Advances in Applied Mechanics Volume 38
Editorial Board Y. C. FUNG DEPARTMENT OF BIOENGINEERING UNIVERSITY OF CALIFORNIA, SAN DIEGO LA JOLLA, CALIFORNIA PAUL GERMAIN ACADEMIE DES SCIENCES PARIS, FRANCE C.-S. YIH (Editor, 1971-1982) JOHN H. HUTCHINSON (Editor, 1983-1997)
Contributors to Volume 38 PAOLO MARIA MARIANO CHANG-LIN TIEN PIN TONG JIAN-GANG WENG THEODORE YAOTSU WU TONG-YI ZHANG MINGHAO ZHAO
A D V A NC E S IN
APPLIED MECHANICS Edited by Erik van der Giessen
Theodore Y. Wu
DELFT UNIVERSITY OF T E C H N O L O G Y DELFT, THE N E T H E R L A N D S
DIVISION OF E N G I N E E R I N G AND APPLIED SCIENCE CALIFORNIA I N S T I T U T E OF T E C H N O L O G Y PASADENA, CALIFORNIA
VOLUME 38
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Contents CONTRIBUTORS PREFACE
vii ix
Multifield Theories in Mechanics of Solids Paolo Maria Mariano I. II. III. IV. V. VI. VII. VIII. IX.
Introduction Configurations and Balance of Interactions Elastic Materials with Substructure Balance in Presence of Discontinuity Surfaces Constitutive Restrictions Evolution of Defects and Interfaces in Materials with Substructure Crack Propagation in Materials with Substructure Latent Substructures Examples of Specific Cases Acknowledgments References
2 9 26 33 38 42 53 68 73 88 88
Molecular Dynamics Simulation of Nanoscale Interfacial Phenomena in Fluids Chang-Lin Tien and Jian-Gang Weng I. II. III. IV. V. VI. VII. VIII.
Introduction Molecular Dynamics Simulation Techniques Liquid-Vapor Interfaces Liquid-Liquid Interfaces Liquid-Solid Interfaces Three-Phase Systems Other Interfacial Phenomena Concluding Remarks Acknowledgments References
96 97 103 122 130 136 139 140 141 141
Contents
vi
Fracture of Piezoelectric Ceramics Tong-Yi Zhang, Minghao Zhao, and Pin Tong I. II. III. IV. V. VI. VII. VIII. IX. X.
Introduction Basic Equations Two-Dimensional Electroelastic Problems and Stroh's Formalism Piezoelectric Dislocation and Green's Function Conductive Cracks Interface Cracks Three-Dimensional Electroelastic Problems Nonlinear Approaches Experimental Observations and Failure Criteria Concluding Remarks Acknowledgments References
148 152 162 186 199 209 220 236 255 274 279 279
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion Theodore Yaotsu Wu I. II. III. IV. V. VI. VII. VIII. IX.
Introduction Subdivisions of Hydrodynamic Theories for Aquatic and Aerial Locomotion Resistive Theory of Aquatic Locomotion Classical Slender-Body Theory of Fish Locomotion A Unified Approach to Nonlinear Theory of Flexible Lifting-Surface Locomotion A Unified Nonlinear Theory of Two-Dimensional Flexible Lifting-Surface Locomotion On Experimental Differentiation between Thrust and Drag in Fish Locomotion Scale Effects in Energetics of Aquatic Locomotion Conclusion and Outlook Acknowledgments References
291 296 300 301
333 338 347 350 350
AUTHOR INDEX
355
SUBJECT INDEX
363
314 316
List of Contributors
Numbersin parenthesesindicatethe pageson whichthe authors' contributionsbegin. PAOLO MARIA MARIANO (1), Dipartimento di Ingegneria Strutturale e Geotecnica, Universith di Roma "La Sapienza," 00184 Rome, Italy CHANG-LINTIEN(95), Department of Mechanical Engineering, University of California, Berkeley, California 94720 PIN TONG (147), Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China JIAN-GANGWENG(95), Department of Mechanical Engineering, University of California, Berkeley, California 94720 THEODORE Y. WU (291), Division of Engineering and Applied Science,
California Institute of Technology, Pasadena, California 91125 TONG-YIZHANG(147), Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China MINGHAO ZHAO (147), Department of Mechanical Engineering, Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
vii
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Preface
The basic intent of Advances in Applied Mechanics, as laid down by founding editors Theodore von K~rm~n and Richard von Mises, is to serve as a forum for informative, expository, and up-to-date accounts of an area of mechanics, for experts and nonexperts alike, interested in their own as well as unrelated fields. Our first concern is that they may be read with interest, profit, stimulation, and perhaps even enjoyment, as put by our former editor, Chia-Shun Yih. This volume of the serial consists of four chapters addressing a variety of topics of basic interest and current activity. An important driving force for advances in solid mechanics is the need to incorporate better and more detailed descriptions of the material behavior. Because engineered materials in particular owe their properties to internal microstructure, much work has been and still is focused on coupling constitutive models with microstructural descriptions. Classically, this is done by introduction of appropriate internal variables, but this is not always sufficient. Therefore, more intricate continuum theories that couple the motion of the solid to other field variables have been developed in recent years. The article by Mariano gives a unified formulation of such multifield, and often nonlocal, theories. It gives a formal treatment of, for example, mechanical stress and configurational forces that drive microstructure evolution. Although written with a certain mathematical rigor, the article is both motivated by practical problems and illustrated by particular known descriptions. Recent advances in studies of nanoscale interfacial phenomena in fluids have opened up a new frontier of fundamental flow science and technology. The exceedingly short lengths and time scales involved present many challenges to experimental and numerical approaches; this is especially true for interfacial phenomena because the surface effects dominate due to the large surface-to-volume ratio in nanoscale configurations. The chapter by Tien and Weng provides an expository survey of the recent development of molecular dynamics (MD) simulations of these phenomena involving all sorts of interfaces, including those consisting of ix
x
Preface
different liquids, vapors, solids, and three-phase systems, in which the physical effects of surface tension, surfactants, diffusive transport, thermal boundaries, and sonoluminescence can all play a role. Through these cases, the success and further development of MD simulation are extensively discussed. Along another frontier, piezoelectric materials made in the form of polycrystalline ceramics are facing rapidly expanded applications to the manufacture of microdevices used in smart structures, microelectronics, and microelectromechanical (MEM) systems. Piezoelectric ceramics have remarkable properties in being chemically inert, immune to ambient conditions, and very quick in conversion between mechanical and electrical energy along mechanical and electrical axes that can be precisely oriented in relation to the arbitrary shape of the ceramics through manufacturing. The article by Zhang, Zhao, and Tong reviews the recent advances in our understanding of the mechanical and fracture properties of piezoelectric ceramics for their significance and importance to mechanical-electrical energy conversion, a process that intrinsically involves comprehensive constitutive and thermodynamic relations, as well as for fulfilling some practical applications to high-tech development. The multidisciplinary subject of aquatic and aerial animal locomotion has renewed strong interest in at least two aspects: achieving a nonlinear theory modeling the hydrodynamic mechanisms underlying the locomotion as observed, and further applications, with the control mechanics incorporated, to developing new robotics, possibly with a biomimetic approach. For both objectives, success in the first is paramount. The article by Wu surveys the advances in hydrodynamic theories modeling aquatic and aerial locomotion at low, intermediate, and high Reynolds numbers, performed by elongated animals and those using lifting surfaces of large aspect ratio for undulatory propulsion. In addition, a new nonlinear theory for evaluating propulsion by a two-dimensional flexible lifting surface moving along an arbitrary trajectory and with motions of arbitrary amplitude is presented. It is intended for general applications to lifting surfaces of large aspect ratio and further extension to three-dimensional configurations. With the publication of the present volume of this serial, it gives us great pleasure to extend our warmest welcome to Professor Hassan Aref of the University of Illinois at Urbana-Champaign, a very distinguished fluid dynamicist and a very highly regarded leader in our profession of Applied Mechanics. Aref will hereby take over T.Y.W.'s responsibility of the co-editorship of Advances in Applied Mechanics. TYW wishes to acknowledge with deep appreciation the benefits of working with John Hutchinson earlier and with Erik van der Giessen more recently for their rewarding editorial teamwork. Warm thanks are due from TYW to the
Preface
xi
members of the present Editorial Board for their valuable help and to the authors for contributing their stimulating articles to this serial during the years of his tenure as co-editor, and to Academic Press for its splendid cooperation. His years with the serial have naturally nourished in him an attachment to it, and seeing that it is now in excellent hands is to him a gratifying comfort. ERIK VAN DER GIESSEN AND THEODORE Y. W u
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A D V A N C E S IN A P P L I E D M E C H A N I C S , V O L U M E 38
Multifield Theories in Mechanics of Solids* PAOLO MARIA MARIANO Dipartimento di lngegneria Strutturale e Geotecnica,' Universitgt di Roma "La Sapienza," 00184 Rome, Italy
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Structure of This Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 2 5
II. Configurations and Balance of Interactions . . . . . . . . . . . . . . . . . . . A. Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Measures of Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Balance of Interactions from the Invariance of Outer Power . . . . E. Effects of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16 21
III. Elastic Materials with Substructure . . . . . . . . . . . . . . . . . . . . . . . . A. Variational Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . B. S o m e Properties of Lagrangian Densities . . . . . . . . . . . . . . . . . . C. Influence of the Substructure on the Decay of Elastic Energy . . .
26 26 27 30
IV. Balance in Presence of Discontinuity Surfaces . . . . . . . . . . . . . . . . A. Interfaces: G e o m e t r i c Characterization . . . . . . . . . . . . . . . . . . . B. Balance at Discontinuity Surfaces . . . . . . . . . . . . . . . . . . . . . . .
33 33 35
V. Constitutive Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Constitutive Restrictions in the Bulk . . . . . . . . . . . . . . . . . . . . . B. Constitutive Restrictions at Discontinuity Surfaces . . . . . . . . . . . VI. Evolution of Defects and Interfaces in Materials with S u b s t r u c t u r e . . A. Configurational Forces in the Bulk . . . . . . . . . . . . . . . . . . . . . . B. Configurational Forces on a Discontinuity Surface . . . . . . . . . . . VII. Crack Propagation in Materials with Substructure . . . . . . . . . . . . . . A. Kinematics of Planar M o v i n g Cracks . . . . . . . . . . . . . . . . . . . . B. Balance of Standard and Substructural Interactions at the T i p . . . C. Effects of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Tip Balance of Configurational Forces . . . . . . . . . . . . . . . . . . . . E. C o n s e q u e n c e s of the Mechanical Dissipation Inequality . . . . . . . E Driving Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. A Modified Expression of J Integral . . . . . . . . . . . . . . . . . . . . . H. Energy Dissipated in the Process Zone . . . . . . . . . . . . . . . . . . .
9 9
11 12
38 39 40 42 43 46 53 54 56 57 59 61 63 65 66
*To M. M., for simple and, at the same time, complicated reasons.
ISBN 0-12-002038-6
ADVANCES IN APPLIED MECHANICS, VOL. 38 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISSN 0065-2165/01 $35.00
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VIII. Latent Substructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Second-Gradient Theories as Special Cases of Latent Substructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Examples of Specific Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Material with Voids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Two-Phase (or Multiphase) Materials . . . . . . . . . . . . . . . . . . . . C. Cosserat Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Micromorphic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Nematic Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E Ferroelectric Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Microcracked Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68 70 73 74 75 76 78 80 81 83
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
I. Introduction A. GENERAL INTRODUCTION The term multifield theories indicates the wide range of models in which some graphic fields must be introduced to describe the influence of material substructures on the gross mechanical behavior of solids. In a certain sense these fields (order parameters, also called phase fields, microstructural fields, microdisplacements, microdeformations, etc., in different special cases) are models of the material structure. Solidification of metal alloys and their possible shape memory, damage states and evolution, and the influence of long chains of macromolecules on the behavior of polymers are examples of physical phenomena of great interest in engineering practice that can be analyzed fruitfully with the help of multifield theories. Usually, in continuum mechanics only the placement within the Euclidean space is assigned to each material patch (volume element), then changes in relative placements are evaluated to measure the crowding and the shearing of material patches (i.e., the deformation). In this way, however, the features of material texture, or, more generally, substructure, are overlooked. The starting idea of multifield theories is to assign to each material patch P at least one pair (x, qo) in which x(P) is the placement of P within the Euclidean space and qo(P) (order parameter l) furnishes information on the substructural configuration of the patch. i Of course, the word order is only conventional, qo can also describe disordered arrangements of macromolecules or crystalline grains. The term orderparameter arises from statistical physics,
Multifield Theories in Mechanics of Solids
3
R e m a r k 1 Many choices of qo can be made; some of them are considered natural, whereas others are of convenience. They depend on the physical circumstances and must be specified each time. 9 The simplest choice is to consider the field qo to be scalar valued. For example, qo may represent the void volume fraction in porous solids or the volume fraction of a certain material phase in a two-phase material, as in the case of austenite-martensite mixtures. Such a choice of the order parameter is not unique for this physical situation; it may be not sufficient, for example, to describe in some detail the directional distribution of grains of austenite or martensite at each point, and tensor-valued order parameters must be introduced. A scalar-order parameter may be also used in the case of mixtures of two fluids or to describe solidification phenomena. 9 qO can be a vector. Liquid crystals are the classical example of bodies modeled by such a special choice of the order parameter. Other situations in which the vector choice is made are direct models of rods and shells. A shell can be represented roughly speaking by the middle surface and a field of unit vectors orthogonal to it in the reference configuration (a Cosserat surface). Analogously, triads of vectors may be used to represent the behavior of cross sections of rods or of stratified rocks, such as gneiss. Vectororder parameters are also useful tools in describing the mechanics of defective crystals (they can be identified with the optical axes of the crystal) or microcrack systems (in this case, they may represent vectors orthogonal to plane microcracks or the perturbation of the displacement field induced by the presence of such defects), ferroelectrics, and magnetostrictive solids. 9 If the material patch is characterized by large molecules undergoing homogeneous deformations, second-order tensor-valued order parameters may be used, a choice that also describes the dipole approximation of some distribution of directional data; for example, the distribution of microcracks. Moreover, such a second-order tensor can represent the Nye's tensor in dislocated continua. 9 Distributions of microcracks can be also described by their quadrupole approximation. A fourth-order tensor must be introduced with the role of order parameter.
in which Landau introduced order parameters to study second-order phase transitions consisting of abrupt changes of symmetry in solids.
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Many other examples can be made, and some are presented in Section IX in a more detailed way. The order parameters quoted in the previous remark are elements of some finite dimensional manifold, here indicated with AA and usually considered compact and without boundary..A4 can be also infinite dimensional, in which case the order parameter field assigns to each material patch an entire distribution function that can represent the geometry of the material substructure or can be the distribution (not necessarily canonical) of different levels of energy within the patch. In the following discussion, I refer the developments presented in this article to the case in which .A/[ is finite dimensional. The characteristic features of each special model depend on the choice of M , and hence on the mathematical properties of it. The physical meaning (or, for instance, interpretation) of these properties should be specified each time, although they are often considered to be convenient devices only. For example, the metric on M determines the quadratic part of the kinetic energy (if any) associated to the order parameter, and therefore to the substructure of the material. In the case of crystalline materials (e.g., such a prominent role of kinetic energy can be recognized only at very high frequencies; however, although the case is rare, such a kinetic energy should be considered in these regimes and determines the metric of M . Basically, the order parameter is considered to be an observable quantity. An external spatial observer must take two different measures to evaluate both the position of each material patch and information on its substructure. In this way, x and ~ together characterize the physical configuration of the solid. Interactions are associated with ~: They are substructural interactions and depend on the nature of the material substructure. These interactions develop explicit power in the rate of the order parameter and perhaps of its gradient, and must be balanced. Consequently, new balance equations arise in addition to those of Cauchy and represent the balance of some sort of generalized momentum and moment of momentum. The latter balance implies an expression of the skew part of Cauchy stress in terms of substructural measures of interaction. The representation of the substructural interactions is a delicate problem. When the gradient of ~o can be evaluated in a covariant way through a connection (hopefully with a physical significance) on A/l, these interactions can be represented by appropriate tensors called microstress and self-force. This terminology is conventional only and evokes special situations in which these tensors are really "perturbations" (in some sense) of the macroscopic stress tensor and also represent some kind of internal forces. In addition, the question of the connection (by which the covariant X7~ can be expressed) is delicate from a conceptual point of view. There are situations in which a physically significant connection can clearly be recognized (e.g., for
Multifield Theories in Mechanics of Solids
5
nematic liquid crystals); however, there are other situations in which this is not so. The choice of the connection, in fact, influences not only the explicit representation of the gradient of r but also the representation of the power. When many connections can be defined indifferently, it is necessary to require the invariance of the power with respect to their choice. Such a requirement allows one to obtain results analogous (in terms of the structure of the balance equations) to the results assured by the existence of a natural and physically significant connection. In general, however, the interactions are represented by general functionals. This situation also occurs when nonlocal effects due to material substructure are considered in both time (memory) and space. Nonlocality can be represented by integrals in time or space. When these functionals must be introduced, retardation (memory) or myopia (space) theorems help develop them (to within some material constant) in terms of differential operators of the fields involved. In this case, one accounts for weak nonlocalities only. Alternatively, one can assign to each material patch P only its placement x(P) in the three-dimensional Euclidean space and decide to introduce internal variables describing microstructural effects. These variables are nonobservable objects by definition; no balance equations are associated with them. The derivatives of the free energy with respect to the internal variables (and, possibly, with respect to their gradients) are in fact not genuine interactions, but are rather only affinities that must satisfy only the second law of thermodynamics, and need not be balanced because they do not develop explicit mechanical power. However, models with internal variables can be derived from multifield theories by appropriate internal constraints. In this case, the substructure becomes latent. Only a possible kinetics (evolution rule) is associated to it. From a mathematical point of view, the assignment of a kinetics to qp, without considering the balance of genuine interactions, is tantamount to take some initial value of qo--say, corresponding to some point qo* on M - - a n d to give a rule selecting elements of the tangent space of M at qa*. Finally, the order parameter field can be chosen to be a stochastic field taking values on the manifold M . In this way, not only the descriptor of the substructure is associated with each material patch, but also the probability that such substructure is really present.
B. STRUCTURE OF THIS ARTICLE The aim of this article is to show that the multifield description of continua is a flexible framework to study many physical situations in which the analysis of substructures is important for both practical and theoretical reasons. The analytical
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tools (and, of course, the difficulties that one tackles when using them) appear to be necessary to describe with certain detail the behavior of material substructures. The general theory is presented first, followed by special cases and applications. The description of configurations and the deduction of balance equations for bodies lacking in discontinuity surfaces are in Section II. Section III is dedicated to the special case in which the behavior of the substructure is elastic, in which case the force exerted on an inclusion in the body by the surrounding medium is deduced from an appropriate version of Noether's theorem accounting for the order parameter field. The influence of the substructure on the axial decay of energy in linear elastic cylinders is also discussed. Section IV deals with the derivation of thermomechanical balance at discontinuity surfaces (interfaces) that are endowed by their own measures of interactions. Surface stress, surface microstress, and self-force are defined on the interfaces. Constitutive restrictions arising from a mechanical version the second law of thermodynamics and involving the measures of substructural interactions are treated in Section V. Section VI is dedicated to the analysis of the influence of substructures on configurational forces that drive the evolution of interfaces. The kinetic equation for interfaces is deduced from the balance of configurational forces and is expressed in terms of a generalized expression of Eshelby tensor. In Section VII, special attention is given to the evaluation of the influence of material substructures on macrocrack propagation. In Section VIII, the case in which the substructure becomes latent in presence of appropriate internal constraints is discussed. Finally, Section IX deals with the application of the general theory to special cases.
BIBLIOGRAPHIC NOTE The "Th6orie des corps d6formables" of Cosserat and Cosserat (1909) is the first known historical example of a special case of multifield theories treated systematically, even though the germinal idea was formulated by Voigt (1887). As is well known, Cosserat and Cosserat's point of view consists of considering each material patch as a rigid body (possibly described by its peculiar triad of vectors) that can rotate independently of the neighboring patches. Couple stresses are associated with these additional degrees of freedom 2 and are balanced. In 1958, Ericksen and Truesdell gave new insight to Cosserat and Cosserat's theory. Following the Cosserats, they consider such a theory to be a suitable tool to describe the 2They are considered "additional" degrees of freedom because, in the classic case of Cauchy materials, each point has only three degrees of freedom.
Multifield Theories in Mechanics of Solids
7
mechanics of rods and shells, which they represent as lines and surfaces, respectively, endowed at each point by triads of mutually perpendicular vectors (the triads describe the behavior of sections). Ericksen and Truesdell's (1958) seminal paper constitutes a generalization of the Cosserats' (1909) ideas because they consider such vectors (the order parameters) to be stretchable (for other contributions to the general theory of Cosserat and Cosserats' materials, see also Mindlin, 1965a,b; Toupin, 1964; Truesdell and Toupin, 1960; Aero and Kuvshinskii, 1960; Grioli, 1960; Mindlin and Tiersten, 1963; Marsden and Hughes, 1983; Povstenko, 1994; Epstein and de Leon, 1998; for related computational techniques, see also Simo et al., 1992). Such an approach to the mechanics of elastic structures have been used in many works since 1958 (see, for example, Amman, 1972, 1995; Amman and Marlow, 1993; Green et al., 1965; Green and Laws, 1966; DeSilva and Whitman, 1969, 1971; Ericksen, 1970; Naghdi, 1972; Simo and Vu-Quoc, 1988; Villaggio, 1997; for computational and stability aspects, see also Fox and Simo, 1992; Simo and Fox, 1989; Simo, et al., 1988, 1989, 1990). Various suggestions to adopt the Cosserats' scheme to describe dislocated structures in crystalline solids have been discussed. Triads of vector-order parameters have also been used by Davini (1986) to introduce a continuum theory of defective crystals (see also Davini and Parry, 1991). Within different settings, vector-order parameters have also been used by Ericksen to describe the behavior of macromolecules within a body (1960, 1962a) and to begin the modern continuum theory of liquid crystals ( 1961, 1962b,c, 1991) that has been further developed by Capriz (1988, 1994,) Capriz and Biscari (1994), and Virga (1994). Mindlin (1964) considers each material patch to be an elementary cell ("interpreted as a molecule of a polymer, a crystallite of a polycrystal or a grain of a granular material," p. 51) that can deform independently of the surrounding medium. A second-order symmetric tensor-valued order parameter is assigned to each cell. Such a continuum is usually called micromorphic (see also Grioli, 1960, 1990; Mindlin, 1965b; Mindlin and Tiersten, 1963; Eringen, 1992, 2000). The approaches of Ericksen and Truesdell (1958), Grioli (1960, 1990) Mindlin (1964) and Toupin (1964, 1965) are special cases of continua with affine structure (see Capriz and Podio-Guidugli, 1976, 1977; Capriz et al., 1982). Higher-order micromorphic continua have been introduced by Green and Rivlin (1964) and further discussed by Germain (1973). A proposal of a nonlocal theory of micropolar continua can be found in Eringen (1973, 1976). For further studies, see Wang and Dhaliwal (1993). Scalar-order parameters were used in 1972 by Goodman and Cowin for describing granular flows. In 1979, Nunziato and Cowin used an analogous approach to
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Paolo Maria Mariano
introduce a nonlinear theory of elastic porous materials (further studies concerning this topic can be found in Cowin and Nunziato, 1983; Cowin, 1985; Dhaliwal and Wang, 1994; Nunziato and Walsh, 1978; Diaconita, 1987; Fr6mond and Nicolas, 1990; Mariano and Bemardini, 1998). Scalar-valued order parameters have been also used to describe phenomena of recrystallization (Gurtin and Lusk, 1999), general solid-solid phase transitions (e.g., Colli et al., 1990; Fr6mond, 1987; Fried and Gurtin, 1993, 1994, 1999; Fried and Grach, 1997), solidification phenomena (Anderson et al., 2000), and isotropic damage evolution (Markov, 1995; Fr6mond and Nedjar, 1996; Fr6mond et al., 1999). Anisotropic damage has also been studied from the point of view of multifield theories by Augusti and Mariano (1999; see also Mariano and Augusti, 1998; Mariano, 1999, and references therein) by using tensor- or vector-valued order parameters. Models with scalar-valued order parameters can be considered special cases of materials with "spherical" structure (Capriz and Podio-Guidugli, 1981). On the basis of classic Lagrangian dynamics of systems of particles, a first attempt to construct general framework for continua with substructures, at least in the case of holonomic-order parameters, has been proposed (Capriz and Podio-Guidugli, 1983). Capriz (1985) introduced the concept of latent microstructures, proving that some higher-order gradient theories of continua can be considered to be multifield theories with appropriate internal constraints. In 1989, Capriz proposed a general theory that is useful for establishing order parameter-based models of continua with substructure. This work opened the way to many theoretical questions, some of which are discussed in Capriz and Giovine (1997a, 1997b), Binz et al. (1998), Segev (1994, 2000), Capriz and Virga (1994), and Mariano and Capriz (2001). In 1990, Capriz and Virga adapted Noll's axioms on interactions in continuous bodies to account for self-interactions among microstructures described by order parameters in linear spaces. In principle, one may think that a body with a fine distribution of voids (or vacancies or microcracks) can be obtained from a mathematical point of view as a limit of a sequence of bodies. In other words, one takes the region occupied by the body in some configuration and at each step of the sequence considers different sets of discontinuities, assigned (perhaps) with some rules. Then one calculates the limit of the sequence and accepts the limit region of the Euclidean space obtained as a reasonable picture of the original finely fractured body. These limit processes are analogous to those used to reach the optimal shape of bodies under some optimum conditions (Kohn and Strang, 1986a,b,c). In 1993, Del Piero and Owen showed that an appropriate fabric tensor describing the influence of microcracks on the macroscopic deformation can be obtained as a consequence
Multifield Theories in Mechanics of Solids
9
of limit of sequences of bodies and corresponding deformations. In 2000, Del Piero and Owen showed that even the vector-order parameter describing liquid crystals (according to Ericksen's theory of nematic liquid crystalsmsee Ericksen, 1962b,c; Capriz, 1988, 1996) can be obtained with a procedure involving the limit of bodies. General results on the evolution of discontinuity surfaces and related configurational forces in continua with substructures during solid-solid phase transitions or during the evolutions of defects have been obtained (Mariano, 2000a, 2001).
II. Configurations and Balance of Interactions
A. CONFIGURATIONS As suggested in Section I, the complete placement of a material body B is described by mappings of the type K ' B ~ g3 x Ad
(1)
assigning to each material patch P of B the pair (placement, order parameter). Of course, g3 is the three-dimensional Euclidean point space, whereas Ad is the collection of all possible configurations of the substructure and is considered a finite-dimensional differentiable compact manifold without boundary. The mapping
KE3 "B ~ g3
(2)
assigning to each material patch its placement, defines the apparent configuration, (i.e., a representation of the body in which the substructure is forgotten). Moreover, KM'B
--+ .A4
(3)
defines the order parameter mapping. In this way, each K is a pair (Kg3, K.A4). For future use, an apparent reference configuration Kg3 is considered, with I(E3(B) being indicated by/3. It is assumed that B is a bounded connected regular region 3 (a fit region) of the Euclidean space and is endowed with a coordinate system {X}. 3Details about the minimal topological requirements necessary for/3 to develop continuum theories can be found in Noll and Virga (1988) and Del Piero and Owen (2000). For the purposes of this Section, one may thinkmroughly speaking--/3 as a bounded connected set that coincides with the interior of its closure and is endowed with a surface-like boundary with well-defined unit normal to within a finite number of corners and edges.
10
Paolo Maria Mariano
For each K, the placement field is indicated by x(.) = (Ke3 o
~-l E~ j (.)
(4)
and the order-parameter field is indicated by
More precisely, given X 9 13, x(X) is the placement of a material patch P resting at X in/3, whereas qo(X) is a descriptor of the substructure of the same patch. At each X, the order parameter qo(X) is an element of.M, and .M itself is a nonlinear manifold, in most cases. 4 For each K, it is assumed that 9 x(/3) is also a fit region 9 x(.) is a one-to-one mapping of 13 into ~3 and is continuous and piecewise continuously differentiable 9 the gradient of deformation Vx, indicated with F, is such that detF > 0; that is, x(.) is orientation preserving 9r
is continuous and piecewise continuously differentiable on/3
In this way the space of the configurations is the collection s of pairs (x(.), qo(.)), each one deriving from the corresponding K. s may be endowed by the structure of a manifold; its tangent space is indicated by T ff, whereas the cotangent space is indicated by T*s Example 1 A useful example to illustrate the statements presented previously is the direct modeling of plates. To this aim, consider an orthogonal frame of reference in s namely {Oele2e}, with O the origin, and a compact bounded set A in the plane ele2. If {X*} is a coordinate system in ele2, with X* = XTel + X~e2, the complete reference configuration of the plate is given by the set
X*+~elX*eA,~ 9 -~,~
(6)
where h is the thickness of the plate. In this picture, A is the apparent configuration. 4Some authors embed .M in a linear space and work consequently, using the handy properties of linear spaces. Their reasoning is based on the hypothesis that .M is finite-dimensional. Each finite-dimensional manifold may, in fact, be embedded into an appropriate linear space (Withney's theorem). However, such an embedding is not unique, and the question of what embedding is phisically significant is completely open. The only clear general results on this point of view are in Capriz and Virga (1990).
Multifield Theories in Mechanics of Solids
11
To define the deformed configuration of the plate, two fields must be defined on A 9 x(.)" A ~ ~3, identifying the placement in ~3 of the points of the "midplane" of the deformed plate; it maps A onto a surface x(A) in ,f3 9 qO(') assigning to each X* a vector t belonging to the unit sphere S: = {t 6 ~3llt I -- 1}; i.e., ~(-) 9(A) ~ S 2 Therefore, in this case, A// coincides with S 2. It is assumed that x(A) is a regular surface in ~.3 and (xl • x 2 ) . t > 0
(7)
everywhere in A where the fields are defined. In (7), x l(X*) and x z(X*) are tangent vectors of x(A) at x(X*); in particular, x,1 is the partial derivative of x with respect to X~ and X,e has the analogous meaning. Condition (7) imposes that t is never tangent to x(A) and excludes the physically unreasonable situation of infinite shearing deformations. Finally, provided the validity of (7), the current complete configuration of the plate is given by the set
x(X*)+~tlX*eA,
teS 2,~
-~,
(8)
Because t need not be normal to the deformed middle surface x(A), shear deformations are allowed in this description of plates. Moreover, the assumption that t is of unitary length precludes thickness changes and cannot take into account initial variable thickness. To account for thickness stretch and initial variable thickness, it suffices to require only Itl > 0 (not the more stringent condition Itl = 1).
B. MOTIONS
Motions are time-parameterized curves (xt, qPt) in the space of configurations, and at a given instant t ~ [0, d], the current placement and the order parameter, respectively, associated to each X ~ B are given by x(X, t)
~o(X, t)
(9)
Moreover, the velocity fields are given by ~(-, .)
~(., .)
(10)
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Paolo Maria Mariano
where the dot over x and qo means time derivative. Of course, qb belongs to the tangent space of .A4, and the pair (~(., t), ~(., t)) belongs to Tff. In the following, Vel indicates the set of pairs of fields (:~, ~). Consider two different observers differing by a rotation described by a proper orthogonal tensor Q with corresponding vector q (i.e., Q = exp(oq), (where e is Ricci's three-dimensional permutation indicator). These two observers evaluate two different values of qoBfor example, qOq and qp--connected by the following relation:
I
t.pq -- qg) -~- --~-q q=O
q + o(Iql)
(11)
If a time-parameterized family of rotations q(t) is now considered, inserting q(t) in (11) and evaluating the time derivative, it follows that, to within higher-order terms, ~q = d~oq dq
cl
(12)
q=0
where/1 is the angular velocity. The term (dqoq/dq)lq=0 is indicated in the sequel of this article with .,4; it is an operator mapping vectors of R 3 into elements of the tangent space of M . In terms of coordinates, ,4 is of the form .147, in which Greek indices denote (here and in the following discussion) the components of the atlas of coordinates on M , whereas Latin indices denote the coordinates in ,s In its matrix representation, .,4 is a (dim .A4 x 3) matrix (three columns and a number of lines equal to the dimension of AA). In the mathematical parlance, ,4 is the infinitesimal generator of the action of the orthogonal group S0(3) on .A4. With these premises, one can say that velocity fields (10) are rigid (and one indicates them with/~R and ~R) if XR - - c(t) +/1 • (x - x0);
r
-- ,,'d-Cl
(13)
C. MEASURES OF INTERACTION
Granted the possibility of defining a covariant gradient of the order-parameter field, indicated with Vqp, a set Jl (C) whose elements are of the form
(x, F, ~a, XT~a)
(14)
may be built up: in the geometric parlance, it is the first jet bundle on the manifold (space of the configurations).
