFM ), the terms corresponding to nⱮ0 are in general omitted to ensure the strain energy density and displacements are bounded at the crack tip. Some researchers have nevertheless found the special use of these terms in determining weight functions, complementary state (II) in 共2.13兲, or elastic T -stresses 关141–143兴, and have called these higher order singular terms or outside expansions 关36 –38,91–93兴. Mathematically, Eq. 共2.15兲 shows a feature similar to that of classical orthogonal series, but the condition (k⫹l⫽0) and the nontrivial value in 共2.15兲 make it slightly different. For convenience, 共2.15兲 is termed here as the pseudoorthogonal property, to show its major feature and to distinguish it from the orthogonal series 关145,146兴. As previously mentioned, in Russian literature, it has also been called the biorthogonal property 关77,78兴. Using Rice’s complete EEF for a semi-infinite interface crack 关36,160,161兴, Chen and Hasebe 关37兴 concluded that the pseudo-orthogonal property is also valid in dissimilar elastic materials 共see Fig. 1兲. Rice’s complete EEF 关36兴 can be written as:
1 共 z 兲 ⫽e ⫺ z 1/2⫺i f 共 z 兲 ⫹D 1 g 共 z 兲
共 z苸R 1 兲
1 共 z 兲 ⫽e z 1/2⫹i¯f 共 z 兲 ⫺D 1¯g 共 z 兲
共 z苸R 1 兲
2 共 z 兲 ⫽e z 1/2⫺i f 共 z 兲 ⫹D 2 g 共 z 兲
共 z苸R 2 兲
2 共 z 兲 ⫽e
⫺ 1/2⫹i¯
f 共 z 兲 ⫺D 2¯g 共 z 兲
z
共 z苸R 2 兲
f 共 z 兲⫽
a
⬁
兺
n⫽⫺⬁
e nz n
B ⌫⫽⌫1⫹⌫2 ⫽ ⫽
(2.16d)
冕
再
⌫1⫹⌫2
H 1 ⫽⫺
关 u (i ␣ ) (i j ) ⫺u (i  ) (i ␣j ) 兴 n j ds
0
共 n⫹m⫹1⫽0 兲
H1
共 n⫹m⫹1⫽0 兲
where 共 C 1 ⫹C 2 兲
1 2
冋
Re
a n¯b m n⫹1/2⫺i
(2.16g)
册
(2.16h)
Secondly, if u (i ␣ ) and (i ␣j ) are induced from one pair of terms in the second summation of 共2.16c兲 with an integer n in 共2.16d兲 and u (i  ) and (i j ) are induced from another pair of terms in the second summation of 共2.16c兲 with an integer m in 共2.16d兲, then the reduced Bueckner integral over the closed contour ⌫⫽⌫ 1 ⫹⌫ 2 has the following property 共Fig. 1兲:
⫽
(2.16c)
n zn 兺 n⫽⫺⬁ n⫺1/2⫺i
g共 z 兲⫽
where and are shear and bulk moduli of the material, respectively. The pseudo-orthogonal property found by Chen and Hasebe 关37兴 for the terms in Eq. 共2.16c兲 is stated below in detail. Firstly, if u (i ␣ ) and (i ␣j ) are induced from one pair of terms in the first summation of 共2.16c兲 with an integer n in 共2.16d兲 and u (i  ) and (i j ) are induced from another pair of terms in the first summation of 共2.16c兲 with an integer m in 共2.16d兲, then the reduced Bueckner integral over a closed contour ⌫⫽⌫ 1 ⫹⌫ 2 has the following property 共Fig. 1兲:
B ⌫⫽⌫1⫹⌫2 ⫽
where (z) and (z) are Muskhelishvili complex potentials, subscripts 1 and 2 refer to the upper 共material #1兲 and lower half planes 共material #2兲, respectively 共Fig. 1兲, and f and g are associated analytical functions defined as: ⬁
Appl Mech Rev vol 56, no 5, September 2003
冕
再
⌫1⫹⌫2
关 u (i ␣ ) (i j ) ⫺u (i  ) (i ␣j ) 兴 n j ds
0
共 n⫹m⫹1⫽0 兲
H2
共 n⫹m⫹1⫽0 兲
(2.17a)
where H 2 ⫽⫺
4nC 1 C 2 Re关 e n¯f m 兴 1 2 共 C 1 ⫹C 2 兲
(2.17b)
Thirdly, all other combinations, for example, if u (i ␣ ) and (i ␣j ) are induced from one pair of terms in the first summation of 共2.16c兲 with an integer n and u (i  ) and (i j ) are induced from one pair of terms in the second summation of 共2.16c兲 with an integer m, then the reduced Bueckner integral always vanishes.
Here, a n and e n are complex coefficients, and C 1⫽ 1⫹ 1 2 C 2⫽ 2⫹ 2 1 ⫽
(2.16e)
冉 冊
C1 1 ln 2 C2
D 1 ⫽2 1
2 ⫹1 C 1 ⫹C 2
D 2 ⫽2 2
1 ⫹1 C 1 ⫹C 2
(2.16f)
Fig. 1 A semi-infinite interface crack in plane dissimilar bimaterials provided by Chen 关63兴
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Chen and Lu: Developments and applications of invariant integrals
It has been established that the pseudo-orthogonal property 共or, equivalently, the biorthogonal property兲 can be used to establish invariant integrals and weight functions for many crack problems. The key is to select a suitable complementary stress-displacement field 共II兲 in 共2.13兲 that satisfies traction-free conditions on crack surfaces. Different selections of field 共II兲 will lead to different kinds of invariant integrals, a feat very difficult to achieve by using Eshelby’s energy momentum tensor 关2,3兴. Another advantage of the pseudo-orthogonal property is that the induced invariant integrals can distinguish between the portions contributed separately by crack opening and crack sliding. Furthermore, this property is found to be valid in various cases where material anisotropy and dissimilar anisotropic materials are taken into account 关82,83,145,146兴. In the following, this topic will no longer be discussed for the sake of brevity, although it will be seen in the forthcoming sections dealing with functional materials that the pseudo-orthogonal property is important for distinctly evaluating the SIFs and electric displacement intensity factor (EDIF). The purpose of presenting this property in this review article is mainly to advocate its importance and applications in crack problems, such that new applications may be found in other functional materials such as shape-memory alloys and ferromagnets. Before proceeding, it must be pointed out that the above approach has a major deficiency. That is, the weight functions established by using Bueckner’s work conjugate integral and the pseudo-orthogonal property are related to the complete Williams EEF in each special case. Although this is straightforward for cracks in piezoelectric materials 关82,83兴, it is not clear that such EEF can always be found in other functional materials. For example, to the authors’ knowledge, no complete EEF analysis for a plane crack in ferromagnets exists in the open literature, although the J -integral expression for soft ferromagnets can be found in Sabir and Maugin 关73兴. In the next section, it will be shown that two-state integrals 关84兴, as direct extensions of the J-, M -, and L -integrals, have no such deficiency: they are independent of the use of the complete Williams EEF. Finally, it should be mentioned that although Bueckner 关18兴 was apparently the first to establish the concept of the work conjugate integral, Hong and Stern 关149兴 were perhaps the first to apply Betti’s reciprocal work theorem with known auxiliary fields to solve interfacial cracks. Again, we note that, in addition to Chen 关76兴, Mazya and Plamenevskij 关77,78兴 also found the pseudo-orthogonal property 共2.15兲 in a general form for elliptical boundary value problems and used it to develop effective methods for determining SIFs. 2.4 Basic formulation of the two-state integrals To study mixed-mode fracture, the key is how to separately determine mode I and mode II SIFs. Gallagher 关163兴 in 1978 reviewed the numerical methods for obtaining single mode SIFs in homogeneous materials. The most effective may be the virtual crack extension method proposed firstly by Rice 关4,5兴 and Parks 关164兴 and studied subsequently by a number of researchers 关162兴. However, as is true for all
521
methods of evaluating the J -integral or ERR, this technique must be augmented if it is to be used to establish stress intensities in mixed-mode fracture. Certainly, the stress intensities in mixed-mode cases could be separately evaluated by using the renowned singular element at a crack tip 关125兴, but this is not convenient when adopting the commonly programmed finite element methods. Instead, it is possible to define associated path independent integrals for this purpose. Stern et al 关12兴 and Ishikawa et al 关24兴 developed such integrals for homogeneous cases, by taking the symmetric and antisymmetric parts 共about the crack plane兲 of the planar displacement, strain, and stress fields and using them separately in path independent integrals. One integral can be manipulated to give K I and the other to provide K II . As noted later by Bui 关157兴, the virtual crack extension method can be used to evaluate path independent integrals. However, the approach of Ishikawa et al 关24兴 and Bui 关157兴 cannot be used for cracks located at bimaterial interfaces. This follows from the fact that the displacements solution in one material does not satisfy the governing equations in the other, and hence the symmetric and antisymmetric parts of the displacements are invalid in both materials 关165–167兴. A significant advance in this area is the introduction of two-state integrals established by Chen and Shield 关84兴 for homogeneous cases and subsequently extended by many researchers in various kinds of mixed-mode crack problems. For example, following Stern et al 关12兴, Yau et al 关86兴 developed a technique for obtaining separate modes by using the M 1 -integral of Chen and Shield 关84兴. The goal of this approach is to overcome the shortcoming of the J -integral in mixed-mode fracture, and to evaluate separate values of crack tip parameters from which mixed-mode crack stability, kinking/deflection, etc, could be predicted. In the remainder of this section, we will focus on the work of Chen and Shield 关84兴 and illustrate, for the same crack configuration, how to evaluate the J -integral in two different stress-displacement fields. The idea of Chen and Shield 关84兴 appears similar to that of Bueckner 关18兴 who also introduced two stressdisplacement states in order to apply Betti’s reciprocal theorem 关75兴. However, Chen and Shield’s method is much simpler, requiring only a simple summation of two J -integrals corresponding to two different solutions for a given crack geometry: Betti’s reciprocal theorem is not needed and no special treatment of the complementary field is required 关84兴. Let uA and uB be two displacement fields representing two solutions to two different boundary value problems 共for the same crack configuration兲, and let the associated J -integrals be denoted by J A and J B , respectively. A very useful result, ie, an additional integral, can be deduced by simply adding up the two solutions. Indeed, when the two displacement fields are summed to give uC , the resulting value of J as obtained by Chen and Shield 关84兴 in homogeneous cases is: J C ⫽J A ⫹J B ⫹J AB
(2.18)
where J AB is an additional integral representing the mutual interaction between the two states, given by
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Chen and Lu: Developments and applications of invariant integrals
J AB ⫽
冕 ⌫
A B i j i j dx 2 ⫺
冉
n i Bi j
冊
u Aj u Bj ⫹n i Ai j ds x1 x1
(2.19)
Originally, Chen and Shield 关84兴 used the symbol M 1 rather than J AB , since the two-state M -integral 关90兴 was yet to be established at that time. Here and throughout, the M 1 -integral will be denoted by J AB to clarify its difference from the M -integral and to distinguish it from M AB defined in Section 2.6. It has been established that the additional integral J AB is path independent and, for a plane crack, is related to crack tip SIFs by J AB ⫽
2 共 K K ⫹K IIA K IIB 兲 E ⬘ IA IB
(2.20)
where the subscripts A and B refer to the states uA and uB , respectively. Practically, state uA and uB can be chosen as the real and auxiliary displacement fields, respectively, both equilibrium fields satisfying the traction-free condition along crack surfaces. Using a commonly programmed finite element method with no regard to singular elements at the crack tip, one can obtain numerical results for stresses and displacements of state uA , but not the separate values of crack tip SIFs. The remaining task is to establish a suitable auxiliary state uB , a process similar to that of Bueckner 关18兴. However, instead of using Betti’s reciprocal theorem and finding a complementary state from weight functions derived by complex potentials 关159兴, Chen and Shield 关84兴 directly found that Eq. 共2.20兲, as a new path independent integral, provides a simple way from which the distinct values of K IA and K IIA can be determined. For example, if uB is chosen to be an existing, purely mode I solution with K IIB ⫽0 and K IB ⫽0, Eq. 共2.20兲 implies that there is a simple relation between the unknown K IA and the known K IB , given that J AB is already evaluated by the finite element method. Once K IA is known, K IIA can be calculated from J A . In summary, the two-state integral approach is very simple and effective, requiring only a simple summation of two J -integrals corresponding to two different solutions; no crack-tip singular element nor Betti’s reciprocal theorem is needed. Moreover, this technique does not depend upon the complete Williams EEF as well as the complex potential theorem. An interesting result is obtained when a third J -integral is deduced from the summation of the two integrals. That is, adding together the two solutions leads to a third field for which the J -integral can be found. The value of this J differs by J AB from the sum of the J values for individual solutions. Thus, when one of the solutions is of a single mode with known magnitude, the value of J AB can be used to determine one of the SIFs for the other solution. That a simple summation of two equilibrium states could lead to so useful results is pleasantly surprising, as evidenced by further investigations of Ishikawa et al 关24兴, Yau et al 关86兴, Bui 关157兴, Wang and Yau 关168兴, and Matos et al 关35兴 in earlier years, and Im and Kim 关90兴, Jeon and Kim 关93兴, and Kim et al 关147兴 in more recent years.