Multifield Theories in Mechanics of Solids
13
Analogously, J1 (Vel) indicates the set whose elements are of the type
(~5)
(~, F, ~, V~)
The power 79 is defined here as a real-valued functional on ,71 (Vel); that is,
79:Jl(Vel) ~ IR
(16)
and accounts for both ordinary and substructural interactions. In addition, it is assumed that, as usual, 79 is additively decomposed into external and internal contributions, for any part B* of the body T't3, = 79~xt - 79~t
(17)
where with the term part of B indicates any subset of B that is also a fit region. The basic problem is thus the representation of T'~t and T'~ t. Assuming the validity of (16), it is necessary to introduce measures of interaction acting on all the elements of (15) and developing power on them. Taking this into account, the following expressions for T'bxt and 79~t are assumed to hold for any part 13" of B:
~, (t 9i + 7- 9qb) dA
~int
(s.~ +z.
dV + ft~ ( T . F + S . ~7~)dV
(18) (19)
where the measures of interaction in (18) and (19) have the following meaning: 9b
external bulk forces
9 /3
external bulk interactions on the substructure
9
boundary traction
9 7"
generalized boundary traction associated with the substructure
9s
zero stress
9z
internal self-force
9T
first Piola-Kirchhoff stress tensor
9S
microstress tensor
From this point, volume, area, and line differentials (namely dV, dA, and dl) are omitted in the integrals to render the formulas more schematic. Of course the
14
Paolo Maria Mariano
reader will understand immediately the kind of differential he needs to use in developing explicit calculations by looking directly to the set on which the integral is calculated. Cauchy's theorem assures that Tn - t
on 0/3*
(20)
where n is the outward unit normal to the boundary of/3*, indicated with 0/3*; moreover, 5 S n - 7-
on 0/3*
(21)
One can write (19) when it is possible to define a connection by which the gradient of qo may be evaluated in covariant manner. This allows for the decomposition of, the substructural contributions to the power in terms of densities/3, qb, z . qb, and S . V gb. However, such a contribution could be expressed by some general complicated functionals of qo, qb, and their spatial derivatives, or could even disappear, as in the case of internal variable schemes. Note that, if necessary, one may define the power on the second jet bundle of ~s or the third, and so on. In (21), the nature of 7- is that of a generalized boundary traction that should be assigned at the external boundary of the overall body where S n - ~'. However, some microstructures, such as porous or microcracked solids, do not allow prescribed boundary data of the type ~'. A pore (or a microcrack) is determined by the surrounding medium and not by itself; therefore, it does not exist at the boundary (pores and microcracks can be considered "virtual" substructures). Hence the boundary data must be specified through a constitutive prescription, or are obtained as a result of some limit procedure based on shrinking some boundary layer at the boundary of the body. However, 7- can be completely prescribed as boundary datum in the case of liquid crystals or other material substructures. Analogous reasonings hold in the case of Dirichlet data (i.e., when values of the order parameter must be assigned on the boundary of the body). It may be that in some situations boundary layer data must be accounted for; however, the treatment of these situations is almost completely open. 5The existence of the microstress tensor by a Cauchy-like theorem is discussed by Capriz and Virga (1990), but in the case in which it is assumed that the manifold .A,4is embedded in some linear space. Probably a more general proof can be made by following the variational procedure in Fosdick and Virga (1989) for Cauchy continua because such a procedure can underline the need for the existence of the covariant gradient of ~o. A general proof of the existence of the microstress tensor on the basis of geometric measure theory might obtained by using the results of Degiovanni et al. (1999). A generalized Cauchy's theorem on manifolds has been obtained by Segev (2000).
Multifield Theories in Mechanics of Solids
15
Balance equations can be deduced from (17), (18), and (19) by assuming (as axioms) that at the equilibrium 9 the overall power T'B, vanishes for any choice of the velocity fields :~, qb and of any part B* of B 9 the internal power T'~t vanishes for any choice of rigid velocity fields and of any part 13" of/3 To this aim, equations (20) and (21) must be used together with Gauss theorem. In applying such a theorem, it is assumed that discontinuity surfaces, lines, and points for the fields involved in the integrals are absent. After some calculation, the overall power can be written in the following form:
T'B, = ft3, ((b
- s + DivT). i + (/3 - z + DivS). qb)
(22)
The requirement that ~e, = 0 or any choice of the pair (~:, qb) and of the part B* of B implies that b - s + DivT = 0
in B
(23)
/3 - z + DivS = 0
in B
(24)
Equation (23) is the standard Cauchy's balance to within the zero stress that must be formally introduced as a consequence of (16); equation (24) is the balance of substructural interactions, a sort of generalized balance of momentum. General information on the structure of s and T are obtained by exploiting the axiom requiring that T'~t vanishes when it is calculated on rigid velocity fields [defined by (13)] and on any part of B. By applying such an axiom, in fact, it follows that
eTF r = ATz +
(25)
inB
s=0
(vAT)S
in B
(26)
Of course, by (25), equation (23) reduces to the well-known balance b + DivT = 0
in 13
(27)
In addition, note that when the influence of the material substructure is negligible and order parameters are not considered, equation (26) reduces to skw(TF r) = 0
(28)
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Paolo Maria Mariano
which is the standard property of symmetry of Cauchy stress tensor TF T [skw (.) extracts the skew-symmetric part of its argument]. Previous balances hold in general dissipative situations and must be supplemented by appropriate constitutive equations.
D. BALANCE OF INTERACTIONS FROM THE INVARIANCE OF OUTER POWER
The procedure discussed in Section II.C has been exploited in different cases to derive balance equations. One may question, however, if such a procedure needs too many hypotheses that may be relaxed. In particular, one might find a more general procedure in which the expression of inner power follows as a consequence and is not an axiom. This would be desirable because, in principle, it is possible to evaluate the outer power by experiments only. In Cauchy's solids, Noll's procedure requiring the invariance of the outer power with respect to changes of spatial observers allows one to obtain standard balance of forces and the symmetry of Cauchy stress. An analogous procedure can be followed in the case of materials with substructure. It underlines some delicate questions that may emerge when one selects some order parameter and tries to write balance equations. Basically, one writes the outer power in (18), taking into account (20) and (21), as
extf ~. --
(b. ~ +/3.
*
~b) +
~*
( T n . ~ + S n . ~b)
(29)
and requires the invariance of 7J~' with respect to all changes of observer. This is a request of invariance with respect to Galilean and rotational changes of spatial observers. Changes of spatial observers are typically given by ~* -- ~ + c(t) + ~l(t) x (x - x0)
(30)
(p* -- (p + A q ( t )
(31)
where ~* and @* are the fields evaluated by the observer after its change and e(t) is the translational velocity. As is common in multifield theories, in writing (31), it is assumed that rigid translations have no effects on the values of r Such an assumption applies even in the case in which the order parameter is a microdisplacement because it is always a "relative" microdisplacement. Galilean changes of observers are obtained from (30) and (31) with the choice ~ - 0, ~1 - 0, whereas rotational changes of observers are characterized by ~ - 0, ii - 0.
Multifield Theories in Mechanics of Solids
17
The requirement of invariance of (29) under transformations (30) and (31) implies
b + f ~, Tn)+ q. (f, , (x • b +
+ f ~, (x x Tn + AT$n))--0 (32)
for any choice of c and Cl and the part/3". The arbitrariness of c and ~1implies
f b+f Tn-0 *
dB*
(33)
O*
, ] 0 O*
Equation (33) is the standard integral balance of forces. The arbitrariness of/3* and the application of Gauss theorem imply from (33) that b + DivT = 0
(35)
From (34), taking into account the validity of (35), the following relation follows from the arbitrariness of/3* and the application of Gauss theorem: e T F T = .AT/3 + Div(.A TS)
(36)
or, by developing the divergence, e T F T - AT/3 + ATDivS + (VA T) S
(37)
To assure the validity of (37), two conditions must be satisfied. It is necessary that all the elements that are not multiplied directly by AT be equal to the product of A T with some generic element z of the cotangent space T*.A4 of .A4 (/3 and DivS are elements of the cotangent space of.A/l); that is, the existence ofz e T*.A// is necessary such that
ATz = e T F T - ( v A T ) s
in/3
(38)
which coincides with (26) and represents the generalized balance of "couples." Equation (37) can be thus written as
.AT (3 + Div8 - z) = 0
(39)
and is satisfied when the term in parentheses belongs to the null space of the linear operator A T6 /3 + Div8 - z 6 null space of AT
6AT is a matrix with three lines and a number of columns equal to dim .AA.
(40)
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Paolo Maria Mariano
which is the second condition. In general, the null space of .Ar is the space orthogonal to the range of ,,4 (range of ,,4)• = null space of .,4r
(41)
when a concept of orthogonality is available on T.A4. The range of A at each qo is by definition a subset of the tangent space of .A4 (namely, T.A/[) at the same qo or it is coincident with the whole T.A4 at go. When the range of ,4 is coincident with the whole tangent space of .A4 at qo, the elements of its orthogonal in T.A/[ reduce to the singleton {0}; then the term/3 + DivS - z must be equal to the sole element of the orthogonal "of range of A." In other words,/3 + DivS - z vanishes identically /3 + DivS - z = 0
(42)
There are counterexamples in which the range of A does not coincide with the whole tangent space of .A4 at a certain qo. The prominent counterexample is the case in which the order parameter qo coincides with a stretchable vector cl; this happens in models of shells with through-the-thickness shear or of microcracked bodies. In this case, ,A = ocl, with o Ricci's tensor. It is obvious that at 0 = 0, the range of A coincides with the singleton {0}. Another counterexample is the model of porous bodies in which the order parameter is the scalar void volume fraction; in this case, .A4 coincides with some interval [0, a] of the real axis, and A vanishes identically. This follows by considering that for porous bodies, the value of qo remains unchanged by rigid rotations: then qgq = 99 and the derivative (dqgq/dq) Iq=o vanishes identically. Models of microcracked bodies also make use of second-order symmetric tensor-valued order parameters ~ij and the operator ,,4 coincides with (Oijr~rk -1- Eirerjk). Even in this case, at E = 0, the range of A coincides with the singleton {0}. In the case in which the range of ,A does not cover T.A/[ at any qo, the argument leading to (42) is not exhaustive, and one must write the differential inclusion (40) as
/3 + DivS - z' = - z "
(43)
where z' satisfies (38) and z" belongs to the null space of A r (i.e., A r z '' = 0). Consequently, (42) still holds, taking for z the difference z ' - z". This property cannot be derived by using the procedure in Section II.C. Example 2 An example of the occurrence of a term like z" in the balance of substructural interactions can be found in the theory of liquid crystals. The usual order parameter of liquid crystals is a vector d belonging to the unit sphere S 2 in/t~3.
Multifield Theories in Mechanics of Solids
19
A coincides with od, which can be also written as d • In balancing substructural interactions, it is required that the covector/3 + DivS - z at each material patch be parallel to the "averaged" direction d of the rodlike molecules of the liquid crystal at the same patch. Therefore, in this case, the balance of substructural interactions is written as
tic + divSC _ z c = c~d
(44)
with c~ some scalar constant and the operator div is calculated in the current configuration (see Ericksen, 1991). It is evident that otd is - z " because d x d = 0. Of course, (44) is written in the current configuration because one deals with a liquid. Therefore, tic, Sc, and z c are measures of interaction in the current configuration. The request of invariance under changes of spatial observers could perhaps not be conclusive in some cases (thanks to the arbitrariness of z'), and more stringent requests of invariance could be required to justify completely the balance equations. A possible way consists in requiting the invariance with respect to all possible representations of each type of substructure. Suggestions for such an invariance follow from the knowledge of situations in which some of the components of the order parameter can be chosen arbitrarily. If one requires in fact that some scalar function k of ~ - - a s , for example, the substructural kinetic e n e r g y - - b e invariant under the action of the rotation group S0(3), it is necessary that
.A T(O(,,k(~)) = 0
(45)
This is a system of partial differential equations in 2(dim .M) + 1 variables. Following a theorem of Tricomi (1954), it is possible to show that previous system admits 2(dim .M) + 1 - char A independent variables as solutions, where char.A is the characteristic of .,4 because .A is a matrix with three columns and a number of lines equal to dim A/[. Such independent variables can be chosen arbitrarily by an external observer. Under such a suggestion, one could think to map .A4 into other manifolds A/" by Cr-diffeomorphisms (r > 1), indicated with Jr, and call these diffeomorphisms 7 representations of the substructure when dim .M = dim A/'. If two elements on .M are related by a rigid rotation, then the corresponding elements on A/" (through the mapping zr) must be related by a rigid rotation as well. To each Jr : . M --+ A/', a mapping Trr is associated; it maps elements of the tangent space of M (i.e., TA4) into elements of the tangent space of A/" 7Roughly, a diffeomorphism is a one-to-one differentiable mapping such that its inverse is differentiable as well.
20
Paolo Maria Mariano
(i.e., TA/'). The mapping TJr may be then described by appropriate Jacobian matrices J, such that if qb is an element of TA/[, then Jqb belongs to TA/'. In other words, defining qo' - zr(qo), it follows that qb' - ~ and Jl - dqo'/dqo. The requirement of invariance of T'~xt under all possible representations of the substructure reduces to the invariance under changes qb --+ Jqb. To render T'~xt invariant under changes qb ~ ~ , it is necessary that
(46) The arbitrariness of qb and the application of Gauss theorem reduce (46) to (I)T(/3 + DivS) + (VJT)S = 0
(47)
where (I) = I - J. The validity of (47) implies the existence of some element i of the cotangent space of .A4 such that (I)Ti + (V2 T) S = 0. Thus (47) reduces to (I)v (/3 + DivS - ~) = 0 and the term in parentheses must vanish identically as a result of the arbitrariness of,II, thus of (I). However, this conclusion is only formal because the condition ~T~ + (VjIT)S = 0 may imply that S vanishes identically, as a result of the arbitrariness of dl, so as to obtain a reduced balance of the form /3 = 0
(48)
Among other things, such a conclusion could induce doubts about the expression of the external power 79~xt. There are situations in which nonlocal terms could appear in the expression of 79~,t through some general functional that may be expanded in series by means of some "myopia" theorems formally analogous to the theorems of "fading memory" used in standard continua with memory. In the former case, such theorems reduce the nonlocal influence of material patches, surrounding the one assigned, to a rather weak nonlocality in space, whereas in the latter, theorems of "fading memory" cut the influence of the events in the past on the present event (the nonlocality is thus in time). In any case, the validity of balances (35), (38), and (42) implies the proposition in the following. Proposition 1
By virtue of the balance equations (35), (38), and (42), it follows
that
(49)
and the last integral in (49) takes the name of internal power and is indicated with 79~t.
Multifield Theories in Mechanics of Solids
21
Note that (49) was used in Section II.C as an axiom [see (18) and (19)] to deduce balance equations (24), (26), and (27) by means of the "virtual power procedure." In this section, another procedure has been followed; only the explicit expression of the external power is assumed as an axiom, then the invariance with respect to changes of external observers is requested. This procedure is in essence more general than that in of Section II.C because it underline the need of elements of the type z" and the expression of the internal power has been deduced in this section as a theorem.
E. EFFECTS OF INERTIA One possible way to account for the effects of inertia pertaining to both the macroscopic motion and possible internal vibrations of the substructures consists of decomposing the volume forces b and/3 in their inertial (in) and noninertial (ni) parts as b = b in - [ - b ni
(50)
j~ _. j~ in ..[_ j~n i
(51)
and in assuming that, for any part B* of/3, d {kinetic energy of 13"} +
~ ) __ 0
(52)
rate of kinetic energy + power of inertial forces -- 0
(53)
dt
(b in x + j~in
9
In this way, one assumes the validity of the balance
and may interpret (53) as a constitutive prescription on the explicit expression of inertial terms once an expression of the kinetic energy has been selected. Although the kinetic energy density associated with the macroscopic motion is proportional to I/~l2, in fact, the kinetic contribution of substructures may have some complicated structure that depends on each special model and, in a certain sense, has constitutive nature. The kinetic energy of 13" is given by 1
ft3, (-~pi~. ~ + k(qo, (p))
(54)
where l p/~./~ is the standard kinetic energy density of material particles and
22
Paolo Maria Mariano
k(qo, qb) is the contribution of the substructures of the material to the kinetic energy density. The term k (., 9) is a nonnegative function such that k(., 0) = 0
(55)
02~k -r 0
(56)
where, here and in what follows, 0y means partial derivative with respect to the argument "y". Really, the symbol 0 was previously used before letters indicating sets. When 0 precedes a letter indicating a set (e.g., 0B), it indicates the boundary of B, whereas when 0 precedes any function, it means partial derivative. Moreover, k (-, 9) must be frame indifferent; that is, taking into account (11), it is necessary that k(qo, qb) - k(qo + Aq + o(Iql), qb + A/I + o(Iql))
(57)
for every q. The invariance condition (57) implies O~kA + O(ok(O~oA)(o - 0
(58)
Equation (58) can be obtained by developing in series the right-side term of (57) around k(qo, ~). Substituting (54) in (52) and taking into account that (52) must hold for any part B*, it follows that b in = -p:~ -
~-~-,geX - ,9~X
)
(59) (60)
where X is called substructural kinetic coenergy density and is such that k -- O~X. O -
X
(61)
In the mathematical parlance, the kinetic coenergy is the Legendre transform with respect to ~ of the substructural kinetic energy density k. As a consequence of (59) and (60), the balances (35) and (42) become DivT +
b ni - - p ~
DivS - z + ~ni _
d dt
O(oX -- O~X
(62) (63)
It is noted that inertial effects ~in associated to the substructure of materials are very minute and not perceivable unless the substructure itself oscillates at high frequencies, as indicated by some experiments on the scattering of phonons within the lattices of crystalline materials, or on liquid crystals.
23
Multifield Theories in M e c h a n i c s o f Solids
The explicit expression of k is of constitutive nature; a possible quadratic form, 8 like the simplest one given by 1_ k - ~ D ( ~ , ~)
(64)
determines the metric on .M. In (64), the brackets and the comma (i.e., (.,-)), indicate the scalar product on TAd, the tangent space of Ad, and /) is some appropriate constant chosen to adjust eventually physical dimensions. When k is constitutively prescribed, the kinetic coenergy density can be obtained by solving the partial differential equation (61), whose solution X(qO, ~) is the sum of a special solution of the complete equation Xs and the homogeneous solution Xh, corresponding to k -- 0:
(65)
X = Xs + Xh
The explicit solution of (61) can be found in (Capriz and Giovine, 1997a). If k is homogeneous of second degree (hsd) in qb, then it coincides with ,~s. Finally, the inertial contributions of the material substructures can be written as follows:
[~in ( d d = -
- ~ 04, Xs - O~ X, + --~ Or
-- 0,r Xh
)
(66)
where the term d d-t 0r Xh - 0,r Xh is powerless, that is,
(d
O~Xh - O~,Xh
)
90 -- 0
(67)
(68)
When k = O, the solution of equation (6 l) prescribes that the kinetic coenergy density X from (61) can be, at most, linear in O; in this case,/3 i" can also be, at most, linear in O, and situations of parabolic evolution can arise, as in the case of magnetostrictive solids or ferroelectrics. When one considers a relative velocity Oret of the order parameter and writes Orel + .,4(:1instead of the absolute velocity qb, it is possible to underline the presence 8Quadratic expressions for the substructural kinetic energy density can be found, for example, in the case of direct models of plates (Antman, 1995), multifield descriptions of ferroelectric (Da~i and Mariano, 2001), or microcracked bodies (Mariano, 1999) when microdisplacements are considered as order parameters. More general expressions must be considered when, for example, there is an intrinsic limit velocity for the propagation of perturbations in the materials. For example, in crystalline materials the dislocations cannot propagate with a velocity greater than the velocity of sound.
Paolo Maria Mariano
24 of a centrifugal term
a 2~ X ( ( O~Ail)(Ai~))
(69)
( O2cpX) .,461
(70)
- 2(0~,(.A/I)) TO~g
(71)
an entrainment term
and a Coriolis term
associated to substructural dynamics. 1 When k is a quadratic form in ~b (e.g., k(~, ~b) - 5~b. N~b, with some constant tensor N) and the velocities are only of rigid rotational type (/1 x x and A/l), l the total kinetic energy of the body becomes 5/1. (J -+- H)(1 where J is the standard moment of inertia given by f8 P (]x]21 - x | x) and H - f8 ATN'A" The final expression of the kinetic energy seems not to be compatible with the classical dynamics of rigid bodies. This paradox can be eliminated by imposing that
k(t,p, ~ ) -
k(qo, ~rel)-
Moreover, there are cases in which F is involved in the kinetic energy density. This is the case of liquid crystals when one chooses as order parameter an observerindependent vector given by F -l d, instead of an element d of the unit sphere S2 in R 3. A complete theory of liquid crystals that accounts for this has not yet been developed. Inertial interactions may also involve spatial derivatives of the acceleration fields. In this case, possible values of the order parameter are subjected to some internal constraint, so the substructure becomes latent (see Section VIII). A prominent example is given by models of capillary phenomena described by Korteweg's fluids, in which the order parameter is scalar and is coincident with det F (see Capriz, 1985). R e m a r k 1 Let the invariance of the kinetic energy density be required under Galilean changes of observers. Let also the invariance of the overall energy (sum of internal and kinetic energy) be required under rotational changes of observers. As a consequence, after some calculations not entirely trivial, the inertial contribution of the substructures results in d dtOCpk - O~k
(72)
When k is homogeneous of second degree (hsd) in ~, or when X is so, (72) may substitute the term in parentheses in (60). In this case, the two expressions of the inertial contributions are equivalent to within powerless terms. When the
Multifield Theories in Mechanics of Solids
25
substructural kinetic energy density requires more complicated expressions than hsd, the problem of the equivalence of the two procedures just explained is far from being completely clarified. Cases different from hsd may be necessary when a limit speed of the substructural perturbations (thus of ~) must be emphasized; invariance with respect to rules of changes in observers more general than the action of the rotation group could be necessary, and the consequences of the relevant gauge invariances examined. Remark 3 When cases in which the balance of substructural interactions reduces to/3 = 0 occur, from (51) and (60) it is possible to write only a kinetic equation of the form
t~ni__ d oqx(u q~rel) _ oqx(~o, q~grel) dt O0rel 0r
(73)
An example of absence of microstress in models of bodies with substructure is the theory of liquid with bubbles presented in Kiselev et al. (1999).
BIBLIOGRAPHIC NOTE Balance equations (24) and (26) are Capriz's (1989), their presentation in Section II.D follows that of Mariano (2000a). Mathematical details on the topics in Section II.D can be found in Di Carlo (1996). Additional remarks on the geometric nature of microstresses and self-forces can be found in Segev (1994, 2000) and Capriz and Giovine (1997a,b). Details on the examples before Remark 2 can be found in Capriz (1989, 2000). For a derivation of balance equations, see also Capriz and Podio-Guidugli (1983) and Capriz and Virga (1990, 1994). Discussions on the derivation of balance equations by means of virtual power arguments (involving the assumption a priori of the expression of the inner power)can be found in Germain (1973), but they are referred only to order parameters that are higher-order perturbations of the displacement field, then only first- and higher-order micromorphic materials are treated (see also Maugin, 1990). The procedure to obtain balance equations for substructural interactions on the basis of the invariance of the external power 7~! under changes of observers has been developed in Capriz and Virga (1994), Capriz (2000), and Mariano and Capriz (2001). Noll's classic procedure of invariance of external power can be found in the 1973 article. "Myopia" theorems for spatial nonlocalities and their applications to Cauchy's continua are in Capriz and Giovine (2000), whereas analogous "fading memory" theorems are in Coleman (1971).
Paolo Maria Mariano
26
All details about the relations between the substructural kinetic energy and the substructural kinetic coenergy are in Capriz and Giovine (1997a).
III. Elastic Materials with Substructure A. VARIATIONALCHARACTERIZATION
In this section, only hyperelastic materials with substructure (or, with some abuse, just "elastic") are examined. They are characterized by the existence of an elastic energy density, indicated with w, such that 6f,. w=f,.
(T 9~iF + S 93(Vqo) + z 96qo)
(74)
for all parts B* of/3, where ~ indicates the variation operator. 9 A consequence of this definition is the possibility to write the measures of interactions in terms of F, V qO, qO, once the expression of w has been selected. In general, in fact, one can assume w = t~(F, qo, XTqo)
(75)
Consequently, by calculating the variation of the first integral in (74), it follows that
f ( ( O v w - T). ~F + (~v~w - S)-~(XTqo) + (O~w - z). 6qO) = 0
(76)
Because variations can be chosen arbitrarily in (76), the following constitutive restrictions hold
Remark 4 structure:
0vw = T
(77)
Ov~w = S
(78)
O~ow = z
(79)
Consider the following special case of an elastic material with sub-
9 ~ni vanishes identically 9 t~ in
is only of powerless type and is given by Bqb, with B an appropriate tensor
9Of course, equation (74) can be written in terms of velocity fields.
Multifield Theories in Mechanics of Solids
27
9 w - t~(F, qo), thus the weakly non-local contribution of the order parameter due to the gradient of qo is neglected The balance (42) reduces to
B~o- O~w
(80)
and the self-force 0~,w becomes powerless. The order parameter assumes the character of an internal variable not satisfying balance of interactions that develop explicit power. Equation (80) is coincident with the evolution rule of an elastic internal variable. An analogous reduction of multifield models to internal variable ones can be obtained in nonconservative cases, as is shown in some parts of the following subsections.
B. SOME PROPERTIES OF LAGRANGIAN DENSITIES Before discussing some properties of Lagrangian densities for elastic materials with substructure, it is convenient to introduce a special symbol for a product between tensorial quantities that will be of future use both here and in following sections. This symbol is _,. The product ,_ is here defined as _," Lin(It~3, T.A4) x Lin(I~ 3, T*.A4) -+ Lin(It~3, R .3)
(81)
where Lin (It~3, •,3) is the space of linear forms associating three-dimensional covectors (belonging to the dual of R 3, which is usually indicated with R .3 and identified with R 3) to vectors in ]t~3 and Lin (It~3, T*.A4) is the space of linear forms o n R 3 taking values on the cotangent bundle ~~ T*.A4. Then, taking Vqo and the microstress S [which is a linear form associating elements of the cotangent space of.A//to vectors in ~;~3 thus an element of Lin(R 3, T'A//)], one writes by definition (Vqo r_,S)n 9v - S n . (Vqp)v
(82)
for any choice of vectors n and v. Note that when the order parameter is scalar valued, the product_, coincides with the standard tensor product | On the contrary, when qo is not scalar valued, the meaning of_, is not the one of a dyadic product. For example, if qo is a third-order ij tensor with components ~k ), one has (Vqo T,_S)jl n j 1) l - - Si~nJ(~rqo)i~l)vI. l~ cotangent bundle of.AAis the space of linear forms on the elements of the tangent space of .A.4.
Paolo Maria Mariano
28
Another product, indicated with +, is also of future use. It is defined as ,i, : T*.AA • Lin(R 3, T M )
~ ]~,3
(83)
thus its result is a covector, l~ Consequently, the products z,i,~'qo and/3+~'qo are covectors. For example, if qo is a fourth order tensor with components ~0/,nJ,,)one mn
ij
has (z+Vqo)l = zij (Vq~ Lagrangian densities for conservative dynamics in multifield theories are of the form
s -- s
x, ~, F, qo, qb, ~Tqo)
(84)
Of course, s depends on the metric y on the referential configuration 13 and on the metric on .A4 through the quadratic part of the kinetic energy k. Granted some regularity properties of the Lagrange density function, ~2 the following Euler-Lagrange equations hold: d
dtO~s - Oxs + DiVOFE -- 0
(85)
dtOCps - 0~,s + DivOv~,s - 0
(86)
d
To obtain a more compact form of equations (85) and (86), let the four-dimensional gradient V 4 be introduced. It is defined by
V4 -- ( ~ )
(87)
The four-dimensional divergence is indicated with V 4. , and then, by indicating with H and fI the derivatives 0V4xs and 0v4~,s respectively, equations (85) and (86) can be written as
V 4. lI - Oxl~ = 0
(88)
V 4" 1=I - 0qo,~ --" 0
(89)
I I With the same symbol +, a product + : Lin(R 3, T*.A/[) x Lin(R 3, Lin(R 3, T.M)) --> ]1~,3 is also indicated. It is not an abuse of notation, because the two products have the same structure. In this way, the product S+VV~0 determines a vector of I~3. 12See, for example, Renardy and Rogers (1993) for the discussion of regularity properties for s in the case of Cauchy materials. The extension to multifield theories is left to the reader as an exersize.
Multifield Theories in Mechanics of Solids
29
Let 1) be defined now by I~ -- s 4 -- V4x r FI -- V 4 (~T ,__1=i
(90)
where 14 is the four-dimensional unit tensor. Tensor 1) has physical dimensions of an energy density and is expressed in material coordinates. I) is also of second order, thus of the type ]?s, with indices r, ~ running 0, 1, 2, 3 (the coordinate 0 being the time). In the following, I? indicates the spacelike part of 1), is of the type ]?m with m, n running 1, 2, 3 and is given by P - s Proposition 2
FrT-
Vq~r_,S
(91)
If the equations of motion hold, then 4-r V~Ps + s
-0
(92)
where the comma as subscript represents in (92) the explicit partial derivative with respect to the coordinates. The proof of this proposition is based on the calculation of the derivative O~])~ and on a lemma stating that oy.....z ; - - ~ 1P m"
(93)
The proof of (93) is rather technical and can be found in (Mariano, 2000a) as well as the complete proof of previous proposition. An important Corollary is the following:
When the body is homogeneous and inertial effects can be neglected, for any closed sufficiently smooth surface within it, Corollary 3
fc,
osed surface
Pn - fc,
osed surface
( w I - F T T - vqoT . S ) n -- O
(94)
Of course, n is the outward normal to the surface. The proof of Corollary 3 follows from the simple application of Gauss theorem. The integral (94) is the extended version of Rice's (1968) integral to multifield theories. Its importance in the study of crack propagation is explained in Section VII. The second-order tensor in the integrand, namely ( w I - F T T V qoT*,.q), is a modified version of Eshelby tensor that holds for elastic continua with substructure. If one inserts an inclusion in an elastic homogeneous body with substructure and defines the force ~ exerted on the inclusion by the surrounding medium
30
Paolo Maria Mariano
(following Eshelby, 1975) as the integral on a region 13" containing the inclusion in its interior, namely, -- -- f/3* /~,i
(95)
then, by (92) and (94)), in absence of inertial effects, one obtains - f ( w l - F T T - Vqpr_,S)n Ja B*
(96)
C. INFLUENCE OF THE SUBSTRUCTURE ON THE DECAY OF ELASTIC ENERGY
One of the central results of the linearized theory of elasticity is the proof of the longitudinal decay of elastic energy in cylinders loaded at one base only by equilibrated force systems. This phenomenon is usually known as Saint-Venant's effect. Here the analogous in multifield theories is shown and the influence of the material substructure on such a decay is evaluated. In this section, in which linearized situations are treated, reference and current configurations are identified with each other and the relevant measures of interaction are denoted with T, ,9, 2. The displacement field u - x(X) - X is introduced for convenience, and F replaced by Vu in the constitutive relations. Let (0X1X2X3) be an orthogonal coordinate system and D an open-bounded compact region in the plane X l X2. The body considered here is a semi-infinite cylinder if2 - / ) x [0, +cx~). In the following, n3 indicates the outward unit normal at /) (i.e., n 3 - - e 3 , with e3 the unit vector along X3), whereas nL indicates the outward unit normal at the lateral boundary 0/) • [0, +c~)./)(~'3) indicates the cross section at S(3 E [0, +c~), whereas f2(2"3, l) -/3(2"3) • [Yf3, Y(3 + l]. Moreover, it is understood in the following that
fa (.)=limf (X3)
l'---~~
(-'~3 ,/)
(.)