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2.5 Application of two-state integrals in interface crack problems Based on Chen and Shield’s two-state integral 关84兴, Matos et al 关35兴 presented a numerical method to calculate intensity * , and K III * , for an interface crack. After factors, K I* , K II obtaining the J -integral by the virtual crack extension method originally proposed by Parks 关164兴, individual stress intensities are determined from further calculations of J perturbed by small load increments. The calculations are carried out by FEM with minimal extra computations, in contrast to those for the boundary value problem. Very accurate results have been obtained for a crack located at the interface of a bimaterial 关35兴. In particular, a comparison was made with SIFs obtained by computing J with the virtual crack extension method but separating the modes through the displacement ratio on the crack surfaces 关35兴. Both techniques work well with fine finite element meshes but the results suggest that the method that relies entirely on J -integral evaluations can be used to give reliable results even for coarse meshes. The purpose of Matos et al’s work 关35兴 is to avoid treating the well-known oscillatory characteristic singularity at the tip of an interface crack 关36,160,161,165–167兴 during numerical calculations. Although many researchers before Matos et al 关35兴 have treated interface cracks by using invariant integrals 关33,34,87,168兴, here we only focus on the work of Matos et al 关35兴 as it is a direct extension of the twostate integral method of Chen and Shield 关84兴 and the virtual crack extension method of Parks 关164兴 for homogeneous materials. The J -integral and the additional integral J AB for bimaterials can be written as J⫽
1 * 兲 2 ⫹ 共 K IIA * 兲2兴 关共 K IA H
J AB ⫽ where
2 * K IIB * 兲 共 K * K * ⫹K IIA H IA IB
冉
1 1 1 1 ⫽ ⫹ H 2 E ⬘1 E 2⬘
冊冒
cosh2 共 兲
(2.21)
(2.22)
(2.23)
where is the oscillatory index representing the mismatch between two dissimilar elastic materials 共Fig. 1兲, and E ⬘1 and E ⬘2 denote the generalized Young’s moduli for phase 1 and phase 2 of the bimaterial, with E ⬘j ⫽E j ( j⫽1,2) in plane stress and E ⬘j ⫽E j /(1⫺ 2j ) in plane strain, being the Pois* do not son ratio. It is important to note that K I* and K II represent opening and shear modes 关36,160,161,165–167兴. While the oscillatory singular nature at the tip of an interface crack dictates that the mode ratio varies with the distance from the crack tip, the wavelength of the oscillations decreases as the crack tip is approached. Wang and Yau 关168兴 first applied this technique to bimaterials by using existing displacement fields and the J AB integral for interfacial cracks to evaluate crack-tip parameters in other solutions of bimaterial cracks. Since J AB has J -like character, it can be evaluated by the virtual crack extension method 关164兴, with highly accurate results obtained at low
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Chen and Lu: Developments and applications of invariant integrals
computational cost. For example, Shih and Asaro 关87兴 used * from finite an equivalent method to calculate K I* and K II element solutions. However, Matos et al 关35兴 emphasized that, in their approach, there is an ambiguity associated with the dimensionality and phase of the complex stress intensity factor K * * 关36兴. Indeed, the extension of this approach to ⫽K I* ⫹iK II bimaterial cases is not straightforward, since the mode I and mode II stress intensity factors of an interface crack are inherently coupled due to the oscillatory singularity 关36,160,161,165–167兴. In general, one cannot directly find a complementary field for the same crack configuration for * vanishes. However, as the oscillation index of which K IIB commonly used bimaterials is always much smaller than unity, an approximate complementary field can be used. In order to extend the Chen-Shield method 关84兴 to treat more complicated problems such as interface cracks, Matos et al 关35兴 developed an alternative method by combining the Chen-Shield method with Parks’ virtual crack extension method 关164兴: J⫽G⫽⫺
冉 冊 U a
1 ⬇⫺ 兵 u n 其 T 共 兵 关 S 兴 / a 兲 兵 u n 其 2 F
(2.24)
where 兵 u n 其 contains as elements the nodal degrees of freedom and 关 S 兴 is the stiffness matrix. First, solve the problem by FEM to find 兵 u n 其 , from which J can be computed by the virtual crack extension technique 共2.24兲. Then, add to 兵 u n 其 the displacements 兵 ⌬u n 其 1 for the same problem except that * ⫽0 and K I* ⫽⌬K I* . This set of displacements can repreK II sent any problem desired, although the field is actually needed only for the nodes associated with the distorted ring of elements as shown in Fig. 2. Consequently, the asymptotic crack tip displacements can be used everywhere as a suitable field, given by ⌬u1j ⫽
⌬K 1* 2G j
冑
r e f 共 r, ,, j 兲 2 共 1⫹e 2 兲 1
j ⌬u11 ⫽
* ⌬K II 2G j
冑
r e f 共 r, ,, j 兲 2 共 1⫹e 2 兲 11
523
(2.25d)
where f11 is another known function 关35兴. The result of this new calculation, J⫹⌬ 11J, can then be used to obtain
*⫽ K II
H ⌬ 11J 1 * ⫺ ⌬K II * 2 2 ⌬K II
(2.25e)
* can be computed directly from 共2.21兲 and Alternatively, K II 共2.23兲 given a prior calculation of J and K I* . The method developed by Matos et al 关35兴 does not depend on the use of Betti’s reciprocal theorem 关75兴 and Bueckner’s work conjugate integral 关18兴 or any complex potential complementary fields, but it does depend on the proper use of the virtual crack extension method 关164兴. On the other hand, the computation of 共2.25a兲 or 共2.25d兲 is de* ⫽0 pendent upon the added auxiliary field 兵 ⌬u n 其 1 with K II * ⫽⌬K II* . and K I* ⫽⌬K I* or 兵 ⌬u n 其 11 with K 1* ⫽0 and K II * , in general, do not represent opening and Since K I* and K II shear modes 关35兴, these added auxiliary fields can only be introduced approximately. Although the weight function method developed by Chen and Hasebe 关37兴 has no such restrictions, it does need the complex potential complementary field, which is inconvenient to use with the displacement-based finite element method. However, to calculate the elastic T -stress that influences the size and shape of the near-tip process zone as well as the two-parameter fracture criterion, the works of Sham et al 关141–143兴 and Chen and Hasebe 关37,38,145兴 reveal that the complex potential method, with higher order singular terms and Bueckner’s integral, provides a powerful tool. In passing, we note that the two-state L -integral 关89兴 and M -integral 关90兴 can be proposed in the way similar to the
(2.25a)
where ( j⫽1,2) is the material index and f1 is a known function introduced by Matos et al 关35兴. The vector 兵 ⌬u n 其 1 is obtained by evaluating 共2.25a兲 at the required nodes. With the value of 关 S 兴 / a already computed during the evaluation of J, the calculation in 共2.24兲 is repeated with the vector 兵 u n 其 substituted by 兵 u n 其 ⫹ 兵 ⌬u n 其 1 . From 共2.21兲, 共2.22兲, and 共2.23兲, it can be shown that the result of this calculation, J ⫹⌬ 1 J, can be expressed as: ⌬ 1 J⫽
1 共 ⌬K 1* 2 ⫹2K 1* ⌬K 1* 兲 H
(2.25b)
or, equivalently, K 1* ⫽
H ⌬ 1J 1 ⫺ ⌬K 1* 2 ⌬K 1* 2
(2.25c)
The second term on the right hand side of 共2.25c兲 can be 2 neglected if ⌬K * 1 ⰆH⌬ 1 J. If desired, this procedure can be repeated for an added * ⫽⌬K II* . The correvector 兵 ⌬u n 其 11 such that K * 1 ⫽0 and K II sponding displacements are
Fig. 2
The distorted ring of elements provided by Matos et al 关61兴
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Chen and Lu: Developments and applications of invariant integrals
two-state J integral. In the forthcoming section, the main applications of the two-state M -integral will be reviewed. The basic concept of two-state integrals can easily be extended to treat mixed-mode fracture in functional materials such as piezoelectric ceramics, although little research has been carried out in this area. 2.6 Two-state M -integral and its applications Recently, Im and Kim 关90兴 applied the two-state M -integral to compute the intensity of the singular near-tip field around the vertex of a generic composite wedge, which includes free edges, crack terminating at a material interface, and reentrant edges of a thin film. The eigenfunction expansion was used together with an energetic argument associated with the M -integral to show that a complementary scheme is effective in finding the complete eigenfunction expansion, including both the dominant singular terms and the higher order terms. The method of Im and Kim 关90兴 is highly efficient and yet simple to use: the near-tip information for the singular elastic boundary layer can be extracted from the far-field data without having to deploy singular finite elements for the wedge vertex. An exemplary case is illustrated by the reentrant edge of a thin-film segment bonded to a substrate. The local stress intensity at the reentrant vertex is obtained in terms of the shear stress intensity based upon the membrane model for the film-substrate structure. Based on the original concept of the two-state J -integral 关84兴, the two-state M -integral can be introduced by two independent elastic states A and B as well as their summation state C: M C ⫽M A ⫹M B ⫹M AB
(2.26)
where M AB is an additional integral accounting for the mutual interaction between the two states, given by: M AB ⫽
冕
⌫
冉
Ai j Bi j n k x k ds⫺ n i Bi j
冊
u Aj u Bj ⫹n i Ai j x ds xk xk k (2.27)
Since A and B are equilibrium states and the area integral version of the contour integral 共2.27兲 vanishes for the domain devoid of singularities, it is easy to prove that the additional integral M AB is path independent 共in Section 4 we will examine the change in the J - and M -integrals when the integration domain completely encloses all singularities兲. For a generic isotropic composite wedge, Im and Kim 关90兴 found that the energetic property of the M -integral places a restriction upon the structure of asymptotic solutions in the eigenfunction series to ensure that the M -integral is path independent. It turns out that this makes the present generic wedge problem easily tractable by using the twostate M -integral. Following Chen and Shield 关84兴, Im and Kim 关90兴 discussed how to apply the two-state M -integral to calculate singularity intensities for singular elastic boundary layers in the aforementioned class of wedges. The success of this approach is crucially linked to the existence of auxiliary solutions in the form of complementary solutions. The weight functions for special cases of the generic wedges are given in Sham and Bueckner 关141兴 and Wu and Chang 关169兴.
Appl Mech Rev vol 56, no 5, September 2003
The proof provided by Im and Kim 关90兴 elucidates that the weight function for each eigenfunction has the same form as the eigenfunction with a different eigenvalue. They concluded that the auxiliary field, used by Sinclair et al 关39兴 in applying Betti’s reciprocal theorem for computing stress intensity at sharp notches, can be obtained in the form of this complementary eigenfunction. In conjunction with a straightforward displacement-based FEM , Im and Kim 关90兴 demonstrated the application of this approach for the reentrant edge of a thin film, and concluded that the scheme based on the two-state M -integral is straightforward and simple, providing an efficient and robust tool for solving elastic boundary layer problems associated with generic composite wedges. Choi and Earmme 关89兴 employed the two-state L -integral to calculate stress intensities of circular arc-shaped cracks, where the two-state J -integral appears not so effective. The two-state J-, M -, and L -integrals constitute the three conservation laws for two equilibrium states, termed the twostate conservation integrals 关84兴. The two-state conservation laws, in conjunction with FEM , provide an effective tool for calculating stress intensities and T -stresses for cracks, stress singularities of generic wedges, and dislocation strength 关90兴. Of great significance is that this method is capable of extracting near-tip information directly from far-field deformation 关141–143兴, which is a major advantage over singular finite elements 关123兴 and other special techniques such as the boundary collocation method 关124兴, the singular hybrid FEM 关125兴, and the enriched FEM 关126兴. 2.7 M -integral analysis for Zener crack Zener 关170兴 proposed that a dislocation pileup concentrates stress, which may cause a crack to nucleate. Any crystallographic discontinuity 共eg, a grain or phase boundary兲 may act as an obstacle to block dislocation gliding. In a pair of bonded solids, when the interface blocks dislocations, the crack can nucleate either on the interface, or in one of the solids. Zener’s crack model has been extensively studied in the past 50 years, as reviewed by Cottrell 关171兴 and more recently by Cherepanov 关172兴 and Fan 关173兴. The main result has been the crack energy release rate, obtained by solving elasticity boundary value problems for various special cases. However, if the two solids have dissimilar elastic properties, only the case where the crack lies on the interface has so far been solved. Suo 关174兴 calculated the energy release rate by using the path-independent M -integral 共proposed originally by Knowles and Sternberg 关11兴, ie, Eq. 共2.9兲兲, following the procedure of Freund 关15兴, and gave several examples of cracks in an isotropic and homogeneous solid. Suo’s work presents a new application of the M -integral with regard to Zener’s crack. Earlier, Kubo 关175兴 had already applied the M -integral to cracks in dissimilar elastic materials, although his rather brief study received little recognition in the literature. If the solid is homogeneous along rays from the coordinate origin and if the closed contour C 0 encloses no singularity, then M ⫽0; that is, the integral M is path-independent. For a pair of dissimilar solids bonded on a flat interface, placing the coordinate origin at any point on the interface satisfies the homogeneity requirement. On the contrary, if C 0
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Chen and Lu: Developments and applications of invariant integrals
encloses a singularity, M may not vanish. For example, Rice 关31兴 showed that the M -integral evaluated over a closed curve around a dislocation equals the pre-logarithmic factor of the dislocation energy. Consider a dislocation of Burgers vector b lying on the interface between a pair of solids 共Fig. 3兲. Place the coordinate origin at the dislocation and select a curve C around the dislocation. The M -integral is then given by: M⫽
1 T ⫺1 b H b 2
(2.28a)
where H is a positive-definite Hermitian matrix, and can be calculated once the elastic constants of the two solids are given 关176兴. The location and shape of the curve C do not affect the value of M . Furthermore, M is unaffected by the presence of an external boundary or other singularities in the solids, so long as C encloses no other singularities than the dislocation, where the singular stress field due to the dislocation prevails over the stress fields induced by other sources. It has been shown that, for a pair of anisotropic solids with a fixed relative orientation, under an in-plane coordinate rotation, H transforms like a second-order tensor. Consequently, if the interface rotates with the coordinate while the Burgers vector b remains fixed relative to the solids, the number bT H⫺1 b is invariant, and so is the value of M . Figure 4 illustrates a flat interface between a pair of semi-infinite solids. The interface blocks N dislocations, each of the Burgers vector b. A crack of length l lies either on the interface or in one of the solids. The dislocations all glide into the crack and blunt one of the crack tips. The other tip advances either along the interface or within one of the solids. For the convenience of calculating the energy release rate G at the advancing crack tip, the origin of the coordinates is placed at the point where the dislocations are blocked. Due to its pathindependence, the M -integral evaluated over any curve enclosing the dislocation-crack complex has the same value. First look at curve C 1 far away from the complex. At a distance far from the dislocations, r→⬁, all the dislocations behave collectively like a single super-dislocation having the Burgers vector Nb. The stress field due to the superdislocation decays as 1/r, but the modification due to the presence of the crack decays as 1/r 2 . Consequently, the M -integral evaluated over C 1 has the same value as that evalu-
Fig. 3 A dislocation of Burgers vector b lying on the interface between a pair of solids provided by Suo 关196兴
525
ated over an isolated interface dislocation of the Burgers vector Nb, and hence Eq. 共2.28a兲 is still applicable when b is replaced by Nb. Next look at curve C 2 tightly surrounding the dislocation-crack complex. The stress field near the origin is now less singular than 1/r, so that the small circle around the blunted crack tip at the origin does not contribute to M ; also, M is not affected by the traction-free crack surfaces. On the other hand, it can be shown that the M -integral evaluated over the small circle surrounding the advancing crack tip equals lG 关15兴. Finally, equating the M -integral evaluated over C 1 and C 2 , one arrives at: G⫽
N 2 T ⫺1 b H b 2l
(2.28b)
For the case where the crack lies on the interface, Fan 关173兴 solved the elasticity boundary value problem with the complex variable method, and calculated the energy release rate from the stress field. The results reported in 关177兴 agree well with Eq. 共2.28b兲. Furthermore, Suo 关174兴 demonstrated that the same equation is applicable to a crack located within one of the solids. Let ␥ 1 , ␥ 2 , and ␥ i represent the surface energy 共per unit area兲 of solid 1, solid 2, and the interface, respectively. When the crack advances, the surface energy per unit area increases by ⌫, with ⌫⫽ ␥ 1 if the crack is in solid 1, ⌫⫽ ␥ 2 if in solid 2, and ⌫⫽ ␥ 1 ⫹ ␥ 2 ⫺ ␥ i if on the interface. A crack can nucleate/propagate if its energy release rate compensates the increase in surface energy, ie, N 2 T ⫺1 b H b⫽⌫ 2l
(2.28c)
Note that the length l of the crack 共Fig. 4兲 scales with N 2 , everything else being fixed. It is also possible to predict crack orientation from this model, as the surface energy ⌫ depends on the crystalline orientation. As previously demonstrated, once the relative orientations of the two solids and the slip plane are fixed, the factor bT H⫺1 b is invariant with the rotation of the interface or the crack. Consequently, once the relative orientations of the two solids and the slip plane are fixed, the crack orientation is entirely determined by the anisotropy of ⌫. According to this model, the anisotropy of elastic constants plays no role in determining crack orientation.
Fig. 4 A crack of length l lies either on the interface or in one of the solids provided by Suo 关196兴
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Chen and Lu: Developments and applications of invariant integrals
Equations 共2.28a兲, 共2.28b兲, and 共2.28c兲 reveal that the M -integral can be combined with dislocation theory to describe microcrack nucleation on or out of an interface. This approach can be extended to treat Zener cracks in two-phase piezoelectric materials, eg, microcracking along particle/ matrix interfaces 关83,96,117,178 –182兴, or piezoelectric composites 关183兴 共eg, BaTiO 3 / PTZ⫺4 or BaTiO 3 / Polymer composites兲. After summarizing the major developments of invariant integrals in single crack problems for traditional structural materials, we are now ready to review and comment on the extensions of the basic concepts of invariant integrals to functional materials, multiple interacting crack problems, and microcrack damage problems. 3 APPLICATION OF INVARIANT INTEGRALS IN FUNCTIONAL MATERIALS In this section we summarize the main applications of invariant integrals in functional materials derived in recent years, with focus placed on discussing features unique to functional materials. Advanced functional materials such as ferroelectric materials, shape memory alloys, and ferromagnets are increasingly being used in smart structures and adaptive structures. Among the most common choices for sensors and actuators are piezoelectric ceramics such as lead zirconate titanate ( PZT), barium titanate (BaTiO 2 ), lead lanthanum zirconate-titanate ( PLZT), and, more recently, piezocomposites (BaTiO 2 / Polymer or BaTiO 2 / PZT). The recent surge in the development of smart devices/structures and the constant drive for more ambitious applications calls attention to the long-term reliability of functional materials. It is known that crack-like defects are abundant in such materials and have been the subject of intensive study in the past decades. Beginning with Parton 关184兴, Cherepanov 关8兴, and Deeg 关185兴, the theory of fracture for electroelastic materials has been systematically developed 共see eg, Pak 关55,56,186兴, Sosa 关187,188兴, Suo et al 关60兴, Pak and Tobin 关189兴, Tobin and Pak 关190兴, Dunn 关191兴, Hao and Shen 关192兴, Park and Sun 关61,62兴, Sosa and Khutoryabsky 关193兴, Chung and Ting 关194兴, Shindo et al 关195,177兴, Deng and Meguid 关178兴, Heyer et al 关196兴, Park et al 关197兴, Xu and Rajapakse 关198 – 200兴, Chen and Han 关115,116兴, Han and Chen 关114兴, Xu et al 关201兴, Tan et al 关202兴, Chen and Tian 关181兴, Zeng and Rajapakse 关203兴, Rajapakse and Zeng 关204兴, Chen and Lu 关118兴, Jiang and Sun 关205兴, McMeeking 关57,206 –209兴, Zhou et al 关210兴, and Gao and coworkers 关63–70兴兲. Meanwhile, as a general method, invariant integrals have been extended to treat cracks in functional materials 关73,74兴. Although various methods can be used to derive invariant integrals for these materials, for brevity, only those based on Eshelby’s energy momentum tensor 关2,3兴 and Bueckner’s work conjugate integral 关18兴 will be reviewed below. 3.1 Energy momentum tensor in piezoelectric materials Pak and Herrmann 关53,54兴 and Pak 关55,56兴 proposed the energy momentum tensor and the relative J k -integral formulation for piezoelectrics containing a crack. The derivations start from Hamilton’s principle 关211兴:
Appl Mech Rev vol 56, no 5, September 2003
␦
冕
V
共 ⫺H 兲 d v ⫹
冕
S
共 T i ␦ u i ⫺q ␦ 兲 dS⫽0
(3.1)
where H is electric enthalpy density, T i is applied surface traction, u i is displacement, q is applied surface charge, and is electric potential related to electric field E by: E k ⫽⫺ ,k
(3.2)
The variational formulation 共3.1兲 is developed for a static linear piezoelectric material occupying region V bounded by surface S. The electric enthalpy density is: 1 1 H 共 i j ,E j 兲 ⫽ C i jkl i j kl ⫺ ¯ i j E i E j ⫺e ikl kl E i 2 2
(3.3)
where C i jkl are elastic moduli measured in a constant electric field, ¯ i j are dielectric constants measured at constant strain, e ikl are piezoelectric constants, E i is electric field, and i j is strain: 1 i j ⫽ 共 u i, j ⫹u j,i 兲 2
(3.4)
The first term in 共3.3兲 is the energy stored during deformation, the second term is the energy stored in the electric field, and the last term is the interaction energy between mechanical quantities and electric quantities. The variational formulation 共3.1兲 provides the elementary equations for a linear piezoelectric:
i j, j ⫽0
共 mechanical governing field equation兲
(3.5) D i, j⫽0 ¯i i j n j ⫽T D i n i ⫽⫺q
i j⫽
共 electric governing field equation兲
(3.6)
共 mechanical boundary condition兲
(3.7)
共 electric boundary condition兲
H ⫽C i jkl kl ⫺e ki j E k ij
(3.8)
共continuous equations兲
H ⫽e ikl kl ⫹ ¯ ik E k D i ⫽⫺ Ei
(3.9)
where i j is stress, n i is unit normal vector, and D i is electric displacement vector. We note that, with regard to Eqs. 共3.1– 3.8兲, Pak and Herrmann 关53,54兴 and Pak 关55,56兴 only studied static electro-elastic problems and did not consider the effect of external electric field. Although this formulation is, strictly speaking, incomplete, it does not inhibit subsequent discussions on invariant integrals. Following the procedure proposed by Eshelby 关2,3兴 for a static elastic continuum, Pak 关55,56兴 simply took the electric enthalpy density to be the Lagrangian density and differentiated it with respect to the spatial coordinate x k :
冉 冊
H H 共 x i ;u i, j ,E i 兲 ⫽ xk xk
⫹ exp
H H u i, jk ⫹ E u i, j E i i,k (3.10)
where ( H/ x k ) exp denotes the explicit dependence of H on x k with the remaining dependent and independent variables
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Chen and Lu: Developments and applications of invariant integrals
held constant. With the help from field Eqs. 共3.5兲 and 共3.6兲 and constitutive Eq. 共3.9兲, Eq. 共3.10兲 can be rewritten as
冉 冊 H xk
⫽ 关 H ␦ ik ⫺ i j u i,k ⫹D j E k 兴
(3.11)
exp
where ␦ ik is the Kronecker delta. If there are no discontinuities in V and the material is homogeneous, one has:
冉 冊 H xk
⫽0
(3.12a)
exp
or 关 H ␦ jk ⫺ i j u i,k ⫹D j E k 兴 ⫽0
(3.12b)
Apart from the electric quantities, the bracketed terms in 共3.12b兲 are identical to the well-known energy momentum tensor of Eshelby 关2,3兴. Integrating the divergence of the energy momentum tensor over V and making use of the divergence theorem, one gets the following invariant integrals for 3D problems: J k⫽
冕
S
关 H ␦ jk ⫺ i j u i,k ⫹D j E k 兴 n j dS
共 k⫽1,2,3 兲
冖 冋冉 C
⫺ J 2⫽
冖 冋冉 C
冉
冉
1 up dx ⫺n i ip ds 2 ij ij 2 x1
1 ds D i E i dx 2 ⫹n i D i 2 x1
冊册
1 up ⫺ i j i j dx 1 ⫺n I Ip ds 2 x2
1 ⫺ ⫺ D i E i dx 1 ⫹n i D i ds 2 x2
冊册
The role of J -integral in piezoelectric fracture has always been of most interest. Pak 关55,56兴 noticed the J -integral 共the ERR or the crack extension force兲 as formulated by 共3.14a兲 has a feature significantly different from that of the original J -integral 关4,5兴. For anti-shear 共mode III兲 fracture of an insulating crack in PZT⫺5H, Pak found that J is always negative in the absence of any mechanical load. This conclusion is far apart from the traditional understanding of the ERR, which is induced from a unit crack tip advance 关4,5,16兴. Pak 关55,56兴 thence concluded that, at a given mechanical load, the presence of an electric load could retard crack growth. However, his conclusion is based on two assumptions: i) the crack surfaces are charge-free; and ii) the critical value of J J IC , can be used as a piezoelectric fracture criterion just as in traditional brittle materials 关162兴. However, these assumptions are, in general, not satisfied simultaneously. For example, McMeeking 关57兴 used the same J -integral analysis to study electrically induced mechanical stresses in dielectrics with permeable cracks, and obtained different results from those by Pak 关55,56兴. This controversial topic will be discussed further in Section 3.3.
(3.13)
The J k -integral vector of 共3.13兲 is identical to that derived by Pak and Herrmann 关53,54兴 for elastic dielectric materials. For a semi-infinite crack in a 2D piezoelectric, 共3.13兲 has the following explicit form 关55,56,60兴 J 1 ⫽J⫽
527
冊
(3.14a)
冊 (3.14b)
where C refers to a closed contour starting from one point on the lower surface of the crack and ending at another point on the upper surface of the crack, and i j , i j , u p , D i , E i , and are the stresses, strains, displacements, electric displacements, electric field components, and electric potential, respectively. Here, the integer indices i, j, p vary between 1 and 2, n i is the outer normal to C, and (x 1 ,x 2 ) represent the 2D Cartesian coordinate system. Compared with the original interpretation of the J k -integral vector 共2.8兲 by Budiansky and Rice 关9兴 and Knowles and Sternberg 关11兴 for non-piezoelectric materials, the first two terms in either Eq. 共3.14a兲 or 共3.14b兲 contributed by mechanical quantities are identical to those in 共2.8兲, while the last two terms are additional ones introduced to account for the contribution from electric quantities. In other words, although the electric and mechanical quantities are inherently coupled in the constitutive equations 共see Eq. 共3.9兲; also see, eg, Deeg 关185兴, Sosa 关187,188兴, Suo et al 关60兴兲, they are not directly linked together in Eqs. 共3.14a,b兲.
3.2 Maugin’s formulations Following Pak and Herrmann 关53,54兴, Maugin and Epstein 关58兴 and Dascalu and Maugin 关59兴 extended Eshelby’s theory 关2,3兴 to electroelasticity. They constructed Eshelby’s energy momentum tensor within the exact nonlinear theory of deformable dielectrics, without involving the Maxwell stress of free electric fields. When the electroelastic body is made of the same material at all points, the electroelastic energy momentum is shown to satisfy a remarkable differential identity involving the torsion of the material connection. The main outcome of Maugin and Epstein’s analysis 关58兴 is that the electroelastic energy momentum thus obtained can only depend on true material fields. As a consequence of their very definition of uniformity and inhomogeneity, and contrary to the apparent results of Pak and Herrmann 关53,54兴, the free electric part of the so-called Maxwell stress tensor cannot contribute to the electroelastic energy momentum tensor. With a quasi-linear approximation, they pointed out that the material force thus defined leads to the notion of a path-independent integral which should be useful in studying cracks in electrodeformable ceramics. Meanwhile, various extensions and generalizations to the ERR concept have been carried out. For example, using a Griffith-type energy approach, Dascalu and Maugin 关59兴 obtained two energy release rates due to quasi-static crack propagation in electroelastic solids. An identity for the electrical quantities was established, permitting the construction of an electromechanical energy momentum that only depends on true material fields for the calculation of crack extension force. A natural question arises in this situation: Are the two integrals constructed separately in 关53,54兴 and 关59兴 different? And if they are, what are their separate roles in crack propagation? Dascalu and Maugin 关59兴 demonstrated that these two integrals are indeed different, and their difference represents the 共free兲 electric field contribution to crack extension. Since the relation between the J -integral and the rate of energy re-
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Chen and Lu: Developments and applications of invariant integrals
leased during crack extension is crucially important in any fracture theory, Dascalu and Maugin 关59兴 expressed the two electroelastic integrals as energy release rates. They claimed that these expressions clearly show the influence they have on crack extension. However, at that time, the experimental study of piezoelectric fracture was yet to be systematically performed so that they could not be aware of the different roles of invariant integrals in traditional materials and in electroelastic solids. We will come back to this topic in Section 3.3. The main difference between Pak and Herrmann 关53,54兴 and Dascalu and Maugin 关59兴 is as follows. Dascalu and Maugin 关59兴 studied energy release rates and pathindependent integrals in electroelastic crack propagation, and derived the following two kinds of J -integral: J共 ⌫ 兲⫽
冕
⌫
F 兲 n K ⫺⌽ ,1D K n K 兴 ds 关 Wn 1 ⫺u i,1共 T Ki ⫹T Ki
(3.15) J *共 ⌫ 兲 ⫽
冕
⌫
关 ⌺n 1 ⫺C 1M T M K n K ⫺⌽ ,1⌸ K n K 兴 ds
(3.16)
Here, the impermeable crack model was imposed, ie, crack surfaces are traction and charge free: D K⫾ n K ⫽0 F ⫾ 兲 n K ⫽0 共 T Ki ⫹T Ki
(3.17)
where the summation of the normal first Piola-Kirchhoff stress and the normal Maxwell stress vanishes. The first integral, 共3.15兲, is identical to that derived by Pak and Herrmann 关53,54兴 even though different notations were used. On the other hand, based upon the second integral, 共3.16兲, Dascalu and Maugin 关59兴 proposed a new electroelastic energy momentum tensor, as:
* ⫽⌺ ␦ KL ⫺C LM T M K ⫺⌽ ,L ⌸ K b KL
(3.18)
for which
* ⫽0 b KL
(3.19)
To ensure that J * is path independent, another mechanicalelectric boundary condition must be satisfied: ⌸ K⫾ n K ⫽0 共 T Ki 兲 ⫾ n K ⫽0
(3.20a)
For an antiplane crack, the difference between the two integrals in a dielectric with induced piezoelectricity is 关59兴: J 共 ⌫ 兲 ⫺J * ⫽⫺
共 KE兲2 2
(3.20b)
where K E is the electric displacement intensity factor. Equation 共3.20b兲 clearly shows that ERR is an even function of the electric field, and hence the electric field may have a negative contribution to ERR. Physically, this implies that electric energy is absorbed, rather than released, when the crack extends. 共We note that this phenomenon has al-
Appl Mech Rev vol 56, no 5, September 2003
ready been observed by Pak 关55,56兴 as well as a number of other researchers.兲 However, there was no experimental data to support Dascalu and Maugin’s fracture estimation, and they did not study electric conditions different from the impermeable crack model. As will be seen in the next section, whether or not the J -integral could be used as a fracture criterion has been a controversial topic ever since 1990: some have even used chaos to describe the contentious situation. 3.3 Piezoelectric fracture criterion In this section we review the current understanding of fracture criterion for piezoelectric ceramics, with special attention on three propositions. First, as a fracture criterion for an impermeable crack, Park and Sun 关61,62,205兴 proposed the mechanical strain energy release rate (M SERR), ie, the mechanical part of the J -integral 共3.14a兲, by neglecting its electric part. Second, for mixed-mode fracture, McMeeking 关207,208兴 and Heyer et al 关196兴 proposed to use the mode mixity as a fracture criterion for permeable cracks. Third, based on the Dugdale model 关71兴, Gao and coworkers 关63– 70兴 performed a series of multi-scale investigations and proposed the global-local J -integral concept for both impermeable and permeable cracks, taking into account electric nonlinearity, electric yielding, and microstructural modeling. Since only the first and third are associated with invariant integrals, the controversial results with these two criteria will be discussed in detail. The extension of linear elastic fracture mechanics to treat piezoelectric fracture has often been regarded as rather natural and straightforward, as done by Cherepanov 关8兴 and more recently by Suo 关60兴, Sosa 关188兴, and Pak 关186兴. According to linear piezoelectric fracture theory, the presence of an electric field, either positive or negative, should always elevate the failure stress 共ie, an even dependence between failure load and applied electric field兲. This theory has nevertheless been questioned by a number of researchers, as it is sometimes in contradiction with experimental observations. For instance, with compact tension and three-point bending tests on PZT⫺4 specimens poled perpendicular to the crack plane, Park and Sun 关62兴 measured the load to failure under mode I conditions. They found that the critical load decreases in the presence of a positive electric loading 共namely, an electric field applied along the poling axis兲, and increases if the electric loading applied is negative 共namely, a field aligned in the opposite direction兲. Tobin and Pak 关190兴 carried out Vickers indentation tests, and observed cracks both parallel and perpendicular to the poling axis in PZT⫺8. Under the application of an electric field along the poling direction, cracks perpendicular to the axis were found to be longer for a positive field, and shorter for a negative field, in comparison with the corresponding cracks under zero electric field. They further observed that crack growth parallel to the poling was unaffected by the presence of an electric field. Obviously, transversely isotropic piezoelectrics with cracks normal to the poling axis exhibit an odd functional dependence of failure load on the applied electric field. Moreover, stable crack growth perpendicular to the poling direction under purely electric loading has been observed under high-
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Chen and Lu: Developments and applications of invariant integrals
cycle 关190兴 as well as low-cycle 关212,213兴 fatigue on PZT and PLZT samples, even though the J -integral (ERR) calculated from 共3.14a兲 is negative. The apparent discrepancy between experiment and theory has long hindered the development of piezoelectric fracture theory, instigating numerous attempts to embellish the linear fracture mechanics theory. One common critique of the standard approach is that the assumption of insulated crack faces proposed first by Deeg 关185兴 and adopted subsequently by many others 关55,56,61,62,186 –188兴 is not strictly valid for most of the experimental conditions. For example, once a crack opens it is generally filled with air, whose permittivity is three orders of magnitude lower than that of common piezoelectric ceramics, but nevertheless nonzero. Consequently, the effect of imposing a permeable boundary condition, proposed more than 36 years ago by Parton 关184兴, on piezoelectric fracture has brought much attention in recent years 共see, eg, Pak and Topin 关189兴, Dunn 关191兴, Hao and Shen 关192兴, Sosa and Khutoryansky 关193兴, Chung and Ting 关194兴, Shindo 关177,195兴, Xu and Rajapakse 关198 –200兴, and Rajapakse and Zeng 关204兴兲. A great advance in this area is owed to Park and Sun 关61,62兴 who first pointed out that the J -integral of 共3.14a兲 cannot be used as a fracture criterion for insulating cracks. Instead, they proposed that fracture is a purely mechanical process, so the mechanical part of the J -integral, named J m or G m 共the mechanical part of the energy release rate, abbreviated here as M SERR, excluding the electric part兲, is the dominant parameter governing piezoelectric fracture. Park and Sun’s M SERR concept is backed by their own experiments 关61,62兴. Kumar and Singh 关214兴 pointed out that Park and Sun’s numerical results 关61兴, unfortunately, contain some errors. Balke et al 关215兴 performed a detailed numerical study and found that, despite of these numerical errors, Park and Sun’s statement 关61,62兴 remains valid, namely, M SERR is a better fracture parameter than ERR or J -integral 共3.14a兲. Nevertheless, the arbitrarily defined M SERR is purely empirical, based on such common understanding that, from the macroscopic point of view, fracture is a purely mechanical process. This view was later challenged by Gao et al 关64,65兴, arguing that fracture is a multiscale process and that the electric part of J could not be simply omitted. More recently, Jiang and Sun 关205兴 performed an analysis of indentation cracking in piezoelectrics by taking inelastic deformation into account, and confirmed Park and Sun’s conclusion. They claimed that the M SERR criterion would be valid if, during experiments, the piezoelectric specimen is put in insulating oil rather than in air. Gao and coworkers 关63–70兴 adopted the famous Dugdale model 关71兴 for elastoplastic fracture in piezoelectric fracture and made a series of multi-scale investigations. They proposed the global-local J -integral concept for both impermeable and permeable cracks, taking into account the effects of electric nonlinearity, electric yielding, and microstructural modeling. However, even though the local J -integral formulation derived by Gao and coworkers shows clearly a positive influence of electric field on fracture 共rather than a nega-
529
tive one in the global J -integral兲, it is still an even function of electric field as opposed to the odd function given in Park and Sun 关61,62兴. Experimentally, it has been found that piezoelectric fracture strength is an odd function of the electric field: either a positive electric field promotes crack propagation and a negative electric field impedes crack propagation, or conversely, a positive electric field impedes crack propagation and a negative electric field promotes crack propagation. By using Park and Sun’s M SERR criterion 关61,62兴, Chen and Han 关115,116兴 studied the crack shielding problem in piezoelectric ceramics. They found that there is a wastage when the remote J -integral transmits from infinity across the microcracking zone to the macrocrack tip, because microcracks in the process zone not only reduce the effective moduli of the material and release the residual stresses, but also disturb the near-tip electric field, and hence provide another source of shielding. Chen and Han 关115,116兴 further found that different microstructural models in the process zone lead to different conclusions regarding the role of an applied electric field. However, no matter how microcracks are distributed in the process zone, Chen and Han’s results show that fracture strength is always an odd function of the applied electric field. At present, it is widely recognized that the current understanding of piezoelectric fracture criterion is that, due to the existence of applied electric field, the role played by invariant integrals such as the J -integral is quite different from that in classical non-piezoelectric materials unless some modifications are made 关61–70兴. Even for two well addressed works by Park and Sun 关61,62兴 and by Gao et al 关63– 66兴, respectively, McMeeking 关209兴 recently pointed out that some of the theoretical treatments introduce features which are hard to justify, such as an ad hoc neglect of the electric contribution to ERR in Park and Sun 关61,62兴 or the apparent dissipation of energy by the saturation of the electrical polarization in Gao and coworkers 关63– 66兴.