(97)
provided the existence of the limit. With these premises, the following assumptions apply" 1. External volume forces vanish: b - 0,/3 = 0; then DivT = 0
(98)
Div,9- ~ = 0
(99)
Multifield Theories in Mechanics o f Solids
31
2. The lateral boundary 0 b x [0, + e c ) is traction flee; boundary tractions are applied to b only and are self-equilibrated in the sense that Tn3 - 0
(100)
Sn3 - 0
(101)
x x 'l'n3 + ,,4T8n3) -- 0
(102)
fo fb( Moreover, it is assumed that lim f
J D(X3)
X3 ---~~
(103)
(1"n3 9u + ,Sn3 9q)) = 0
3. The elastic energy w(Vu, q0, Vqo) is a positive definite quadratic form in its variables. By indicating with y the triplet (Vu, qO, Vqo) in a way such that yl
= VII,
Y2 -- go, Y3 = V qo, the elastic energy density can be written as 1 w(Vu, qo, Vqo) -- -~aijYiYj
(104)
where aij is the ijth element of a matrix a expressed by
/
og~
a = (aij) =
Ogumlo 0g~
og .mlo
/
og mlo]
with o indicating a stress-flee state considered to be a natural (or reference) state of the body. Matrix a is such that aij = aji. It is assumed that there exists a > 0 such that aij Yi Yj <_ a ~'i f/i for any pair of constant ~7's. Assumption 3 also implies that w must be sufficiently regular to assure the validity of Taylor expansion around some "natural" state o, at least up to the second order, and that residual stresses are not accounted for; that is, it is assumed that Wo = 0;
8vuWlo = 0;
O~,wlo = 0;
Ov~oWlo= 0
(105)
32
Paolo Maria Mariano
Let U(X3) be the elastic energy of the part of the cylinder given by f2(X3) D(X3) x [X3, nt-oo). It is defined by
U(X3)- [ 113 J~ (x3)
(106)
The following proposition holds. It explains Saint-Venant's effect in elastic materials with substructure and is the main result of this section. Proposition 4
If the fields Tn3, ,.~n3, u, qp are square integrable, then
U ( X 3 ) < U ( 0 ) e x p ( X 3 - 1~ )_ _
(107)
for any 1 > O, X3 > I. The scalar )~ can be interpreted as the lowest non-zero characteristic value of the free vibrations of f2( f(3, l) with quadratic substructural kinetic energy.
When the ratio a/)~ is greater than the analogous ratio in the material considered without substructure, the presence of material substructures within an elastic body has a stabilizing effect and decreases the decay length of the elastic energy. Another kind of decay can be studied: the radial decay (Knops-Villaggio's effect) related to boundary conditions imposing that both u and qp vanish at the lateral boundary. It is possible to prove that the elastic energy of a cylindrical annulus at a given distance from/), of variable height and whose other surface coincides with the lateral surface of the cylinder, decays to zero algebraically at most.
BIBLIOGRAPHIC NOTE Section III.B is based on the first part of Mariano (2000a). With reference to Section III.C, the proof of Saint-Venant's decay in continua with substructure follows basically Toupin's (1965) proof in Cauchy's continua. Analogous proofs in the special cases of Cosserat continua and micromorphic continua have been developed in Berglund (1977) and Batra (1983), respectively. The Knops-Villaggio effect in the case of Cauchy's materials is proved in their 1998 article, whereas its possible discussion within the setting of multifield theories is in Mariano (2000b), along with the proof of Proposition 4.
Multifield Theories in Mechanics of Solids
33
IV. Balance in Presence of Discontinuity Surfaces The presence of discontinuity surfaces (also called interface) is a recurrent phenomenon in solids with material substructure. Interfaces between fluid and solid or solid and solid are evident in solidification phenomena and occur also in second-order phase-transitions between austenite and martensite. This section discusses how interactions can be balanced across discontinuity surfaces and at the junctions among them. Each discontinuity surface may or may not be considered to be endowed with its, own structure. In the former case, surface measures of interaction must be introduced because they take into account classic surface tension and a surface tension as a result of the material substructure.
A. INTERFACES: GEOMETRIC CHARACTERIZATION Within the body in its reference placement/3, an interface E is considered to be an oriented surface defined by E _= {X E cll3, f ( X ) =
0},
(~o8)
where the function f is smooth on/3. The orientation of Z is given with the normal vector field in, defined by m = V f / I V f l (Figure 1). Given any continuous and differentiable vector field e defined on Z, it is possible to define its surface gradient Vz by considering parametrized curves t on E and
V
FIG. 1. Geometric characterization of interfaces.
34
Paolo Maria Mariano
applying the chain rule e -- (Vze)i where i is the derivative of t with respect to the parametrization of the curve. The curvature tensor L of E is defined as the opposite of the surface gradient of the normal vector field, namely, L = -Vzm
(109)
whereas the overall curvature E is the trace of L KS = - t r ( V z m ) = - D i v z m
(110)
The previous formula allows one to define the surface divergence as the trace of the surface gradient. On each subsurface of E enclosed in a genetic part B* of B, a vector field u on the curve OB*NE is defined; it belongs to the tangent space of E at OB* A E m that is, v is the normal to 0B*N E in the direction of the tangent of E (see Figure 1). Given any field e that is continuous on B \ E, the limits e+ = lime(X + em, t), e---~0
X 6E
(111)
define the jump [e] through [e] = e + - e -
(112)
when the difference makes sense. Given two such fields--for example, el and ezmthen, with some significance assigned to the product, [ele2] = [el](e2) + (el)[e2]
(ll3)
where (e) is the average given by (e)-
1 ~(e + + e - )
(114)
If the fields F and V q0 have the properties just mentioned for e, then a discontinuity surface E is called coherent if [F](I - m | m) = 0
(115)
[Vqo](l - m | m) = 0
(116)
where I is the second-order unit tensor, whereas (I - m | m) is the projector on E and condition (116) is taken here only for convenience in the sequel of this article. In the following, it is convenient to indicate with F the surface gradient of x, namely, F = V~x - (F)(I - m | m)
(117)
Multifield Theories in Mechanics of Solids
35
while with N the surface gradient of qo, namely N = Vzqo = (Vqo)(I - in | m)
(118)
Interfaces across which the order parameter is continuous are considered only in the sequel of this article; discontinuities are accounted for the gradients of the order parameter itself. Cases in which the order parameter itself can be discontinuous across E may occur. However, because qO is in general the element of a nonlinear manifold, the possible difference qo+ - qo- may not make sense. For example, if .A4 coincides with the unit sphere in the example in Section II.A, the difference of two arbitrary elements of such a sphere may not belong to the sphere itself. However, one can define the jump [qo] of qo even on a nonlinear manifold .A// when it is possible to find some group that acts on .A// as the translation group; that is, there exists on .A4 a given element @ such that it is possible to reach any other element of .A/[ by multiplying @ by elements of the group. In this case, the jump [qo] can be defined through the difference of the elements of the group. Alternatively, one may embed .A4 into a linear space by Whitney's theorem, even if the embedding is not unique. When qo is continuous across E, the jumps [~] and [V~o] are always defined because both qb and XTqoare elements of linear spaces.
B. B A L A N C E AT DISCONTINUITY SURFACES
1. Discontinuity Surfaces without Own Structure The simplest way to characterize interfaces is to imagine them only as purely mathematical surfaces free of any form of own structure. Balances of measures of interaction can be derived easily in this case. First, one must consider the integral form of (35) and (42) and for a generic part B*, crossed by the discontinuity surface E (i.e., B* A E -r 0), one writes
fBTn-0. b+f ~.
fB(~--Z)+] *
f ,J0 B*
(119)
Sn--0
(120)
In equations (119) and (120), the bulk interactions b,/3, and z are assumed to be continuous throughout B*, whereas stress measures T and S may suffer jumps at E. Balances at E; can be obtained from (119) and (120) in different ways. Probably the simplest one consists of shrinking B* at the discontinuity surface E with a limit procedure prescribing that B* --+ B* A E. In making this, the bulk integrals vanish as a result of the continuity of their integrands, whereas the integrals on the boundary OB* reduce to integrals on B* A E of the jumps of their integrands.
36
Paolo Maria Mariano
By translating the previous words into symbols, it follows that
f,
ft3 0 b--+ , ,
0
as/3* ~ / 3 * n yz
(121)
Consequently, (119) and (120) reduce to f
IT]in - 0
(122)
[31m - 0
(123)
~*NZ
f
13*nz
The arbitrariness of/3* implies [T]m = 0
on Z
(124)
[S]m = 0
on Z
(125)
Proposition 5 In absence o f standard and substructural surface tensions on a discontinuity surface ]E, the standard and generalized tractions (t = Tm and 7" = 8 m , respectively) are continuous across Z. As a consequence of the invariance of the lower, equation (125) should be written as .,4 T [S]m -- 0. Thereof the assumption in Proposition 5 that surface tensions are absent implies that even surface self-forces of the type ~" such that .AT~" -- 0 vanish identically. 2. Discontinuity Surfaces with Own Structure
More complicated is the case in which ~ is endowed with its own structure. Then ]E may posses standard surface tension measured by means of a surface referential stress T and a surface tension induced by the material substructure and measured through a surface referential microstress S. By definition, T(I - in | m ) = T, S ( I - m |
m ) = S.
Let/3* be a genetic part of/3 intersected by the discontinuity surface ~:, as in Section IV.B.1, T v and St,, are, respectively, standard and generalized tractions at the curve 0/3" n Z (i.e., at the intersection of Y: with the boundary 0/3" of/3"). Vector u has been defined in Section IV.A: At each X belonging to 0/3* n Z, u(X) is the normal at the curve 0/3" N Y: at X in the direction of the tangent of Z at X. With these premises, the external power is written here as t3, --
*
(b. ~ + / 3 . ~) +
f
~*
(Tn. ~ + S n . ~)
+ f ( T u . :~+ + •u. ~b+) da /3*OZ
(126)
Multifield Theories in Mechanics of Solids
37
By shrinking/3* at the discontinuity surface E as in Section IV.B. 1, 79~xt reduces to the external power "Dext ~*NE developed on E namely, r
T'~,t -+ T'~,tnr.
as 13" -+ 13" n E
(127)
where, thanks to the previously mentioned hypotheses of continuity of bulk interactions b and/3, 79~;'n$ =fB*n~: ([Tm /']
+ [Sm" ~]) + L t3* n:c (Tu
9~+ + ~u 9~ ~:)
(128)
Balance equations at the discontinuity surfaces can be obtained from (128) by applying the same procedure of Section II.D, that is, by requiring the invariance of 79ext B*NE with respect to all changes of observers After some calculations in which one accounts for the arbitrariness of k, ~,/3*, and uses Gauss theorem on the surface E, one finds the necessity of the existence of an element 3 of the cotangent space of .AA, called surface self-force, such that
ATz
=
eTF
T -
(vzAT)s
on E
(129)
and the validity of the following balance equations: [T]m + Divx T = 0 [S]m + DivES - 3 = 0
on E on E
(130) (131)
Equation (129) represents a sort of generalized "balance of couples" on E, whereas (130) and (131) are balances of interactions on E. The surface self-force must be considered to be the difference of two terms 3' and 3" such that 3' satisfies (129) and 3" belongs to the null space of .AT, that is.,
A r3'' = O. R e m a r k 5 Note that, in absence of surface substructural interactions ,9 and z, i.e. in absence of substructure, the standard surface Cauchy stress TF T is symmetric, i.e. ~F T = F~ T.
Proposition 6 By virtue of balance equations (35), (38), (42), (129), (130), and (131), it follows that
f (b 9 + ~ 9~) + f
~ (Tn 9k + Sn 9~)
BNE
+ f (/T/m. r*J+/S/m. ~e~)+ f(T. F + ~. N +~. e~)
(~32)
Paolo Maria Mariano
38
The last integral of equation (136) is thus the internal power "Dint in presence of a discontinuity surface endowed with its own structure within the body.
BIBLIOGRAPHIC NOTE Balances of interactions at interfaces free of own structure have been obtained in Capriz and Virga (1994). Special forms of balances (124) and (125) suitable for micromorphic (or higher-order micromorphic) continua are in Germain (1973). General balances of interactions at interfaces endowed with own structure have been obtained in Mariano (2000a). The matter of discontinuity surfaces in standard Cauchy continua is treated extensively in scientific literature (see, for example, Abeyaratne and Knowles, 1990, 1991, 1994; Pence, 1992; James, 1983; Gurtin, 1995, 2000, and references therein). Of course, in the case of Cauchy continua (see references mentioned previously), equation (131) is not present, and equation (129) reduces to the requirement of symmetry of the surface Cauchy's stress TF r. The scientific literature on discontinuity surfaces in standard Cauchy continua includes cases in which surface measures of interaction are considered as well as cases in which they are not. For example, the derivation of equation (130) and detailed comments on it can be found in Gurtin (1995, 2000).
V. Constitutive Restrictions Usually, in multifield theories one postulates that the first second laws of thermodynamics can be written in a form analogous to the one of Cauchy continua once one considers the extra power developed by the substructural measures of interaction. First, one formally assumes for any part 13" of 13 at each instant t the existence of the internal energy E1 (/3"), the entropy Ht(]3*), the heat flux Qt(]3*, ~,e), and the entropy flux QtI-I(B*, B *e) exchanged by B* with its exterior B *e. After the "natural" assumption that E1 and Ht are time differentiable, one then writes for the first law of thermodynamics d
d t E t ( B *) - Qt(13* ' 13*e) + "l)ext13* ,-
(133)
and for the second law d
d t H t ( B , ) >_ QH(B, ' ~,e)
(134)
(heat sources and entropy sources are neglected here for the sake of simplicity).
Multifield Theories in Mechanics of Solids
39
By introducing Helmoltz free energy density as {free energy density} = {internal energy density} - {temperature} {entropy density}
(135)
the second law may be written in terms of free energy, and is a tool that allows one to derive constitutive restrictions on the measures of interaction once the external power ~S)ext has been substituted by the internal power "pint. Then one must represent explicitly all the elements of (133) and (134). Basically one introduces bulk densities for internal energy and entropy; however, surface internal energy and entropy densities (together with surface heat and entropy fluxes) can be introduced in the presence of interfaces when relevant to describe different physical phenomena. Here thermodynamic phenomena related to variations of temperature are not treated (no details on the various explicit expressions of (133) and (134) are then given) and an isothermal version of the second law of thermodynamics is considered. It is called here mechanical dissipation inequality and prescribes that d
m dt
{free energy of 13"} - "Dint ,-t3, -< 0
(136)
A. CONSTITUTIVE RESTRICTIONS IN THE BULK In the bulk, it is assumed that {free energy of B*} - ft3, ~
(137)
where 7t is the bulk free-energy density. Consequently, the mechanical dissipation inequality becomes d--~ , ~ -
,(T'~'+z'q~+S'V~b)-<0
(138)
Usually, one assumes for ~ a constitutive structure of the form - ~(F, ~o, V~o)
(139)
By calculating the time derivative of the free energy in the mechanical dissipation inequality and collecting terms, it follows that
ft3((Ov~ - T)-~" + ,
1
z). q, +
- S)- V~b) _< 0
(140)
In principle, given any triplet (F, ~o, V~o) representing the isothermal state of any material patch placed at X, one can arbitrarily choose velocity fields F, ~b and
40
Paolo Maria Mariano
~'q~ from (F, qo, Vg~). Consequently, since the integrand of (140) is linear in the velocity fields, the following proposition follows. Proposition 7 interaction:
The following constitutive restictions hold for bulk measures of T = 0v~(F, ~, Vq~)
(141)
z = 0~p(F, qa, Vqp)
(142)
$ = 0v~,~p(F, qo, Vqo)
(143)
Other expressions for #r could be assumed; for example one may choose #r = ~(F, ~, Vqo, F, ~b, Vgb). By inserting this expression in (138) and calculating the time derivative, one obtains t~,((0r~ - T). ~" +
- z).
+
+
- S ) . Vqb) 0
(144)
dr3 ,
Since one can arbitrarily choose in principle both velocity and acceleration fields, and the integrand is linear in the acceleration fileds, one obtains 0~.1# = 0, 0~ ~p = 0, 0v~ ~P = 0 that reduce ~p to the expression (139).
B.
C O N S T I T U T I V E R E S T R I C T I O N S AT D I S C O N T I N U I T Y S U R F A C E S
When a discontinuity surface E endowed with its own structure crosses the body, a surface excess energy must be accounted for. It assures the stability of the surface itself, in some sense its existence as a "thin layer" within the body. In this case, one writes {free energy of t3*} - f~, o + f B * n z 4~
(145)
where 4~ represents a surface free-energy density (i.e., the excess energy at the interface). Consequently, the mechanical dissipation inequality changes as
dt
*
V, +
*NZ
~
)
-
*
( T . ~" + z. ~, + S . V~,)
, ~ (
m. [~l + <S>m . [ ~ l ) - f ~ , ~ (T 9F + ~ 9Iq + 3 9~,•
_< 0 (146)
Multifield Theories in Mechanics of Solids
41
where, in writing (146), one uses the result in Proposition 6. By developing the time derivative of the integral ft3, ~P and taking into account the results of Proposition 7, the interfacial mechanical dissipation inequality follows:
afu
dt
*n~
fu *nz (
- { (ql". I~ + S. fil + 3" ~• Jr3*NE
< 0
(147)
Usually, one may accept for 4~ the following constitutive structure:
4, - ~(F, ~, N)
(148)
underlining in this way that the excess energy at the interface depends on the bulk deformational and substructural state immediately next to it. By developing the time derivative of 4~, the interfacial mechanical dissipation inequality (147) reduces to
f~
*NZ
((aF4, - 7 ) . I~ + (a~4, - 3)" r177+ (aN4, - s). lq)
- f ((T)m. [k] + (S)m. [~]) < 0 Jr3*NE
(149)
Given any triplet (F, qa+, N) representing the isothermal state of any material patch at the interface, it is possible in principle to choose arbitrarily velocity fields F, ~+, Iq from (F, qo+, N). Because the integrand of previous inequality is linear in the velocity fields, the following proposition holds.
Proposition 8 of interaction:
The following constitutive restrictions hold for surface measures qI' = OF4~(F, qo, N)
(150)
3 = 0~,#~(F, qp, N)
(151)
S = 0N4~(F, q#, N)
(152)
The arbitrariness of E implies also that (T)m. [k] + (S)m. [qb] > 0
at E
(153)
Once the explicit expression of the surface free energy has been selected, the surface measures of interaction follow from (150) to (152). Representation theorems for such energy expressions would be desiderable; they could assure that the derived measures of interaction possess the properties assigned by definition.
42
Paolo Maria Mariano
A research program with the aim of describing (or better, representing) energies such as 4~(F, qg, N) for interfaces is still far from being developed.
BIBLIOGRAPHIC NOTE
A general discussion on the expression of the principles of thermodynamics in multifield theories together with the derivation of bulk constitutive restrictions in absence of interfaces is in Capriz (1989). Standard references on thermodynamics of continua are Truesdell (1984), Coleman and Owen (1974), Ericksen (1998a), Silhav3~ (1997). The validity of a general mechanical dissipation inequality such as (136) has been proved in Mariano (1998) by requiting the boundedness from below the action functional along state transformations.
VI. Evolution of Defects and Interfaces in Materials with Substructure
When a body B is subjected to standard deformations, a material patch at X is mapped into x(X, t), the placement in the current configuration ~t at the instant t. Conversely, by the inverse motion x-l(X, t), one may associate the actual placement x of the material patch with its original place X. The picture becomes more complicated when the integrity of the body is altered by, for example, the evolution of defects or phase interfaces. When one observes such an evolution, one may describe it by taking time-varying parts of the reference configuration B. In the following, R(t) indicates any time-varying part of B. The loss of integrity of the body due to the evolution of defects or phase interfaces is basically considered as an additional "independent" kinematics in the reference configuration; this kinematics is associated to R(t). The time variation of R(t) generates new interactions between R(t) itself and the rest of the body; they drive or obstruct the motion of R(t) and are measured by means of a set of additional forces called configurationalforces. These forces are material forces in the sense that they live (or better, they are generated) in the reference configuration. In the standard picture of deformative processes, configurational forces do not exist. The first Piola-Kirchhoff stress T and the referential body forces b (as well as the referential measures of substructural interaction/3, z, S) are only mathematical pictures in B of the real interactions induced by deformation in the current configuration. They are obtained, in fact, by means of pull-back of the measures of interaction arising in the current configuration into B, which is considered fixed permanently.
Multifield Theories in Mechanics of Solids
43
When some part of 13 varies in time, the configurational forces generated by its evolution are measured through second-order tensors and internal forces. These tensors are introduced first as undeterminate terms that develop extra power in the kinematics of R(t), then are expressed explicitly in terms of standard and substructural measures of interaction. Inertial effects are neglected in this section for the sake of simplicity.
A. CONFIGURATIONAL FORCES IN THE BULK
It is assumed that the boundary 0 R(t) of R(t) is a regular surface in R 3, endowed with a outward unit normal n, and parametrized by a pair of parameters Ul and u2 in a way such that
X ~ OR(t)----> X -
X(ul,
U2,
t)
(154)
The velocity v of 0 R(t) in the reference configuration is ^
u -- 0 t X ( U l , //2, t)
(155)
Only the normal component V -- v 9n of v is independent of the parametrization (Ul, u2, t), and this property is crucial in the following. Let e be any continuous field of the time t and the place X. The time derivative of e following 0 R(t) is indicated with e ~ and defined by eO = ~ d e(]~(u i, u2, t'), t')lt,= t
dt'
(156)
and is the time derivative of e along trajectories crossing 0 R(t) at t' = t. When the previous definition is applied to x(X, t) and qo(X, t), it follows that x~ =
~ + Fv
(157)
qo~ = qb + (Vqo)v
(158)
The configurational interactions generated by the motion of R (t) in the reference configuration/3 are measured by means of 9 a
second-order tensor ~' representing the configurational stress
9 an internal configurational body force g 9 an external configurational body force e
44
Paolo Maria Mariano
The balance of bulk configurational interactions and their expression in terms of standard and substructural measures of interaction are shown in the following discussion. To obtain these results, first consider material observers that measure events in the reference (material) configuration. In evaluating the power -,pext - R ( t ) developed during the migration of R (t) in/3, a fixed material observer writes
ext_f R(t)
(b. z~+ / 3 . ~) + (t)
(Tn. x ~ + S n . ~p~ + ~ n . v)
(159)
R(t)
Forces g and e develop no power because they act on bulk points of/3 that are considered fixed when evaluated by a fixed material observer. Bulk interactions b and/3 develop power because they are mathematical pictures in/3 of interactions that in the current configuration act on points that move with respect to the material observer through x (X, t). When one considers a material observer moving with constant velocity v*, such an observer sees g and e as migrating with velocity v* relative to him; so g and e ,pext, mov. obs. evaluated by develop an extra power density (g + e) 9v*. The power --R~t) the moving observer can be thus written as R(,)mo
.
--
(b. ~ + / 3 . ~) + (t)
f.
(g + e). v* (t)
+ f (Tn. x ~ + S n . ~ ~ + ~ n . (v + v*)) J~ R(t)
(160)
However, a classic axiom of mechanics requires that the power be independent of the observer, namely, j~)ext
R~t)
~
,E)ext, mov. obs.
--R~t)
VV*
(161)
To satisfy this requirement, taking into account the arbitrariness of v*, it is necessary and sufficient that
f
R(t)
n+f, (g+e)-0 (t)
(162)
which is the integral balance of configurational forces on R (t). The arbitrariness of R (t) and the application of Gauss theorem imply DivY + g + e = 0
in 13
which is the local balance of configurational forces.
(163)
Multifield Theories in Mechanics of Solids
45
By taking into account the explicit expressions (157) and (158) of x ~ and qo~ the power "l:3ext --R(t) can be also written as 'Qext--fR R(t)
(t)
(b./~+/3.~)+f
R(t)
(Tn.~+Sn.qb)
+ f (~ + F r T + Vqor.S)n 9v Jo R(t)
(164)
As anticipated at the beginning of this section, only the component of v normal to OR(t) is independent of the parametrization of OR(t), whereas the tangential component of v depends on the parametrization itself. Physical plausibility requires that -"lgext - R ( t ) be independent of the parametrization of 0R(t). To satisfy this requirement, it is necessary that (~ + F r T + ~7qor*S) n be simply a vector normal to 0 R(t). In this way, it takes only the normal component of v when it is multiplied by v. Consequently, it is necessary and sufficient that + FTT + ~7~or,_S = wl
(165)
where co is a scalar function that is undetermined at this stage and I is the unit tensor. To determine co, the mechanical dissipation inequality is used. It requires here that md
"19ext ; ~---n(t)-
(166)
This inequality is conceptually analogous to the inequality (136). The sole difference is that here the part R of 13 varies in time. As a consequence of the timevariation of R, the following transport theorem holds:
"f,
-'~
(t)
(t)
++f
(167)
R(t)
where V is the normal component of v: V - v- n. By taking into account (165) and (167), the mechanical dissipation inequality may be written as
f,
(t)
R(t)
(t)
(b.zt+/3.@)-f~
R(t)
(Tn.~:+Sn.qb)<0
(168) This inequality must hold for any mechanical process involving R(t), that is, it must hold for any choice of velocity fields. Thus, the following identity must hold c o - 7z
(169)
Paolo Maria Mariano
46
Finally, this implies from (165) that ?-
fl-
F T T - VqoV._S
(170)
In this way it is shown that the configurational stress IF' can be expressed in terms of the free energy density, 7t and the stress measures T and S. When the setting is conservative, 7t reduces to the strain energy density w and has thus the expression of the generalized Eshelby tensor obtained in Section III. Here, the way followed to obtain the explicit expression of ~ assures that such an expression holds even for dissipative mechanical processes. This result guarantees the possibility of using the expression of ~ to describe dissipative processes like the evolution of macrocracks. As a result of the validity of the balance of configurational forces and standard and substructural balances of interactions, the expression of IP implies that g - -VTz + T 9V F + S+VVqo + z+Vqo
(171)
e - - F T b - Vqo T+/3
(172)
B. CONFIGURATIONAL FORCES ON A DISCONTINUITY SURFACE Let E(t) be a moving interface within R (t) of normal m (t), as in Section IV. The boundary 0 R(t) N E(t) of R(t) n E(t) is a curve that can be parametrized by some parameter u in a way such that X 6 0 R(t) n E(t) is given by X - X(u, t). Thus, the velocity field u(X, t) of the curve OR(t) n E(t) is given by u(X, t) -- OtX(u, t)
(173)
Because E(t) moves within the body, points of it are described by X 6 E(t) so that X - X(vl, v2, t), where (vl, v2) is a parametrization different from (ul, u2) on OR(t). Consequently, the velocity ~ of E(t) may be defined as f(X, t)
-- OtX(Ul,
1)2,
t)
(174)
The attention is focused in the following discussion on normal velocities -~ - U m . Note that u-
~ + U0v
(175)
where, as in Section IV, u is the normal of the curve OR(t) N E(t) directed along the tangent to E(t) at OR(t) n E(t) and Uo = u . u. Two different time derivatives may be defined: the first is the time derivative following the curve 0 R(t) n E(t), and the second is the time derivative following
Multifield Theories in Mechanics o f Solids
47
E(t). Let e be any continuous field of the place X and the time t; the former type of time derivative is indicated with e ~, the latter with e -~. Velocity fields following the curve 0 R(t) N E(t) are given by x ~ =/~+ + F•
= (/~) + (F)u
qo~" : ~ + + Vqo+u : (~> + (Vqo)u
(176)
(177)
Velocity fields following E(t) are given by x ~ = ~+ + F+~ = (~) + (F)~
(178)
qo-~ = qb+ + V q o •
(179)
(~) + (Vqo)~
The following relations may be derived easily from previous definitions after some algebra: x ~" = x -~ + UoFu
(180)
qo~ = qo~- + Uo N u
(181)
The case in which a moving interface E(t) crosses R(t) introduces the need for additional configurational forces strictly related to the peculiar kinematics of the interface. They represent the forces driving or obstructing the interface: their balance allows one to derive a complete evolution equation for the interface itself. Surface configurational interactions are measured by means of 9 a surface configurational stress C 9 a surface internal configurational force gz The stress C is a second-order tensor that maps vectors tangent to E(t) into vectors of R 3. In evaluating the power "19ext --R(t) on R(t) taking into account the structure at the interface, a fixed material observer measures an extra power due to C v . u and then writes in the present situation
extf.
R(t)
:
(t)
( b . x --}-/~. ~ ) --}-
( T n . x ~ + S n . q o ~ + IPn. v) R(t)
f + ] ( C u . u + ql'u. x c> + S t , . qoc>) Ja R(t)AE(t)
(182)
Taking into account the definitions of x ~ qo~ x ~, qo~', and the results on bulk
Paolo Maria Mariano
48
TQext may --R(t)
be written as
:~ + / 3 . ~) +
fOR(t)(Tn./~
configurational forces,
~ )R(t) e x t-f R (t)(b.
+ S n . qb + ~OV)
+ f ( T u . ~+ + S u . qb+ + (C + F+r• + Vqo+r_,S)u 9u) da R(t)AE(t) (183) Because TQext --R(t) must be independent of the parametrization of E as previously accepted in treating bulk configurational forces, here it needs to be independent even of the parametrization of the curve 0 R (t) 71 E (t). In other words, if t represents the tangent field to 0 R(t) @ E(t), because the component o f u tangential to 0 R(t) N E(t) depends on the parametrization of the curve, to assure independence of the parametrization of the curve 0 R(t) N E(t), it is necessary and sufficient that f~
(C + F+rqF + V ~ +r_,s)u 9t - 0
R(t)AE(t)
(184)
However, C may be decomposed into the sum of its tangential component Ctan and normal component m | e to the interface" Ctan maps tangent vectors to E into vectors tangent to the deformed interface, whereas e - Cm. Thanks to the arbitrariness of R(t), (184) reduces to (C + F+rqI ' + Vqp+r*S)u 9t - 0. In other words, the projection of the vector (C + F+rqr + Vqp + r * s ) u on E must vanish; in symbols, one writes that (I - m @ m)(C + F + r 7 + Vqp + r * g ) u - 0, which yields
Ctan + [::T~ + NT_,S __ coz(l - m | m)
(185)
with cozc a scalar function that is undetermined at this stage. Consequently, it follows that C + (F)r'II' + (Vqo)r_, S - cor~(I - m | m) + m | c
(186)
where c - (C + (F)rql" + (V~)r_,S)m. The following formula connects vectors e and c" e - c - T r (F)m - S(Vqo)m
(187)
When one writes the power as perceived by a moving material observer one must consider the contribution of the internal force gr.. As in the case of bulk configurational forces, the requirement of invariance of the power with respect to changes from fixed to moving observers and vice versa implies the validity of the following integral balance of configurational forces that holds in presence of
Multifield Theories in Mechanics of Solids
49
moving interfaces within the body: 0 R(t)
(t)
R(t)nz(t)
(t)nZ(t)
It is assumed that g and e are continuous on/3, whereas IP may suffer jumps at the interface. Consequently, by shrinking R(t) to R(t) n X(t), the arbitrariness of R(t) implies the following balance of configurational forces at the interface: on ]E
[I?]m + DivzC + gz = 0
(189)
The normal component of the interface balance of configurational forces plays an important role: It allows one to write the evolution equation for the interface itself. To obtain such a normal component, it is necessary only to multiply previous balance by the normal in at ]E. First one must consider that DivzC = Div~Ctan + Divz(m | e) and that 9 m.
DivzCtan
Cta n 9t - -
--
T Divz (Ctan m)
-
Ctan 9V z m
- D i v z ( C r ( I - In | m)m) +
Cta n . t
9 m . Divz (m | c) = Divz c As in Section IV, L is the curvature tensor of Z such that its trace/C = tr L is the overall curvature. With these premises, the normal configurational balance of forces can be written as Ill. [~]nl
+ Ctan " t -+-
Divz;z + gz = 0
on E
(190)
where gz = m . g z . To obtain an explicit evolution equation for the interface, it is necessary to exploit some consequences of the mechanical dissipation inequality written here considering the contribution of configurational forces as [cf. eq. (146)]
'(f,
dt
(t)
(t)nz~t)
vext < 0
(191)
Standard transport theorems prescribe that
dt dt
(t)
(t)
R~t)
(t)NZ(t)
(192) (193)
(t)nz(t)
(t)nZ(t)
where 4~-~ is the time derivative of 4~ following Z(t).