3.4
Bueckner work conjugate integral in piezoelectrics
Although Bueckner’s work conjugate integral was established in traditional materials based on Betti’s reciprocal theorem, Ma and Chen 关82,83兴 found that it could also be utilized in materials with coupled mechanical-electric fields. Physically, this is because Betti’s theorem holds in any linear coupled mechanical-electric field 关184,211,216 –220兴. However, the extension of Bueckner’s integral to treat crack problems in coupling fields, eg, in linear piezoelectrics, is neither simple nor straightforward, as illustrated below. Using Stroh’s complex potential theorem 关221–225兴 and Suo’s compact representations 关60兴, Ma and Chen 关83兴 obtained the Williams EEF for a semi-infinite crack in dissimilar piezoelectric materials 共see Fig. 1 but consider the two half plane as two dissimilar piezoelectric materials兲:
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Chen and Lu: Developments and applications of invariant integrals
f⬘1 共 z 兲 ⫽
⬁
1 2 冑2 z
i B⫺1 1 共 I⫹i  兲 Y共 z ,z 兲
兺
n⫽0
an z n ⫹B⫺1 1 共 I⫹ ␣ 兲
Appl Mech Rev vol 56, no 5, September 2003
Thus, the vectors of stress and electric displacement intensity factors, which uniquely characterize the singular field, can be defined by 关224兴
⬁
⫻
f⬘2 共 z 兲 ⫽
兺
n⫽0
共 In upper half plane #1兲
ibn z n
(3.21a)
⬁
1 2 冑2 z
i B⫺1 2 共 I⫺i  兲 Y共 z ,z 兲
兺
n⫽0
an z
n
⫹B⫺1 2 共 I⫺ ␣ 兲
⬁
⫻
兺
n⫽0
共 In lower half plane #2 兲
ibn z n
(3.21b)
or
f1 共 z 兲 ⫽
⬀
1 2 冑2 z
B⫺1 1 共 I⫹i  兲
兺
n⫽0
⬀
⫹B⫺1 1 共 I⫹ ␣ 兲
兺
n⫽0
ibn
X共 z i ,z ,n 兲 an z n⫹1
z n⫹1 n⫹1
共 In upper half plane #1 兲
f2 共 z 兲 ⫽
(3.21c)
⬀
1 2 冑2 z
B⫺1 2 共 I⫺i  兲
兺 X共 z i ,z ,n 兲 an z n⫹1 n⫽0
⬀
⫹B⫺1 2 共 I⫺ ␣ 兲
兺
n⫽0
ibn
z n⫹1 n⫹1
共 In upper half plane #2兲
(3.21d)
where an and bn are real vectors with a0 representing the strength of crack tip singularity, and the oscillating function X is defined by X共 z i ,z k ,m 兲 ⫽D1
z i z ⫺i ⫹D2 1 1 m⫹ ⫹i m⫹ ⫺i 2 2
⫹D3
1
冑2 x 1
k Y共 x i 1 ,x 1 兲 g共 x 1 兲
冦
¯ Tk B ¯ k ⫹B ¯ Tk A ¯ k ⫽I ATk Bk ⫹BTk Ak ⫽A T ¯ ¯T T ¯ ¯T Bk Ak ⫹Bk Ak ⫽Ak Bk ⫹Ak Bk ⫽I ¯ kA ¯ Tk ⫽Bk BTk ⫹B ¯ kB ¯ Tk ⫽0 Ak ATk ⫹A ¯ k ⫹BTk A ¯ k ⫽A ¯ Tk Bk ⫹B ¯ Tk Ak ⫽0 ATk B
(3.24)
(3.25)
where subscript k⫽1,2 is the material index, Ak and Bk denote the 4⫻4 complex matrices defined by Ting 关222兴 and Suo et al 关60兴 in the 4D Stroh’s complex potential formulations, over bar denotes complex conjugate, and superscript T denotes matrix transformation. Bueckner’s work conjugate integral in dissimilar piezoelectric materials can then be defined from Betti’s reciprocal theorem as 关218兴:
兺
(3.22) ⫹
冋冕
⌫k
k⫽1
(3.23)
冑2 xY共 x ⫺i ,x ⫺k 兲 T2 共 x 兲
⫺k where K⫽ 关 K 1 K 2 K 3 K 4 兴 T. Since Y(x ⫺i 1 ,x 1 ) and T2 (x 1 ) are real, K is also real. The intensity factor K may be considered as an extension of the elastic version proposed by Wu 关223兴 and Qu and Li 关225兴. Note that, on one hand, although K as defined in 共3.24兲 does not have the proper dimension, it does provide a unique characterization of the crack tip state. On the other hand, the coefficient b0 in 共3.21a,b,c,d兲 represents a stress acting parallel to the crack surface 共ie, 11 and D 1 ), and is referred to as the generalized T-stress for homogeneous materials 关36兴. Ting 关222兴 obtained the following important orthogonal property of complex matrices and their conjugates, valid in both material I and material II as proved by Deng and Meguid 关178兴:
B⫽
Note that, in 共3.22兲, two real coefficients and are introduced for dissimilar piezoelectric materials rather than one single oscillation index for traditional elastic bimaterials 关36,160,161,165,166兴. It should be emphasized that Stroh’s theorem and Suo’s compact form 共3.21a,b,c,d兲 have four distinct complex variables in each half plane. Their separation becomes a key problem when making the forthcoming manipulations to establish the Bueckner integral 关82,83兴. The oscillating singular stress field along the bounded interface near a crack tip is given by: T2 共 x 1 兲 ⫽
x→0 ⫹
2
⫺
z z ⫹D4 1 1 m⫹ ⫹ m⫹ ⫺ 2 2
K⫽ lim
冕
⌫k
共 u ␣i i j ⫺u i ␣i j 兲 n j ds
共 ␣ D i ⫺  D ␣i 兲 n i ds
册
共 i⫽1,2,3; j⫽1,2兲
(3.26) where superscripts ␣ and  refer to two mechanical-electric fields in an anisotropic piezoelectric material. By using Suo’s compact form 关60兴, Eq. 共3.26兲 can be rewritten as 2
B⫽
兺
k⫽1
冕
⌫k
共 u ␣i i j ⫺u i ␣i j 兲 n j ds
(3.27)
where i⫽(1,2,3,4), j⫽(1,2), u 4 ⫽ , 41⫽D 1 , and 42 ⫽D 2 . Alternatively, by using resultant forces and resultant displacement acting on the contour ⌫ k , Bueckner’s integral 共3.27兲 can be expressed as 关83兴:
Appl Mech Rev vol 56, no 5, September 2003
兺 冕⌫ 共 u ␣i dT i ⫺u i dT ␣i 兲 k⫽1
Chen and Lu: Developments and applications of invariant integrals
2
B⫽
(3.28)
k
B⫽2 Re
In order to separate the four distinct complex variables for each half plane involved in 共3.27兲 or 共3.28兲, Ma and Chen 关82,83兴 made use of the orthogonal property 共3.25兲 and in this way obtained the following pseudo-orthogonal property
再兺 冕 2
B⫽2 Re
k⫽1
⫽
冦
W2 ⫽
⌫k
T
关 fk dfk␣ 兴
1 ⬘ Im共 W1兲 a n ⫺ am 4
⫽
再
再
531
兺 冕⌫ 关 fk dfk␣其 k⫽1 2
T
k
1 T ⫺ am Im共 W3 兲 an 4
m⫹n⫽0
T 2 bm W4bn
m⫹n⫽⫺1
(3.29b)
where
冎
W1⫽
DT1 共 ␣ˆ 1 ⫹ ˆ 1 兲 D2 1 m⫹ ⫹i 2
n⫹m⫹1⫽0
⬘ W2b n 2bm
m⫹n⫹1⫽⫺1
0
others cases
⫹
(3.29a)
⫹
DT3 共 ␣ˆ 1 ⫹ ˆ 1 兲 D4 1 m⫹ ⫹ 2
DT2 共 ␣ˆ 1 ⫹ ˆ 1 兲 D1 1 m⫹ ⫺i 2 ⫹
DT4 共 ␣ˆ 1 ⫹ ˆ 1 兲 D3 1 m⫹ ⫺ 2
⫺1 ⫺T ⫺1 T 共 I⫹ ␣ 兲 T Im共 B⫺T 1 B1 兲共 I⫹ ␣ 兲 ⫹ 共 I⫺ ␣ 兲 Im共 B2 B2 兲共 I⫺ ␣ 兲
(3.30b)
m⫹1
W3 ⫽DT1 共 ␣ˆ 1 ⫹ ˆ 1 兲 D2⫹DT2 共 ␣ˆ 1 ⫹ ˆ 1 兲 D1⫹DT3 共 ␣ˆ 1 ⫹ ˆ 1 兲 D4 ⫹DT4 共 ␣ˆ 1 ⫹ ˆ 1 兲 D3
(3.30c)
⫺1 W4 ⫽ 共 I⫹ ␣ 兲 T Im共 B⫺T 1 B1 兲共 I⫹ ␣ 兲 ⫺1 ⫹ 共 I⫺ ␣ 兲 T Im共 B⫺T 2 B2 兲共 I⫺ ␣ 兲
(3.30d)
for which the generalized Dundurs parameters are defined as 关224兴: ⫺1 ␣ˆ 1 ⫽ 共 I⫺i  兲 TB⫺T 1 B1 共 I⫺i  兲
(3.31a)
⫺1 ˆ 1 ⫽ 共 I⫹i  兲 TB⫺T 1 B1 共 I⫹i  兲
(3.31b)
convenient for separating crack tip parameters in linear functional materials than Bueckner’s integral, because there is no need to use Betti’s reciprocal theorem and the complete EEF. That is, if the J - and M -integrals are defined in a linear functional material 共eg, shape memory alloy or ferromagnet兲, the two-state J - and M -integrals can be directly defined without any other conditions attached, as they would be in elastic materials 关84兴.
3.5
Equations 共3.29a兲 and 共3.29b兲 show that the pseudoorthogonal property (biorthogonal property) 关60,76 –78兴 found in elastic materials is still valid in mechanical-electric coupling problems, providing a powerful tool to establish weight functions for separately evaluating crack tip parameters such as SIFs and EDIF 关83兴. The above manipulations indicate that this property and its reduced weight functions are both associated with the complete Williams EEF 共3.21兲. Some researchers argue that the complete EEF is not needed and that direct manipulations can be given in piezoelectric crack problems, from which the bi-orthogonality condition is obtained much easier without making complex argument separations. However, when weight functions need to be introduced to separately calculate SIFs, EDIF, and T terms, such manipulations should still be combined with the detailed EEF, at least several major terms of the EEF 共3.21a,b,c,d兲. This is true for the pseudo-orthogonal property, but not the reduced weight functions, as the latter are definitely associated with the complete EEF. Consequently, two-state integrals are much more
(3.30a)
Invariant integrals in soft ferromagnets
In the context of magnetoelasticity, it is probably Shindo 关226 –228兴 who first applied invariant integrals to study the local behavior of fields with both mechanical and magnetic nature in the vicinity of a planar or penny-shaped crack. More recently, this subject has been active for magnetizable elastic materials. Sabir and Maugin 关73兴 derived the ERR 共the J -integral兲 based on the rotationally invariant 共finitestrain兲 quasi-magnetostatic theory of elastic paramagnets and soft ferromagnets, for which neither magnetic ordering nor spin effects need be introduced. They constructed the corresponding invariant integrals, and showed that these lead to essentially the same results as the canonical field-theoretic approach using the notions of Eshelby’s theorem 关2,3兴. The limitation of Sabir and Maugin’s work 关73兴 is that the magnetoelastic bodies considered are weakly magnetizable, and hysteresis and spin-ordering effects are not present. More involved cases require a much more elaborate framework, and a carefully established, rotationally-invariant theory of magnetoelasticity is needed. This theory has been put forward by various researchers, and has been covered at length in research monographs such as 关216,217兴. For convenience,
532
Chen and Lu: Developments and applications of invariant integrals
Sabir and Maugin 关73兴 followed the presentation of Abd-Alla and Maugin 关46兴, originally aimed at studies in nonlinear magnetoacoustics. Considering a piecewise smooth non-self-intersecting path ⌫ which begins and ends on the crack and surrounds the tip, Sabir and Maugin 关73兴 established two kinds of pathindependent integrals. The first is: J共 ⌫ 兲⫽
冕兵 ⌫
ˆ "N兲 , 1 其 d⌫ ¯ E ⫹TH 兲 "u, 1 ⫺ 共 B WN 1 ⫺N"共 T (3.32a)
if the boundary conditions along the crack faces ⫾ of a straight crack are ˆ ⫾ ⫽0, N"M
¯ E 兲 ⫾ ⫽0 N"共 T
(3.32b)
where B is magnetic induction, H is magnetic field, M is magnetization per unit volume in the current configuration, T is surface stress 共the first Piola-Kirchhoff stress兲, TE and TH are additionally defined stresses, N 1 ⫽N"E1 , is magnetostatic potential, and E1 is unit vector oriented along the positive direction of the X 1 axis, while the semi-infinite crack is located along the negative part of the X 1 axis. The second integral is: J *共 ⌫ 兲 ⫽
冕兵 ⌫
ˆ "N兲 , 1其 d⌫ ¯ E "u, 1⫺ 共 M ⌺N 1 ⫺N"T
(3.32c)
if the boundary conditions along the crack faces ⫾ of a straight crack are N"Bˆ⫾ ⫽0,
¯ E ⫹TH 兲 ⫾ ⫽0 N"共 T
(3.32d)
Sabir and Maugin 关73兴 applied the two invariant integrals 共3.32c,d兲 to the simple case of an antiplane crack for which only the out-of-plane displacement is nonzero, while the small induced magnetic field h is parallel to the plane of the figure. In this case, the singular part of the solution in the neighborhood of the crack is given by: h 1 ⫽⫺
sin , 冑2 r 2
w, 1 ⫽⫺
KH
sin , 2 冑2 r KS
h 2⫽
KH
冑2 r
w, 2 ⫽
cos
KS
冑2 r
2
cos
(3.32e)
2
1 J * ⫽ 兵 共 K S 兲 2 ⫺ ␥ 共 K H 兲 2 其 ⫹bH 0 K H K S 2
3.6 Energy momentum tensor in other functional materials Recently, based on the concept of the Eshelby energy momentum tensor 关2,3兴, Wang and Shen 关72兴 derived the J k -vector and the M -integral for a linear electro-magnetoelastic material. From the classical electromagnetic field and elasticity theory 共Parton and Kudryavtsev 关219兴, Mikhailov and Parton 关220兴兲, the state function, free energy f , F and complementary free energy g, G for such material can be expressed as: 1 mm f 共 i j ,E i ,B i 兲 ⫽ 共 ␣ ss ⫺ ␣ ee i j E iE j⫺ ␣ i j B iB j 兲 2 i jkl i j kl sm em ⫺ ␣ se i jkl i j E k ⫺ ␣ i jk i j B k ⫺ ␣ i j E i B j
(3.33a) 1 ee mm g 共 i j ,D i ,H i 兲 ⫽ 共 ⫺ ss i jkl i j kl ⫺ i j D i D j ⫺ i j H i H j 兲 2 sm em ⫺ se i jk i j D k ⫺ i jk i j H k ⫺ i j D i H j
(3.33b)
(3.32g) and (3.32h)
so that their difference is 1 ⌬J⫽J⫺J * ⫽⫺ 共 K H 兲 2 ⫹H 0 K H K S 2
where , , and ␥ are material constants of magnetoelastic bodies. If the bias magnetic field is switched off, then the last contribution in 共3.32i兲 vanishes and ⌬J, now a negative quantity, is due solely to the induced magnetic field. This conclusion is similar to that for an impermeable crack obtained by Pak 关55兴 in piezoelectric materials. Sabir and Maugin 关73兴 concluded that this is tantamount to saying that the magnetic field has a negative contribution to the energy release rate, so that its presence is beneficial from the point of view of fracture toughness. As we have already discussed in Section 3.3, this conclusion may be doubtful since a similar conclusion by Pak 关55兴 has been shown incorrect in piezoelectric materials for electrically induced crack arrestment. Furthermore, Park and Sun 关61,62兴 pointed out that the J -integral proposed by Pak 关55兴 is not suitable as a fracture criterion. In summary, neither 共3.32g兲 nor 共3.32h兲 could be directly used as a fracture criterion in magnetoelastic bodies, unless modifications similar to those for piezoelectric materials are carried out. It is noticed that, due to the considerable complexities of fracture in magnetoelastic materials, all investigations published thus far in the open literature are theoretical, with no experimental observations supporting the predictions. Further theoretical and experimental investigations are absolutely needed.
(3.32f)
where K H is the magnetic field intensity factor and K S is the strain intensity factor. Clearly, the singularity orders are the same as those obtained by Shindo 关226,227兴. Sabir and Maugin 关73兴 related the energy release rates to the strain and magnetic field intensity factors by: 1 ¯ H 0K HK S J⫽ 兵 共 K S 兲 2 ⫺ 共 K H 兲 2 其 ⫹b 2
Appl Mech Rev vol 56, no 5, September 2003
1 mm F 共 i j ,E i ,H i 兲 ⫽ 共  ss ⫺  ee i j E iE j⫹  i j H iH j 兲 2 i jkl i j kl sm em ⫺  se i jk i j E k ⫹  i jk i j B k ⫺  i j E i H j
(3.32i)
(3.33c)
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Chen and Lu: Developments and applications of invariant integrals
1 ee mm G 共 i j ,D i ,B i 兲 ⫽ 共 ⫺ ss i jkl i j kl ⫹ i j E i E j ⫺ i j B i B j 兲 2
M⫽
sm em ⫺ se i jk i j E k ⫺ i jk i j B k ⫺ i j E i B j
i j⫽
f f f , D i ⫽⫺ , H i ⫽⫺ ij Ei Bi
(3.34a)
f f f , E i⫽ , B i⫽ i j Di Hi
(3.34b)
i j ⫽⫺
F F F i j⫽ , D i ⫽⫺ , B i⫽ ij Ei Hi i j ⫽⫺
G G G , E i⫽ , B i ⫽⫺ i j Di Hi
共 f 兲 ,k ⫽
(3.34d)
f f f ui, jk ⫺ ,ik ⫹ e A ij Ei B i ilm m,lk
共 f 兲 ,k ⫺ 共 i j ui,k ⫹D j ,k ⫹ei jm H m Ai,k 兲 , j ⫽0
(3.35)
(3.36)
Its integration form is
C
共 f ␦ k j ⫺ i j u i,k ⫺D j ,k ⫺e i jm H m A i, j 兲 n j ds⫽0 (3.37)
Note that Eqs. 共3.36兲 and 共3.37兲 hold only if the constitutive relation 共3.34a兲 is available and the effect of an external electric field is not involved. Moreover, they are limited to the quadratic energy. We briefly review the procedure of Wang and Shen 关72兴 and show that, in order to establish invariant integrals in more general cases, the key step is to redefine the Eshelby energy-momentum tensor 关2,3兴, as: P1k j ⫽ f ␦ k j ⫺ i j ui,k ⫺D j ,k ⫺ei jm H m Ai,k
(3.38)
When only considering an electric field, 共3.38兲 simplifies to: 1 P1k j ⫽D j E k ⫺ E i D i ␦ k j 2
(3.39)
This energy-momentum tensor has also been called the Maxwell stress tensor 关219,220兴. Without going into details, the invariant integrals formulated by Wang and Shen 关72兴 on the basis of 共3.38兲 are: J k⫽
冖
C
J k⫽
关 f ␦ k j ⫺ i j u i,k ⫺D j ,k ⫺e i jm H m A i, j 兴 n j ds
(3.40)
1 2
冖 冖
C
C
⫺ (3.34c)
关 x k f ⫺x j 共 ik u i, j ⫹D k , j ⫹e ikm H m A i, j 兲兴 n k ds
冖
C
共 ik ⫹D k ⫹e ikm H m A i 兲 n k ds
(3.41)
For a piezoelectric material with no magneto-effects, Eqs. 共3.40兲 and 共3.41兲 simplify to:
M⫽
Using the constitutive relation 共3.33a兲 and the fundamental equations of the material, Wang and Shen 关72兴 found the following conservation equation:
冖
C
⫺
(3.33d) where the coefficients ␣兵其, 兵其, 兵其, 兵其 denote the characteristic features of the material 关219,220兴, with
冖
533
关 f ␦ k j ⫺ i j u i,k ⫺D j ,k 兴 n j ds
(3.42)
关 x k f ⫺x j 共 ik u i, j ⫹D k , j 兲兴 n k ds
1 2
冖
C
共 ik u i ⫹D k 兲 n k ds
(3.43)
Equation 共3.42兲 represent the J k -vector for piezoelectric materials, which coincides well with Eqs. 共3.14a,b兲 derived originally by Pak 关55,56兴 for single crack problems and extended by Chen and Lu 关118兴 to many crack interacting problems. With the electromagnetic field neglected, Eqs. 共3.42兲 and 共3.43兲 are in agreement with those derived by Knowles and Sternberg 关11兴, Budiansky and Rice 关9兴, and Fletcher 关23兴. It should be mentioned that, strictly speaking, Wang and Shen’s work 关72兴 is incomplete, as they only considered static electro-magneto-elastic problems and ignored the effect of an external electric field. No application to practical crack problems has been carried out, and experimental study is needed to validate the theory. 4 APPLICATION OF INVARIANT INTEGRALS IN MICROCRACK DAMAGE PROBLEMS Traditionally, invariant integrals and microcrack damage are two distinct research areas in Solid Mechanics. While the development of invariant integrals has been reviewed in previous sections, various theories regarding microcrack damage can be found in Chaboche 关229,230兴, Jun and Chen 关231,232兴, Jun and Lee 关233兴, Kachanov and Laures 关234兴, Kachanov 关235兴, Kachanov 关236,237兴, Krajcinovic 关238兴, Lu and Chow 关239–241兴, among many others. In most cases the former was restricted to single crack problems in the general framework of Fracture Mechanics, whereas the study of the latter did not involve invariant integrals, except for a few works by Hutchinson 关49兴, Ortiz 关50–52兴, and Chow and Lu 关241兴. In these works, the J -integral was applied to study the shielding of a macrocrack by a cloud of microcracks, assuming that the global J -integral is identically equal to the local J -integral when the contour enclosing the microcrack zone surrounding the macrocrack tip becomes infinitely small and surrounds the tip only. Perhaps the only attempt in the open literature to uncover the inherent relations between invariant integrals and microcrack damage was made by Chen 关119,120兴. He noticed that, on one hand, from the physical point of view, microcrack damage is an energy dissipation process mainly owing to the decrease of effective elastic moduli and the release of re-
534
Chen and Lu: Developments and applications of invariant integrals
sidual stresses. On the other hand, the J k -vector, M -integral and L -integral represent energy release rates corresponding to the formation of a crack and its propagation and/or rotation. Does there exist any relation between these two? If such a relation does exist, what is the relation and how can it be derived? If there exists no such relation, then why? To answer these questions, we will review the work of Chen and coworkers 关119–122兴 who applied invariant integrals in many interacting crack problems and hence provided a bridge connecting the two distinct research areas. 4.1 Conservation laws of the J k -vector in many interacting crack problems Consider an infinite plane elastic solid with many interacting cracks as shown in Fig. 5. Assume that all cracks are stationary under a given remote uniform loading, ie, no crack growth is allowed 共see, eg, Jun and Chen 关231,232兴兲. Applying the J k -integral vector to treat the interacting problem, Chen 关119兴 established the following conservation laws: Both components of the J k -vector, defined in a global coordinate system of an infinite brittle solid, vanish if the closed contour chosen to calculate the vector encloses all the cracks (or voids, inclusions, etc) or, equivalently, there is no other discontinuity outside the closed contour. Mathematically, this can be written as N
J 1⫽
兺
l⫽1
N
J (l) 1 ⫽
N
J 2⫽
兺
l⫽1
兺 共 J 1(l)* cos l ⫺J 2(l)* sin l 兲 ⫽0
(4.1a)
l⫽1 N
J (l) 2 ⫽
兺 共 J 1(l)* sin l ⫹J 2(l)* cos l 兲 ⫽0
(4.1b)
l⫽1
(l) where J k , J (l) k , and J k (k⫽1,2 and l⫽1,2,...N) are defined * separately in the global system (x 1 ,x 2 ) and local system (l) (l) (x 1 * ,x 2 * ), the latter oriented by an angle l to the former 共see Fig. 5兲. It should be emphasized that Eqs. 共4.1a,b兲 are not directly covered by Eshelby’s energy momentum tensor theory 关1–3兴, even though they are based on the original J k -vector 关9–11兴. This is because the conservation laws
Appl Mech Rev vol 56, no 5, September 2003
共4.1a,b兲 are associated with a closed contour enclosing all singularities completely, whereas Eshelby’s theory 关1–3兴 is valid without enclosing any singularities. It appears that the J k -vector conservation laws have not been well accepted by some researchers. There are two groups of scholars, with whom the authors have exchanged ideas on the topic, who either suspected the validity of 共4.1a,b兲 or claimed they have little usefulness. The first group argued that since the SIFs at different crack tips have different values due to crack interaction, the vanishing of the J k -vector is not warranted 共unless the system of cracks is symmetrical to both global axes兲. However, Chen’s analytical and numerical results 关119兴 show that, due to remote uniform loading and the asymptotic nature of stresses and strains, 共4.1a,b兲 always hold irrespective of how many cracks exist in the system and whether the crack system is symmetrical with respect to the global coordinates or not. The second group argued that although the conservation laws 共4.1a,b兲 are valid, they are trivial and do not shed any new light. They further added that, because there are no singularities outside the closed contour, the closed contour could actually be considered to enclose the outside infinite region without singularities and hence, according to Eshelby’s conservation laws, the J k -integral vector should vanish. If this assertion is correct, then all other invariant integrals like the M - and L -integrals should also vanish. This is nevertheless not true because, as discussed by Chen 关119,120兴 and also pointed out by Suo 关174兴 in Eqs. 共2.28a,b兲, the M - and L -integrals do not vanish when the closed contour encloses all singularities. Their values are dependent upon how the cracks 共singularities兲 are arranged in the material 关119,120兴. This means that Eshelby’s conservation laws derived for the cases without singularities cannot directly cover the situation with many interacting singularities. In other words, under the present laws some integrals like J k do vanish and others like M - and L -integrals do not, although their formulations are all based on Budiansky and Rice 关9兴 for single crack problems. We will discuss below how to use the non-vanishing M and L to describe the damage level induced by multiple interacting cracks. 4.2 Independence of M -integral from origin selection Chen 关119,120兴 extended the original definition of the M -integral to treat the problem shown in Fig. 5. For the l -th (l) crack in the local system (x (l) 1 ,x 2 ), it is M (l) ⫽
冖
C (l)
(l) (l) (l) (l) 共 wx (l) i n i ⫺T l •u l,i •x i 兲 ds
共 l⫽1,2,¯ ,N 兲
(4.2)
(l)
The value of M can be calculated by using Freund’s formulation associated with the crack tip SIFs 关15兴, as: M (l) ⫽
⫹1 (l) 2 (l) 2 (l) 2 (l) 2 兲 ⫹ 共 K IIR 兲 ⫹ 共 K IL 兲 ⫹ 共 K IIL 兲 兴al 关共 K IR 8 (4.3)
Fig. 5 Many interacting cracks in an infinite plane elastic solid provided by Chen 关18,19兴
where is the shear modulus of the brittle solid, ⫽3 ⫺4 for plane strain, is Poisson’s ratio, subscripts R and L denote SIFs at the right tip and left tip of the l -th microc-
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Chen and Lu: Developments and applications of invariant integrals
rack, subscripts I and II denote Mode I and Mode II fracture, and a l refers to the half length of the l -th microcrack. When the M -integral is evaluated in the global system, it is found that the total contribution of the cracks calculated in the global system is not equal to the sum of individual contributions calculated in each local system, namely, N
M 共 x 1 ,x 2 兲 ⫽
兺 M (l)共 x (l)1 ,x (l)2 兲 l⫽1
(4.4)
(l) x 1 ⫽ (l) 1 ⫹x 1 (l) x 2 ⫽ (l) 2 ⫹x 2
共 l⫽1,2,...N 兲
(4.5)
Obviously, the difficulty in extending the original definition of M to treat multiple interacting crack problems is how to translate all the local coordinate systems to a global system. A question arises naturally as to whether or not the value of M depends on coordinate shifts. 共As a scalar, the M -integral is independent of coordinate rotation.兲 Accounting for coordinate shifts from local systems to the global system, Chen 关119,120兴 found that N
M⫽
兺 M (l)共 x 1 ,x 2 兲
l⫽1
再冖 兺再冖 N
⫽
兺
l⫽1
Cl
共 wx i n i ⫺T l u l,i x i 兲 ds
冎
N
⫽
l⫽1
Cl
(l) (l) (l) 关 w 共 x (l) i ⫹ i 兲 n i ⫺T l u l,i 共 x i ⫹ i 兲兴 ds
N
⫽
兺 兵M
冎
N
(l)
l⫽1
(l) 共 x (l) 1 ,x 2 兲 其 ⫹
兺 兵 (l)1 J (l)1 ⫹ (l)2 J (l)2 其
l⫽1
⫽M N ⫹M A
(4.6)
where
兺 兵 M (l)共 x (l)1 ,x (l)2 兲 其 l⫽1
(4.7a)
兺
l⫽1
(4.8)
where 1 and 2 denote coordinate shifts. In the new global coordinate system (x 01 ,x 02), the value of the M -integral is denoted by M 0 and is given by 关119兴
冖
M 0⫽
C
共 wx 0i n i ⫺T l u l,i x 0i 兲 ds
N
⫽
兺
(k) (k) (k) (k) (k) 兵 M (K) 共 x (k) 1 ,x 2 兲 ⫹ 共 1 ⫹ 1 兲 J 1 ⫹ 共 2 ⫹ 2 兲 J 2 其
k⫽1
N
⫽M ⫹ 1
兺
k⫽1
N
J (k) 1 ⫹2
兺
k⫽1
J (k) 2
(4.9)
By the conservation laws 共4.1a,b兲, the last two terms on the right hand side of 共4.9兲 vanish. Consequently, the M -integral is independent of the origin selection of the global coordinates. In fact, Eq. 共4.9兲 coincides well with the argument by Budiansky and Rice 关9兴 for a single crack. They pointed out that M from a physical point of view, represents the energy release rate induced from the uniform expansion of the single crack. Simultaneous uniform expansion is certainly possible for a system of strongly interacting cracks conceived as a single super defect. As a result, its energy release rate does not depend on the origin selection of the global coordinate system. It has been established that the same conclusion applies to the L -integral, with 关119兴: L⫽
兺 L (l)共 x (l)1 ,x (l)2 兲
(l) (l) (l) 兵 (l) 1 J1 ⫹2 J2 其
(4.7b)
Consequently, the M -integral defined in the global system can be divided into two distinct parts, ie, Eqs. 共4.7a兲 and 共4.7b兲. The first, denoted by M N , is a summation of the contributions induced from each crack in its corresponding local system, while the second, denoted by M A , is another summation of the contributions induced from the inner product between the J k -vector of each crack and the position (l) vector ( (l) 1 , 2 ) of the origin of each local system. For convenience, Chen 关119兴 called M N and M A as the net and additional parts of M , respectively. Equation 共4.6兲 poses significant trouble for the evaluation of M in multiple interacting crack problems. Some believe that the value of M is dependent upon the origin selection of
(4.10)
l⫽1
L 共 x 1 ,x 2 兲 ⫽L N ⫹L A ⫽L 0 共 x 01 ,x 02兲
N
M A⫽
x 1 ⫽x 01⫺ 1
N
N
M N⫽
the global coordinate system (x 1 ,x 2 ), since different selec(l) tions of the origin lead to different values of (l) 1 and 2 according to 共4.5兲. However, Chen’s analysis 关119兴, backed by numerical calculations, has shown that this is not the case. With reference to Fig. 5, let the global system (x 1 ,x 2 ) be translated to another (x 01 ,x 02):
x 2 ⫽x 02⫺ 2
where the shift from each local system to the global system is
535
(4.11)
N
L N⫽
兺 L (l)共 x (l)1 ,x (l)2 兲
(4.12)
l⫽1 N
L A⫽
兺 共 (l)2 J (l)1 ⫺ (l)1 J (l)2 兲
(4.13)
l⫽1
where L N and L A are the net and additional parts of the L -integral, whereas L and L 0 are calculated in the two different global coordinate systems (x 1 ,x 2 ) and (x 01 ,x 02), respectively. Equation 共4.11兲 coincides well with another argument by Budiansky and Rice 关9兴 and Herrmann and Herrmann 关10兴 for a single crack. They argued that L from a physical point of view, represents the energy release rate induced from crack rotation. As a single super defect, a system of
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cracks can also rotate simultaneously: the associated energy release rate does not depend on the selection of the global coordinate system. All the manipulations performed above can be repeated for more complicated cases where, besides interacting cracks, other discontinuities such as inclusions and voids coexist in a finite region of an infinite plane. Equations 共4.1a,b兲, 共4.6兲, and 共4.11兲, with N denoting the total number of discontinuities, still hold. Although the contributions induced from other discontinuities are quite different from those induced from cracks, the conservation laws derived in this section remain valid.