R(t)NZ(t)
PaoloMariaMariano
50
As a tool for developing future calculations, note that from (175), (178), (179), and (185), it follows that fa R(t)NE(t) (Cv. u + T v .
-~
R(t)nE(t)
X t>
coEua+f
+
R(t)nE(t)
~o~') (Cu.~+Tu.x-~+~;u.qo
-~)
(194)
By substituting (183), (192), (193), and (194) into (191), the mechanical dissipation inequality reduces to
fR(t)~-~(t)n~(t,[~P]U+L(t,nr~(t)(4~--c~ICU) R(t)nE(t)
(t)
R(t)
R(t)nE(t)
(195) This inequality must hold for any choice of the velocity fields, and for any choice of Ua. This implies as a first result wE -- 4'
(196)
The identity (196) allows one to express the tangential part of the surface configurational stress explicitly in terms of standard and substructural measures of interaction and the surface free energy, namely, Ctan - -
4~(I - in | m) - FTT - NT,_S
(197)
By shrinking R(t)to R(t) N E(t), as a result of the regularity properties assumed (and declared) in previous sections and to (196), in the limit R(t) --+ R(t) N E(t), the inequality (195) reduces to ([Tin. 5r + [ S m . ~]) (ONE
-{
Jo R(t)NE(t)
(t)nz(t)
(t)NY,(t)
(Cu. ~ + Tu. x -~ + Su. qo-~) _< 0
(198)
To reduce the inequality (198) further, one may use the following lemma, which is valid under the assumptions declared in this section up to this point.
Multifield Lemma
Theories
in M e c h a n i c s
51
of Solids
9
f
( C u . ~ + T u . x -~ + S t , . ~ o -<)
R(t)nE(t) f
(ql'. F -~ + S . N -< + 3" (~o) -< - [ T m . ~] - [,Sm. ~])
f
([~p]U + U g z
dR(t)nE(t) dR(t)NlE(t)
+ wzU/C + c . m -~)
(199)
The proof of this lemma is rather long and somewhat tedious. It is based substantially on the definitions (176)-(179), the use of Gauss theorem, and the auxiliary results 9 V z ~ - - V z ( U m ) -- - m | V z U + U V z m -- - m | m -~ - UL; 9
x -~ 9Divz'~' -- - [ T m .
~] - U m .
[FTT]m;
9
~o-~ 9D i v z ~ - 3" (~o) -~ - [ S m . ~b] - U m .
[V~pr,_S]m;
9 ql'- V z x -< = T . (F) -< - (T T ( F ) m ) . m -< - u(FTqr) 9L; 9 S . Vzqo -< = S . (V~o) -< - (S r m) 9m -< - u(NT_,s) 9L. The results in this list are consequences of previous definitions in Section VI and the balances at the interface derived in Section IV. By introducing (199) into (198) and taking into account (196), one obtains the integral dissipation inequality at the interface
L(t)nE(t) ~)~- -~-L(t)nz(t) UgZ + f
dR(t)nz(t)
(c. m -~ - T . F -~ - 3" (~o) -~ - ~" N -~) 5 0
(200)
The arbitrariness of R ( t ) implies the following local version of the mechanical dissipation inequality, namely, ck <- + e . m <- - T . F <- - 3 . (~o) ~- -
S . N <- + U g z
<_0
(201)
Two results follow from (201). First, it is possible to generalize the results of Proposition 8 to take into account the possible anisotropy of ~2(t). To this aim, one may choose a constitutive relation of the form 4~ = ~(m, F, q0, N)
(202)
52
Paolo Maria Mariano
where the dependence on the normal m accounts for the anisotropy of the interface itself. In further steps, one calculates the time derivative 4~-~ of 4~ following E(t), inserts it into (201) (by collecting analogous terms) and observes that, given any state (m, F, qo, N), velocity fields m -~, F -~, (qo)-~, and N -~ may be chosen arbitrarily from (m, F, qo, N). Consequently, because (201) must hold for any choice of velocity fields, the following relations must hold:
"IP - aFt(m, F, ~, N)
(203)
,~ = a ~ ( m , F, ~, N)
(204)
,~ -
aN~(m,
(205)
c -
- a m P ( m , F, ~ , N)
F, qo, N )
(206)
where, relative to the results in Proposition 8, there is here an explicit dependence on the normal of the interface and vector c represents the force generated by the anisotropy of E during the motion. Consequently, as a result of (203)-(207), the mechanical dissipation inequality reduces to gz U _< 0
(207)
gz - - ~ z ( m , F, ~o, N, U)U
(208)
A solution of (207) is given by
where gz is a positive scalar function of arguments that are the same as in (203)(206) plus the normal velocity U. Note that gz is the normal component of the dissipative force gz driving the interface [inequality (207) states the dissipative nature of gz]. Because gz is dissipative, it depends on the velocity U and may depend on the other state variables: gz must be assigned by a constitutive prescription. When one assigns ~z, one determines the evolution of E(t). The evolution equation of E(t) is, in fact, the normal configurational balance (190) once one substitutes in (190) the explicit expression of ~ [identity (170)], the expression of Cta n [identity (197)], the constitutive restriction (206) [taking into account (187)] and finally (208), which is the fundamental ingredient for writing the evolution equation. Remark 6 With the previous results, it is possible to prove that the tangential component of the configurational force balance (189) is identically satisfied once gz is purely normal" namely, once gz(I - m | m) - 0.
Multifield Theories in Mechanics of Solids
53
BIBLIOGRAPHIC NOTE A rather general treatment of the influence of material substructures on the expression of bulk and surface configurational forces can be found in Mariano (2000a). Some of the proofs discussed here are developed following different methods in some aspects from the ones of the previously mentioned paper. Results on the influence of scalar order parameters on the influence of bulk configurational forces can be found in Fried and Gurtin (1993, 1994, 1999) and Fried and Grach (1997). A detailed treatment of configurational forces in standard Cauchy materials can be found in Gurtin (1993, 1995, 2000), Gurtin and Struthers (1990), Angenent and Gurtin (1989), Abeyaratne and Knowles (1991), Gurtin and Podio-Guidugli (1996), Maugin (1993, 1995), and references therein. The term configurational is due to Nabarro (1985) (see Ericksen, 1998). The seminal ideas on this subject are in Eshelby (1951, 1975).
VII. Crack Propagation in Materials with Substructure One application of the framework of configurational forces discussed in Section VI is the analysis of the propagation of macrocracks. Evolving cracks may be considered to be evolving curves (for two-dimensional bodies) in the reference configuration. By inverse motion, in fact, one could pull back the actual opened and deformed crack into a curve in the reference configuration, which is a picture of the crack taken as closed. Of course, when the crack evolves in the actual configuration, its picture in the reference configuration evolves in time: configurational forces driving the crack must be considered. By using some of the results presented in the previous discussion, it is possible to discuss the influence of material substructures on the force driving the crack tip. Experiments show the influence of material texture on crack propagation (see Herrmann and Roux, 1990). When one models some material as a Cauchy continuum, such an influence is considered only indirectly, through constitutive equations, and often models of the local cohesive structure at the tip of the crack are proposed. They offer interesting special solutions; however, these models can be difficult to manage when one considers the crack propagation to be a global bifurcation problem, because in this case it is often necessary to know the overall distribution of material texture rather than the situation near the crack tip. In the framework of multifield theories, one may derive the driving force at the crack tip in a way accounting for the explicit influence of the perturbations induced
Paolo Maria Mariano
54
on standard interactions by material substructures. This is the result shown in this section, in which for simplicity, all calculations are restricted to two-dimensional bodies.
A. KINEMATICSOF PLANAR MOVING CRACKS
In this section, the body B is considered as an open compact region of the two-dimensional point space ,re. A crack in B is identified as a regular curve C represented by a function r : [0, ~] ---> B (with [0, ~] some bounded interval of the real axis ~) defined as follows:
{r
' ( 0 , ~ ] --+ B r(0) ~ 0B
(209)
This definition states that one considers the crack as a curve in B starting from the boundary and ending within/3, without intersecting further on the boundary of/3. The crack tip is, of course, r(g), and is indicated by Z; it is a point in oc2 belonging to the interior of/3. The curve C is also characterized by its tangent vector t - 0~r, 5 ~ (0, g) and the lateral normal m (thus such that t. m = 0). When the crack evolves in a time interval [0, d], C is a function C(t) of time, defined as
C(t) = {r(s, t) E 1315 E [0, ~], t E [0, d], r(0, t) ~ 0B, r(0, t) -----r(0, 0), Vt} (210) with the condition
C(tl) c C(t2), u
< t2
(211)
It is assumed that C evolves without intersecting further on the boundary of B: the tip Z becomes a function of time Z(t) and Z(t) 6 B, whenever t 6 [0, d]. In other words, one imagines studying the crack propagation during a time interval in which the crack does not completely cut the body B (Figure 2). The velocity of the crack tip Vtip is defined as dZ Vtip = dt
(212)
When one assigns the tangent of C at the tip Z--namely, t (Z(t))--one assigns the direction of propagation of the crack. It is thus possible to write Vtip = f'(t)t (Z(t)). From this definition, it follows that m(X) 9Vtip ~ O, as X --+ Z(t). Special parts of B are of future use; in particular, it is necessary to consider a disc D centered at the tip. The boundary 0 D of the disc intersects C at only one point
Multifield Theories in Mechanics of Solids
55
11
B t(z(o) FIG. 2. Geometric characterization of a cracked body. denoted with X A . The outward unit normal at 8 D is indicated by n, whereas the radius of D is indicated by O. When the crack evolves, the disc is considered to be time varying to follow the evolution of the crack; thus D and XA become D(t) and XA (t). Let f be the velocity of the curve C(t). The velocity of XA (t) can be written as ra(t) -~ Xa(t) -- Ua(t)ta(t), where/~a is the scalar amplitude of the velocity and ta is the tangent of C(t) at the point XA. Because t (Z(t)) is prescribed, it is assumed that ta(t) ~ t (Z(t)) Cta(t) - ~ V(t)
as O ~ 0 as O --~ 0
(213) (214)
Let e represent any continuous field of the place X and the time t. The time derivative of e following the crack tip is defined as e~(X, t) = 8te(Y, t)lv=x-z(t)
(215)
where Y denotes a genetic point in s By applying this definition to x and qo, the velocity fields following the tip can be written as
x <>- ~ + Fvtip qD<>= ~ + (V ~)u
(216) (217)
Assume now the existence of a tip velocity field ~tip and a tip order parameter ~lti p when the crack is deformed. To be more precise, first assume that if X 6 C and x(X, t) is the position of X at the instant t, then x(X, t) ~ x(Z(t), t) and qo(X, t ) ~ qo(Z(t), t), as X ~ Z(t) uniformly in time. The assumption of the existence of Vtip and Vitip i s tantamount to requiting that x<>(X, t) ~ Vtip and 9:,<>(X, t) ~ @tip, as X ~ Z(t) uniformly in time. rate
Paolo Maria Mariano
56
The boundary of the time-varying disc D(t) may be parametrized by some parameter v in such a way that X 6 0 D ( t ) =~ X - fK(v, t). The velocity vo ofthe boundary 0 D(t) of the disc is given by VD(X, t) = OtfK(v, t)
(218)
Only the normal component of v D - - n a m e l y , / 2 -- VD.n--is independent of the parametrization of 0 D(t). Velocity fields x ~ and ~~ following OD(t) may be defined in the same way as in (157) and (158) as X~ -- ~ + FVD
(219)
qa~ - qb + (Vqp)VD
(220)
The physics of the crack imposes a kinematical condition, namely a condition of impenetrability" [x] 9m >_ 0
(221)
In other words, when the body deforms, the margin of the cracks do not penetrate one into another. In presence of a crack, the mapping x(.) is no longer pointwise one-to-one and becomes a piecewise bijection" x(.) is one-to-one to within the curve C; that is, a set of zero volume measure.
B. BALANCE OF STANDARD AND SUBSTRUCTURAL INTERACTIONS AT THE TIP The following assumptions on standard and substructural measures of interaction apply: 9 bulk measures of interaction b and/3 are continuous on/3 9 the self-force z is continuous on/3 9 stresses T and S may be singular at the crack tip and suffer jumps across C; they are also continuous and continuously differentiable outside C 9 surfaces stresses 7i" and S and surface self-force 3 are not considered Two new measures of interactions are assumed acting at the crack tip; they represent external actions acting at the crack tip as "concentrated f o r c e s "
Multifield Theories in Mechanics of Solids 9 tip force
57
btip
9 tip substructural measure of interaction
~tip
The previous assumptions guarantee that balance equations (35), (38), and (42) hold in the bulk, and that balance equations (128) and (129) hold on C. Additional balance equations hold at the tip; they may be derived by writing the integral balances of interactions with respect to the disc D and shrinking the disc to the tip. Integral balances accounting for tip interactions are
fo
b+f
D
(Tn) + btip -- 0
fD(~ -- Z)-l- fo (Sn)-~- ~tip --
(222)
(223)
To write them, it is necessary only to add btip and Otip to (119) and (120). Because T and 8 may be singular at the crack tip, by shrinking the disc to the tip--that is, letting O ~ 0 u t h e s e balances reduce to
where
btip + ftip Tn - 0
at tip
(224)
/3tip + ftip 8n - 0
at tip
(225)
ftip must be interpreted as limo_,o fo"
Remark 7 The assumption that the pair (x <>, ~o<>)tends to (Vtip, ~Ntip) a s X ~ Z(t) implies that
retip) - 0
(226)
Sn . (~<> - fVt~p) = 0
(227)
L T n . (x ~ -
fp
C. EFFECTS OF INERTIA
To evaluate the effects of inertia on standard and substructural balances one may follow the technique adopted in Section II. To this aim, first assume the following
Paolo Maria Mariano
58
decompositions of tip interactions into inertial and noninertial components: bti p - blnp -~- btnip
(228)
fl tnip
(229)
fl tip -- fl iTp +
These decompositions must be used together with relations b = b in + b ni and f l _ flin+ flni. The noninertial part bt'~/p of btip is a possible standard external force applied at the tip of the crack, whereas flt~iipcan be interpreted as a possible external substructural action at the tip. The terms bt'~/pand fltnip are not essential, and are quoted here only for the sake of completeness. On the contrary, the inertial terms may be identified explicitly. To obtain this identification, it is first necessary to evaluate the production of standard momentum and the production of substructural momentum, indicated with pr and pr., respectively, during the evolution of the D(t). Because the disc D evolves in the reference configuration, there is an inflow of momentum across OD(t). The inflow of standard momentum is -faD(t)pi~bl, whereas the inflow of substructural momentum may expressed as -f~o(t)OcpX Lt, where L/ is the normal velocity of OD(t) (i.e., /d = yD. n). Consequently, one may write by definition
dfo p -f
prpr.
~
---
(t)
(t)
(230)
D(t)
(t)
D(t)
The basic step to identify bitp and fli~ is to consider a balance analogous to (53) and write
blnp +
fo
(t)
bin -- --pr
I' I~ITP + I t~in -- --]Or. dD (t)
(232)
(233)
By shrinking D(t) to the tip, the assumptions of regularity of bulk measures of interactions b and fl and the regularity properties of motions imply
fD(t) bin; fo(t) flin; m
d f
dt Jo(t)
p~:
d -~fD(t) ~ --fo(,)~
all tend to0
asD--+Z
(234)
Multifield Theories in Mechanics of Solids Because the previously listed terms vanish in the limit D ~ identifications hold because v o 9n = U --+
Vtip 9n)
bi~p - ft/p(pfc)(vt/p, n) =
as D ~
ftip(p~ |
59 Z, the following
Z
n)Vtip
~'~p : ftip(~X)(Vtip "n) ~ ~.p(~,x Q n)Vtip
(235)
(236)
As a consequence, the tip balances (224) and (225) reduce to
Remark 8
btn.p+ ftipTn : - ftt.p(px | n)vtip
at tip
(237)
tip "3t- Sn -- -- fti p (O(oX Q n)Vtip t~niftl.p
at tip
(238)
Physical plausibility suggests that standard and generalized tractions
Tn and Sn are bounded by up to the tip as D ~ Z. When bt~i%a n d fltnp vanish identically (as usual), the hypothesis of boundedness of the stresses implies fi Tn -- O ==~fi n| P
P
. Sn=O==~ fi n| p
(239)
P
D. TIP BALANCE OF CONFIGURATIONAL FORCES
When the crack evolves, its mathematical picture C varies in time and configurational forces intervene to obstruct or drive the crack. The following assumptions apply to the framework developed in Section VI:
9 ~ may be singular at the crack tip and suffer jumps across C; it is also continuous and continuously differentiable outside C 9 g and e are continuous and on I~ 9 C is now a vector c and is continuous along C 9 gz is continuous on C
Paolo Maria Mariano
60
New configurational forces are associated with the tip and are peculiar of tip singularity. They are 9 internal tip configurational force gtip 9 external tip configurational force etip of inertial nature The integral balance of configurational forces (188) on D must be written here accounting for the tip configurational forces. It may be deduced from work invariance arguments such as those, discussed in, Section VI and is expressed by
D(t)
IPn+ fo (t)
(g+e)+fo
(t)NC(t)
(gz)
+ gtip -k- etip -- CA - - 0
(240)
By shrinking D at the tip of the crack, equation (240) reduces to (241)
gtip + etip -- Ctip + ftip ]?n -- 0
which is the configurational tip balance. It is now necessary to characterize the tip configurational forces explicitly. First, the attention is focused on etip, which is of inertial nature. To derive an explicit expression for it in terms of velocity fields, one may follow a procedure analogous to that discussed in Section II.A and may derive etip by requiring that the rate of kinetic energy E of D(t) plus the power of all inertial forces on D(t) vanish identically. In symbols, one writes
d ~ + fo 0 -- dt
(bin "x + t~in "(P) + btip" in reap + ,~tip" in ,Tvtip+ etip 9Vtip
(t)
(242)
Because D(t) varies in time, in computing the rate of the kinetic energy it is necessary to consider its inflow through the boundary of D(t), because this inflow is due to the motion of D(t) itself. Then the rate of kinetic energy of D(t) is given by dt
~
fo(1 (t)
)
2(P~" ~) + k(~, @) -
D(t)
(p~. ~) + k(~, @) H (243)
The next step is the introduction of (243), (235), and (236) into (242) and the shrinkage of D(t) at the tip Z. Note that f~o(t)(.)H ~ ftip(.)(vtip, n) because
Multifield Theories in Mechanics of Solids
61
D(t) ~ Z(t). At the end of calculations, one finds that etip " u
-- u
" ftip (~(,o~, " x)-'F- k(qo, qo)) n
- vti p
f
Vtip)n-- u
. ftip(OCpX . Wtip)n
(244)
To reduce (244) further, one defines the relative kinetic energy ~.rel as ~.rel-1 1 1 "~plX-~ltip 12 and observes that ~.rel- -~p~Itip " Vtip -- "~pX" X - pX" reap. Moreover, Ytip PVtip " Vtip -- limos0 fad Pretip " ~r but retip is independent of spatial coordinates, then lim0~0 fad pretip 9retip -- lim0~0 retip 9r4tipfao P -- 0 because the density of mass p is a continuous function. By taking into account these auxiliary results on the relative kinetic energy, and because (244) must hold for any choice of the velocity of the tip, it follows that
etip--ftt.pl~reln-+-ftipk(qo,@)n-ftip(O(oX.f~Ctip)n
(245)
The identity (245) characterizes completely the external configurational force etip and specifies its inertial nature.
E. CONSEQUENCES OF THE MECHANICAL DISSIPATION INEQUALITY To give some characterization of the internal configurational tip force gtip, it is necessary to exploit the mechanical dissipation inequality, which is now written as
d re{free energy of D(t)} - {power developed on D(t)} < 0
dt
(246)
A line free energy density 4~ along the crack C is considered besides the bulk free energy density ~p. The line free energy accounts for surface tension c along the faces of the crack. It is assumed that ~p is continuous on/3 and may suffer jumps at C, whereas 4~ is continuous along C. The line free energy density 4~ does not depend on the time t. From (203) to (206), 4~ may depend only on the normal m of C because standard and substructural surface measures of interaction are not considered here. However, m does not depend on t because there is no motion of C along its normal. Possible phenomena of aging are not considered; thus 4~ does not depend explicitly on t. With these assumptions, d--td{free energy of D(t)} -- --~
(t) ~ + -~
(t)nc(t) 4~
(247)
62
Paolo Maria Mariano
Because the normal velocity of C is zero, the transport theorem (167) holds and is written here as --
(t)
dt
r =
(t)
r +
D(t)
eL/
(248)
whereas for the line free energy one finds -dt
(t)NC(t)
~) -- ~tip ~/ -- ~)A bl A
(249)
because the integral in (249) is a line integral. By inserting (248) and (249) into (247) and writing explicitly the power developed on D ( t ) , it follows that fD
(t)
~-s .qt_ f
-fo
(t)
D(t)
(!~r~[) '~ ~)tip (/ -- ~)A~IA
(b"/~ +/3" r
-- btip 9~r
fo
D(t)
(Tn. k + S n .
r + CU)
- ~tip " fVtip - etip 9Vtip + U AIA 9CA < 0
(250)
The last integral in (250) is an alternative manner to write f
D(t)
(Tn.
Xo
+ Sn.qo + I?n. vo) o
and the expression used in (250) can easily be obtained by using (219), (220), and (170) [see also the analogous integral in (183)]. A first result can be obtained directly from (250). Because the inequality (250) must hold for any choice of the velocity fields and--among othersmfor any choice of/~a, as a result of the arbitrariness of D ( t ) , the following identity must hold:
= t. c
(251)
it characterizes completely the tangent part of the configurational stress c: this tangential part is the surface tension along the faces of the crack. By shrinking D ( t ) at the tip, one obtains from (250) the tip mechanical dissipation inequality: ~)tip V -- btip " Vtip - ~tip " Wtip - etip " u
- ft, Tn. ~ - ft/pSn 9~, _< 0
-
lP(Vtip . n )
(252)
63
Multifield Theories in Mechanics of Solids
The use of the tip configurational balance (241) and the identity (251) allow one to reduce the tip mechanical dissipation inequality (252) to
ftip 1 .fti,o fti o o
gtip " Vtip -- btip
" Vtip -- tOtip "~ltip
--
l[f(Vtip
"
n)
Note that from (170) it follows trivially that
so that inequality (253) reduces to gtip 9vtip - btip 9Vtip - ,Otip 9Wtip - ftip T n . (~, + Fvtip) - ftip,..~n . ( ~ -4c-(~Tqo)Vtip) < O
(255)
From (216) and (217) and the assumption previously stated that (x<>, qo<>) (Vtip, wtip) as X ~ Z, it follows that ft.pTn'(~'+Fvtip)+fttipSn'(~(Vq~
Sn
(256) By substituting (256) into (255) and using the tip balances (232) and (233), the mechanical dissipation inequality reduces to gtip " Vtip < 0
(257)
which characterizes the dissipative nature of the internal configurational force gtip. It is necessary to prescribe constitutively only gtip rather than the energy release rate (as usual in technical literature). The constitutive prescription of gtip is subjected only to the condition (257).
F. DRIVING FORCE
Previous results allow one to derive an expression of the driving force at the tip of the crack, which accounts for the presence of material substructures and the interactions they generate (and consequently the expression of the energy release rate during crack propagation). To simplify the developments in the following, it is first assumed that btnp and/3t]p vanish identically.
64
Paolo M a r i a M a r i a n o
By using the identities (170) and (245), the tip balance of configurational forces can be written as ftip((lP "at- ~.rel "-[- k ( ~ , (a) - O~X 9qVtip) I - F r T - ( V ~ ) r * S ) n - (.tip -- -grip
(258) The integral in (258) is indicated here with jm for compactness of notation, jm represents the tip traction exerted by the material on an infinitesimal neighborhood around the tip of the crack. The component of the tip balance of configurational forces along the direction of propagation of the crack is obtained by multiplying the balance (258) by t (Z(t)) t ( Z ( t ) ) . jm -
t (Z(t))
9 C,ip -
-gtip.
t (Z(t))
(259)
By denoting with Jm the product t (Z(t)) 9jm and using (251), equation (259) may be written as (260)
Jm -- Ck,ip -- --grip" t (Z(t))
In equation (260), the term grip 9t(Z(t)) is the internal force exerted by molecular bonds that opposes motion of the tip, whereas f - Jm -dPtip -- O(oX " W t i p )
t (z(t)) 9f,/p((~ +
ereZ +
~:(~,, ~)
I - FTT + ( v ~ ) T * S ) n -
~)tip
(261)
is the driving force at the tip accounting for the influence of material substructures. Because Vtip = ~'(t) t (Z(t)), the internal dissipation inequality (257) and the tip balance (258) imply f'~' >_ 0
(262)
which represents a version of the internal dissipation inequality. From (262), when the crack grows (i.e., when V > 0), 9 the driving force must be nonnegative f >_ 0
(263)
9 the tip traction must form an acute angle with the direction of propagation
t(Z(t)), j,, >_ 4~tip > 0
(264)
Multifield Theories in Mechanics of Solids
65
The results (263) and (264) coincide with the analogous results in Cauchy continua.
G. A MODIFIED EXPRESSION OF J INTEGRAL
As a consequence of (239), the tip traction jm reduces to jm --
(( ~ + ~Pls 1 2 + k(qo, ~) ) I - F T T - ( V ~ ) T,,S ) n
(265)
Consequently, it follows that Jm V -- jm " Vtip = jm " V ( t ) t (Z(t))
= ftip ( T n . i~+Sn. ~ + (Tz + ~ Pl~lZ + k(~p, (o)) (Vtip . n))
(266)
The product Jm V is the flow of energy into an infinitesimal neighborhood of the crack tip. Jm is thus the dynamic energy release rate accounting for substructures because it has the physical dimensions of an energy. When the influence of substructures is not considered, Jm coincides with t(Z(t)).
' )
~p + ~Pl~l 2 I - FTT
)n
which is the standard dynamic energy release rate in Cauchy continua (see, e.g., Freund, 1990; Gurtin, 2000; Maugin, 1992). If inertial effects are absent, the tip traction jm reduces to its "quasi-static", counterpart jm,qs given by
jm,qs -- ftip(~l - FTT-(Vqo)T ,__S)n -- ftip~n
(267)
If the body forces b and ~ are absent, the material is homogeneous [in particular, ~ = ~(F, qo, Vqo), so ~p does not depend explicitly on X], the faces of the crack are free of standard and substructural tractions (T• = 0; S • = 0), the "quasi-static" energy release rate Jm,qs - t (Z(t)) . jm,qs i$ path independent; that is, Proposition 10
Jm,qs -- t
(Z(t)) . fr Pn
(268)
where F is any closed, regular, nonintersecting path beginning and ending at the crack.
66
Paolo Maria Mariano To sketch the proof of previous proposition, let intF denote the closed region
of,f2 with boundary 1-'. By Gauss theorem, fr ]?n = fintF Div]]3) + fintI'nC []l~]m" By using the configurational force balance (163), Div? may be substituted by the sum - (g + e). The absence of body forces implies e = 0 [see (172)], whereas the homogeneity of the material implies g - 0 [one calculates the gradient of 7t, inserts it into (171), and uses Proposition 7]. In addition, by using (170), one realizes that the hypothesis concerning the faces of the crack implies that [/~]m = [~]m. As a consequence,
t (Z(t)) 9f~trnc [~p]m -- f~trnc [~p] t 9m - 0 and the validity of Proposition 10 is proved.
H. ENERGY DISSIPATED IN THE PROCESS ZONE
When a crack propagates in an elastic-plastic material, a critical zone around the crack tip occurs: the process zone. It is highly unstable (in the sense that any increment of the loads may alter its coherence even drastically), in a certain sense "fragmented," so that one may doubt that basic axioms of continuum mechanics (e.g., the continuity of the material) do not work well within it. When one evaluates the energy dissipated during the evolution of the crack, hence of the process zone, one realizes that J integral is no longer sufficient to describe the energy dissipated into the process zone and other path integrals must be introduced. Let P indicate the process zone around the crack tip. Assume that the boundary 0 P of P is a closed regular curve without self-intersection that admits normal n. During the evolution of the crack, P is considered time dependent [i.e., P = P(t)]. The curve 0 P(t) is described by some function X = X(vp, t), with Ve an appropriate parameter along 0 P(t). The velocity Vp of 0 P is given by ve(X, t) -OtX(ve, t). In a local frame {X*} of coordinates centered at the tip Z, the velocity Vp may be written as V p = Vp,tr
+ CI X X* + a ' X * + a
where Vp,tr and/1 are the rigid translational and rotational components, respectively; a'X* is the component of the velocity associated to the self-similar
Multifield Theories in Mechanics of Solids
67
expansion of P(t); and d is the component associated to the distortion of the process zone. Of course Vp,tr, (~, and a* are independent of the space coordinates. Proposition 11
The energy dissipated into the process zone, ~(P), is given by dp(p)
--
YP,tr
" jm(P)
+/1" L + a*M + I
(269)
where i n ( P ) --
1 (( ~pl:~l 1 2 ) 1 (( 1 ) ( ( ,~pl:~l) 2 + k(~o, ~) I 1 (( 1 ) r +
L =
!/r + ~PlRI 2 + k(qo, qb) I - FTT - (gqo)r_,s
P
M =
r +
n x X*
- FTT - (Vqo) T_,S n . X*
P
I =
) ) ) )
+ k(cp, ~) I - F r T - (Vqo)~_,S n
P
r + ~plfr 2 + k(qo, qb) I - F r T - (Vqo)r_,S
P
- f (Tn. ~+Sn-~b) aa P
(270) (271) (272)
n-a (273)
Note that when the results of this subsection are applied to the direct modeling of plates (i.e., to Cosserat surfaces) it is possible to express both J integral and the basic laws of the evolution of cracks (224) and (225) directly in terms of normal and shear stresses and bending moments. This result allows one to obtain a strong reduction of the computational burden in numerical calculations involving cracks that cut the thickness of the plate completely.
BIBLIOGRAPHIC N O T E
This section is based mainly on some unpublished notes of the writer. Proposition 11 may be proved by adapting to the present situation the general results of Proposition 5 in Mariano (2000a); a special case of Proposition 11 can be found in Mariano (1995). In the case of Cauchy materials, the evolution of cracks has been treated with the framework of configurational forces in Gurtin and Podio-Guidugli (1996), Gurtin (2000) and Maugin (1992), and some classic results on crack propagation (see Freund, 1990) on fracture have been reobtained within such a theoretical setting.
68
Paolo Maria Mariano
Detailed discussions on the process zone around the crack tip can be found in Aoki et al. (1981, 1984), Curtin and Futamura (1990), Hutchinson (1987), Freund and Hutchinson (1985) and Lam and Freund (1985). The concept of energy release rate has been introduced in Atkinson and Eshelby, (1968) and Freund (1972), whereas the original motivation of the J integral can be found in Rice (1968).