4.3 Description of microcrack damage based on the M -integral Damage mechanics has been a topic under intensive study, and is beyond the scope of this review. To evaluate damage level, the effective medium theory due to Kachanov 关235兴 has been widely used 共see, eg, Mori and Tanaka 关242兴, Jun and Chen 关231,232兴, Lu and Chow 关239,241兴兲. The focus of this section is to provide an alternative description of microcrack damage based on the M -integral. Following Jun and Chen 关231,232兴, it is assumed that all microcracks with a characteristic length from 10 microns to 100 microns are fully open, stationary, and not intersecting with each other, although crack density and crack spacing can vary from one place to another so that strong microcrack interaction may occur. The crack interacting problem shown in Fig. 5 could be easily solved 共see, eg, Gross 关243兴 and Chen 关244兴兲 to obtain stresses, displacements, and SIFs of each crack. Let the SIFs at both tips of the l -th microcrack be denoted by (l) (l) (l) (l) ,K IIR ,K IL ,K IIL (l⫽1,2,¯ ,N). Even though the numK IR (l) (l) (l) (l) ,K IIR ,K IL ,K IIL ber of microcracks may be large and K IR
Fig. 6 Four regularly distributed cracks in an infinite plane elastic solid provided by Chen 关18,19兴
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Table 1. Normalized values of M N and M A , and their summation, M or M 0 , and the total potential energy, 2U, owing to the formation of the four microcracks shown in Fig. 6
(°)
MN
MA
M
M0
2U
MT
0 6 10 16 30 46 60 76 90
3.5686 3.5442 3.5006 3.3931 2.9405 2.1080 1.2493 0.4479 0.1860
⫺0.0828 ⫺0.0875 ⫺0.0955 ⫺0.1128 ⫺0.1571 ⫺0.1596 ⫺0.0972 ⫺0.0006 0.0371
3.4858 3.4567 3.4051 3.2803 2.7834 1.9484 1.1521 0.4473 0.2231
3.4858 3.4567 3.4051 3.2803 2.7834 1.9484 1.1521 0.4473 0.2231
3.4858 3.4567 3.4051 3.2803 2.7834 1.9484 1.1521 0.4473 0.2231
3.7321 3.6942 3.6276 3.4689 2.8660 1.9396 1.1340 0.4707 0.2680
may be heavily influenced by neighboring cracks, a modern personal computer is usually sufficient for the purpose of numerical computations. Physically, for a single crack, it is well known that the M -integral is twice the change of the total potential energy (CT PE). Chen 关119兴 found that this is also valid for multiple interacting cracks. Consider, as an example, a system with four regularly distributed cracks in an infinite plane elastic solid as shown in Fig. 6. Numerical results 关119兴 to verify the validity of Eqs. 共4.6兲, 共4.9兲, and 共2.12a兲 are listed in Table 1. From an energy balance point of view, this implies that the change of total potential energy due to the formation of a system of cracks could be described by using a single phenomenological parameter, the M -integral. An apparent application of 共4.6兲 is shown in Fig. 7, where the normalized M -integral is plotted as a function of the remote loading angle for four regularly distributed, strongly interacting cracks. Of great interest is the inherent relation between the M -integral and the effective elastic moduli of the microcracking solid. It is seen from Fig. 7 that the maximum of M is coincident with the direction along which the largest reduction of effective moduli occurs ( ⫽90°), while the minimum of M corresponds to the direction along which the
Fig. 7 Variable tendencies of the normalized M -integral against the loading angle for the four regularly distributed cracks provided by Chen 关18,19兴
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smallest reduction of effective elastic moduli occurs ( ⫽0°). This example shows clearly that the M -integral may be used to describe the microcrack damage level. Furthermore, Fig. 8 depicts a system with twenty randomly distributed cracks. The corresponding numerical results of the M -integral against the loading angle are shown in Fig. 9, where M N , M A , and M T represent the net and additional parts of the integral, and its value obtained with the Taylor model by entirely neglecting the interaction among the cracks. It is seen from Fig. 9 that M is not sensitive to the direction of remote loading, as expected, because a microcrack-weakened solid exhibits an isotropic behavior if the microcracks are sufficiently randomly distributed, although its effective elastic moduli are significantly reduced. In passing, we note that, most recently, Chang and Chien 关245兴 also suggested to use the M -integral as a fracture parameter for description of the degradation of material and/or structural integrity caused by irreversible evolution of multiple defects. 4.4 Relation between M - and L -integrals For a single crack, the relation between the M - and L -integrals is given by Eq. 共2.12c兲, namely, the L -integral is just the negative half of the partial derivative of the M -integral with respect to crack rotation angle ␥. In this section we question whether this holds under strongly crack interacting situations, where the change of total potential energy due to the formation of a cloud of cracks is 关139兴:
兺 冕⫺a i2⬁ 共 l 兲 •⌬u (l)i 共 x 1(l)* 兲 dx 1(l)* l⫽1 N
U⫽
a
(4.14)
⬁ (l) is the uniform stress at the location of the l -th Here, i2 is the displacement jump across the l -th crack, and ⌬u (l) i crack. Obviously, ⌬u (l) i is disturbed by neighboring cracks due to interaction 共see Fig. 5兲. The study of Chen 关119兴 confirms the validity of Eqs. 共2.12a,c兲 for multiple interacting cracks. Physically, because a system of cracks can be simply considered as a single
Fig. 8 Twenty randomly distributed cracks in an infinite plane elastic solid provided by Chen 关18,19兴
537
complicated and irregular defect, both 共2.12b兲 and 共2.12c兲 should hold. Therefore, M and L are not two distinct integrals. The fact that the energy release rate induced from the simultaneous rotation of a single super defect 共including many cracks兲 leads directly to 共2.12c兲. For practical applications, the technique developed by King and Herrmann 关25兴 may provide an effective way to nondestructively measure M. The present reviewers do not agree with the following claim by some researchers: ‘‘Fracture is a property, which happens in a local region and must be characterized by its local trait. As an extended principle of Saint-Venant, it is not possible to make a judgment about local fracture by means of the global integrals.’’ It is well known that fracture initiation and the associated fracture toughness can be measured by means of the famous compliance method well addressed in the literature 共see, eg, ASTM standards 1990 关162兴兲, which only depends on the change of the stress-displacement field away from the crack, without any treatment of near-tip singularity. This is a typical example to illuminate that, even though fracture initiates in a local region surrounding the crack tip, it can be described by an integral such as J calculated along a closed contour far away from the tip 共eg, the outside boundary of a simple tension or three-point bending specimen兲. A small change ⌬a in crack length causes changes in the outside stress-displacement fields and hence the total potential energy ⌸, with J⫽⫺(d⌸/da) providing an important physical parameter. In this case, the principle of Saint-Venant should be adopted with caution, since the crack length increase ⌬a and the induced change of total potential energy ⌬⌸ have the same order of magnitude so that ⌬⌸/⌬a is a nonzero constant. The study of King and Herrmann 关25兴 provides another example to corroborate the above view. They measured J and M by using the displacement and stress data far way from the crack tip. Their work was not only confirmed by results
Fig. 9 Variable tendencies of the normalized M -integral against the loading angle for the twenty cracks randomly distributed provided by Chen 关18,19兴
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measured with other destructive techniques but also appreciated by Kanninen and Popelar 关16兴. It should be emphasized that, physically, J represents the energy release rate per unit crack advance while M represents the change of total potential energy due to the formation of a crack. This is the fundamental reason why a global integral like J and M can be used to describe crack stability, at least under small-scale yielding (SSY ) conditions, without characterizing, in detail, the local trait such as plasticity and finite deformation in the vicinity of the crack tip. The same conclusion applies to multiple interacting cracks, because the physical significance of J and M remains unchanged in these situations 关119兴. Nevertheless, only stationary damage with fully open microcracks has so far been studied by using the M -integral description 关119–122兴. Besides the loss of effective elastic moduli studied by Jun and Chen 关231兴 and discussed in 关119,120兴, damage stability or growth with invariant integrals is another important topic yet to be addressed 关232兴. Although Chen’s work 关119,120兴 did build a bridge betweem the two distinct research fields of solid mechanics—the conservation laws and the continuum damage mechanics—more investigations on damage and its evolution, both theoretically and experimentally, are needed. Furthermore, practical 3D damage problems 关233,234兴 such as rocks in response to differential compressive stresses with the sliding of crack surfaces 关246,247兴 need to be studied to demonstrate the validity of using M or L as a damage measure. 5 THE ROLE OF HIGHER ORDER SINGULAR TERMS The key concept of linear elastic fracture mechanics (LEFM ) is Irwin’s stress intensity factor (SIF) 关248兴, commonly found in the asymptotic stress field at a crack tip from Williams’ eigenfunction expansion form (EEF) 关249,250兴. Customarily, to describe the crack tip field under the assumption of SSY , only two terms in EEF need to be determined for a given crack configuration and body geometry, corresponding separately to the inverse square root singularity 共or the oscillating singularity for an interface crack兲 and the elastic T term—a uniform stress acting parallel to the crack plane 关251兴. This approach has been branded the twoparameter fracture theory formulated in terms of SIFs and the T -stress. The higher order singular terms (HOSTs) such as r ⫺3/2, r ⫺5/2, etc in the complete Williams EEF were always omitted for brittle materials, because strain energy as well as displacement in the near-tip region are bounded. However, as pointed out explicitly by Rice 关251兴, if there exists a nonlinear zone around the crack tip 共Fig. 10兲, the complete solution in the elastically deforming material outside a circle of radius greater than the greatest extent of the nonlinear zone should include the higher order singular terms. Furthermore, choosing the path in the definition of J to be a large circle surrounding the crack tip and considering the limit as the radius of the circle approaches infinity, Rice 关251兴 found that J takes on the same value it would have if there were no nonlinear zone and the material responded elastically everywhere, and there is no T -effect on J. The existence of HOSTs outside the nonlinear zone was also
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pointed out by Edmunds and Willis 关252,253兴 and was implicitly present in a few exact solutions by Dugdale 关71兴, Rice 关254兴, and Hult and McClintock 关255兴. Edmunds and Willis 关253兴 performed elastic-plastic finite element computations to extract the complete Williams type expansion coefficients. More recently, Hui and Ruina 关91兴 found that HOSTs do exist in the elastic field outside the nonlinear zone 共which may be inelastic, finite deformation, discontinuous, etc兲, since the complete Williams EEF, including the HOSTs, is not assumed to be applicable to the crack tip itself: rather, it is applied only at a distance from the crack tip. After performing lengthy manipulations and introducing an annular region around the crack tip for antiplane fracture 共mode III兲, they derived an explicit formulation of J with HOSTs taken into account. There are two additional contributions to J due to the existence of the nonlinear zone. The first is due to changes in local SIFs and the second is attributable to the interaction of HOSTs with nonsingular terms in the complete Williams expansion forms. Chen and Hasebe 关92兴 and Jeon and Im 关93兴 subsequently applied the general results to mode I and mode II fracture. 5.1 Hui-Ruina’s investigation and remarks Hui and Ruina 关91兴 reexamined the role of HOSTs in characterizing the near-tip stress field outside a nonlinear process zone 共Fig. 10兲. They found that the J -integral defined in an annular region outside the nonlinear zone could be divided into two parts. The first is the so-called local J -integral 共or local ERR) contributed to by the well-known r ⫺1/2 singularity, but influenced by the existence of the nonlinear zone. The second is the additional contribution due to mutual interaction between HOSTs and higher order regular terms. For antiplane fracture, the classical formulation in purely elastic cases is 2 /2 J⫽K III
(5.1)
Fig. 10 An annular region outside of the nonlinear zone surrounding a crack tip provided by Hui and Ruina 关35兴
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where K III is the mode III SIF. By introducing an annular region outside a nonlinear zone and accounting for HOSTs, Hui and Ruina 关91兴 obtained ⬁
J⫽ 共 R 2a / 兲共 d 0 兲 ⫹2
兺
n⫽0
d n⫹1 e n
(5.2)
where R is the outer radius of the annular region, a is shear loading level, d 0 , d n⫹1 , and e n are introduced to account for the influence of HOSTs, and is the shear modulus. Equation 共5.2兲 reveals that there are two additional contributions to J due to the existence of the nonlinear zone. The first is caused by changes in the local SIF, which is of the order O( /R) with and R being the inner and outer radii of the annular. The second is induced by the interaction of HOSTs with nonsingular terms, which is also of the order O( /R). Hui and Ruina’s work 关91兴 is of great significance in LEFM . As a nonlinear zone around a crack tip exists for almost all engineering materials, both nonsingular terms and HOSTs in the complete Williams EEF should be considered when evaluating J, from which the validity of SSY for a practical crack problem can be judged. 5.2 Chen-Hasebe’s investigation and remarks Following Hui and Ruina 关91兴, Chen and Hasebe 关92兴 used the pseudo orthogonal property of the Williams EEF to reexamine the role of HOSTs, and derived explicit J -formulations for antiplane cracks, plane cracks, interfacial cracks, and anisotropic plane cracks. For an antiplane crack: ⬁
2 /共 2 兲⫺ J⫽K III
兺
k⫽1
k 共 k⫹2 兲 A ⫺k A k⫹2
(5.3)
where A ⫺k and A k⫹2 are complex coefficients of the HOSTs and regular terms in the complete EEF, respectively. For a plane crack: ⬁
共 1⫹ 兲 1⫹ 2 2 J⫽ 兲⫹ 共 K I ⫹K II 共 ⫺1 兲 k k 共 k⫹2 兲 8 2 k⫽1 ⫻Re关 A ⫺k ¯A k⫹2 兴
gon and Hancock 关257兴 and Al-Ani and Hancock 关258兴 used a modified boundary formulation of SSY , and proposed to use SIF and T as an alternative set of fracture parameters for hardening and non-hardening materials. Betagon and Hancock 关257兴 compared the stress field associated with the positive and negative T -stress in the plastic zone with the classical HRR field. Jeon and Im 关93兴 combined Chen and Shield’s two-state J - and M -integrals 关84兴 with Hui and Ruina’s concept of HOSTs 关91兴 for plane elastic-plastic fracture, and found that J comprises only the contributions from the mutual interaction between all complementary pairs of eigenfields. The same applies to M , with a slightly different definition of the complementary pair. Particularly, it is found that the interaction of higher order singularities with nonsingular higher order eigenfields generates an extra configurational force, in addition to the energy release rate 共or J) resulting from the inverse square root singularity. This additional J -value is associated with the translation of the plastic zone alone, the crack tip being fixed. Numerical examples illustrate that the effect of HOSTs is negligible in terms of J when the plastic zone size is small, but in the case of large scale yielding, HOSTs make a difference in the plastic zone configuration through the interaction between singular and nonsingular terms. To represent J and M in terms of two-state conservation integrals, Jeon and Im 关93兴 contemplate a given stress state outside the elastic-plastic zone as a summation of all eigenstates with eigenvalues ␦ n ⫽⫺⬁⬃⬁. The J - and M -integrals can then be written in terms of the mutual interaction between two eigenfields with eigenvalues ␦ n and ␦ cn , respectively. For definiteness, ␦ n (n⫽0,⫺1,⫺2,...) are taken as the eigenvalues of the singular eigenfield, ie, ␦ 0 ⫽⫺1/2, ␦ ⫺1 ⫽⫺1,␦ ⫺2 ⫽⫺3/2,..., with ␦ cn given by ⫺1⫺ ␦ n for J( ␦ n , ␦ cn ) and by ⫺2⫺ ␦ n for M ( ␦ n , ␦ cn ). The results are: ⫺⬁
J⫽J 共 ⫺1/2,⫺1/2兲 ⫹
兺
兺
n⫽⫺1
5.3 Jeon-Im’s work and remarks Jeon and Im 关93兴 utilized two-state conservation laws, in conjunction with finite element analysis, to obtain the complete Williams eigenfunction series for elastic-plastic cracks, including not only the inverse square root singularity and T -stress but also higher order singular and nonsingular terms. Eigenfunction series has been extensively utilized to examine solutions of crack problems. Particularly, the so-called two-parameter fracture theories, formulated in terms of SIFs and T -stress, have been popular for elastic-plastic crack problems. Bilby et al 关256兴 performed a large deformation analysis for a non-hardening material, and examined the T-effect using a modified boundary layer formulation. Beta-
J 共 ␦ n , ␦ cn 兲
(5.5)
⫺⬁
(5.4)
where A ⫺k and A k⫹2 are complex coefficients of the HOSTs and regular terms in the complete EEF, respectively, and K I and K II are the local SIFs.
539
M ⫽M 共 ⫺1/2,⫺1/2兲 ⫹
兺
n⫽⫺2
M 共 ␦ n , ␦ cn 兲
(5.6)
where J(⫺1/2,⫺1/2)⬅J 0 is the J -integral associated with the inverse square root singularity ␦ 0 ⫽⫺1/2, ie, J 0 ⫽( ⫹1)/8 K I2 under mode I fracture. If there are no concentrated line force or dislocations at the crack tip, the logarithmic terms for ␦ ⫺1 ⫽⫺1 related to rigid body motion disappear, and hence the first term in 共5.6兲 disappears in summation. Equations 共5.5兲 and 共5.6兲 show that the contribution to J and M by the eigenfunction of eigenvalue ␦ n is generated from the mutual interaction between this eigenfield and its complementary eigenfield; no interaction occurs between two eigenfields 共in terms of J and M ) if they are not complementary to each other. This conclusion can be achieved straightforwardly by using the pseudo-orthogonal property 关76,92兴 or biorthogonal property 关77–79兴.
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For qualitative interpretation of J and M as energetic forces for an elasto-plastic crack, the smallest circular domain including the plastic zone, with its center at the crack tip, may be viewed as an inhomogeneity embedded in the elastic material exterior to it. The J -integral can be interpreted as ERR when all singularities originating from the crack tip and the circular fictitious inhomogeneity translate in position relative to a material body. Particularly, J 0 , related to the translation of ␦ 0 ⫽⫺1/2 singularity, represents the extension of the crack but with no translation of the cir⫺⬁ J( ␦ n , ␦ cn ) cular inhomogeneity. The summation term 兺 n⫽⫺1 in 共5.5兲, associated with the translation of HOSTs, is the energetic force governing the translation of the inhomogeneity 共ie, the plastic zone兲 while the crack tip is fixed. The growth of an elastic-plastic crack inherently accompanies the corresponding movement of the crack tip plastic zone. As a plastic zone develops near the crack tip, the overall specimen compliance increases so that J 0 will change according to a given traction or displacement boundary condition. In addition to J 0 , plastic deformation has the mechanism of creating another term in J. This additional term, ⫺⬁ represented by 兺 n⫽⫺1 J( ␦ n , ␦ cn )⫽J⫺J 0 , stems from the mutual interaction between higher order singular eigenfields and their complementary nonsingular eigenfields. As a conse⫺⬁ J( ␦ n , ␦ cn ) may be quence, while J 0 is always positive, 兺 n⫽⫺1 either positive or negative depending upon the loading and overall specimen geometry. This will be verified via the numerical examples to follow. In the same spirit, the M -integral may be interpreted as the ERR associated with the uniform expansion of the fictitious inhomogeneity via transformation of elastic properties outside the plastic zone. If no plastic zone exists near the
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crack tip, as in purely elastic cases, the HOSTs disappear, and M vanishes when the origin of coordinates is located at the crack tip. Figures 11共a兲 and 共b兲 show two panels under plane deformation 关93兴: the single edge notched tension panel (SENT) and the middle tension panel (M T). Numerical results for both specimens are presented in Figs. 12共a兲 and 共b兲, with the contribution percentage of HOSTs plotted separately as a function of a兲 applied loading and b兲 hardening exponent. Figure 12共a兲 demonstrates that, as the load 0 is increased, J⫺J 0 normalized by J 共which increases with increasing 0 ) decreases for the SENT panel and increases for the M T panel. Thus, it accounts for the difference between the two in the image effect of overall specimen configuration including both loading and exterior boundary 关93兴. For the SENT panel, the higher order contribution leads to a difference in (J⫺J 0 )/J larger than 10% when 0 is increased to over 30% larger than the uniaxial yield stress y (⫽E/300) 关93兴. However, for the M T panel, the difference in (J⫺J 0 )/J is always less than 10% even though 0 is 45% larger than y . From Fig. 12共b兲 it is seen that, as the hardening exponent 1/m increases, the higher order contribution (J⫺J 0 )/J increases for the SENT panel and decreases for the M T panel 关93兴. However, its value is small for both panels, indicating that the higher order contribution is not sensitive to variations in m. Jeon and Im’s work 关93兴 provides a quantitative estimation of the accuracy of the SSY assumption, as noted by Hui and Ruina 关91兴. That is, for a given specimen geometry and loading configuration, the SSY condition is met if the influence of HOSTs on J is less than a prescribed engineering
Fig. 11 a) Single edge notched tension panel (SENT) and b) middle tension panel (M T) provided by Jeon and Im 关37兴
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tolerance, say 10%. In contrast, the classical empirical estimation of plastic zone size relative to crack length is not always sufficient to ensure SSY . 5.4 Summary The current understanding on higher order singular terms in the complete Williams expansion form can summarized as follows: 1兲 If a cracked body experiences purely elastic deformation 共eg, glass兲, HOSTs should be omitted entirely to ensure strain energy density and displacements are bounded. This is the classical case in LEFM . 2兲 If there exists a nonlinear zone surrounding the crack tip 共eg, ductile metal兲, HOSTs should be taken into account to evaluate additional contributions to J unless the SSY assumption is actually met. 3兲 There are two additional contributions to J due to the presence of the nonlinear zone. The first is due to the change in local SIFs, and the second is attributable to the interaction of HOSTs with nonsingular terms in the complete Williams expansion form. 4兲 The study of Hui and Ruina 关91兴, Chen and Hasebe 关92兴 and Jeon and Im 关93兴 provides detailed notes, from which
Fig. 12 The contribution percentage of the HOSTs to the J -integral versus: a) the applied loading and b) the hardening exponents provided by Jeon and Im 关37兴
541
questions such as why K plays a dominant role under SSY conditions and when higher order singular terms could be omitted can be answered. 5兲 The J -integral for elastic-plastic cracks comprises mutual interactions between two eigenstates that are complementary to each other, ie, the so-called pseudo-orthogonal property 关76兴 or biorthogonal property 关77– 81兴. In other words, there are no interactions between an arbitrary pair of eigenstates unless they are complementary to each other. 6兲 The empirical estimation of the plastic zone size relative to crack length is not always sufficient to ensure the validity of SSY . Rather, SSY is met if the influence of HOSTs on J is less than a given engineering tolerance. 6 APPLICATION OF INVARIANT INTEGRALS IN 3D CRACK PROBLEMS Three-dimensional cracks are commonly found in practical structures, eg, surface cracks and penny-shaped cracks. The analysis of 3D crack initiation and propagation is, in general, much more complicated than that for a 2D crack. Many mathematical tools such as the integral transform method 关259,260兴 have been proposed to calculate the intensities along a 3D crack front. Meanwhile, invariant integrals have also been playing a significant role for this purpose. Bakker 关30兴 presented a detailed description of the 3D J -integral. Lorenzi 关28兴 studied the ERR and J -integral for 3D crack configurations. Bui 关157兴 developed a technique using the J -integral associated with mode I and mode II, where the symmetric and antisymmetric parts of the planar displacement, strain, and stress fields about the crack plane are separated. Nikishkov and Atluri 关128,129兴 developed a domain integral approach to calculate mixed-mode stress intensity factors for planar 3D cracks. Dodds and Read 关131兴 presented experimental and numerical results of the J -integral for a surface flaw. Rice 关31,32兴 established the conservation laws and weight function theory for 3D elastic cracks. Sharobeam and Landes 关154兴 developed a single specimen approach to evaluate J for a semi-elliptical surface crack. Nakamura and Parks 关155兴 and Nakamura 关156兴 employed the two-state integral approach to determine mixedmode stress intensity factors for 3D interface cracks. Kuo 关150兴 studied the use of a path-independent line integral for axisymmetrical cracks with nonaxisymmetric loading. Shih and Asalo 关87兴, Nahta and Moran 关153兴, and Gozs et al 关148兴 developed domain integrals, used asymptotic auxiliary fields to decompose the SIFs in mixed-mode cracks, and proposed a method to evaluate the divergence term in the two-state integral. Kim et al 关147兴 developed a mode decomposition technique for 3D mixed-mode cracks via two-state integrals. Other applications of invariant integrals to 3D cracks can be found in Shivakumar and Raju 关130兴, Huber et al 关132兴, Murakami and Sato 关29兴, Aliabadi 关133兴, and Rigby and Aliabadi 关134,158兴, among others. This section aims to review two main advances of 3D invariant integrals, and to present a basic understanding of mode decomposition techniques. Section 6.1 studies the equivalent domain integral for 3D mixed-mode fracture
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关133,134,158兴. In Section 3.2, the mode decomposition of 3D mixed-mode cracks via two-state integrals developed by Kim et al 关147兴 is described. In Section 6.3 some other developments are reviewed and discussed. Comments on the difference between different techniques and their potential applications in functional materials with 3D cracks are presented. 6.1 Equivalent domain integral for 3D mixed-mode fracture The J -integral, in conjunction with the displacement-based FEM , has been directly applied to distinguish mixed-mode fracture along the front of a surface crack by various researchers. However, the evaluation of surface integrals with FEM is rather cumbersome. This has led to the modification of J to a domain integral originally proposed by Li et al 关151兴 and Nikishkov and Atluri 关128,129兴, in which J is multiplied by a simple function called S 共see, eg, Lorenzi 关28兴兲. This method is known as the equivalent domain integral, and is computationally appealing as the domain integral can be accurately and easily obtained with FEM . For example, the equivalent domain integral has been applied to mixed-mode fracture problems by Nikishkov and Atluri 关128,129兴 and Shivakumar and Raju 关130兴 by using the decomposition method 共see Ishikawa et al 关24兴兲. In the decomposition method, mode I, II, and III J -integrals are directly obtained from mode I, II, and II stresses and displacements. The application of the J -integral method to BEM was originally presented by Aliabadi 关133兴 for 2D problems and by Rigby and Aliabadi 关134,158兴 for 3D problems. Meanwhile, Huber et al 关132兴 presented the formulation for problems involving mode I and III. The BEM analysis is ideally suited to the evaluation of J, since the required stress, strain, and derivatives of strain are accurately obtained at internal points in an elastic body even with a surface crack. These internal point solutions utilize boundary integral equations and, hence, no discretization of the domain is required. The J -integral is then calculated by integrating stress, strain, and its derivative products, found at the internal points, along a contour in a plane perpendicular to the crack front and also over the area enclosed by the contour. Consequently, the J -integral is accurately calculated without altering the surface mesh. In order to obtain mode I, II, and III J -integrals, the stress and strain products as well as stress and derivative of strain products are combined from points symmetric to the crack plane. These integrals are then integrated over the symmetric contour and area enclosed by that contour to yield two parts of the J -integral: one comprised of symmetric elastic fields (J S ) and the other comprised of anti-symmetric elastic fields (J AS ). The J S integral is equal to the mode I J -integral, whereas the J AS integral contains both mode II and III J -integrals. The decomposition method is further developed to decouple mode II and III stresses, stains, and derivatives of strains. Finally, the separated stress intensity factors are obtained directly from mode I, II, and II J -integrals. Rigby and Aliabadi 关134,158兴 presented the proper derivation of the decomposition method for the mixed-mode J -integral. They pointed out that the equation used in previous
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studies 共see, eg, 关128 –132兴兲 is incorrect. They found an area integral for mode II and III J -integrals different from that quoted in Rigby and Aliabady 关134,158兴 and Huber et al 关132兴. For elastic or nonlinear elastic materials, the 3D J -integral is derived from the energy momentum tensor of Eshelby 关2,3兴. Eshelby’s energy momentum tensor, denoted here by P k j 共Amestoy et al 关127兴兲, is: P k j⫽ b W ␦ k j⫺ i j共 u i / x k 兲c
(6.1)
where ␦ k j is the Kronecker delta, i j and u i , with (i, j,k ⫽1,2,3), are stresses and displacements, and W is the strain energy density in linear elastostatics: W⫽
冕
i j
0
i j d i j ,
共 i, j⫽1,2,3 兲
(6.2)
1 i j ⫽ 共 u j,i ⫹u i, j 兲 2
(6.3)
Here, i j are strains, and u i, j denote derivatives of displacements with respect to the 3D coordinates x j . Differentiating W with respect to x k gives
冉
W u i, j u j,i ⫽ij ⫹ xk xk xk
冊
i, j,k⫽1,2,3
(6.4)
Equation 共6.3兲 together the equilibrium i j / x j ⫽0 yield
冋 冉
冊 冉
W 1 ui u j ⫽ ij ⫹ ij xk 2 x j xk xi xk
冊册 冉 ⫽
冊
ui ij x j xk (6.5)
which can be rewritten in a more compact form as
Pkj ⫽ 关 W ␦ k j ⫺ i j 共 u i / x k 兲兴 ⫽0 x j x j
(6.6)
It is obvious that Eq. 共6.6兲 associated with the derivative of Eshelby’s energy momentum tensor 共6.1兲 is different from the original definition of the J -integral associated with the tensor itself. Many researchers 关87,128 –134兴 found that a new invariant integral, called the domain integral, can be introduced from the conservation law 共6.6兲. Consider a cross section of a crack shown in Fig. 13共a,b兲, and focus on the region ⍀(C⫺C ), ie, the area delimited by the contours C C and the crack surface ␥. Integrating P k j over any area ⍀(C⫺C ) in the plane x 3 ⫽0 and excluding the crack singularity gives:
冕
关 W ␦ k j ⫺ i j 共 u i / x k 兲兴 d⍀⫽0 ⍀(C⫺C) x j
(6.7)
Making use of Green’s theorem
冕
⌫
Tdx 1 ⫹Qdx 2 ⫽
冕 冉 ⍀
冊
Q T ⫺ d⍀ x1 x2
(6.8)
and the relations dx 1 ⫽⫺n 2 d⌫, dx 2 ⫽n 1 d⌫ with ⌫⫽C ⫹C ⫹ ␥ , one obtains from 共6.7兲:
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冕
⌫
Chen and Lu: Developments and applications of invariant integrals
关 Wn k ⫺ i j 共 u i / x k 兲 n j 兴 d⌫
⫺
冕
共 i3 共 u i / x k 兲 d⍀⫽0 ⍀(C⫺C) x 3
J k共 s 兲 ⫽ ⫽
(6.9)
C⫹ ␥
⫽
冕 冕
冉
i3 共 u i / x k 兲 d⍀ ⍀(C) x 3
C
⫺
J 1共 s 兲 ⫽
关 Wn k ⫺ i j 共 u i / x k 兲 n j 兴 d⌫
冕
共 i3 共 u i / x k 兲 d⍀ x ⍀(C) 3
关 Wn k ⫺ i j 共 u i / x k 兲 n j 兴 d⌫
C⫹ ␥
关 Wn k ⫺ i j 共 u i / x k 兲 n j 兴 d⌫
冕
共 i3 共 u i / x k 兲 d⍀ ⍀(C) x 3
(6.11)
where ⌫ is identical to contour C except that it proceeds in the counterclockwise direction. Note that J k (s) is dependent on the position of crack front s, and is defined on the plane x 3 ⫽0. With k⫽1 for a traction free crack, the contour integrand over crack surfaces ␥ is zero, because T i ⫽ i j n j ⫽0 and n 1 ⫽0. Then, from 共6.11兲 one obtains:
关 Wn k ⫺ i j 共 u i / x k 兲 n j 兴 d⌫
⫺
⌫
⫺
Equation 共6.9兲 can be reordered as
冕
冕 冕
543
⫽
(6.10)
Consider C to be a circular contour of radius . As →0, the area term on the right hand side of 共6.10兲 tends to zero 共Dobbs and Read 关131兴兲. Consequently, the J k (s) -integral can be defined as:
冕 冕
⌫
C
⫺
关 Wn 1 ⫺ i j 共 u i / x 1 兲 n j 兴 d⌫
关 Wn 1 ⫺ i j 共 u i / x 1 兲 n j 兴 d⌫
冕
共 i3 共 u i / x 1 兲 d⍀ ⍀(C) x 3
(6.12)
Consider the case where the contour is held constant. The right hand side of 共6.12兲 is then constant for any contour C ie, the right hand side is path-area independent rather than path independent 关4,5,6,7兴. The path-area independency of J 1 (s) is limited to small regions around position s on the crack front. If parts of the path are distant from the crack front, then J 1 (s) is influenced by the singular fields of those points neighboring s on the crack front. In comparison, the J -integral is path-area independent in a global sense, ie, the total strength of the singularities of the whole crack front is independent of the surfaces enclosing it 关130,131兴. Early in 1974, Cherepanov 关8兴 recognized that, for each of the three modes of fracture, there is a corresponding J -integral denoted by J m (m⫽I,II,III) and defined by: J m⫽
冕
⌫
m 关 W mn 1⫺ m i j 共 u i / x 1 兲 n j 兴 d⌫
(6.13)
where ⌫ is a contour of vanishing radius normal to the plane x 3 ⫽0 which proceeds in the counterclockwise direcm tion, m i j and u i are the mode m stresses and displacements, i j m m and W ⫽ 兰 0 i j d m i j . They are related to the J k -vector by
Fig. 13 (a,b) A cross section in a 3D surface crack provided by Dobbs and Read 关27兴 or Rigby and Aliabadi 关31兴
J 1 ⫽J I ⫹J II ⫹J III
(6.14)
J 2 ⫽⫺2 冑J I J II
(6.15)
6.2 Mode decomposition of 3D mixed-mode cracks via two-state integrals Recently, based on the two-state integral concept, Kim et al 关147兴 proposed a mode decomposition technique for 3D mixed-mode cracks. Since they presented a typical example on how to extend Chen-Shield’s method to treat 3D crack problems, their numerical scheme to obtain individual SIFs of an axisymmetric crack and a 3D mixed-mode crack is described below.