VIII. Latent Substructures Material substructures are called latent when there is a set of holonomic or anholonomic constraints relating the order parameter to the descriptors of the macroscopic motion and deformation. In defining the concept of latence, Capriz (1985) writes: "I say that the microstructure is latent when, though its effects are felt in the balance equations, all relevant quantities can be expressed in terms of geometric and kinematic quantities pertaining to apparent placements" (p. 49). First, one assumes that 9 there is no substructural inertia: k(qp, qb) - 0 9 substructural bulk interactions are absent:/3 = 0 An immediate consequence is that the balance of substructural interactions (63) reduces to DivS = z
(274)
Consequently, the generalized balance of couples (38) changes in eTF T = Div(.A TS)
(275)
whereas the density of internal power of substructural interactions changes in z. ~ + S . ~'~b - Div(S r qb)
(276)
It has a divergence form only and the product sT ~b can be interpreted as a substructural flux of power that corresponds to the interstitial work flux, which is necessary to consider in the special case of higher gradient elastic materials (as shown in the following). Another crucial assumption is the following: 9 the substructural flux of power is objective
Multifield Theories in Mechanics o f Solids
69
This condition requires that ST ~b must not change when qb changes into ~ + A~I (i.e., S T ~ -- S r (qb + .A~I) for any choice of the rigid rotational velocity el)- This implies that ST A = 0
(277)
Condition (277) further reduces the balance of couples to TF T = FT T
(278)
which is the standard symmetry condition of symmetry Cauchy stress TF T. The last assumption that defines completely latent substructures is the following: 9 the order parameter is constrained by a set of frictionless holonomic and anholonomic constraints that express it in terms of the deformation gradient F and, perhaps, of its gradients R e m a r k 9 When Div(S ~ ~b) -- 0 for some special choice of the order parameter, substructural interactions become powerless. In this case, the order parameter appears on constitutive equations only, and an evolution equation for it must be considered instead of the balance of substructural interactions. This is another case in which multifield theories reduce to internal variable schemes. R e m a r k 10 The balance of substructural measures of interactions gives rise to evolution equations for the order parameter even in situations more general than those occurring in the case of latent substructures. The constitutive prescriptions of Proposition 7 are peculiar of thermodynamic equilibrium, or of a reasonable 13 neighborhood of it. In principle, nonequilibrium parts of the interaction measures, depending on the velocity fields, could be considered along nonequilibrium thermodynamic processes. This happens, for example, when viscosity phenomena occur. An interesting case occurs when one considers a decomposition of the self-force z into its equilibrium and nonequilibrium components. The equilibrium part of z is given by (142), whereas the nonequilibrium part z ne may depend on the rate of the order parameter, that is, z -- 0~o~(F, qo, Vq~) + z"e(F, qa,Vq~; qb)
(279)
The nonequilibrium part of the self-force z ne is dissipative and is such that z "e. qb > 0 13The physical meaning of reasonable is currently a matter of open discussion.
(280)
70
Paolo M a r i a M a r i a n o
A solution of (280) is
(281)
Zne --zne~o
with i ne an appropriate definite positive tensor (possibly scalar in some special model) such that ine is a function
i n e : zne(F, qo, Vr
~0)
(282)
In common special cases, a decomposed free energy of the form
-- ~l(V, ~) + @2(qO,V~) [with 61(I, qg) -- 0, I the unit tensor] may be selected and
(283)
lp2 chosen as
1
~ 2 = -~b V qo . V qo + cr(qo)
(284)
with b an appropriate constant and cr(qo) a double-well coarse-grained potential, as in cases of solidification or solid-to-solid phase transitions. When this happens, the bulk balance of substructural interactions (274), as a result of (143), changes into
A ~ - b A q o - 0~0o'(~o) - 0~o~l(F, ~)
(285)
which is a generalized form of the Ginzburg-Landau equation. When, in fact, both A and qO are scalar valued and the body does not undergo deformations, (285) reduces to A~b = bAqo - 0~ocr(~o)
(286)
which is the standard Ginzburg-Landau equation with kinetic coefficient A.
A. SECOND-GRADIENT THEORIES AS SPECIAL CASES OF LATENT SUBSTRUCTURES An important case of latence is characterized by the internal frictionless constraint = ~(F)
(287)
which expresses the order parameter as a function of the macroscopic gradient of deformation F. From (287), by time differentiation one obtains that along the motion qb = ( ~ ) F "
= (~)(grad~)F
(288)
Multifield Theories in Mechanics of Solids
71
where grad indicates the gradient calculated with respect to x. When the velocity i is rigid [see (13)] gradi = e/l
(289)
@R --- e (OqF~) Fc 1
(290)
and (288) changes into
consequently, from (12) it follows that (291)
A = e(o~)F
This relation reduces (277) to (292)
F = o
eS r (~)
which implies that the third-order tensor A, whose elements are given by (293)
t~AB C -- ( s T ) ~ A ( O F ~ ) i B F i C
is symmetric in the last indices, that is, (294)
AABC = AACB
In the standard treatment of internal constraints [such as (287)] in Cauchy continua, it is necessary todecompose the stress T in its "active" and "reactive" parts, indicated with T and T, respectively (as standard in scientific literature on internal constraints). The latter is assumed to be powerless. Analogously, here it is prescribed that T=
r
+T;
a
z--z+z;
r
S=
~
+
(295)
~.F+~.~+~.V~=0 v~', r
(296)
and
By substituting (288) into (296), one finds two conditions. The first condition is that r
+ ~(aF~ + S(V(aF~)) = 0
(297)
The second condition is that third-order tensor S(0v~) is symmetric in the last two indices
(298)
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Paolo Maria Mariano
Another basic assumption here is that the free energy density 7r has the following structure: A
~p - ~p (F, V F )
(299)
When one uses the mechanical dissipation inequality to obtain constitutive restrictions on the measures of interaction, one finds that (140) reduces to
f
,
+ +
v + (~vv~P
vv) _< o (300)
Given any state (F, V F ) , velocity fields ~" and V~" can be chosen arbitrarily from (F, VF). This arbitrariness implies that a
a
I" + ~ + V(SOF~) -- (VS)OF~ -- 01~
(301)
a
S Ov~ - Ovv ~
(302)
By inserting (297) and (301) in (299) and using (302), one proves the validity of the following proposition"
Proposition 12 (Capriz, 1985) -- Ov~ - Div(Ovr~) - Div(F skw(OvF~F-l))
(303)
or, in components, Tiaa -- OFiAff/ -- (O(VF)iA, O ) , , --(Fis(O(VF)j,c ff/ Faj' -- O(VF)jBAO ) F ~ I ) , c
(304)
where capital indices refer to the reference configuration, whereas the other indices refer to the current configuration. Note that (302) is the standard constitutive restriction of second-gradient elastic materials, which are thus special cases of continua with latent substructure (i.e., special cases of multifield theories).
BIBLIOGRAPHIC NOTE
This subsection is based on Capriz (1985). Many other remarks on latent substructures can be found in Capriz (1989), whereas the special case of smectic liquid
Multifield Theories in Mechanics of Solids
73
crystals is treated in Capriz (1994). The standard theory of second-gradient Cauchy materials can be found in Dunn and Serrin (1985), where the necessity of the introduction of a rate of supply of mechanical energy, called interstitial working in the balance of energy, is proved to be necessary to eliminate the incompatibility with the second law of thermodynamics shown in Gurtin (1965) for these models of materials. The fundamental result of Capriz (1985) is that the interstitial working is not an object whose existence is assumed without any explicit reference to some types of interactions; rather, it is a consequence of the existence of substructural interactions due to a substructure that generates the oscillations of deformations that are measured by VF.
IX. Examples of Specific Cases The framework discussed in previous sections allows one to describe many material substructures. Detailed special theories can be found in the references listed at the end of Section I. Here some prominent examples are summarized briefly. To build up any special multifield theory describing some particular phenomenon, one must 9 choose a suitable order parameter and then M to model the substructure of the material 9 choose an appropriate form of the free energy ~p 9 evaluate the possible occurrence of latence of substructures induced by the need of some internal constraints motivated by the physical experience After these steps, the constitutive equations for the measures of interaction follow from Proposition 7 and, in the presence of discontinuity surfaces, are supplemented by the results in Proposition 8. In this way, one can write field equations (35), (39), and (46) [with the addition of (129), (130), and (131) in presence of interfaces] in terms of x(.) and qo(-) and then attempts to solve them. In particular, when one chooses to solve (35), (39), and (43) by means of finite element schemes and then must analize integral (relaxed) forms of the field equations, one obtains stiffness matrices more articulated than those of Cauchy continua and an array containing both the components of the placement (or the displacement) of each material patch and the components of the order parameter.
74
Paolo Maria Mariano A. MATERIAL WITH VOIDS
The model of materials with voids is the simplest multifield model. When pores are finely distributed throughout the body, one way to describe them is to choose the order parameter qo as a scalar that associates to each point X the void volume fraction of the material patch at X. In this case, .Ad reduces to the interval of the real axis [0, 1] and A is identically zero [see discussions before (43)]. Substructural bulk measures of interaction/3 and z reduce to scalar, whereas the microstress S becomes a vector as a consequence of Proposition 7. An interesting case is the one of linear elastic materials with voids (or elastic porous materials), with perhaps some damping effects in the pores. By indicating with e the infinitesimal strain tensor e = symVu (u is the displacement), linear constitutive equations relevant for this case (and written with respect to a reference state free of stress) can be obtained by taking for the free energy ~p a quadratic form in e, qO, and Vq9 and applying Proposition 7. In addition, one can consider small viscous effects due to the surface tension at pores by using the procedure discussed in Remark 8 before (283). At the end of calculations, one obtains for elastic porous materials the following constitutive relations with damping: g-,(1)
,,-,(2)
1"ij --Cijhk~hk + ,-.ijkq),k + Gij (D
(305)
-- C(3)~ - C(4)~ - Ci(5)~ij - C~ 6)q),i
(306)
t.~i _ ,,-,(7) c ij qg,i -Jr-_(8) I-.ijk~jk -Jr- C~9)q)
(307)
where 1", ~, and S represent (as in Section III) linearized measures of interaction; Cijhk is the usual stiffness tensor; and C (i) are appropriate constitutive constants. Of course, ~0,i denotes the derivative Ox, q). In the isotropic case, (305)-(307) reduce to
Tij = ~.~ij~,hh + 21zeij -I- ~ ( l ) ~ i j
(308)
= _ ~ ( 2 ) ~ _ ~:(3)q9 _ ~(l)6h h
(309) (310)
~--~i = ~(4)q9,i
where ~ij is the unit tensor and X and # are the standard Lam6 constants and are related to the other constants by the following inequalities: /z>O; 3X + 2/z ~> O;
~(4)/>0;
~(3)>~0
(3X + 2/z)~ <3) >~ 12~
(311) (312)
Multifield Theories in Mechanics of Solids
75
which assure uniqueness and weak stability of solution of the dynamic balance equations. When one analyzes the dynamics arising from (62) and (63) with constitutive equations (308)-(310), one finds the validity of the proposition in the following. Proposition 13 (Cowin and Nunziato, 1983) Transverse waves propagate at a constant speed without affecting the porosity of the material and without attenuation. Longitudinal acoustic waves are both attenuated and dispersed as a result of the changes in material porosity that accompany the wave. Note that the damping term in (306) arises from a nonequilibrium component z ne of the self-force. In this sense, these linear materials are not purely elastic.
BIBLIOGRAPHICNOTE The nonlinear theory of materials with voids has been introduced by Nunziato and Cowin (1979). Proposition 13 is discussed in Cowin and Nunziato (1983). Further studies on this topic can be found in Cowin (1985), Dhaliwal and Wang (1994), Nunziato and Walsh (1978), Diaconita (1987), Fr6mond and Nicolas (1990), and Mariano and Bernardini (1998).
B. TwO-PHASE (OR MULTIPHASE)MATERIALS In two-phase materials, possible phase transitions may change one phase into another. A scalar-order parameter may be used to associate to each point X information on the phases; that is, it may associate to X the volume fraction of one phase, or it may be the indicator of same phase. In the latter case, ~0 is zero if there is not a certain phase and is equal to 1 if the contrary is true. Then interfaces (across which phase transition occurs) are the boundaries of sets that admit ~0 as an indicator function. When ~0 is interpreted as the fraction of a certain phase, .A4 reduces to [0, 1], and the interfaces are considered to be smeared throughout the whole body. Transition layers are identified throughout the body as the thin layers in which V~o undergoes large oscillations. The use of such a kind of order parameter allows one to approximate the nonconvex variational problems arising in the evaluation of two phase bodies. If order parameters are not introduced, one manages, in fact,
76
Paolo Maria Mariano
potentials that have two wells (one well for each phase). During the evolution of phase transitions, the order parameter is ruled by an evolution law, such as (186), which derives from the balance of substructural interactions. In the case of multiphase materials, one may choose as order parameter a list ~ = (991 . . . . .
qgN)
(313)
whose entries take values on [0, 1] and are subjected to the constraint N Z(/9 i -
(314)
1
i-I
thus only N - 1 are independent. In this case, .A4 becomes the cube [0,1] x . . . x [0,1]
Ntimes
(315)
BIBLIOGRAPHIC NOTE
Two-phase materials have been treated with the help of scalar-order parameters in Penrose and Fife (1990), Colli et al. (1990) Fr6mond (1987), Fried and Gurtin (1993, 1994, 1999), and Fried and Grach (1997), and references therein; the original idea of using order parameters to describe phase transitions is from L. D. Landau.
C. COSSERAT CONTINUA
Each patch is considered to be a rigid body that can rotate independently from the neighboring patches. Such a rotation can be described by using as order parameter an orthogonal tensor Q (X, t). In this case, M reduces to the space of orthogonal tensors Orth + such that QTQ = I and det Q = 1. Simple calculations show that z" is always zero because .A4 coincides with Orth +. The strain tensor E, which in Cauchy solids is given by I ( F T F - I), is here given by E = 2MTM- M T - M
(316)
where 1
M - :(QTFZ
I)
(317)
Multifield Theories in Mechanics of Solids
77
In the setting of small deformations, the infinitesimal strain tensor e becomes = V'u - eq
(318)
with q a vector describing the rotation. If one chooses the free energy as a function of F, Q, VQ, by applying Proposition 7, one realizes that z is a second-order tensor, S a third-order tensor, and, consequently,/3 is a second-order tensor. Moreover, after a straightforward calculation, one realizes that ,4 is a third-order tensor given by .A - - e Q ( . A i j k eijh Qhk), and then (39) becomes -
e(jOQ r - zQ T + (DivS)Q r)
-
0
-
(319)
It is then possible to put 1
- ~e(/3Q T)
(320)
-- ~e(zQ r + S V Q v)
(321)
1
,.~- IoSQT 2
(322)
where/3 and ~ are vectors whose components are 1
~i - -~Oijkl~jlO T
(323)
Zi = ~Oijk (Zjl O~ + Silm Qrlm,k)
(324)
1
whereas S is a second-order tensor given by 1
8ij -- -~OijkSklm O/rm
(325)
The generalized balance of couples (38) reduces to 1 e T F r = e~ 2
(326)
which eliminates the self-force from the balance of substructural interactions. Consequently, the balance of substructural interactions becomes /3 - 1 T F r + Div,~ - 0 2
(327)
To account for inertial effects, one must consider the right-side term of (63), in which the kinetic coenergy X density must depend only on the product Q v Q to be objective. Simple algebra allows one to write tensor Q T 0 in terms of a vector
78
Paolo Maria Mariano
a -- e ( Q TI)), and X may depend thus only on a. In this way, the explicit dynamic expression of (327) becomes
/ 3 - ~ T1 F T + D i v , ~ -
( p OX(a))" Oct
(328)
The framework of Cosserat materials is widely used to model structural elastic elements such as beams, plates, and shells. In addition, some proposals for a Cosserat plasticity have been developed with the aim of analizing certain aspects of localization of deformations.
BIBLIOGRAPHIC NOTE
A detailed list of references on Cosserat materials is given at the end of Section I. Here I followed some notes in Capriz (1989), Ericksen and Truesdell (1958), Antman (1972), and Villaggio (1997). For Cosserat plasticity, see Steinmann, (1994).
D. MICROMORPHICMATERIALS The model of micromorphic materials is useful to describe the influence of large molecules on gross mechanical behavior, as happens in some polymers. Each material patch is considered to be a cell that may undergo additional deformations independently of the neighboring patches. The order parameter qo is a second-order invertible tensor (the gradient of microdeformations) and A4 reduces to Lin + [i.e., the space of linear forms between vector spaces], with positive determinant. By using Proposition 7, one realizes that the self-force z becomes a second-order tensor whereas the microstress S is a third-order tensor. Nonlinear measures of deformation are (cf. Capriz, 1989)
E--~I(FTF-I); M- ~(qoVF-l); ~-qo -~Vqo
(329)
By assuming that r = Qg~ after rigid changes of spatial observers (where Q is an orthogonal tensor describing the rotation of the observers), one obtains that A is a third-order tensor given by eg~, or in indices, ~ i j k = eijl~Olk. Of course, because ~o ~ Lin § it never vanishes; then z" in (43) is identically zero, as in the case of Cosserat continua.
Multifield Theories in Mechanics of Solids
79
An interesting aspect of micromorphic material appears evident in the setting of small deformations. It is possible to introduce the relative infinitesimal strain ~r as er = V u -
qo
(330)
and associate to each patch the state (e, er, Vqo). In this way, the internal power becomes f13 (T " ~ "~- Z " ~r -'~"S " V qo)
(331)
,
If one assumes an internal constraint prescribing that ~r = 0
(332)
the inner power (329) reduces to
fB
(T.e+S.
VVu)
(333)
,
which is the inner power that one writes when dealing with second-gradient materials. From the multifeld scheme of micromorphic materials with the internal constraint (332), strain gradient plasticity has been formulated to account for length scale-dependent effects in plasticity, such as the Hall-Petch effect (the strength of polycristalline aggregates increases with decreasing grain size) or the experimental results showing strong size effects in the case of torsion tests on copper wires. Basically, one realizes that the presence of VV u in (333) accounts for weak nonlocal effects as a result of the latence induced by the constraint (332). However, each multifield theory is "weakly nonlocal" in nuce as a result of the presence of the gradient of the order parameter in the list of constitutive variables.
BIBLIOGRAPHIC NOTE
Micromorphic materials have been introduced in Mindlin (1964; see also Grioli, 1960, 1990; Bofill and Quintanilla, 1995; Mindlin, 1965b; Mindlin and Tiersten, 1963; Eringen, 1992, 2000; Capriz and Podio-Guidugli, 1976). For the strain gradient plasticity, see Fleck and Hutchinson (1997).
Paolo Maria Mariano
80
E. NEMATICLIQUID CRYSTALS Nematic liquid crystals are characterized by the presence of rodlike molecules smeared throughout the liquid. A commonly accepted choice of the order parameter is a second-order tensor Do representing a second-order approximation of an unknown orientation distribution function 14 of the molecules 9 In some sense, DD(X) represents the macroscopic average of the orientation of the molecules in the patch at X. Do is also such that
DD -- DTD
trDo = 1
(334)
and has nonnegative eigenvalues. Usually, the deviatoric part D of Do is adopted instead of Do, and one has D = Do - lI. Nematic liquid crystals are characterized by the coincidence of two of the three eigenvalues of D, which is represented in the form
D=~' (
d|
')
9 d.d-1
9
2' < ~ < 1
(335)
The scalar ~"is interpreted as degree of orientation. Vector d (which is the averaged direction of the molecules of each material element) and the parameter ~" are taken as order parameters. Consequently, one has from Proposition 7 measures of interaction z0 and So associated to d (which is the averaged direction of the molecules of each material element) and measures of interactions zc and Sc associated to ~'. There are two manifolds A/l; the first manifold coincides with the unit sphere in It~3 and the second manifold is the interval [ - 21, 1]. Two different substructural balances of the same form of (42) must be satisfied. As pointed out in Example 2 in Section II.D, because one deals with a liquid, one must have measures of interaction in the current c c C configuration and denotes them with/3~, z~, $~,/3~, z~, S~. In this way, one writes two balances of substructural interactions [because one has at his disposal two order parameters; see (44)] /3~ - z~ + divS~ - otd
(336)
/3~ - z~ + divS} - 0
(337)
the Cauchy balance of standard interactions divT C + b' = 0
(338)
where T r is Cauchy stress and b c denotes bulk forces in the current configuration, and only one balance of torques because r on [ - ~1, 1] vanishes identically skw (T ~ + z~ | d + (gradd) rS~) - 0 14An analogous choice can be made in the case of microcracked bodies.
(339)
Multifield Theories in Mechanics of Solids
81
where grad is calculated with respect to coordinates in the current configuration. _ct! m 0. Note that in (337), zc The free energy depends on cl and its gradient and ot and its gradient, in addition to the measures of deformation. The use of Proposition 7 allows one to obtain constitutive prescription after pushing forward the relevant constitutive variables. The degree of orientation is not the sole characteristic parameter that can be derived from D. To describe the possible emergence of optic biaxiality of the nematic, two other parameters must be used and may be derived from D. In particular, one realizes that the tensor Do generates an ellipsoid. Then the degree of prolation dg and the degree of triaxiality dt are useful parameters to describe the nematic distribution and are defined by I
dg -- -~
(340)
(3)~i - 1 ) I
dt =
6~/-3
I)~i - ~.i+11
(341)
i=1
where ~-i a r e the eigenvalues of Do, i = 1, 2, 3. They can also be considered to be order parameters.
BIBLIOGRAPHIC NOTE
The theory of liquid crystals with variable degree of orientation has been introduced by Ericksen (1991) and the derivation of balance equations as well as the constitutive restrictions can be found in his paper. The degree of prolation dg and the degree of triaxiality dt have been introduced by Capriz and Biscari (1994).
F. FERROELECTRIC SOLIDS
Some crystalline materials as barium-titanate experience spontaneous polarization associated to the rearrangements at the crystalline temperature called Curie temperature. To describe polarization, one may choose as order parameter a vector p of ~3 such that [Pl < Ps with Ps a material constant. In this way, M becomes the ball {p 6/~3 I Ip[ < Ps}. In writing the overall power 79 , one must consider not only the external power j'Qext and the inner power TAint of mechanical interaction, but also a self-power TAself of electrical nature. Then, 79 = j-Qext _ ~')int _ TAself. In particular, with reference to
Paolo Maria Mariano
82
any part B~' of the current configuration ~t of the ferroelectric body, one writes
pext(B'~)-faB: ( b C ' v + j O c ' o ) + f
B~ ( (
Tc -- 2(P" 1
n)2 ) n . v +
SCn 90 ) (342)
/,
~s)int(]3t) -- [
dB
79self(Bt) --
(T C. gradv +
(pC(grade)p. v + ;
S c.
gradl~ + z c. 1~)
fie. P)Jr f
l
~ ( p - n ) 2 n 9v B;
(343) (344)
where v is the velocity ~, e is the electric field, and the superscript c indicates that the relevant fields are taken in the current configuration. Moreover, one may disregard gravitational forces and consider the noninertial parts of bulk interactions b c and/3 c to be measures of the electrostatic interactions of B t with its exterior. Taking into account such a hypothesis, the procedure of Section II.C allows one to obtain balance equations in the current configuration divT C + pC(grade)p = divSC - zC +
PCe --
pC~r
Prop + B p
(345) (346)
where Pm is a material constant and B is an appropriate second-order tensor such that B p . p = O. In the reference configuration, the balances (344) and (345) become DivT + p(grade)p = p~
(347)
DivS - z + pe - (det F)(pmli -+- BI~)
(348)
T = (det F)TCF - r
(349)
S = (det F)SCF - r
(350)
z = (det F)z c
(351)
F)p c
(352)
where
p = (det
To derive constitutive equations one chooses in this case ~ / - - it(F, p) + W ( p ) + 1 ff2Vp 9V p
(353)
where 9 is a material constant and s is such that Lt(Q, .) = 0
(354)
Multifield Theories in Mechanics of Solids
83
for any orthogonal tensor Q. Taking into account (353), Proposition 7 allows one to derive constitutive equations. During the polarization, the walls of ferroelectric domains evolve; in addition, the coupling of mechanical deformation and electric polarization can cooperate and generate localization of domain walls on some "macroscopic" discontinuity surface that may evolve and can be considered as an "emerging" domain wall. To study the evolution of standard and emerging domain walls, one can use the general results of Section VI, taking into account that here the configurational stress I? assumes the form ~ = ~pl - F r T - (Vp) TS. In the case of rigid ferroelectrics, I? reduces to
I~rigid--
W ( p ) + ~ff V p . Vp
I-
ff V p ) r V p
(355)
and the evolution equation (190) reduces to an equation describing an isotropic motion by curvature (see Dav] and Mariano, 2001) which fits, for example, the experimental results on barium titanate.
BIBLIOGRAPHIC NOTE
The model of ferroelectric materials summarized in Section IX.F is discussed in Dav] (2001) and in Dav] and Mariano (2001) and allows one to deduce expressions for the evolution rules for domain walls and comers in accord with experimental evidences. Basic remarks on ferroelectrics can be found in Jona and Shirane (1962), Little (1955), Hwang et al. (1995), Rosakis and Jiang (1995), and references therein.
G. MICROCRACKEDMATERIALS In bodies free of microcracks, any given deformation maps each patch at X into a point x(X) of the Euclidean space and the displacement u(X) from X to x(X) is u(X) = x(X) - X. When microcracks are diffused within the body, under the same given deformation the patch at X undergoes a displacement u + d, where d is the kinematical perturbation due to the enlargement or the closure of microcracks. At each x, d(x) is a relative displacement; it is the difference between the real placement x'(X) of the patch at X and the theoretical placement x(X) occurring at X if microcracks were absent i.e. d = x' - x. It is thus possible to take the vector d as order parameter; therefore, A//reduces to I~3. By using Proposition 7, one realizes that the self-force z is a vector and the microstress S is a second-order tensor.
84
Paolo M a r i a M a r i a n o
The standard gradient of deformation F is given by F - I + Vu (where I is the unit tensor), whereas the overall gradient of deformation Ftot is given by Ftot F+Vd. Overall measures of deformations associated to Ftot are the overall right CauchyGreen Ctot given by Ctot --FtotFtot T and the overall deformation tensor Etot = 1 ~ (Cto, - I). Two linearized measures of deformation may be derived by taking IVul << 1 and IVcll << 1. They are indicated with eu and ed and defined by eu - symVu
ed -- symVd
(356)
Balance equations may be deduced in the manner indicated in Section II. When one considers the equilibrium of a microcracked body without the help of order parameters, one writes the energy and tries to minimize it over a domain endowed with an unknown set of discontinuities for the displacement field: the microcracks. Then one must find the solution of the variational problem
minlftsw(X,Vu)+fG(ts)r(X,[u]| }
(357)
where w(., .) is the bulk strain energy density, r(., .) is the surface energy density of microcracks, and nm is the normal to each microcrack face. Problem (357) has two unknowns: the displacement field u(. ) and the set G(/3) of microcracks. Thus the equilibrium problem becomes a free boundary variational problem and is very difficult to solve. By using the just described order parameter, one may regularize this free boundary problem and reduce it to a more simple one. In particular, it is possible to "approximate" (357) with the problem
min{fw(x, vu, ct,vcl)]
(358)
where/3 is considered to be free of microcracks and cl represents the kinematical perturbation induced to u by microcracks. When irreversible behavior occurs and the microcracks may grow, one considers the free energy r instead of w and may use the procedure in Section II.D and Proposition 7 to write balance equations of the form Div(OvCr) + b = 0
(359)
Div(OvdTr) -- 0d~ = 0
(360)
s k w ( O v ~ F r + OdTZ|
cl + Vd T0vd~) = 0
(361)
where Ov~ is the Piola-Kirchhoff stress T, 0vo r is the referential microstress S, and Od~/ris the self-force z'. A basic problem is thus to find an explicit expression for ~r (i.e., constitutive equations for the measures of interaction). One way to obtain
Multifield Theories in Mechanics of Solids
85
constitutive equations is to use an identification procedure based on the equivalence of the density of power in the continuum and the power of a cell of a periodic discrete model of the microcracked body. This procedure may always be used when the order parameter has the physical meaning of a displacement, a rotation, or a deformation, provided a suitable choice of the discrete model. The physics of microcracked bodies suggests the adoption (at least in the case of elastic behavior) of a discrete model made of two periodic lattices connected by elastic links. The first lattice (macrolattice) is made of indistinct material points (spheres) and describes the material at molecular level. The second lattice (mesolattice) is made of empty shells (lakes) connected to one another by elastic links and describes the material at the mesoscopic level of microcracks; each shell is forced to deform along a plane orthogonal to its major axis only. This is a case of "virtual" substructures because the stiffness of each shell is the one of the material surrounding each microcrack. Because the lattice is assumed to be periodic, the attention is focused on a cell of the lattice: a representative volume element (RVE). Chosen the RVE of the discrete system, it has N spheres and M shells; in the macrolattice of the RVE there are L links: LN is the number of interlattice links, and LM is the number of links in the mesolattice. It is assumed that the elastic links in the RVE can carry only axial forces: ti denotes the force in the ith link of the macrolattice, zt the force in the lth interlattice link, Zj the force in the jth link in the mesolattice, and the force on the shell located at h is indicated with z0h. Spheres located at a and b in the RVE undergo displacements u a and u b, whereas a shell located at h (or at k) undergoes a relative displacement clh (or clk) of its margins along the plane of deformation. In this way, appropriate measures of deformation in the discrete RVE are Oh, clh - clk, Illa -- d h, and u a - u b, where the last three measures of deformation make sense only when the spheres at a and b and the shells at h and k are connected. There are two steps to obtain constitutive equations for the measures of interaction in the continuum from the discrete model in the case of infinitesimal deformations.
1. First one equalizes the density of the internal work in the continuum with the internal work developed in the discrete RVE and in the case of linearized kinematics writes T . Vu + ~,. cl + , S . V d -
1 Vol(RVE)
LN
M
l=l
h=l
i=1
LM j=l
ti 9(u a - u b)
) \
(362)
Paolo Maria Mariano
86
where L is the number of links of the macrolattice in the RVE, LN is the numbers of interlattice links, M is the number of shells, and LM is the number of links in the mesolattice. 2. Then one must write the measures of deformation in the discrete RVE in terms of the ones of the continuum. A reasonable choice is to take U a --" U ( X ) +
Vu(x)(a - x)
(363)
d h -- O(X) +
Vd(x)(h
(364)
-
x)
with x a point in the RVE chosen such that ua
-
u b =
O h -- O k : Ua -- O h :
Vu(x)(a
Vu(x)(a
-
(365)
b)
Vd(x)(h - k) -
x) -
(366)
By inserting (363)-(367) in (361), one obtains
1
Vol(RVE)
i=1
ti | (a - b) + Z
1=1
1
1 Vol(RVE)
h=l
zt | (a - x)
M
Vol(RVE) ~
(367)
V d ( x ) ( h - x)
z~
z0~ | (h - x ) + ~ zj | (h - ~ j=l
)
(368)
(369)
~ z, | (h - x~ t=l
(370) At this point, it is necessary only to choose appropriate constitutive relations for the links in the RVE to find explicitly the measures of interaction in the linearized kinematies T, ~, ,S in terms of Vu, d, Vcl. The simplest choice is ti - IK(ua - ub), Z~ - - Q d h, z l - ]~-]~(Ua - dh), and zj - - D l d h l l d h l ( ( h - k ) / I h - k12), with K, Q, H, D appropriate stiffnesses. Note that the first three constitutive relations are forms of the Hook's law, whereas zj (the interaction between two neighboring shells) is nonlinear is due to the fact that two neighboring microcracks interact like neighboring dislocations located at a distance ( h - k ) / I h - kl 2 from each other. When one inserts the previous constitutive relations into (368)-(370), one finds explicit constitutive equations for the measures of interaction in the continuum. The simplest form, which is valid in the case of linear elastic behavior
Multifield Theories in Mechanics of Solids
87
(microcracks do not grow) and when the material is centrosymmetric, is the following: T = AVu + A'Vd - Cd S - A'Vu + GVcl
(371) (372) (373)
where the fourth-order tensors A, A', C, G have major symmetries and can be computed explicitly by the just described procedure (a detailed calculation of them can be found in Mariano and Stazi, 2001). Consequently, balance equations (359) and (360) become, in the case of linear elastic behavior of the material, b + div(AVu + A'Vd) = 0
(374)
div(A'Vu + GVd) - Cd = 0
(375)
When one solves with standard finite element procedures the equilibrium problem with constitutive equations (371)-(373), one finds as a basic result the occurrence of localization phenomena of the deformation. This result cannot be obtained by using standard linear elasticity in the setting of Cauchy continua and has an immediate experimental evidence (to make an experiment, the reader can take any rectangular sheet of paper, load it with a traction in the middle point of one of its sides and fix only the opposite side; localization occurs). These localization phenomena can be obtained from (374) and (375) as a result of the presence of the self-force Cd and of the gradient of the order parameter. Basically, I conjecture that one may find localization phenomena of the order parameter (i.e., of the substructure) in a wide range of possible multifield models as a result of the analytical structure of the balance of substructural interactions. From a physical point of view, I interpret these localization phenomena of the order parameter as the description of cooperative patterns of material substructures.
BIBLIOGRAPHIC NOTE
Details about this multifield model of microcracked bodies can be found in Mariano (1999), Mariano and Stazi (2001), and Mariano et al. (2001); localization phenomena of deformation within the setting of linearized constitutive equations are described in Mariano and Stazi (2001) and Mariano et al. (2001).
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Paolo Maria Mariano
Acknowledgments I t h a n k G i u l i a n o A u g u s t i , G i a n f r a n c o Capriz, and E r i k van der G i e s s e n for p r o f o u n d and d e t a i l e d d i s c u s s i o n s a b o u t the topics in this article. M y friend and f o r m e r student F u r i o L o r e n z o Stazi was f u n d a m e n t a l in m a k i n g c o r r e c t i o n s and t y p i n g this article.
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Nunziato, J. W., and Cowin, S. C. (1979). A nonlinear theory of elastic materials with voids. Arch. Rat. Mech. Anal. 72, 175-201. Nunziato, J. W., and Walsh, E. K. (1978). On the influence of void compaction and material nonuniformity on the propagation of one-dimensional acceleration waves in granular materials. Arch. Rat. Mech. Anal. 64, 299-316. Pence, T. J. (1992). On the mechanical dissipation of solutions to the Riemann problem for impact involving a two-phase elastic material. Arch. Rat. Mech. Anal. 117, 1-52. Penrose, O., and Fife, P. C. (1990). Thermodynamically consistent models of phase field type for the kinetics of phase transitions. Physica D. 43, 44-62. Povstenko, Y. Z. (1994). Stress functions for continua with couple stresses. J. Elast. 36, 99-116. Renardy, M., and Rogers, R. C. (1993). An introduction to Partial Differential Equations. Springer Verlag, Berlin. Rice, J. R. (1968). Mathematical analysis in the mechanics of fracture. In Fracture 2 (H. Liebowitz, ed.), pp. 191-311, New York, Academic Press. Rosakis, P., and Jiang, Q. (1995). On the morphology of ferroelectric domains. Int. J. Eng. Sci. 33, 1-12. Segev, R. (1994). A geometrical framework for the static of materials with microstructure. Math. Mod. Meth. Appl. Sci. 4, 871-897. Segev, R. (2000). The geometry of Cauchy's fluxes. Arch. Rat. Mech. Anal. 154, 183-198. Silhav2~, M. (1997). The Mechanics and Thermodynamics of Continuous Media. Springer Verlag, Berlin. Simo, J. C., and Fox, D. D. (1989). On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization. Comp. Meth. Appl. Mech. Eng. 72, 267-304. Simo, J. C., and Vu-Quoc, L. (1988). On the dynamics in space of rods undergoing large motions--a geometrically exact approach. Comp. Meth. Appl. Mech. Eng. 66, 125-161. Simo, J. C., Fox, D. D., and Hughes, T. R. J. (1992). Formulations of finite elasticity with independent rotations. Comp. Meth. Appl. Mech. Eng. 95, 277-288. Simo, J. C., Fox, D. D., and Rifai, M. S. (1989). On a stress resultant geometrically exact shell model. II. The linear theory: Computational aspects. Comp. Meth. Appl. Mech. Eng. 73, 53-92. Simo, J. C., Fox, D. D., and Rifai, M. S. (1990). On a stress resultant geometrically exact shell model. III. Computational aspect of the nonlinear theory. Comp. Meth. Appl. Mech. Eng. 79, 21-70. Simo, J. C., Marsden, J. E., and Krishnaprasad, P. S. (1988). The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods and plates. Arch. Rat. Mech. Anal. 104, 125-183. Steinmann, P. (1994). A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity. Int. J. Sol. Struct. 44, 465-495. Toupin, R. A. (1964). Theories of elasticity with couple-stress. Arch. Rat. Mech. Anal. 17, 85-112. Toupin, R. A. (1965). Saint-Venant's principle. Arch. Rat. Mech. Anal. 18, 83-96. Tricomi, E G. (1954). Lezioni sulle Equazioni alle Derivate Parziali. Gheroni, Torino. Truesdell, C. A. (1984). Rational Thermodynamics. Springer-Verlag, Berlin. Truesdell, C. A., and Toupin, R. A. (1960). Classical field theories of mechanics. In Handbuch der Physics. Springer-Verlag, Berlin. Villaggio, P. (1997). Mathematical Models for Elastic Structures. Cambridge University Press, Cambridge. Virga, E. G. (1994). Variational Theories for Liquid Crystals. Chapmann & Hall, London. Voigt, W. (1887). Teoretische Studien tiber Elasticit~itsverh~iltnisse der Krist~ille. Abh. Ges. Wiss., Gottingen, 34. Wang, J., and Dhaliwal, R. S. (1993). On some theorems in the nonlocal theory of micropolar elasticity. Int. J. Sol. Struct. 30, 1331-1338.
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A D V A N C E S IN APPLIED M E C H A N I C S , V O L U M E 38
Molecular Dynamics Simulation of Nanoscale Interfacial Phenomena in Fluids CHANG-LIN TIEN and JIAN-GANG WENG Department of Mechanical Engineering, University of California, Berkeley, California 94720 I. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. M o l e c u l a r D y n a m i c s S i m u l a t i o n T e c h n i q u e s . . . . . . . . . . . . . . . . A. Classical M o l e c u l a r D y n a m i c s S i m u l a t i o n . . . . . . . . . . . . . . . . B. N o n c l a s s i c a l M o l e c u l a r D y n a m i c s S i m u l a t i o n . . . . . . . . . . . . .
96 97 97 103
III. L i q u i d - V a p o r Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. P l a n a r Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Spherical Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. C y l i n d r i c a l Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103 104 113 119
IV. L i q u i d - L i q u i d Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. P l a n a r Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Spherical Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. M i x t u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122 124 128 128
V. L i q u i d - S o l i d Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. P l a n a r Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Spherical Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. C y l i n d r i c a l Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
130 130 134 135
VI. T h r e e - P h a s e S y s t e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. P l a n a r Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Spherical Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. S p r e a d i n g Wetting and C o n t a c t L i n e R e g i o n . . . . . . . . . . . . . . VII. O t h e r Interfacial P h e n o m e n a . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. S o n o l u m i n e s c e n c e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. C o n c l u d i n g R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
136 136 137 138 139 139 140 140
Acknowledgments ...................................
141
References .........................................
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95 ISBN 0-12-002038-6
ADVANCES IN APPLIED MECHANICS, VOL. 38 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISSN 0065-2165/01 $35.00
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Chang-Lin Tien and Jian-Gang Weng
I. Introduction
Molecular dynamics (MD) simulation, the computational method that simulates detailed material behavior by simultaneously solving the equations of motion for a system of molecules interacting with given potentials, has been applied in various disciplines. Such molecular level understanding provides scientists and engineers with tremendous opportunities for the exploration of material properties in different length and time scales. The past several decades have witnessed an explosive growth in literature of MD simulation. Chou et al. (1999) and Maruyama (2000) provide excellent and comprehensive reviews of MD simulation in microscale and nanoscale studies. Interfacial phenomena have attracted researchers' attention for a century. Research in this field has made a significant impact on many industry applications, ranging from macroscale (e.g., power generation and material processing) to microscale and nanoscale (e.g., thermal inkjet printing and liquid microcoolers). However, many fundamental problems still remain unsolved, which may hinder further understanding of interfacial phenomena. For example, insufficient understanding of the curvature effect on surface tension makes it difficult to predict the nucleation temperature in homogeneous nucleation, which has significant influences on condensation, thin film growth, and many other heat transfer and material processing applications. In the past, researchers applied statistical mechanics and statistical thermodynamics methods to solve interfacial problems. These approaches treat the interface as a zero-thickness region in which abrupt changes of material properties occur. Because the interface thickness is in the nanometer range, such simplification is reasonable in studying macroscale phenomena, but it becomes highly questionable in studying nanoscale phenomena. In studies of nanoscale mass, momentum, and energy transport processes, detailed information on molecular structures is often required and MD simulation can become an effective tool to provide that. This chapter reviews recent developments of MD simulation on mechanical aspects of interfacial phenomena in fluids (e.g., surface tension and interface stability), discusses some unsolved problems, and proposes several future directions. All interfaces, based on the different phases involved, are grouped into liquidvapor interfaces, liquid-liquid interfaces, liquid-solid interfaces, three-phase systems, and other interfacial phenomena (i.e., sonoluminescence and surfactants). Because simulation of liquid-vapor interfaces has the longest history, it is therefore emphasized here. In several groups, interfaces are further divided into three subgroups according to their geometric shapes: planar, spherical, and cylindrical. Concluding remarks are then given at the end of this chapter.
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II. Molecular Dynamics Simulation Techniques An attractive feature of MD simulation is that its techniques are simple and it imposes fewer assumptions than other numerical methods. In classical MD simulation, the only material property to be specified beforehand is the intermolecular potential function. In other cases in which quantum effects are important, such as in electron transport or photon absorption, empirical potential functions may be unavailable. Therefore, nonclassical or Quantum Molecular Dynamics (QMD), such as the ab-initio method, is developed. This section reviews basic MD simulation techniques related to the study of interfacial phenomena. The first subsection discusses classical MD simulation and the second subsection discusses nonclassical MD simulation. For more general information on MD simulation techniques, the reader is referred to the books written by Allen and Tildesley (1987) and Haile (1997).
A. CLASSICALMOLECULAR DYNAMICS SIMULATION
Classical MD simulation methods can be divided into equilibrium MD (EMD) and nonequilibrium MD (NEMD). In the past, most work on MD was based on EMD, and NEMD has not been treated thoroughly. However, because it can handle calculation of transport coefficients easily NEMD will receive increasing attention in the future. This subsection reviews several issues on classical MD simulation, including intermolecular potentials, the finite difference method, initial conditions, boundary conditions, equilibrium criteria, simulation ofthermophysical properties, and simulation of transport coefficients.
1. Intermolecular Potentials In classical MD simulation, a molecule moves according to imposed forces on it, which can be expressed by Newton's classical equations of motion
d?)i
mi--d~ - E
Fij
(1)
J
where mi is the mass of a molecule i, ~i the velocity vector, and Z j Fij the summation of all force acting on the molecule i from other molecules. Here, Fij is determined by differential of the intermolecular potential
-. ddp(rij ) Fij --- - drij t'ij
(2)
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Chang-Lin Tien and Jian-Gang Weng
where ~(rij) is the intermolecular potential, rij is the intermolecular distance, and ?ij represents a unit vector pointing from a molecule j to the molecule i. Much MD work uses the Lennard-Jones (LJ) potential, which is a simple twobody potential form and known to be suitable for inert monatomic molecules such as neon, argon, and krypton
_ (~j.j)6] (~(rij ) -- 4e
m rij
(3)
where e is the energy parameter and 0. is the length parameter; these two parameters vary for different types of molecules. MD simulation adopts non-dimensionalized variables to make simulation results more general. Such non-dimensionalization is based on 0., e, and m, so that the reduced parameters (denoted with an asterisk) of temperature, length, time, mass density, pressure, and surface tension are T* = k s T / e , r* = r/0., t* = t(e/m0"2) 1/2, p* = p0"3/m, P* -- P0"3/e, and y* -- v0"z/e, respectively. In this sense, it does not matter which inert monatomic molecules are simulated in the program; however, for ease of physical understanding, most studies assume the "LJ molecules" to be argon. Other types of potential functions are also available to simulate different materials. In simulation of liquid water, one may use ST2, SPC, TIP4P, MYC, and CC potential functions. Tersoff and SW potential forms can model materials with covalent bonds such as carbon and silicon. Refer to Maruyama (2000) for a more detailed description of these models. Other examples include the shell model for ionic solids and the embedded atom method for metals. If several types of molecules exist in the system, the potential parameters between unlike molecules are usually different from those between like molecules. For example, if both types of molecules are modeled by the LJ potential, the potential parameters between molecules of type 1 and type 2 can be expressed as
612 --- ~:~/611e22
(4)
O'12 ~" 70.5(O"11 + 0.22)
(5)
where the subscript 12 refers to the interaction for unlike molecules, subscripts 11 and 22 refer to like molecules, and s~ and 71 are constants. According to LorentzBerthelot combination rules, ~ and r/are 1.0, but different values have also been used (Mecke et al., 1999). Other models are also available to make interaction between unlike molecules different from that of like molecules, such as by introducing a parameter to modify intermolecular attraction, the second term in the LJ potential, between two unlike molecules (Dfaz-Herrera et al., 1999). The potential function for polymers is another interesting example. In simulation of polymers, all
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99
pairs of molecules interact with one potential (e.g., the LJ potential), and bonded neighbors in a chain further interact with the Finitely Extensible Nonlinear Elastic (FENE) potential, expressed as
~FENE(rij ) --
2
Ro J
where k equals 30 and R0 equals 1.5 in a non-dimensionalized system (Kremer and Grest, 1990). MD simulation uses the common convention of a cutoff radius rcut in the calculation of intermolecular potentials and forces, and rcut is defined as the distance beyond which the interaction between molecules is neglected. In a homogeneous system, the choice of the cutoff radius is not very critical because long-range forces from all directions cancel each other. However, in nanoscale interfacial regions where the local density changes by two orders of magnitude over a few nanometers, the choice of the cutoff radius becomes important. Generally, the cutoff radius should be sufficiently large to avoid possible long-range force correction, but it should also be small enough to keep computational time reasonable.
2. Finite Difference Method There are many types of algorithms to advance the differential equations of motion in MD simulation, but the "velocity Verlet" algorithm is usually preferred. Let At be the duration of each time step. During At, the "velocity Verlet" algorithm includes the following four steps:
7i(t + At) = ri(t) -k- At. vi(t) + vi t + ~
-- ~)i(t)-~
2
At2 ~'i(t) 2 mi
(7)
mi
Evaluate Fi(t + At) based on molecule positions at t + At
( 2t) AtFi(t+At) ~)i(t + A t ) = ~)i t + ~ +--2mi
(9) (10)
For other methods, such as the Gear predictor-corrector method, refer to Allen and Tildesley (1987) and Haile (1997).
1O0
Chang-Lin Tien and Jian-Gang Weng 3. Initial Conditions
Here, only the initial conditions for liquid and vapor phases are considered. The solid phase is usually treated as one of the boundaries in interfacial studies (except in studies of melting, in which the solid phase is modeled as a crystalline structure). Initial conditions for liquid and vapor phases can be: (1) a crystalline (solid) structure that melts and further vaporizes as the temperature exceeds the melting point; (2) a combination of different-phase samples that have equilibrated individually; and (3) the final condition from a previous simulation. Generally speaking, for EMD simulations, as long as a final equilibrium state can be reached, different choices of initial conditions lead to the same results.
4. Boundary Conditions Due to the limitation of current computational capacity, the classical MD technique can only simulate a system with about 106 molecules in a period of several nanoseconds. There is generally nothing one can do with the short time scale except avoid simulation of slow reactions. The small length scale limitation, however, can be overcome by implementation of periodic boundary conditions in which the computational domain is replicated throughout space to form an infinite system. According to such periodic boundary conditions, any information outside the computational domain can be inferred from the information within the domain, and thus direct simulations of bulk material properties are possible. It should be noted that this scheme is unable to simulate any phenomena that occur in a scale larger than the domain size (e.g., a wave with a very long wavelength). If solid walls exist in the system and interact with liquid (or gas) molecules, they are sometimes modeled differently from liquid and vapor molecules. In case that dynamics of solid molecules can be ignored, Abraham (1978) presented four models to simulate walls: (1) solid molecules are put into crystalline lattice sites and not allowed to move; (2) the potential between the liquid (or gas) and the wall is infinite when liquid (or gas) molecules are in the wall plane, and zero otherwise; (3) the solid wall is simulated as a semi-infinite medium with a uniform density, so that interaction forces acting on a liquid molecule from wall molecules are integrated throughout the wall and the liquid molecule feels the force only in the direction normal to the wall surface (z direction); and (4) the third model is improved by considering the possibility of finding a solid molecule in a certain distance z.
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101
In many cases, dynamics of solid molecules should be taken into account. To simulate a wall at constant energy, the simplest method is to follow Abraham's first model, but allow molecules to move according to imposed forces. For a wall at a constant temperature, which is frequently encountered in real applications, three simulation methods are available: (1) perform velocity scaling once every few steps; (2) add Langevin noise and frictional terms to the equations of motion (Allen and Tildesley, 1987); and (3) use the "phantom molecules" model (Tully, 1980; Maruyama, 2000), in which the first several layers of solid molecules interacting with liquid (or gas) are modeled as regular molecules, and a layer of "different molecules" (phantom molecules or ghost molecules) represent the other wall molecules that interact with the surface molecules. 5. Equilibrium Criteria In EMD methods, it is important to make sure that the system is at the equilibrium condition before the commencement of statistical calculation of properties. As stated previously, it is impractical to simulate a macroscopically long period. Some criteria, therefore, must be chosen to determine whether the system is at equilibrium. Unfortunately, such choices are still arbitrary in the literature. Maruyama et al. (1994) chose the criteria that both the temperature and the number of vapor molecules settle to constant values in simulations of droplets. Weng et al. (2000) chose the requirement that the local temperature and normal pressure component be constant in liquid film simulations. 6. Simulation of Thermophysical Properties MD simulation programs yield position and velocity of each molecule and intermolecular forces at each time step. Statistical methods must be applied to convert such detailed molecular level information to profiles of thermophysical properties such as local density, pressure components, and local temperature or concentration (if any). Three steps are needed to achieve this: (1) the computational domain is first artificially divided into small parallel slabs (for planar interfaces) or concentric shells (for curved interfaces); (2) properties within these slabs (shells) are calculated at each time step; and (3) at the end, time averaging is performed on these instantaneous values to get local properties. For example, the local temperature can be calculated as -
--
mivZi
(11)
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Chang-Lin Tien and Jian-Gang Weng
where nk is the total number of molecules in slab (or shell) k, kB is the Boltzmann constant, and the brackets ( ) denote a time average of the instantaneous values. Note that the slab or shell should be divided so in a manner that every point within the slab or shell is statistically equivalent.
7. Simulation of Transport Coefficients Two classes of methods, EMD and NEMD, are available to calculate transport coefficients such as viscosity, compressibility, diffusion coefficients, and thermal conductivity. In the first class of EMD methods, integration of time correlation functions of fluctuating quantities yields transport coefficients according to the Green-Kubo formalism,
ff -
f0 ~
dt(A(t + r)A(r))
(12)
Equivalently, one can calculate transport coefficients by applying the "Einstein relation"
1 ((A(t + r ) - A(r)) 2) = 2
t
t---~~
(13)
where ff is a transport coefficient, t the time, A(t) a variable that varies for each ff in study, A(t) a time derivative, r an arbitrarily defined time origin, and the brackets ( ) indicate an average over different r. For example, in determination of self-diffusion coefficients, A(t) represents molecule positions/~(t) and its time derivative is ~'(t). Hence diffusion coefficients are expressed as
Dsetf- -~ i f ~ dt(~)i(t -k- r). vi(r))
(14)
or
1 ([/~i(t + r)-/~i(r)[
O s e l f-- 6
t
2) t---~c~
(15)
The factor 51 in Eqs. (14) and (15) comes from the averaging process in all the three directions. Although the time correlation of only one molecule i is sufficient to determine bulk transport coefficients, as expressed in Eqs. (14) and (15), usually an average over N molecules in the system should be taken to improve statistical accuracy. In the second class of NEMD methods, the system is artificially driven away from equilibrium, and system responses are monitored. Two methods are available to implement the nonequilibrium condition. In the first method, Eqs. (1)
Molecular Dynamics Simulation
103
and (2) are modified to include a perturbation term. Although no physical gradient exists in the system, such a perturbation term creates a thermodynamic flux and integral of time correlation of this flux gives the transport coefficient (Allen and Tildesley, 1987; Hoover, 1993). In the other method, boundary conditions are modified to physically induce a gradient. For example, Lukes et al. (2000) imposed different temperatures on two sides of a solid film to induce a temperature gradient, and obtained thin-film thermal conductivities according to the Fourier law.
B. NONCLASSICALMOLECULAR DYNAMICS SIMULATION Any phenomena that involve electrons or photons cannot be treated easily by classical MD simulation, unless empirical equations can be obtained to describe dynamics of these particles. Usually, interactions among electrons and photons have the quantum mechanical nature and must be obtained by solving the Schr6dinger equation. MD simulations involving these interactions are called Quantum Molecular Dynamics (QMD) and are very useful in many areas, such as in the study of radiation-related processes (Kotake and Kuroki, 1993; Shibahara and Kotake, 1997, 1998). However, limited by computational capacity, QMD can simulate a system with only about 100 molecules. Attracted by the fact that classical MD can treat a relatively large number of molecules, researchers attempted to combine QMD and classical MD in one simulation, and such a technique has been applied successfully in simulating chemical reactions (Singh and Kollman, 1986; Field et al., 1990) and gas/vapor interaction with solid substrates (Carmer et al., 1993; Skokov et al., 1994).
III. Liquid-Vapor Interfaces This section studies planar, spherical, and cylindrical liquid-vapor interfaces. Planar interfaces include a liquid film in a bulk vapor and a vapor film in a bulk liquid, with the main goal of determining surface tension, interface stability, and evaporation and condensation coefficients. Spherical interfaces refer to a droplet in vapor and a bubble in liquid, with the main study being the curvature effect on surface tension. For cylindrical interfaces of liquid jets, the main topics are interface stability and the rupture time. Most MD work simulates LJ molecules, but simulations of other molecules are also available (e.g., Dang and Chang, 1997; Daiguji and Hihara, 1999).
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Chang-Lin Tien and Jian-Gang Weng
FIG. 1. System configuration for a thin liquid film in its vapor.
A. PLANAR INTERFACES 1. Liquid Film in Bulk Vapor Several issues are discussed here: the density profile, local pressure components, local stress, surface tension, film stability, and evaporation and condensation coefficients. Figure 1 shows a typical system configuration with periodic boundary conditions applied in each of the three directions. Here, Ls is the initial film thickness, and usually Lx = Ly.
a. Density Profile Figure 2 depicts a typical density profile, where z* = 0.0 corresponds to the center of the film. The local density falls from the liquid density (~0.771) to the vapor density ('~0.007) within about 1 rim, and these values agree with the experimental data of saturated argon densities at that temperature. The density profile can be expressed as (Rowlinson and Widom, 1982) p(z) -- 0.5(pt + p~) - 0.5(pt - p~)tanh[2(z - ze)/d]
(16)
where Pl and p~ refer to the liquid and vapor densities, respectively. Two important parameters are defined in Eq. (16): the position of the equimolar surface Ze and the interface thickness d, both of which can be obtained by numerical fitting, also
105
Molecular Dynamics Simulation '
0.8 0.6 i
i
o
~
" ~
~
,
i
,
|
,
!
O MD Simulation -- - Fitting Curve Pt = 0.77155 • 0.0016 Pv = 0.00726 + 0.00105 z~ = 3.70422 • 0.00623
Q'0.4 0.2 0.0 0
i
i
i
2
i
|
4
~
=
_
_
_- r . . . .
6
8
_'r
......
10
Z
FIG. 2. Densityprofile for a liquid thin film in vapor at T* -- 0.818.
shown in Fig. 2. Usually, the overall film thickness is considered to be 2Ze. The interface thickness d is expressed as d -- -(Pl
-
Pg)[dP(z)/dz]
(17)
-1
Z--Ze
and has two contributions: mass diffusion (evaporation and condensation) and random perturbations at the interfaces (surface waves). In MD simulation, gravity is usually neglected, so the surface wave considered here is the capillary wave.
b. L o c a l Pressure C o m p o n e n t s
In a bulk medium, the pressure is spatially uniform and can be expressed by the virial theorem V
3V
. . dp'(rij)?ij 97ij
-
PK -- PI
(18)
Here, all the N molecules in a system of volume V are considered in the pressure calculation. Equation (18) indicates that the pressure has two terms: PK considers the contribution from the kinetic motion of molecules, and is always isotropic; PI counts for intermolecular forces and is isotropic only in a system with a uniform density. In the interracial region where a large density gradient exists, the pressure is neither uniform nor isotropic. Two components in the local pressure, the normal pressure component PN (normal to the interface) and the tangential pressure component P r (parallel to the interface), are used to describe the anisotropic nature, and both of themhave two contributions: P N = P N , K -- P N , I , and PT = PT, K - PT, I. In calculation of the local pressure, one cannot use Eq. (18) by just changing the total system volume V to the volume of the local region: PN,K and PT, K can be
106
Chang-Lin Tien and Jian-Gang Weng
calculated in this manner, but PN,I and PT, I cannotnintermolecular forces acting across the boundary of the local region also contribute to the pressure components. The conventional approach defines normal and tangential pressure components as follows:
1 P N ( k ) - PN,K(k)- P g , l ( k ) - (n(k))kBT - ~sl Pr(k) = Pr, K ( k ) - Pr,1(k)=
i,j
i,j -'J rij ~'(rij)
>
-~(x~ rij+ y~) r (rij)
(19)
(20)
where n(k) is the number density in slab k, V~t the volume of slab k(Vsl = LxLyLsl), Lsl the slab thickness, and r the intermolecular potential. The summation Y~i~,j runs over all particle pairs (i, j), of which at least one of the particles is situated in slab k. If only one particle is in slab k, half of the intermolecular force contribution is given to slab k whereas the total contribution is given to slab k if both molecules are in that slab. The contribution of pair (i, j) is Z2jqb'(rij)/rij for PN and (x~ + yZ)r for Pr (Nijmeijer et al., 1988). This approach excludes the contribution from any pair of molecules if both of them are not in that slab. It should be noted that this is a simplified approach and adopted mainly for computational efficiency. According to the theory by Kirkwood and Buff (1949), the intermolecular force contribution to the local normal pressure component is given as the sum over the normal components of all the pair forces acting across a surface element divided by the surface area. The same argument holds for the tangential pressure component (Rowlinson and Widom, 1982). In other words, an intermolecular force contributes to local pressures in all imaginary planes between these two molecules. Here, the intermolecular force is assumed to act in a straight line. Based on these, Weng et al. (2000) modified the conventional method and proposed equations to calculate the pressure components as
PN(k) = PN,K(k)- PN,l(k)-- (n(k))kBT- ~sl 1 Pr(k) = Pr, l,:(k)-PT, l(k) = (n(k))kBT- ~
i,j i,j
\ ri--~ ~ (x 2 + y2) r rij
)> (22)
where a new variable fk,ij represents the ratio of the size of the vertical interval in which Fij is effective in the slab k to Zij. For example, consider the force between
M o l e c u l a r D y n a m i c s Simulation
107
FIG. 3. Schematicof pressure contribution of the pair i, j. the pair (i, j ) in Fig. 3. If neither molecule is in a slab but the line connecting them crosses the slab, such as slab 2, f2,ij -- Lst/zij; if one molecule is in a slab, such as slab 1, fl,ij = L1/zij; and if both molecules are in a slab (not shown in Fig. 3), f k , i j - - 1.0. Figure 4 compares two typical profiles of the normal pressure component calculated by these two methods, in which P~v, P~V,K, and P~v,i are reduced pressures in Eqs. (19) and (21), respectively (Weng et al., 2000). The result from the conventional method has significant fluctuations across the interface, whereas fluctuations disappear in the result from the new method. This smooth profile agrees with the condition that the film is under mechanical equilibrium in the normal direction.
c. Local Stress Local stress is defined as the difference between the normal and tangential pressure components (PN -- PT, or Pr, l - PN,I), and is important because the integral of local stress from the center of the film to the bulk vapor region gives surface tension •
-
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-- P r ) d z
(23)
Thermodynamically, local stress can be understood as the excess energy per volume. Excess energy refers to the difference between the local potential energy at the interfacial region due to the existence of a large density gradient and that of a
Chang-Lin Tien and Jian-Gang Weng
108 (a)
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bulk liquid and vapor mixture at the same density. In a bulk medium, because the pressure is isotropic (and the excess energy is zero), local stress is zero, whereas in the interfacial region local stress is not zero and serves as an indicator of the location of an interface. Therefore, local stress profiles can show how the interfacial region evolves while system parameters (such as temperature and film thickness) change. Figure 5 lists local stress profiles for films with different thickness (Weng et al., 2000). One can observe that as the film thickness decreases, the two interface regions start to overlap; consequently, local stress at the film center increases. When film thickness decreases below a critical value, film rupture occurs. As discussed later, the critical thickness depends on computational domain sizes, surface tension, and the Hamaker constant.
FIG. 5 9 Profiles of P*N , I ' PT,I, * and local stress PN* -- Pr" * The left column is from the conventional method and the right column is from the modified method. The film thickness decreases from S 1 to $4. (Reprinted with permission from Weng, J. G., Park, S. H., Lukes, J. R., and Tien, C. L. (2000). Molecular dynamics investigation of thickness effect on liquid films. Journal of Chemistry and Physics 113, 5917-5923. Copyright 9 2000, the American Institute of Physics.)
110
Chang-Lin Tien and Jian-Gang Weng d. Surface Tension
Since the pioneering work on MD simulation of surface tension by Chapela et al. (1977), simulation techniques have been improved and system size increased. Nijmeijer et al. (1988) used a much larger system to obtain the local stress profile and to study the temperature effect on surface tension. Mecke et al. (1997) proposed a new long-range force correction method and used it to simulate effects of the cutoff radius and temperature on surface tension. All of these studies indicate that surface tension values decrease as the temperature increases or as the cutoff radius decreases. Moreover, Chen (1995) applied the Gaussian model of capillary waves to study the effect of simulation domain sizes on surface tension a y ( L ) - y(cx~)= L2 (24) where y(cxz) refers to the value of surface tension at an infinitely large surface, L is the dimension of the computational domain (here L -- Lx = Ly), and a is a constant (about 2.2 in a non-dimensional system). Because both surface tension and the interface thickness can be related to characteristics of the dominant capillary wave, it is possible to determine surface tension values directly from density profiles. By assuming that the surface wave contribution to the interface thickness can be decoupled from the mass diffusion contribution, Sides et al. (1999) were successful in extracting surface tension values from the density profiles, and their results exhibited a striking agreement with results from the conventional method.
e. Film Stability and Rupture A liquid-vapor interface is always subject to random perturbations that can be treated statistically as capillary waves with different wavelengths. Thermodynamic models indicate that a capillary wave changes the system energy in two ways: (1) it increases the liquid-vapor interface area and then increases the surface energy (defined as the surface tension multiplied by the surface area); (2) it also changes (in this case, decreases) the interaction energy between two vapor phases across the liquid film. Waves with different wavelengths have different contributions to the two changes. For short waves, the first change is dominant, and the system is stable; however, for long waves, the second change is dominant, and the system is destabilized because the corresponding perturbation decreases the system energy and is amplified. At the critical wavelength, the two changes cancel each other. For a film to be stable, the largest wavelength in the system should be smaller than the critical wavelength ,~max ~ ~.cr
(25)
Molecular Dynamics Simulation
111
FIG. 6. Capillarywavepropagation in a liquid-vaporinterface.
For simplicity, a one-dimensional surface wave is considered here. Figure 6 shows the most unstable condition schematically: two synchronic waves propagate along the x direction. In this case, the critical wavelength is
)~cr- L2f,/4zr3y`~ /V A
(26)
where Lf is the film thickness, Yt, the value of liquid-vapor surface tension, and A the Hamaker constant that describes both the interaction between two vapor phases across the liquid film and interaction between two liquid phases across a vapor film (Israelachvili, 1992). Refer to Edwards et al. (1991), Majumdar and Mezic (1998), (1999), and Weng et al. (2000) for detailed derivation. MD simulation can determine the critical wavelength from a totally different approach. The largest wavelength in MD simulation equals the computational domain size due to the imposed periodic boundary conditions: ~ m a x "-- Lx. For the same film thickness, film rupture occurs when Lx is large (Weng et al., 2000). By fixing the film thickness and simultaneously varying Lx and Ly, one can obtain the critical value of Lcr. Any domain sizes larger than Lcr lead to unstable films; any domain sizes smaller than Lcr lead to stable films: Lcr ~- ~cr. The quantitative comparison between results from the classical thermodynamics prediction [Eq. (26)] and MD simulation shows that the value from the classical method is about twice as large as the value from MD. The difference might come from the fact that MD simulation considers a two-dimensional wave (in both x and y directions), but the classical prediction is based on a one-dimensional model.