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Kim et al 关147兴 started with the 3D M - and J -integrals:
冕冉
M⫽
A
冊
1 Wx i n i ⫺T i u i, j x j ⫺ T i u i dA, 2
共 i, j⫽1,2,3 兲
(6.16) where A indicates a surface. For axisymmetric cracks, Eq. 共6.16兲 can be rewritten as 共Kuo 关150兴兲 M ⫽2
冕冉 C
冊
where subscripts ␣ and  denote components in the r⫺z 共or x 1 ⫺x 2 ) plane 共see Fig. 14a,b兲, and C encloses the crack tip 共it may be taken to be either C 1 or C 2 in Fig. 14a,b兲. In the absence of mode III or torsion, the J - and M -integrals are related by 关150兴: (6.18)
MI
1⫺ 2 KI E
M II
1⫺ 2 K II E
2
⫽ 2 r 2c
J II ⫽
2 r 2c
(1)
(6.20)
(2)
where J and J are defined for the target problem and the auxiliary field, respectively, and J (1,2) is given by 1 r 2c
冕
C
(1,2) Fm n m x 1 ds
(6.21a)
(1,2) (2) (2) (1) ⫽W (1,2) x i ␦ im ⫺ 共 (1) Fm ik u i, j ⫹ ik u i, j 兲 x j ␦ km
1 (1) (2) (2) (1) ⫺ 共 kn u k ⫹ kn u k 兲 ␦ nm 2
(6.21b)
for which the two-state strain energy density is: (2) (2) (1) W (1,2) ⫽C i jkl u (1) i, j u k,l ⫽C i jkl u i, j u k,l
(6.21c)
Finally, one has
where J I⫽
J (0) ⫽J (1) ⫹J (2) ⫹J (1,2)
Here, (6.17)
J⫽J I ⫹J II
and an auxiliary field denoted by ‘‘共2兲.’’ Then the J -integral for the resulting state has the following form:
J (1,2) ⫽
1 Wx ␣ n ␣ ⫺T ␣ u ␣ ,  x  ⫺ T ␣ u ␣ x 1 ds, 2
共 ␣ ,  ⫽1,2兲
Appl Mech Rev vol 56, no 5, September 2003
(6.19a)
2
⫽
(6.19b)
In the above relations, r c 共or x 1 ) represents the shortest distance from the z -axis to the crack tip. Following Chen and Shield 关84兴, Kim et al 关147兴 considered two independent elastic states of a penny-shaped crack: the field of the target problem denoted by superscript ‘‘共1兲’’
J (0) ⫽
共 1⫺ 2 兲 兵 关 K I(1) ⫹K I(2) 兴 2 ⫹ 关 K II(1) ⫹K II(2) 兴 2 其 E
J (0) ⫽J (1) ⫹J (2) ⫹
(6.22a)
2 共 1⫺ 2 兲 (1) (2) 兵 K I K I ⫹K II(1) K II(2) 其 E (6.22b)
J (1,2) ⫽
2 共 1⫺ 2 兲 (1) (2) 兵 K I K I ⫹K II(1) K II(2) 其 E
(6.22c)
The J -integral shown in Eqs. 共6.21a兲 and 共6.22c兲 deals with the interaction term only, which is very useful for solving mixed-mode, penny-shaped crack problems. The J (1,2) integral is related to the details of stresses and deformation at the crack tip. However, because of the path-independence nature of this integral, it may be evaluated in a region away from the tip. Equations 共6.21a兲 and 共6.22c兲 provide sufficient information for determining SIFs of a mixed-mode, penny-shaped crack, when a proper known auxiliary field is introduced. Let superscript (2a) denote the solution for an auxiliary elastic field, wherein the body under consideration is in the state of (2a) ⫽0. Then mode I deformation only, ie, K I(2a) ⫽0 and K II Eq. 共6.22c兲 can be simplified as J (1,2a) ⫽
2 共 1⫺ 2 兲 (1) (2a) 兵KI KI 其 E
(6.23)
Consequently, for the target problem, one has K I(1) ⫽
J (1,2a)
(1) ⫽⫾ K II
Fig. 14 (a,b) A mode decomposition for a 3D mixed-mode crack provided by Kuo 关234兴 or Kim et al 关231兴
(6.24a)
2 冑J (2a) 共 1⫺ 2 兲 /E
冑
E 1⫺ 2兲 共
冑
J (1) ⫺
关 J (1,2a) 兴 2 4J (2a)
(6.24b)
Note that the integrals J (1) , J (2a) and J (1,2a) have to be calculated accurately for proper evaluation of 共6.24a,b兲. For a given crack geometry and loading condition, the first two are easily calculated and the last one can be achieved by inte-
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Chen and Lu: Developments and applications of invariant integrals
grating 共6.21a兲 along a properly selected band in the far field utilizing the domain integral expression 共Li et al 关151兴, Nikishkov and Atluri 关128,129兴兲. In the same sprit, a 3D formulation for mixed-mode I, II, and III fracture can be developed 共see Figs. 14a,b兲. Let J(s) denote the EER associated with the self-similar crack growth on the crack front. As the cylindrical surface A shrinks to the crack front, Moran and Shih 关152兴 found: ⫺lim⌫→0 兰 St l k H k j m j dA J共 s 兲⫽ 兰 Lc l k k ds
(6.25)
where ⌫ is the circular contour on the r⫺z plane that represents the cross section of the shrinking surface S t 共see Fig. 14b兲; s denotes the coordinate along the crack front of a 3D crack; l k , short for l k (s), is the crack advance vector 共see Fig. 14b兲. With 共6.20兲, it follows that J (1,2) ⫽
⫺lim⌫→0 兰 St l k H (1,2) k j m j dA 兰 Lc l k k ds
(6.26)
(2) (2) (1) (1,2) ␦ k j ⫺ 共 (1) H (1,2) k j ⫽W i j u i,k ⫹ i j u i,k 兲
(6.27)
where
J (1,2) ⫽
冉
冊
2 共 1⫺ 2 兲 1 (1) (2) K I(1) K I(2) ⫹K II K II ⫹ K (1) K (2) E 1⫺ III III (6.28)
Equations 共6.26兲 and 共6.28兲 provide sufficient information for determining individual stress intensity factors of a mixedmode crack when two known auxiliary solutions are introduced, which are denoted by superscripts ‘‘(2a)’’ and (1) (1) , and K III ‘‘(2b),’’ respectively. The unknowns K I(1) , K II (1,2a) can then be straightforwardly determined from J(1), J , J (2a) , J (1,2b) , and J (2b) . 7 APPLICATION OF INVARIANT INTEGRALS IN NANO-STRUCTURES This final section is devoted to the recent development of invariant integrals in nano-structures and nano-materials 共as another kind of functional materials兲. Micro-electromechanical systems (M EM S) and advanced materials at the nanometer scale are becoming increasingly popular for a wide range of applications. A detailed review on this topic has been given by Ortiz 关135兴. In this section, the focus is placed on the extension of invariant integrals to nanostructures. Chiu and Gao 关136兴 and Gao 关137兴 applied the J -integral to nano-structures with defects 共singularities兲. Chiu and Gao 关136兴 clarified the role of J 2 -integral in rough thin films, whereas Gao 关137兴 studied a heteroepitaxial thin film, ie, a very thin single-crystal layer of one material deposited onto a single-crystal substrate of another material having the same crystalline structure but different lattice spacing. A comprehensive review of this subject can be found in Gao and Nix 关138兴. Typically, the film thickness ranges from a few nanometers to a few micrometers. As the film evolves from a perfectly flat layer into an undulating configuration, the interface monolayer, initially fully covered by the film material, may eventually become partially exposed after the sur-
545
face touches the interface 共Fig. 15兲. During the non-exposure stage, it is found that thin film surface evolution is closely related to the conservation law of the J -integral at the macroscopic scale, as 关137兴: During the evolution, the projected average of strain energy density along the surface remains constant. Mathematically, this can be written as
具 w 共 sur f ace 兲 典 ⫽w O
(7.1)
where the operator 1 2 L→⬁ 共 2L 兲
具 ¯ 典 ⫽ lim
冕 冕 L
L
⫺L
⫺L
共 ¯ 兲 dxdz
(7.2)
defines the projected average of a quantity over the entire x⫺z 共film兲 plane. The constant, ie, the strain energy density, 3D 2D for a biaxially stressed film and w O in w O is denoted as w O a 2D plane-strain problem 关136,137兴. Equation 共7.1兲 involves the application of the J -integral to thin films. Figure 15 shows the contour of the J -integral selected for the film/substrate system. According to the coordinates chosen by Gao 关137兴, the integral should be rewritten in a 3D space, as J⫽
冖冉 ⌫
wdxdz⫺n j i j
ui ds y
冊
(7.3)
Here, it should be emphasized that there are three components of the J k -vector in a 3D space: J k⫽
冖冉 ⌫
wn k ⫺n j i j
ui ds xk
冊
共 k⫽1,2,3 兲 ,
(7.4)
where ⌫ refers to a closed surface, and J 1 ⬅J(x 1 ⫽x,x 2 ⫽y,x 3 ⫽z), see Fig. 15. As in the 2D cases, the J -integral in the 3D space vanishes along any closed surface that encloses no elastic singularities. To prove 共7.1兲, it is convenient to consider first a doubly periodic surface with period 2L and then approach an arbitrary rough surface by extending L to infinity. This is reminiscent of the usual procedure of extending a Fourier series to a Fourier integral. According to the discussion in Section 4, it is critical to choose a suitable integral contour to avoid
Fig. 15 Stress-driven surface evolution in a heteroepitaxial thin film structure provided by Chiu and Gao 关222兴, Gao 关223兴, and Gao and Nix 关224兴
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Chen and Lu: Developments and applications of invariant integrals
the breakdown of the conservation law. Gao 关137兴 applied the integral along a closed contour consisting of a tractionfree rough surface ⌫ sur f ace , two horizontal plane surfaces on ⫹ ⫺ the ⫹ and ⫺ sides of the interface, ⌫ int and ⌫ int 共see similar treatment of an interface by Smelser and Gurtin 关261兴, and Zhao and Chen 关102兴兲, and a plane surface at infinite depth ⌫ ⬁ , all within the period (⫺L⬍x⬍L,⫺L⬍z⬍L). To form a closed contour, a number of vertical plane surfaces parallel to the y -direction can be added, but these surfaces make no net contribution to J because they cancel each other due to their periodicity and opposite senses. Figure 15 illustrates the 2D configuration of such a contour. The contour segment ⌫ ⬁ makes no contribution to J because the stress field vanishes far away from the film. The contribution to J by the traction-free surface is ⫺
冕 冕 L
L
⫺L
⫺L
wdxdz
(7.5)
and that by the interface contours is 共see the same treatment of Zhao and Chen 关102兴兲: ⫹ ⫺ J 共 ⌫ int 兲 ⫹J 共 ⌫ int 兲
⫽
冕 冕冋 L
L
⫺L
⫺L
共 w ⫹ ⫺w ⫺ 兲 ⫺n j ⫹ ij
册
⫺ 共 u⫹ i ⫺u i 兲 dxdz y
(7.6) Equation 共7.6兲 can be simplified as ⫹ ⫺ J 共 ⌫ int 兲 ⫹J 共 ⌫ int 兲 ⫽ 共 2L 兲 2 w 0
⫹0
冕 冕 L
L
⫺L
⫺L
⫹ 关u⫺ x / x⫹ u x / z 兴 dxdz
(7.7) where the second term on the right hand side vanishes due to the periodicity of the displacement field. The conservation of J requires
冕 冕 L
L
⫺L
⫺L
wdxdz⫽ 共 2L 兲 2 w 0
(7.8)
Dividing both sides of 共7.8兲 by (2L) 2 and then letting the length L approach infinity directly yields Eq. 共7.1兲. We thus conclude that the J -integral still plays an important role in nanomechanics, and there is need for future studies on the usefulness of the M - and L -integrals in small structures, especially in applications of the so-called quasicontinuum method 共see Ortiz 关135兴兲.
8 CONCLUDING REMARKS The main conclusions of this review article are summarized as follows: 1兲 Although the basic concepts of invariant integrals have been established and extensively studied over the past four decades, new and important applications are currently being found in solids with multiple interacting cracks or mircocrack damage, in modern functional materials, and in nano-structures.
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2兲 The energy momentum tensor can be straightforwardly extended to establish invariant integrals in electromagneto-elastic or other linear functional materials. It applies a general law of continuum physics that exists in 3D space for any material behavior and any form of energy, eg, finite deformations. However, the resulting invariant integrals always represent energy release rates, from which many crack tip parameters such as SIFs and EDIF cannot be distinguished if there were no other betterments. This appears to be the major drawback of using the energy momentum tensor. This drawback can be overcome by using two-state conservation laws, which provide powerful and simple tools to calculate distinct values of SIFs for different fracture modes. The extension of these laws to treat linear functional materials is needed in the futhure. 3兲 The Bueckner work conjugate integral can be used to establish useful weight functions for solving complicated problems such as crack kinking or deflection in homogeneous piezoelectric materials, and interface cracks in dissimilar piezoelectric materials. However, mathematically, this method is restricted to quadratic energy, and requires complete Williams EEF in the considered material as well as its reduced pseudo orthogonal property, which may significantly restrict its application in functional materials. This property has nevertheless been proved inherent and universal, providing a simple yet very powerful tool for constructing path-independent integrals as well as weight functions that can be used to distinctly calculate dominant fracture parameters, without any special treatment of the crack-tip region. Moreover, this property leads to the important conclusion that the J - and M -integrals are merely two special cases of the Bueckner integral: That is, by using two different complementary states, either J or M can always be reduced from the Bueckner integral. Therefore, besides the energy momentum tensor, Betti’s reciprocal theorem provides another way to establish path-independent integrals and weight functions in functional materials with quadratic energy. 4兲 The roles played by invariant integrals in the fracture of functional materials, eg, piezoelectrics, are quite different from those of purely mechanical materials. At present, the commonly recognized opinion is that direct applications of invariant integrals as fracture criteria are not generally fair unless some betterment is made 关61–70兴. Indeed, these theoretical treatments introduce features which are hard to justify 关209兴 because in the open literature different experimental data confimed different fracture criteria. Further investigations are needed to clarify the existing controversal results and the detailed relations between invariant integrals and fracture criteria. 5兲 The new conservation laws of the J k -vector for solids with microcracking damage, ie, Eqs. 共4.1a,b兲, provide some new concepts of invariant integrals that are quite different from their original definitions 关3,4,9–11兴. These laws not only hold in infinite solids but also in finite solids, provided that the external boundaries are treated as a special kind of discontinuities 共ie, the interface between
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air and solid, or between rigid solid and microcracking solid兲 关120兴. It is proved that only in this way could the analysis of invariant integrals be extended to finite microcracking solids. Besides the well-known continuum damage mechanics 共CDM兲 theory based on effective elastic moduli 共Kachanov 关231,232,235–237兴兲, the M -integral provides a new, perhaps more objective description of microcrack damage. Physically, the M -integral is simply twice the change in total potential energy due to the formation of microcracks, ie, M ⫽2U 关119,120兴. Thus, from phenomenological point of view, the M -integral represents energy dissipation due to microcracking and, hence, may be used as a quantitative damage measure 关119,120兴. The simple relation between the M - and L -integrals, L ⫽⫺ 12 M␥ , holds even for strongly interacting multiple cracks. Thus, the two integrals are not independent, although they represent two different kinds of energy release rates 关9,10兴. New conservation laws of the J k -vector have been established for microcracking piezoelectric materials. Detailed manipulations and numerical results verify their validity under mechanical-electric coupling situations. Although mechanical and electric quantities exhibit totally different behaviors, both can be treated in a similar way by introducing the concepts of generalized stress 共corresponding to electric displacement兲 and generalized displacement 共corresponding to electric potential兲. Substantial further investigations of path-independent integrals are warranted, especially in modern functional materials with defects, eg, piezoelectrics, magnetostrictives, electrostrictives, and shape memory alloys, as well as in nano-structures. Only after doing so, the physical significance and practical applications of invariant integrals for these materials could be recognized and clearly understood. Therefore, the purpose of this article may be considered as reviewing new developments in this old topic and motivating further investigations in the new century.
ACKNOWLEDGMENTS This work was supported partially by the Royal Society KC Wong Award 共2000兲 and the Royal Society Award 共2002兲 for YHC’s visit to University of Cambridge, UK, partially by the Chinese State Key Laboratory Foundation for TJL’s visit to Xi’an Jiao-Tong University, China, and partially by the National Science Foundation of China. YHC wishes to thank Professor John Willis of Cambridge University for constructive discussions. Finally, both authors wish to thank the referees for their critical comments, which have helped to substantially improve the quality of the original manuscript. REFERENCES 关1兴 Eshelby JD 共1956兲, The Continuum Theory of Lattice Defects, Solid State Physics, F Seitz and D Turnbull 共eds兲, Academic Press, New York, 3, 79–141. 关2兴 Eshelby JD 共1970兲, The Energy Momentum Tensor in Continuum Mechanics, Inelastic Behavior of Solids, MF Kanninen 共eds兲, McGrawHill, New York.
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Yi-Heng Chen is a Professor in the School of Civil Engineering and Mechanics at Xi’an Jiaotong University, PR China. He received a D.Eng degree from Xi’an Jiaotong University and a PhD degree from Kyushu University, Japan. He has published over 130 academic journal papers and one book (Kluwer Academic Publishers). His research interests include Fracture and Damage Mechanics, Smart Materials and Structures, Laminated Composite Materials, Microcrack Shielding Effects in Functional Materials, and Conservation Laws in Solid Mechanics. He was an Alexander von Humboldt Fellow of Germany, a Research Fellow at Nagoya Institute of Technology and at Kyushu University of Japan, a DAAD Fellow at Magdeburg University, and a Research Fellow at the Max-Planck Institute at Stuttgart, Germany. He was also a Royal Society Research Fellow at Cambridge University. He became a full Professor in 1990 and has been appointed the Dean of School of Civil Engineering and Mechanics at Xi’an Jiaotong University since 1995.
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Chen and Lu: Developments and applications of invariant integrals
Appl Mech Rev vol 56, no 5, September 2003
Tian Jian Lu is a Reader in Micromechanics at Cambridge University Engineering Department. He obtained a PhD from the University of Hong Kong in deformation and fracture, and another PhD from Harvard University in advanced materials. His current research projects (on Micromechanics, Design Optimization, Smart Materials/Structures, MEMs, Thermal Management, and Noise Control) are sponsored by the UK Government, Royal Society, US Office of Naval Research, and Industry, supporting more than 10 graduate students and post-doctoral researchers. He has held Guest Professorships at several leading Chinese universities, a Visiting Professorship at Princeton University from 2000 to 2001, a Visiting Fellowship at the Max-Planck Institute, Stuttgart, Germany, and consults extensively for the industry. He has published about 90 peer-reviewed journal papers in the field of materials, design and process selection, acoustics, and heat transfer. (Personal website: www2.eng.cam.ac.uk/⬃tjl21/)