112
Chang-Lin Tien and Jian-Gang Weng
Hwang et al. (1998) applied MD simulation to demonstrate the rupture process for a liquid film in vapor and a liquid film on a solid substrate. Simulation results revealed that a large liquid-liquid interaction and a small liquid-solid interaction help to increase the rupture rate, which is reasonable because strong liquid-solid interaction would keep the liquid film on the substrate. Several other interesting issues on liquid film rupture have not yet been addressed. For example, after establishment of a thermodynamic two-dimensional surface wave model [similar to Eq. (26)], determination of the stability condition in MD simulation can be a new way to calculate the Hamaker constant because values of surface tension, film thickness, and critical wavelength are readily known from simulation. However, because direct simulation of the Hamaker constant by MD is not available (Monte Carlo simulation of the Hamaker constant has been established), no comparison can be made between the direct method and the method from the stability simulation.
f. Evaporation and Condensation Coefficients Yasuoka et al. (1994) and Matsumoto et al. (1994) studied evaporation and condensation at the surface of liquid argon and methanol. A strong correlation between evaporation and condensation, termed the molecular exchange phenomenon, has been found in their work in which vapor molecules colliding on the liquid surface knock liquid molecules out to the vapor phase. They also found that temperature dependence on evaporation and condensation behavior is not significant. For other studies in evaporation and condensation coefficients, refer to review articles by Tanasawa (1994) and Carey (1995).
2. Vapor Film in Bulk Liquid After a thorough treatment of liquid films, at first sight it seems redundant to consider the vapor film any further because simulation techniques are similar and results of surface tension, the density profile, and the stability condition should be exactly the same. A detailed investigation on molecular structures of liquid and vapor films, however, reveals an interesting difference. Known from the previous discussion on local stress of liquid film and interface stability, interface overlapping occurs if the film thickness is small. As a result, the center of a film experiences a higher tensile stress, and decreasing the film thickness further breaks the film because the liquid can sustain only a limited stress. Note that the difference between liquid and vapor phases is that vapor cannot sustain any tensile stress. In the vapor film, therefore, interface overlapping should never
Molecular Dynamics Simulation
113
occur. Before two interfacial regions start to "see" each other, interface rupture should occur. It can be concluded, then, that for the same computational domain size, the critical thickness of a vapor film should be larger than that of a liquid film. This conclusion seems to contradict the classical thermodynamics prediction, because for LJ liquids the stability condition depends only on surface tension and the Hamaker constant, both of which are supposed to be identical in these two film cases. In order to solve this discrepancy, detailed analysis and extensive simulation would be required.
B. SPHERICAL INTERFACES
1. Droplets Simulation of droplets started as early as that of liquid films (Rusanov and Brodskaya, 1977). Formation, dynamics, and surface tension of droplets have been the major focus in MD simulation. The temperature effect on surface tension is similar to that in the planar interface, whereas the curvature effect is unique and has been simulated extensively in the past several decades. Disagreement, however, still exists. Figure 7 shows a typical profile of instantaneous molecule position of a droplet viewed from one direction (e.g., x direction) in the computational domain.
FIG. 7. Molecule position for a droplet in the computational domain.
114
Chang-Lin Tien and Jian-Gang Weng a. Droplet Formation and Dynamics
The discussion of droplet formation here considers multiple droplet formation during homogeneous nucleation in the vapor instead of preparation of the initial condition for the liquid-vapor interface. Classically, the nucleation rate is an exponential function of the free-formation energy AG, and A G is expressed as A G = YA + Ag V
(27)
The first term represents the energy required to create a liquid-vapor interface and is always positive, where Y is surface tension and A is the surface area. The second term is usually negative and represents the free energy change between bulk vapor and liquid phases, where Ag is the free energy difference per unit volume and V is the volume. Further derivation that relates A G to the droplet radius R is based on the assumption that the droplet is perfectly spherical. Yasuoka and Matsumoto (1998a-c) simulated homogeneous nucleation in vapor phases of argon and water. They revealed a significant difference between A G(R) from MD simulation (interpreted from the simulated nucleation rate) and classical prediction (with experimental data of Y and Ag). The possible reasons are: (1) the surface tension value in classical prediction is for planar interfaces and is different from that for small droplets due to the curvature effect; (2) the droplet shape is far from spherical; (3) Ag in nanoscale is different from that in macroscale; and (4) the classical theory is based on the infinitely large bulk phase whereas MD simulates the system with a limited number of molecules. Usually, the impact of errors in surface tension is large, and this induces extensive research efforts to study the curvature effect on surface tension. Droplet evaporation simulation reveals another interesting phenomenon. In the classical theory, the evaporation rate is defined as dr/dt. The large droplet evaporates at a rate so that ~ (x t, where R is the droplet radius and t is time. For small droplets, the evaporation rate is constant: R cx t. Long et al. (1996) used MD simulation to confirm this proportionality. b. Density Profile Similar to the planar interface case, the density profile can be fitted with the expression p(r) = 0.5(pt + P v ) - 0.5(pt - Pv) tanh[2(r - Re)/d]
(28)
as shown in Fig. 8. Here, Re is the radius of the equimolar surface and d is the interface thickness. From the comparison between density profiles of a liquid film
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and a droplet, one may find that the vapor density in droplet simulation is usually lower than the vapor density in film simulation and the liquid density is higher. The reason is that the pressure in a droplet is higher than that in the vapor. One may also notice that the local density (and the local pressure, as discussed later) near the droplet center exhibits large fluctuations, which are caused by statistical calculation over a small volume (or surface). Another interesting issue is that if the simulation starts by melting and vaporizing the same crystalline solid in vacuum (as discussed in Section II.A.3), the final vapor density differs for different computational domain sizes. The vapor phase is formed as molecules evaporate from the crystalline structure to the vacuum, which would take quite some time to reach equilibrium. Criteria to determine whether the system is at equilibrium have not yet been established. c. Pressure Tensor
In a spherical system, the pressure decomposes into two components: the normal pressure PN and the tangential pressure PT. Different from the liquid film case, the two pressure components are not independent because of the mechanical equilibrium in the system r d PN(r) PT(r) = Pu(r) + - ~ 2 dr
(29)
116
Chang-Lin Tien and Jian-Gang Weng
Therefore, MD simulation need consider only the normal component because it is easier to simulate than the tangential one. The knowledge of the normal component profile not only gives the tangential component, but yields surface tension as well, as expressed by the Laplace or the Young-Laplace equation
Pl-- P~ = 2?, Rs
(30)
where Rs is the radius of the surface of tension. The pressure at the droplet center, Pl, and the pressure in the bulk vapor phase, P~, can be obtained from the profile of the normal pressure component: Pl -- PN (r = 0) and P~ = PN (r = c~). Note that planar liquid films are at mechanical equilibrium only in the normal direction because external forces are acting in the tangential direction at the film boundary to keep the film flat. Based on the assumption that the intermolecular force is acting on a straight line, Thompson et al. (1984) provided the formula to simulate the normal pressure component
PN(k) = (n(k))kBT
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where the subscript k indicates the local value of the kth shell surface. Note here that the pressure is taken as the surface average, as compared to the case of planar interface, in which the pressure is a volumetric average [in Eqs. (19) and (21)]. The reason is that in a shell with radius from r to r 4- Ar, the normal component of an intermolecular force has a different direction in each concentric surface. If one force crosses a shell surface twice, its contribution to that plane is doubled. Figure 9 shows pressure components for a droplet and a bubble (discussion on the pressure tensor of a bubble is given later) (Park et al., 2000b). Just as in the planar interface case, the local stress is defined as the difference between the normal and tangential components. Fluctuations of the tangential pressure (and therefore local stress) on the liquid side are very large.
d. Surface Tension Classically, Tolman's equation is applied to express the curvature effect on surface tension of spherical interfaces (Tolman, 1949; Thompson et al., 1984) y
= 1--
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where g ~ refers to the surface tension value for planar interfaces and ~ is called the T o l m a n l e n g t h and is defined as the difference between the radius of the equimolar surface and the radius of the surface of tension
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118
Chang-Lin Tien and Jian-Gang Weng
The Tolman length 6 is usually considered to be positive for droplets and negative for bubbles. Other researchers applied the Density Functional Theory (DFT) and found that 3 varies with the droplet radius and becomes negative when the curvature is small or the temperature is high (e.g., Kalikmonov, 1997; Koga et al., 1998; Baidakov and Boltachev, 1999; Bykov and Zeng, 1999; Koga and Zeng, 1999). Two methods are available to calculate surface tension of droplets, both of which require the knowledge of the normal pressure component profile and the surface tension value for planar interfaces. One is from Eqs. (30), (32), and (33) (three equations for three unknowns: y, 3, and Rs, and the radius for the equimolar surface Re can be derived from the density profile). The other way is to integrate the local stress as ?' -- fo
~r
[PN(r) -- Pr(r)]dr
(34)
and Rs is from Eqs. (32) and (33). Neijimeijer et al. (1992) tried this method and found fluctuation in the liquid phase is too large to perform integration. However, it is still possible to integrate the fitting curve (Thompson et al., 1984; Park et al., 2000b). Usually the Tolman length 3 is very small (< 1.0 or) and varies with temperatures. Haye and Bruin (1994) took a different approach to calculate 3 directly from the liquid film simulation and found that 3 is about 0.2 o. However, because of statistical errors, MD simulation has not clearly revealed the temperature and curvature effects on 6. One may find that in Fig. 7, the droplet is not perfectly spherical because of capillary waves at the spherical interface. Similar to the planar interfaces, it should also be possible to calculate surface tension directly from the density profile. Powels et al. (1983) studied the capillary wave contribution to the overall interface thickness, but this contribution was obtained with a known surface tension instead of directly from the density profile.
2. Bubbles a. Bubble Formation and Initial Conditions
Typically, a bubble forms either as the temperature increases (constant pressure condition) or as the pressure decreases (constant temperature or energy condition). In MD simulation, the latter condition is easier to achieve. Kinjo et al. (1999) obtained a bubble by stretching the bulk liquid (in other words, by decreasing the pressure to negative values). The main objective of their work was to study the
Molecular Dynamics Simulation
119
incipient stage of homogeneous nucleation and bubble dynamics, and therefore it was not necessary for the bubble to be stable. In studies of the surface tension or density profile of a bubble, it becomes important to be able to maintain a stable bubble surface. Park et al. (2000a, 2000b) obtained a bubble by taking a liquid sphere away from an otherwise uniform bulk liquid and allowing the liquid to equilibrate itself. They found: (1) the minimum size of the stable bubble that can be obtained was different from that of the droplet; and (2) the condition of maintaining a stable bubble depended on the computational domain size. For a smaller domain, the average number density (defined as the total molecule number divided by the total domain volume) should be lower to guarantee a phase separation; for a larger domain, the average number density can be higher. Such observations indicate that small domains are more stable than large domains, and the density should be decreased more in order to destabilize the liquid phase, which is physically true because the metastable condition is destroyed by perturbations with long wavelengths and in a large system the allowed wavelength is longer. For detailed information of phase stability, refer to the work by Abraham (1979) and Koch and coworkers (1983). Note that even if a phase separation occurs, the resultant "bubble" may not be spherical because the interaction of the bubble in the computational domain with other "bubbles" in its neighboring domains (i.e., its images) results in a vapor column.
b. Surface Tension
The surface tension simulation scheme is same as that used in droplet cases. Figure 9b plots typical profiles of normal and tangential pressure components. Figure 10 shows simulation results for the curvature effect on surface tension of bubbles and droplets, as compared with Tolman's prediction (Park et al., 2000b). It is found that surface tension of droplets follows Tolman's prediction quite well, but surface tension of bubbles does not. The reason for this discrepancy is still unknown.
C. CYLINDRICALINTERFACES
Currently, simulation of liquid jets in vapor studies only the stability condition and the interface rupture characteristics. Classical treatment of liquid jet breakup relies mainly on Rayleigh's studies on linear stability of an infinitely long, circular, incompressible liquid jet. The axisymmetric perturbation can be expressed as (Bogy, 1979) r = a + ae exp(ott - i k z / a )
(35)
Chang-Lin Tien and Jian-Gang Weng
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where r is the "local" radius in the jet, a the average radius, z the direction along the cylinder axis, e a non-dimensionalized perturbation amplitude (different from the energy parameter in the LJ potential), and c~ the wave growth rate. The wave number k is non-dimensionalized according to the radius: k = 2roaN~L, where N is an integer and L is the jet length. In inviscid fluids, the growth rate is given as ll(k)(1 - kZ)k 1/2
?' -
10(k)
(36)
and in highly viscid fluids, it is given as lY - - ~ V ( 1
6a#1
-k
2)
(37)
where V is surface tension of the liquid jet, p the bulk liquid density, #1 the liquid viscosity, and In the nth-order modified Bessel function of the first kind. Despite increasing importance of the liquid jet in industry applications, its MD studies are very limited. Koplik and Banavar (1993) and Kawano (1998) used MD to study the rupture time and compared their results with Rayleigh's theory in Eqs. (36) and (37). Recently, Moseler and Landman (2000) simulated nano liquid jets emitting from a solid nozzle and found that nozzle exit condition has profound effects on the nanojet stability.
Molecular Dynamics Simulation
121
Several issues regarding liquid jets have not been studied sufficiently and are discussed in the following subsections.
1. Stable Cylindrical Interfaces In order to evaluate thermophysical properties such as density profile or surface tension, a stable cylindrical interface should be maintained because rupture occurs in a time scale that is short enough for the MD simulation method to detect. One possible way to accomplish this is to control the length of the liquid jet so that the wave growth rate ot is less than zero (the wave exponentially decays with time). According to Eqs. (36) and (37), the non-dimensional wave number k should be larger than 1 or L < 2Jra.
2. Density Profile and Surface Tension A typical density profile can be found in Kawano (1998) and is reproduced in Fig. 11. Note the y axis is in the log scale. Similar to planar and spherical liquidvapor interfaces, the interface thickness is related to the surface waves. Comparison of density profiles for liquid cylinders with different lengths may reveal the effect of different capillary waves on the perturbation amplitude in each system.
10.000
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0.100
0.010 L
0.001
0
. . . . . . . . . . . . . . . . 2
4
6
r1
8
FIG. 11. Densityprofile of a liquid cylinder in vapor. (Reprinted with permission from Kawano, S. (1998). Molecular dynamics of rupture phenomena in a liquid tread. Ph~'sicsReviewE 55, 3068-3071. Copyright 9 1998, the American Physical Society.)
122
Chang-Lin Tien and Jian-Gang Weng
Surface tension of cylindrical interfaces, as a result of interface instability, has not been addressed by MD simulation. However, in order to obtain accurate values of the wave growth rate, it is necessary to study the curvature effect on surface tension instead of using the surface tension values for planar interfaces. In a cylindrical system, the force can be decomposed into three components along radial (normal), azimuthal, and longitudinal coordinates: Fr, Fr and Fz, respectively. Two are particularly important: the normal component Fr can determine the surface tension, and the longitudinal component Fz can reveal detailed information on tensile stress that the liquid jet experiences when interface rupture occurs. The simulation algorithm has not yet been established.
3. Interface Rupture The ability to control the length of liquid cylinders provides another opportunity to study the interface rupture phenomenon. By gradually increasing the jet length (increase the computational domain size in z direction), one should be able to observe the trend of destabilizing the nanojet. A plot of rupture times with the nanojet length could provide some insights on how the capillary wavelength can change the stability.
IV. Liquid-Liquid Interfaces Liquid-liquid interfacial studies are of great interest to researchers in various fields because many biological, chemical, and physical processes occur at liquidliquid interfaces accompanied with phase separation or transport of electrons, ions, and molecules. Here, surface tension and diffusion coefficients play critical roles, and sufficient understanding of them is highly desirable. Currently, most studies of these processes are still based on macroscopic models or experiments. Molecularlevel details on liquid-liquid interfaces not yet available. By exploring the molecular structure at interfaces, MD simulation can provide a theoretical tool to interpret experimental data, offer new physical insight, and benchmark applicability of existing phenomenologic models. Basic simulation techniques have been presented in previous sections. However, several difficulties that have not been encountered in the study of liquid-vapor interfaces arise. In simulation of liquid-vapor interfaces, presence of space with a dilute vapor phase eases the equilibration process because the liquid phase has enough room to adjust its size or shape, and molecules can evaporate from or condense to the liquid phase freely. As a result, the system pressure is constant and close to zero
Molecular Dynamics Simulation
123
FIG. 12. Systemconfiguration for liquid-liquid interfaces.
(as shown in Fig. 4). In the simulation of liquid-liquid interfaces, in contrast, no space exists for liquid phases to adjust themselves and, consequently, the normal pressure component becomes difficult to control; It not only shows large fluctuations, but also differs significantly for different initial conditions. Solutions for this problem are either to introduce a vapor phase into the system (shown in Fig. 12) or to adopt different simulation schemes (Allen and Tildesley, 1987). Refer to Zhang et al. (1995) for a more detailed discussion. Moreover, in studies of liquid-liquid interfaces, two or more types of molecules are present, and the intermolecular potential model between unlike molecules is slightly artificial.
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Chang-Lin Tien and Jian-Gang Weng
This section reviews studies of various liquid-liquid interfaces. In addition to planar and spherical interfaces, mixtures of two or more liquids are discussed as a result of their significance in real applications.
A. PLANAR INTERFACES The typical setup is a liquid slab sandwiched by two slabs of the other liquid, as shown in Fig. 12, with periodic boundary conditions applied in all three directions. Most studies adopt Figs. 12a and 12c, and different normal pressures in these studies may influence their simulation results. The two liquids can be either immiscible or partially miscible. Several important issues are discussed here: density profiles, diffusion coefficients, surface tension, and interface stability.
1. Density Profile The simulation method of local densiti,~s is same as that in liquid-vapor interface studies. The only difference is that three profiles are required here: the local density of Phase A, the local density of Phase B, and the total density. A typical density profile is shown in Fig. 13 (Dfaz-Herrera et al., 1999). Many factors affect the shape of density profiles, including normal pressure, temperature, and affinity between unlike molecules. Due to the existence of three curves, a clear definition of the interface thickness is not available. Different from the liquid-vapor interfaces, the density profile of Phase A may not show monotonic decrease from its bulk phase region to the region of the other liquid. An oscillatory structure has been revealed by MD simulation (Toxvaerd and Stecki, 1995; Stecki and Toxvaerd, 1995). Density profiles at an interface of two immiscible liquids at low temperature and high pressure are shown in Fig. 14. MD simulation also shows that such oscillation is prominent only at interfaces of small cross-sectional areas because capillary waves with long wavelengths smear it. The cause of such oscillation is similar to the layering process when a liquid approaches a solid (discussed in Section V).
2. Surface Tension Surface tension simulation, again, follows the same procedure as that in liquidvapor interfaces. Simulation results indicate that surface tension values depend on not only system temperature and cross-sectional area, but also normal pressure and interaction between unlike molecules. D/az-Herrera et al. (1999) found
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1
0.8
i 0.6
~ a
0.4
~
0
~,
4
8
2'"
. . . . . . . . . . .
T'=0.827, p ~
12
16
20
FIG. 13. Densityprofiles at a liquid-liquid interface, p* is the total density. (Reprinted with permission from Dfaz-Herrera, E., Alejamdre, J., Ramffez-Santiago,G., and Forstmann, E (1999). Interfacial tension behavior of binary and ternary mixtures of partially miscible Lennard-Jones fluids: A molecular dynamics simulation. Journal of Chemistry and Physics 110, 8084-8089. Copyright 9 1999, the American Institute of Physics.)
non-monotonic behavior of v(T) with respect to the system temperature T in the interface of two partially miscible liquids (Fig. 15), as compared to monotonically decreasing behavior in the liquid-vapor interface. The reason is that the diffusion coefficient usually increases with temperature, which may change the local stress profile in the interface. The relationship between these two parameters has not yet been addressed.
3. Transport Coefficients As stated in previous sections, an interface is a highly anisotropic region, and this leads to the anisotropic behavior in transport coefficients at the interface. It is also expected that values of transport coefficients at the interface will be different from those in the bulk phase. Such differences are prominent in studies of nanoscale systems. Among all the transport coefficients, the diffusion coefficient might be considered the most important because it is closely related to processes of chemical and physical reactions and the density profile across the interface. Let DAB be the diffusion coefficient of molecules A in a bulk phase B. At equilibrium, DAB = DSA.
0.8 0.7
0.2
0.1 n 4.5
5
5.5
6
6.5
z position
7
7.5
6
0.2
15
15.5
16
16.5
17 17.5 porition
16
18.5
19
19.5
2
FIG.14. Oscillatory profiles for (a) density of phase A and (b) total density at liquid-liquid interfaces. The full line is for the domain size L , = 7.5 u, the dashed line for L , = 15 u , and the dotted line for L , = 22.5 u . (Reprinted with permission from Toxvaerd, S., and Stecki, J. (1995). Density profiles at a planar liquid-liquid interface. Journal of Chernisfry and Physics 102, 7163-7168. Copyright 0 1995, the American Institute of Physics.)
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|
127
9
2.2 2.1
f
1.9 1.8
9N = 1 7 2 8
1.7
9N = 2 5 9 2
1.6 1.5 1.4
0.5
l
I
1
,
I
1.5
,
I
T
.2
i
I
2.5
,
I
3
FIG. 15. Temperature effect on surface tension at liquid-liquid interfaces for systems with two different molecule numbers. (Reprinted with permission from Dfaz-Herrera, E., Alejamdre, J., RamfrezSantiago, G., and Forstmann, E (1999). Interfacial tension behavior of binary and ternary mixtures of partially miscible Lennard-Jones fluids: A molecular dynamics simulation. Journal of Chemistry and Physics 110, 8084-8089. Copyright 9 1999, the American Institute of Physics.)
The method to determine DAB can be found in Allen and Tildesley (1987). As with surface tension, one might find DAB to be a function of temperature, normal pressure, and the miscibility of the two components. Another interesting problem is to consider the diffusion process of a third type of molecules across the interface. Benjamin (1991, 1992) used MD to simulate electron and ion transport across the interface of two immiscible liquids. For different types of liquid (e.g., polar or nonpolar) and difference particles (e.g., changed or uncharged), the diffusion process is completely different, and more extensive simulations are required to solve this problem.
4. Interface Stability When a liquid slab is sandwiched by two slabs of another liquid, the stability issue arises. If the material properties (e.g., the dielectric constants and refractive index) are very close, the Hamaker constant is very small [for calculation of the Hamaker constant, refer to Israelachvili (1992)]. According to Eq. (26), if the surface tension value is larger than or comparable to that at liquid-vapor interfaces, the liquid film can be very thin before rupture occurs.
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Chang-Lin Tien and Jian-Gang Weng
This phenomenon might bring about the question of applicability of concepts of the Hamaker constant (or disjoining pressure) and the surface tension when the liquid film is atomically thin.
B. SPHERICAL INTERFACES The spherical liquid-liquid interface refers to the case of a large molecule or a droplet (i.e., a molecular cluster) in a bulk liquid consisting of another type of molecules, a condition frequently encountered in practical engineering applications. The droplet usually adopts a spherical shape in order to minimize its surface energy. Complete understanding of spherical liquid-liquid interfaces involves studies of three problems: droplet formation, growth, and transport. Droplet formation in liquids is similar to that in vapors: the nucleation rate exponentially depends on the formation energy A G that is expressed in Eq. (27). As in spherical liquid-vapor interfaces, the curvature effect on surface tension is important here. However, due to involvement of different molecules (and therefore different interaction potentials) in this study, such an effect is much more complicated. After formation of droplets, how they grow is another concern in engineering processes. Laradji et al. (1996) used large-scale MD simulation to monitor the dynamic growth process of droplets during the final stage of spinodal decomposition, and their simulation results were in agreement with classical prediction: the droplet size R(t) is proportional to t, where t is time. Many parameters may affect such a proportionality, including temperature, intermolecular potentials, and pressure. It has been known that droplets dispersed within another liquid can change the transport coefficients of the host liquid significantly. An example is that shapedeformation at surfaces of dilute droplets results in a macroscopic viscoelasticity in an otherwise continuous Newtonian fluid (Edwards et al., 1991). Effects of droplets on transport coefficients is different if system parameters such as temperature and pressure change, all of which deserve more detailed investigation.
C. MIXTURES
Mixtures of two or more liquids can be categorized further into isotopic and nonisotopic mixtures. In an isotopic mixture, the intermolecular potential between like and unlike molecules is equivalent; the only difference is the mass of each component. If the gravity effect is neglected or the mass differences are small,
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these components achieve an ideal mixing condition. In contrast, in the case of nonisotopic mixtures, things are quite different. Consider a case in which B molecules disperse in a bulk phase consisting of A molecules. If the attraction force between two B molecules is stronger than that between an A molecule and a B molecule, the B molecules tend to bind together and form a small droplet if attraction between unlike molecules is stronger, many A molecules surround a B molecule, consequently changing the structure of the host liquid. For both mixtures, the main concern is how to express its properties as simple functions. For an isotopic mixture, the effective transport coefficient depends on the mass ratio, mole fractions of two components, and the bulk properties of each component. Kerl and Willeke (1999a) simulated these effects on selfdiffusion coefficients. Other properties such as viscosity and thermal conductivity are still unknown. In a nonisotopic system, simulations are more complicated, as is interpretation of simulation results. Kerl and Willeke (1999b) considered the effect of the mass ratio on diffusion coefficients of an equimolar binary mixture. Additional work must be done in exploration of the effects of molecule sizes, mole fractions, intermolecular potentials, and system parameters such as temperature and pressure. In many engineering applications, properties (especially thermodynamic properties such as enthalpy) of a liquid can be represented as functions of its density and temperature. One of the main goals in simulating mixture properties is to explore whether a mixture could be treated as a "pure" liquid with its properties depending on its "effective" density and temperature. Generally, mixture properties can be obtained by two mixing rules: Kay's (mole fraction-weighted) rule and Enskog's rule (Heyes, 1992). For example, for a mixture of different LJ molecules, Kay's rule is expressed as ~x -- ~
Xi(7i
(38)
XiEi
(39)
ximi
(4O)
i
Sx -- Z i
mi -- Z i
where cri and ei are parameters in the LJ potential for molecules i, mi is the molecular mass, and xi is the mole fraction. The effective densities and temperature for this mixture are given as Px = p(~rx Icrl )3
(41)
Tx = T ( s l / S x )
(42)
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Chang-Lin Tien and Jian-Gang Weng
where p and T are the actual (overall) density and temperature in the mixture, and liquid properties are expressed as a function of Px and Tx. Heyes (1992) demonstrated that for some mixtures (Ar-Kr, Ar-CH4, CH4-N2), different mixing rules lead to similar results. More extensive simulation is necessary to solve this problem.
V. Liquid-Solid Interfaces Subjects in liquid-solid interfacial study are a little different from those in liquid-vapor and liquid-liquid interface cases. Here, the main efforts are on nanoscale boundaries and liquid-solid transition. Nano-boundary problems include determination of viscosity, velocity slips, and temperature slips between solid substrates and the liquid. The study of liquid-solid transition considers mass transfer and surface morphology during melting and solidification. Only a few studies address liquid-solid interfacial tension because large fluctuations in the local stress profile exist at the liquid-solid interface.
A.
PLANAR
INTERFACES
1. Nano-Boundary Problems
Boundary conditions are essential in solving momentum transport problems at a fluid-solid interface. Traditionally, a no-slip condition is enforced, and the underlying assumption is that interactions between fluid and solid molecules are strong enough to keep the interface at thermodynamic equilibrium. If such an assumption is not satisfied, the empirical linear Navier boundary condition must be applied V~ ,,all - L~ y A Vx - Vx,J~,id -- Vx,wa, = L s -~z
(43)
where Ls is the constant slip length, ~, the strain rate, x the flow direction, and z the direction along which the velocity gradient exists. In gas-solid interfaces, the equilibrium assumption requires a high frequency of collisions between gas and solid molecules or the Knudsen number K,, - - lmfp/D < 0.01 (Gad-el-Hak, 1999), where lmfp is the mean free path and D is the length scale in the boundary problem (in some literature, the Knudsen number is defined as D/lmfp). For liquid-solid interfaces, however, the macroscopic criteria to determine whether the
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no-slip condition is valid for a specific problem are not well established, and the nonequilibrium Molecular Dynamic (NEMD) method is often applied to study the liquid-solid boundary condition. Detailed molecular-level simulations reveal that three basic boundary conditions exist at solid-liquid interfaces: slip, no-slip, and multilayer locking. The third case occurs when liquid molecules adjacent to the solid substrate are adsorbed to the solid due to strong attraction between liquid and solid molecules and, consequently, they form highly ordered layers and follow the motion of the solid. Because the thickness of these ordered layers is in the nanometer range, the multilocking condition is always considered to be the no-slip condition in macromicro-scale boundary conditions; in nanoscale studies, however, they become critical, and neglecting them could introduce a significant error. One difficulty encountered in studies of flow boundary condition is that the apparent viscosity deviates from its bulk value when the system dimensions shrink to the nanometer range. This deviation is usually accompanied by a change of the molecular structure in liquid (e.g., multilayer locking). Both experimental study (Gee et al., 1990) and numeric simulation (Thompson et al., 1992) indicate a transition from the liquid to solid-like structure when an ultra-thin liquid film is confined between two solid plates. Although it is known that the critical thickness at which liquid structure transitions occur depends on the strength of interactions between liquid and solid molecules, a general guideline to predict such thickness has not been developed. The other difficulty involves inability to determine, from first principles, the condition under which slip, no-slip, or multilocking occurs, and MD simulation is often applied to provide insight into this problem. An interesting example is the simulation work by Thompson and Troian (1997), who simulated a Couette flow and studied velocity slips as a function of interaction parameters between solid and liquid molecules. Their velocity profiles are shown in Fig. 16, in which a significant velocity slip is observed when the density of the wall (pw) is large or the wall-fluid interaction (characterized by LJ parameters 0 "wf and e ~f) is weak. Thompson and Troian (1997) further obtained a general equation that relates Ls to }~
L--~
( c),J2 1-
(44)
where L ~ $ and ~,~ are the scaling values that depend on the intermolecular potential parameters. Note that although it is obtained from the molecular level simulation, Eq. (44) is a general boundary condition and can be applied to solving macroscale problems. Other studies (e.g., Thompson and Robbins, 1990) also reveal the multilayer locking condition in a Couettee flow due to strong solid-fluid interactions.
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Z
9
I
II
I
I
'
'
i
,
~F
0.5
oooOO~J .
e,~li 9
F-wf(Twf
. * ~y~ J~
0 o~ 0
0 0 . 6 1.0 *0.60.754 9 0.2 0.75
_PlWI I
0.5
z/h
FIG. 16. Velocity profile in Couette flow geometry, p~' is the nondimensionalized solid density and cr wf and e w f are LJ parameters for the wall-fluid interaction. (Reprinted with permission from E A. Thompson and S. M. Troian, N a t u r e . A general boundary condition for liquid flow at solid surfaces. 1997;389"360-362. Copyright 9 1997, Macmillan Magazines Limited.)
Many other questions in flow boundaries remain unsolved. Examples include flow boundaries of liquids consisting of diatomic or polyatomic molecules and Hagen-Poiseuille flow. In the former case, not only the characteristics of interaction potential between solid and liquid are different, but other degrees of freedom (rotational and vibrational) contribute to the slip condition as well, complicating the analysis further. The Hagen-Poiseuille flow, a flow in the pipe of a circular cross section, is another flow dominated by viscosity and, therefore, by molecular interactions across solid-liquid interfaces. In boundary conditions for the energy transport equation, an analogous nojump condition is applied: T f l u i d l w a l l - - Twa11.Similar to the no-slip condition in flow boundaries, the no-jump condition occasionally fails to provide accurate description at solid-liquid interfaces. Generally, a temperature jump is always present between two different materials due to boundary scattering of phonons--for example, a contact resistance between two solids. If one of the materials is a fluid, the temperature jump becomes larger. Maruyama and Kimura (1999) and Ohara and
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Suzuki (2000) proved that the temperature jump would be significant in nanoscale studies or when the temperature gradient across the interface is large. However, a general temperature boundary condition, similar to Eq. (44) in flow boundaries, has not been established. When a chemical potential gradient exists across the interface, mass diffusion occurs. Geysermans et al. (2000) simulated the interface between solid copper and liquid aluminum and found the diffusion coefficient of liquid adjacent to the solid is four times as small as that of the bulk liquid. The dramatic decrease of diffusion coefficient, probably due to the multilayer locking process at the solidliquid interface, would lead to a concentration jump and deserve further studies.
2. Liquid-Solid Transition Studies of melting and solidification are of both fundamental and practical interests, especially in the area of semiconductor material processing, because any defect or impunity introduced during crystallization significantly deteriorates microelectronic devices. Usually, two theoretical methods are available to study crystalline growth: the thermodynamic method and the kinetic rate theory. However, neither can provide detailed dynamic information of the crystal facet or the diffusion process of foreign molecules (e.g., oxygen or hydrogen molecules) across the liquid-solid interface. These problems can be solved easily by MD simulation because MD can monitor motion of molecules at any time. Film growth during Liquid Phase Epitaxy (LPE), which involves the precipitation of a crystalline film from a supersaturated melt onto the parent substrate when the system temperature decreases under a threshold value, is one of the most important topics of MD simulation of solid-melt interfaces. It has been known that temperature gradients across solid-melt interface have significant impacts on film growth rates, facet morphology, and the temperature at which epitaxy is initiated (the initiation temperature can be noticeably lower than the liquid-solid coexisting temperature). Landman and coworkers (1988) and Luedtke et al. (1988) conducted interesting research on such effects. Later, Nishihira et al. (2000) found that stress of the grown film also depends on the temperature gradient. The diffusion process of defects and foreign particles is another important problem. Defects refer to the vacancies and interstitials; foreign particles include hydrogen, oxygen, and other impunities (such as sulfur). If the production processes (e.g., LPE, Czochralski pulling, or floating-zone methods) are not well controlled, densities of defects and impunities increase by orders of magnitude. This study differs from the study in the concentration slip condition in that here a temperature
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gradient always exists across the solid-melt interface, which greatly complicates the analysis. Only a few MD studies have been taken to simulate dynamics of foreign particles (Kakimoto et al., 2000) and transport of defects (Ishimaru et al., 1998) in conditions of different temperature gradients. Moreover, during Czochralski pulling or floating-zone purification, macroscopic motion of melt usually exists, and such motion also influences formation of defects and impurities. No studies have yet addressed such an effect.
3. Density Profiles and Interfacial Tension
Many microscale studies on liquid-solid interfaces have revealed that when a liquid phase approaches a crystalline solid, liquid molecules near the solid surface form ordered layers that are similar to the solid structure due to the strong repulsive force between the liquid and solid molecules. Such a layering process is different from the multilayer locking in that the liquid molecules do not necessarily move with the solid. Because liquid molecules are localized in specific regions and other regions have no molecules, large fluctuations are encountered in density profiles, making evaluation of the interfacial thickness impossible. Similarly, fluctuations in local stress make it difficult to determine interfacial tension (Sikkenk et al., 1988; Nijmeijer et al., 1990) because interfacial tension is just an integral of local stress across the interface, according to Eq. (23). Because sufficient understanding of interfacial tension is critical, more extensive simulation or better simulation techniques should be developed in the future.
B. SPHERICALINTERFACES Spherical liquid-solid interfaces are encountered in colloidal liquids. If the particle size is close to the size of the liquid molecules, analytical treatment based on classical statistical and kinetic theories might be problematic due to the increasing importance of layering liquid molecules near the solid surface. Formation and growth of solid particles are similar to those of liquid droplets discussed previously, but the transport process is quite different because in the latter case interface deformation could occur. Two problems are discussed here: drag force and the diffusion coefficient of solid particles. Classically, for a sphere moving in unbounded fluids the drag force can be expressed as Fd = 6rc# R U
(45)
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where/z is viscosity, R the sphere radius, and U the particle velocity. Note here a "no-slip boundary" is assumed. For the sphere moving adjacent to a solid wall, the expression becomes Fd = 6zr# R2 U / h
(46)
where h is the minimal distance between the wall and the sphere. Vergeles et al. (1996) found that as h is small, a velocity slip occurs and Eq. (46) is not applicable, and as the particle size is small, layering of the liquid molecules adjacent to the solid sphere makes the definition of the "sphere radius" in Eqs. (45) and (46) slightly ambiguous. The diffusion coefficient is related to the drag force by Einstein's expression, and can be expressed as D# =
ksT
6zrR
(47)
Simulation techniques are similar to those discussed previously.
C. CYLINDRICALINTERFACES
The Hagen-Poiseuille flow, a flow in a tube with a circular cross section, is one of the most important flow types in industry applications. Classically, the velocity inside the tube follows a parabolic distribution and vanishes at the inner wall of the tube. As the tube radius decreases, the velocity profile is quite different. For a gas flow, the gas mean free path is on the order of 10 nm at atmospheric pressure and room temperature; if the tube size is on the order of 1 micron or the Knudsen number is on the order of 0.01, the slip boundary condition must be considered (Harley et al., 1995). Furthermore, Wu (2000) proved that in addition to the slip condition, the gas compressibility, nonparabolic velocity profiles, and channel deformation due to high pressure in the channel should be considered in order to predict the mass flow rate in micro channels accurately. For a liquid flow, macroscopic criteria to determine the onset condition of velocity are not available, and molecular level simulation must be used. Only a few MD studies are related to the Poiseuille liquid flow; some considered flows in nano tubes (Heinbuch and Fischer, 1989) whereas others simulated two-dimensional Poiseuille flows confined between two solid plates (Koplik et al., 1988; Travis et al., 1997; Travis and Gubbins, 2000). According to simulation results of these studies, the velocity profile deviates dramatically from the classic parabola when the length scale of the flow shrinks to the same order of the molecular radius because the viscosity is no longer uniform (instead, it shows
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large fluctuations). The boundary conditions, slip, no-slip, and multilayer locking, however, have not received enough attention. For example, in most studies, interaction parameters between solid and liquid molecules are at least equal to the the interation parameters between liquid molecules, which seems to guarantee establishment of the no-slip or multilayer locking condition.
VI. Three-Phase Systems The three-phase system refers to the condition in which solid, liquid, and vapor phases are all present. Because of its complexity, it has been less extensively simulated by MD techniques than other interface cases. The main research efforts have been focused on the spreading wetting and contact line region but, as discussed later, interfacial problems in planar and spherical systems are also important and interesting. A. PLANAR INTERFACES
The planar three-phase system includes (1) a liquid film sandwiched by a bulk vapor and a bulk solid and (2) a vapor film by a bulk liquid and a bulk solid. The first case exists in heterogeneous condensation and the second case exists in heterogeneous boiling. The Lithographically-Induced Self-Assembly (LISA) technique recently developed by Chou and Zhuang (1999) also belongs to the second case. The major problem in this interface case is film stability. Similar to the planar liquid-vapor interface case, the stability conditions for a liquid film and a vapor film are different. Using thermodynamics theories, Majumdar and Mezic (1998, 1999) considered stability of a water film on an atomically smooth and hydrophilic substrate and obtained a stability region map. This stability study is similar to that in the case of a liquid film between two vapors except that more interaction potentials are involved, including hydration, electrostatic force, and elastic strain between solid and liquid. With properly given intermolecular potentials between liquid molecules and the solid wall, MD techniques can certainly simulate this condition and then compare simulation results with classical predictions. Since a liquid film can form on a smooth, solid substrate during condensation, it is also reasonable to assume that a vapor film could form on the solid substrate during boiling, called explosive boiling. Yang and Tsutsui (2000) used microfabricated line resistors to study heterogeneous boiling in a bulk liquid. They found that if the heater surface was very smooth and the heat flux was high (~ 106 W/m2), a vapor layer formed first between the bulk liquid and the heater surface (shown
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(a)
FIG. 17. Schematic of (a) electrical heating in liquid and (b) Lithographically-Induced SelfAssembly (LISA).
schematically in Fig. 17a), and as the vapor layer thickness grew, it became unstable and turned into a bubble. Note that this boiling mechanism is different from classical heterogeneous boiling due to the unavailability of nucleation cavities on the heater substrate (Carey, 1992). It is not known what causes the film to break, and both experimental studies and numerical simulations are required in order to solve this problem. With the exchange of the liquid and solid positions, one obtains the condition for LISA (see Fig. 17b). Chou and Zhuang (1999) found when they put a mask (solid) above the polymer (liquid) surface (the vapor phase was in between) and increased the temperature, the polymer formed pillars that reached the mask. In contrast to the previous case, here gravity helps stabilize the interface, and the interface instability was caused primarily by attraction between the solid and the liquid.
B. SPHERICAL INTERFACES A simple example of the three-phase spherical interface case is that a droplet containing a solid core suspends or moves in its vapor, and this condition occurs when a solid particle induces condensation in the vapor phase and a liquid layer
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Chang-Lin Tien and Jian-Gang Weng
consequently covers the solid surface. In this system, the curvature effect on surface tension for both solid-liquid and liquid-vapor interfaces exists; and, because of the small thickness of the liquid layer, these two effects may be highly correlated. Such a problem has not yet been addressed.
C. SPREADINGWETTING AND CONTACT LINE REGION
MD simulation of spreading wetting of a droplet or a bubble on a solid substrate and the contact line region has revealed some unique phenomena. The concept of a contact angle is different from the classical definition: Because several layers of liquid molecules cover the solid substrate due to absorption, usually no vaporsolid interface exists, and neither does the contact line where three phases meet. Yang et al. (1991, 1992) and Maruyama et al. (1998) demonstrated the dynamic evolvement of liquid-vapor interface shape with time. In addition, Maruyama and Kimura (2000) also conducted simulation of a bubble on a solid substrate. Results from these studies lead to the conclusion that the ordered liquid molecules on the solid substrate influenced the spreading wetting significantly. In all MD simulation of spreading wetting, the system temperature is kept constant and uniform. In real applications, however, a temperature difference, called superheat, usually exists between the solid substrate and liquid. Under this condition, the liquid film and the contact line region may have different behavior as a result of evaporation at the liquid-vapor interface. Currently, many interesting problems regarding this subject are still unsolved; for example, there are two totally different arguments regarding whether a significant heat flow occurs in the nanoscale contact line region. MD simulation can be a useful method to understand these issues. The capillary phenomenon refers to the case in which the liquid level in a small tube is different from the level of outside liquid, and it occurs when the contact line region moves up or down in order to achieve equilibrium. In macro- and microlength scales, the capillary phenomenon has been well understood and used in applications (e.g., heat pipes). When the size of the capillary tube reduces to nanoscale, the scenario changes. Two things can happen: (1) the curvature effect of surface tension becomes important, and (2) the bounded liquid molecules near the tube wall further decreases the area of flow path. Ugarte et al. (1996) observed that open carbon nanotubes with inner diameters larger than 4 nm could be filled with molten silver nitrate, whereas carbon nanotubes with inner diameters about 2 nm could not. The authors concluded that this was due to increased solid-liquid interfacial tension caused by increased stress of the tube wall when the inner
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diameter decreases. However, the curvature effect on surface tension as well as the decreased flow path might also contribute to this discrepancy.
VII. Other Interfacial Phenomena This section discusses the interfacial problems that are not included in previous four sections: sonoluminescences (SL) and surfactants. These two problems are generally more complicated and require more detailed physical understanding, more computational power, and sometimes nonclassical MD simulation techniques. However, as nanoscale interfacial study develops, these problems will receive more attention because of their practical importance.
A. SONOLUMINESCENCE
When a small bubble experiences a large-amplitude volume oscillation driven by a sound field, it emits visible photons in every acoustic cycle. This phenomenon is called sonoluminescence (SL). The first single bubble sonoluminescence (SBSL) was observed around 1990 (Gaitan and Crum, 1990), and one of the earliest serious studies was conducted by Barber and Putterman ( 1991). Since then, an intense amount of research work in this area has been published (e.g., Moss et al., 1997; Vuong et al., 1999; Yasui, 1999; Jensen and Brevik, 2000), but thorough and unanimous understanding of the generation mechanism of SL is still not available. Two different theories exist: One theory explains SL based on a shock-wave model, and predicts that the temperature at the bubble center can reach several million degrees, which causes photon emission (Vuong et al., 1999); the other theory indicates that SL is the first macroscopic demonstration of quantum vacuum radiation, in which photons are emitted due to the interaction between the rapidly decelerating dielectric interface and the quantum vacuum (Yasui, 1999; Jensen and Brevik, 2000). The entire experimental apparatus is relatively simple, but the accurate measurement is extremely difficult. Under this condition, theoretical and numerical approaches might be preferred. In SL the input and expected (calculated) system parameters are as follows: the initial pressure is about 1 bar; the applied acoustic pressure is also about 1 bar, and frequency is on the order of 10 kHz; the initial size of bubbles is on the order of 1 #m; the minimum size of bubble is on the order of 0.1 #m; the temperature at the bubble center is on the order of 10,000 K, at the final stage of bubble collapse the time duration is about 50 ps (Moss et al., 1997; Vuong et al., 1999). The
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Chang-Lin Tien and Jian-Gang Weng
macroscopic numerical approach is based on equations of conservation of mass, momentum, and energy for the system. Many simplifications are made, some of which have not been justified. If SL is indeed induced totally by electron dynamics (the second school of theory), MD simulation cannot be used because although quantum MD (QMD) deals with dynamics of electrons and photons, the length and time scales QMD can handle are far too small compared to the scales in SL. For the shock-wave model, however, classical MD simulation can be applied as a benchmark. In this case, MD need only simulate the final period of acoustic oscillation when SL occurs. Here, the time duration is about 50 ps and the bubble size is about 0.1 #m. Note that the bubble size is one order of magnitude larger than what MD can simulate, which may indicate that MD simulation can yield only qualitative results. Pressure oscillation can be realized by changing the boundary size and at the same time keeping the total molecule number constant. This can be achieved by forcing each molecule to move inward or outward for a short distance simultaneously. When the system reaches the steady state, a temperature distribution can be measured and the temperature at the bubble center indicates the possibility of SL. In order to prevent vapor molecules from moving out of the bubble, different intermolecular potentials can be applied to vapor and liquid molecules.
B. SURFACTANTS
Surfactants refers to surface-active materials that are strongly adsorbed at an interface in the form of an oriented monomolecular layer (monolayer) and can dramatically reduce the surface tension of water and other aqueous solutions. They are usually long-chain hydrocarbon molecules and are relatively insoluble in water. Using surfactants actively and efficiently makes it possible to engineer and control the interface. Limited work in MD simulation of surfactants has been published (e.g., Laradji and Mouritsen, 2000), and more work should be done to provide more insights on this subject.
VIII. Concluding Remarks This chapter presents a review of MD simulation of various nanoscale interfacial phenomena in fluids. The liquid-vapor interface simulation is emphasized because publications in this area outnumber publications concerning all of other interfacial studies combined. Despite the extensive efforts, many fundamental problems in this
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area are still not solved satisfactorily; for example, the curvature effect of surface tension and interfacial stability. For interfacial phenomena involving solid materials, the critical issue seems to be how to model solids (thermally and mechanically) in the most realistic way, because very large fluctuations in local properties occur in "bulk" solid and solid-liquid interfaces. Determination of transport coefficients is also important in studies of interfaces because any interface is highly anisotropic and results in large differences between transport coefficients in different directions, and flow, thermal, and diffusion boundary conditions change dramatically. Sonoluminescence and surfactants are very interesting interfacial phenomena that could be simulated by MD methods after some assumptions are made. As stated previously, the advantage that makes MD simulation unique is its capability to obtain molecular-level structures and capture short-time molecular behavior. It should also be noted, however, that in MD simulation the computational domain size is so small and the simulated period is so short that direct comparison of MD results with macroscopic behavior (i.e., experimental observation or classic analysis) is almost impossible. It is, therefore, crucial to apply statistical or scaling theories to bridge this gap. The developments of these "bridges" will not only make MD a versatile tool to calculate micro- nanoscale properties, but also an economical tool to optimize micro- nanosystems.
Acknowledgments The authors thank Professor Arunava Majumdar in Department of Mechanical Engineering, University of California at Berkeley, and Professor Theodore Wu in Department of Engineering Science, California Institute of Technology for valuable discussions and numerous comments during preparation of this chapter, and Professor Seungho Park in Department of Mechanical Engineering, Hongik University, Seoul, Korea, for extensive collaboration in the course of this study. The financial supports from the U.S. Department of Energy and the National Science Foundation are gratefully acknowledged.
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Fracture of Piezoelectric Ceramics qONG-YI ZHANG, MINGHAO ZHAO, and PIN TONG Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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IV. Piezoelectric Dislocation and G r e e n ' s Function . . . . . . . . . . . . . . A. Interaction of a Piezoelectric Screw Dislocation with an Elliptical Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Interaction of a General Piezoelectric Dislocation with an Elliptical Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Extended Line Force on the Elliptical Cavity Surface . . . . . . . . D. Solutions for Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Force on a Piezoelectric Dislocation . . . . . . . . . . . . . . . . . . . .
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V. Conductive Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Uniform Remote Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Antiplane Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Interaction of a Dislocation with a Conductive Crack . . . . . . . .
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VI. Interface Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. I m p e r m e a b l e Interface Cracks . . . . . . . . . . . . . . . . . . . . . . . . B. Other Relevant Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Three-Dimensional Electroelastic Problems . . . . . . . . . . . . . . . . . A. General Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. An Ellipsoidal Cavity under Remote Loading . . . . . . . . . . . . . C. An I m p e r m e a b l e Planar Crack of Arbitrary Shape under Arbitrary Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. A Permeable Planar Crack of Arbitrary Shape under Arbitrary Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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VIII. Nonlinear Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Electrostriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Domain Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Domain Wall Kinetics Model . . . . . . . . . . . . . . . . . . . . . . . . . D. Polarization Saturation Model . . . . . . . . . . . . . . . . . . . . . . . . IX. Experimental Observations and Failure Criteria . . . . . . . . . . . . . . A. Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Failure Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. Introduction
Piezoelectricity is a phenomenon of interaction between mechanical and electrical responses of materials. When a mechanical load is applied to a piezoelectric body, a voltage is induced. Conversely, when a voltage is applied to such a material, the shape and dimensions of the body change. Thus, piezoelectric materials have the ability to convert mechanical energy into electric energy, and the converse is also true. There are classes of crystalline materials possessing such properties. The most widely used piezoelectric materials are ferroelectric. For a crystal composed of ferroelectric material, at temperatures higher than the Curie point, the local positive center of a unit lattice cell coincides with the negative charge center of the cell. In such instances, the material is said to be in the paraelectric phase. When the temperature is lower than the Curie point, spontaneous polarization occurs along crystalline directions, wherein the local positive charge center of a unit cell deviates from the negative charge center. The material is then said to be in the pyroelectric phase because the value of the spontaneous polarization is dependent on temperature. The deviation generates an electric dipole as well as internal local strains. The direction of the spontaneous polarization is called the polar axis. If the direction of the spontaneous polarization in a pyroelectric crystal can be reversed by application of an electric field, the crystal is called a ferroelectric crystal (Jona and Shirane, 1993). The electric dipoles may align only over a region of the crystal, whereas in another region the direction of the spontaneous polarization may be different or even reversed. The region of uniform polarization is called an electric domain. Figure l a shows a schematic depiction of the domain structure of a piezoelectric ceramic, and Fig. 1b shows the microscopic domain structure as revealed through scanning electron microscopy.
Fracture of Piezoelectric Ceramics 180 ~ d o m a i n wall
/ (a)
\ J
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. . . .
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FIG. 1. Domain structure. (a) Schematic depiction and (b) a scanning electron microscopy photograph of an etched fracture surface showing grain and domain structure of PZT-841 ceramics. Many of piezoelectric materials used in contemporary applications are polycrystalline ceramics. Piezoelectric ceramics have quick response. Their physical, chemical, and piezoelectric characteristics can be tailored to specific applications. Hard and dense bulk ceramics can be manufactured in almost any shape and size by the powder metallurgy technique. Ferroelectric thin films can be grown by deposition, such as metal-organic chemical vapor deposition, sol-gel, sputtering, and so on.
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Piezoelectric ceramics are chemically inert and immune to moisture and other atmospheric conditions. In addition, the mechanical and electrical axes of these ceramics can be oriented precisely in relation to the shape of the ceramic. These axes are set during poling, a process that induces piezoelectric properties in polycrystalline ceramics. Before poling, the electric domains in a polycrystalline ceramic are distributed randomly. When a highly static electric field is applied to the polycrystalline ceramic near its Curie point, all domains align along the direction of the applied electric field, which is called the poling field. Then, decreasing the temperature to room temperature while holding the applied poling field causes the aligned domains to freeze, to a large extent, and thus the material exhibits macroscopically piezoelectric responses. It is the poling that gives polycrystalline ceramics their macroscopically piezoelectric properties. The orientation of the static poling field determines the orientation of the mechanical and electric axes of the material. Different poling processes may cause a polycrystalline ferroelectric ceramic to exhibit macroscopically piezoelectric responses in different directions or in a combination of directions. Figure 2a schematically shows a hysteresis loop of the polarization P versus the electric field strength E, whereas Fig. 2b depicts a butterfly loop between the strain, e, and the electric field strength. The relationship of P - E or e - E is generally nonlinear. However, a linear relationship of P - E or e - E holds approximately in certain narrow regimes. The linear constitutive relationship is described in detail later in this article. In addition to the piezoelectric property, ferroelectric ceramics exhibit the pyroelectric effect, whereby the thermal energy can be converted into the electric energy, and the converse is also true. As a result of these unique properties, ferroelectric ceramics are being more widely used in smart structures, microelectronics, and microelectromechanical systems. Clear understanding of the mechanical and fracture properties of ferroelectric ceramics is becoming increasely important as the applications of piezoelectric materials expand rapidly. This article summarizes current knowledge concerning the fracture of piezoelectric ceramics. Attention is confined to fracture mechanics studies, yet experimental results are also examined for comparison with theoretical predictions. Basic equations, including the thermodynamic functions, kinematic equations, static equilibrium equations, principles of piezoelectric virtual work, linear constitutive equations, energy release rates, and the J integral, are introduced in Section II, where we also list the material matrices of commonly used piezoelectric crystals and introduce thermal stresses. Section III presents Stroh's formalism for two-dimensional electroelastic problems and studies an elliptical cavity. To demonstrate physical insights explicitly, antiplane problems are solved in parallel with using Stroh's
Fractureof PiezoelectricCeramics
151
I P Ps
E ,....-
E~
(a)
l
E
Fro. 2. Schematic plots of (a) a hysteresis loop of electric polarization versus electric field strength and (b) a butterfly loop of mechanical strain and electric field strength.
formalism. We examine closely the electric boundary conditions along the crack faces and their influences on the intensity factors and the energy release rate. Section IV describes Green's function for an elliptical cavity and the interaction of a piezoelectric dislocation with an elliptical cavity. An image method, which converges much faster than the Faber series, is used to construct the solution. In Section IV, we also determine the force acting on a piezoelectric dislocation. The result is an extension of the well-known Peach-Koehler formula. Section V establishes the fracture mechanics and Green's functions for conductive cracks. In Section VI, we consider interfacial cracks in piezoelectric ceramics, with special
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attention given to electrically impermeable cracks due to their mathematical simplicity. The linear solution is oscillatory near the crack tips. Three-dimensional electroelastic problems are discussed in Section VII. The electrical boundary conditions along crack faces play an important role also in the three-dimensional fracture mechanics. Four types of nonlinear approaches (i.e., electrostriction, domain switching, domain wall kinetics, and polarization saturation) are introduced in Section VIII. In Section IX, we review experimental observations and failure criteria. Finally, summary remarks are provided in Section X. There are voluminous theoretical studies on the fracture of piezoelectric materials, not all of which can be discussed in this article. The number of experimental studies, although increasing greatly in recent years, is still limited. The experimental results indicate that fracture behaviors under electrical, mechanical, and combined loading are complex and cannot be explained totally by existing theoretical work. Moreover, the experimental data presented by different researchers sometimes contradict. More theoretical and especially more experimental studies are needed to provide a complete understanding of the fracture behaviors of piezoelectric ceramics.
II. Basic Equations Parton and Kudryavtsev (1988) summarized the thermodynamic functions of piezoelectric materials. The internal energy, free energy, electric enthalpy, mechanical enthalpy and full Gibbs energy are described briefly here. The internal energy per unit volume u can be expressed in the differential form using the index notation as du = (YijdEij -']- E i d D i + T d s ,
(2.1)
where s is the entropy per unit volume, T is the absolute temperature, Ei is the electric field strength, Di is the electric displacement, O'ij is the component of the stress tensor, and 8ij is the component of the strain tensor. Hereafter, the repeated indices denote summation, unless special notice is given. The free energy per unit volume f is defined as f = u - Ts.
(2.2)
Then df = r
-Jr-EidDi - s d T .
(2.3)
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The electric enthalpy per unit volume h is defined as (2.4)
h - u - E i D i - T s,
with dh
=
oijdEij
DidEi - sdT.
-
(2.5)
Let the mechanical enthalpy per unit volume w be (2.6)
113 = U - - O ' i j e i j -- T s .
Then dw = -6ijdaij
+ EidDi
- sdT.
(2.7)
The full Gibbs energy per unit volume g is defined as g = u
-~ijeij
-
Ei Di -
Ts,
(2.8)
with dg = - e i j d c r i j
-
DidEi
-sdT.
(2.9)
There are three types of independent field variablesmmechanical, electrical, and thermal--that determine the six thermodynamic functions. The independent m e c h a n i c a l variables can be strains or stresses, whereas the independent electric variables can be electric field strengths or electric displacements. The thermal variable is usually temperature. For constant temperature (i.e., the isothermal condition), Eqs. (2.3), (2.5), (2.7), and (2.9) reduce, respectively, to df
= c r i j d e i j 2r- E i d D i ,
dh = aijdeij
-- D i d E i ,
(2.10) (2.11)
dw
=
-Eijd(Yij
"1- E i d D i ,
(2.12)
dg
=
-6ijdoij
-
DidEi.
(2.13)
For piezoelectric solids that have zero body forces and are free of electric charges, the kinematic and static equilibrium equations are given by
1 8ij ~. -~(ui,j -Jr-uj,i),
oij,j -- O,
(2.14) Ei = -~,i,
Di.i = 0
Tong-Yi Zhang et al.
154
where ui is the displacements, 4~ is the electric potential, and the subscript ,j denotes differentiation with respect to xj. We have the following principles of piezoelectric virtual work of an isothermal process for the domain rI (Zhang, 1994b; Zhang et al., 1998)
f
(tit~Ui-
ck6nr)dF-
fn6f dl-I -O,
(2.15)
0,
where F denotes the boundary of the domain, t is the traction vector along the boundary with the component ti = o'ijn j, ~ = Dj nj is the boundary value of the electric displacement, and n is the unit vector normal to the boundary and outward from the domain. Let us introduce the following four isothermal potential energies:
PF = fr (tiui -- ~r~)dF
-
fn f dl-I,
(2.19)
where the tilde (~) denotes the prescribed quantities on the boundary, which are tractions and electric potential;
Pu -- fr (~iui + g~ck)dF - fn h dFl,
(2.20)
where the prescribed quantities on the boundary are tractions and the boundary value of electric displacement;
Pw=fr(-aiti-~$)dr'-fnwdIq,
(2.21)
where the prescribed quantities on the boundary are displacements and electric potential;
Pc - fr (-~iti + ffrdp)dF - fn g drI,
(2.22)
where the prescribed quantities on the boundary are displacements and the boundary value of electric displacement. Then, the piezoelectric virtual work can be
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155
rewritten as 8PF = fc([iaui - ~Sttr)dF - s
3 f dII = O,
(2.23)
(~PH = fr([iaui + ffrac/))dr - s
ah dFl = 0 ,
(2.24)
aPw = f r ( - a , Sti - a S ~ ) d F - s
aw dFl = O,
(2.25)
8Pc = f r ( - t i i S t i + O a q ~ ) d F - s
agdFl = 0.
(2.26)
If the generalized force P, the generalized displacement A, the generalized voltage V, and the generalized charge Q are used, then the changes in the total free energy F, the total electric enthalpy H, the total mechanical enthalpy W, and the total full Gibbs energy G for the entire sample are given by gF = PdA + VaQ,
(2.27)
dH = PdA - QdV,
(2.28)
+ VdQ,
(2.29)
d G - - A d P - Qd V.
(2.30)
dW = -AdP
Now, consider a piezoelectric material containing a crack. Adding the energy change associated with the crack extension into each of Eqs. (2.27)-(2.30) leads to Jda,
(2.31)
QdV - Jda,
(2.32)
dF = PdA + VdQd H --- P d A dW = -AdP
Jda,
(2.33)
QdV - JdA,
(2.34)
+ VdQ-
d G --- - A d P -
where A is the area of the crack faces and J is the energy release rate for crack propagation defined as
J=
~
A,o_
a,v
= -
-07
P,o
= -
P,V
.
(2.35)
Alternatively, the energy release rate can be evaluated by following Rice's (1968) treatment of elastic fracture mechanics, and is given by J--
OPF
aPH
3A
3A
-
3Pw 3A
-
3Pa 3A "
(2.36)
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156
For two-dimensional problems, if the crack grows along the crack plane (along the x~ axis), the energy release rate can also be calculated from each of the following four path-independent J integrals as long as the integration path encloses the crack tip:
J
iF (fnl --~Yijtljui,1 - - D1Eini)dF,
(2.37)
J - fr (hnl -crijnjui,1 + DiElni)dF,
(2.38)
J - fr (wnl + crij,lnjui - D1Eini)dF,
(2.39)
J -- fr (gnl + crij,lnjui + DiElni)dF.
(2.40)
--
Cherepanov (1974) derived the J integral [Eq. (2.38)] and Pak and Herrmann (1986) presented a similar path-independent integral, which included the Maxwell stresses, for dielectric materials. It is noted that no constitutive equations are involved in the derivation of the piezoelectric virtual work, the energy release rate for crack propagation, and the J integrals. This means that Eqs. (2.37)-(2.40) can be applied to piezoelectric solids with linear or nonlinear constitutive laws. The isothermal linear constitutive equations define the linear relationships between the field variables O'ij , gij , E i , and D i in the form E
(7ij = Cijkll3kl -- ekij E k ,
(2.41)
O i - - eiklEkl + Kike E k ,
where ci~kl is the isothermal elastic constants, e k i j is the isothermal piezoelectric constants, and tcik e is the isothermal dielectric constants at constant strains. In Eq. (2.41), the stresses and the electric displacements are expressed in terms of the strains and the electric field strength. The constitutive equations can also be written in the form E
p
Eij --" Sijkl(Ykl @ dki j Di --
Ek,
(2.42)
d~takt + K i ~ E k ,
where siejkt is the isothermal elastic compliance constants at constant electric field strengths, xi~ is the isothermal dielectric constants at constant stresses, a n d dkPij
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157
is the isothermal piezoelectric moduli. The relationships between the material constants in the index notation are E E __ r CijklSklmn
'
dPmij m emklSklij, E
(2.43)
Kika ~ Kike -~- eij m d~jm ,
where r i s the Kronecker delta. We can also take thermal stresses into account. Iesan and Scalia (1996) discussed thermoelastostatic deformations in detail for conventional solids. In thermoelastostatics, thermal stresses are treated as "body forces." In the absence of time dependence, the entropy does not change with time. Temperature is controlled by the steady state heat transfer equation, which ignores the temperature change induced by the mechanical and electrical fields. We name this approach the pseudoisothermal approach because the temperature field is calculated from heat transfer only, and not affected by the mechanical and electrical fields. Once the temperature field is available, the mechanical and electrical fields can be calculated in a manner similar to the isothermal process. In the widely used pseudoisothermal approach, the temperature distribution is calculated from the steady state heat transfer equations
hi
hi,i = O,
= --(TlijO),j,
(2.44)
where hi is the ith component of the heat fluxes, l']ij is the heat conduction coefficients, 0 - T - To and To is a reference temperature. For a given system, solving Eq. (2.44) with appropriate boundary conditions yields the temperature field. In this case, we may define the pseudoisothermal stresses and pseudoisothermal electric displacements as O-i*j
- -
O-ij Af_ ~. iEjjO
- -
CijklSkl -- ekij E k , E
(2.45)
D~ -- Di - pEo -- eiktekt + Ki~Ek, which satisfy the equilibrium equations
9
Di,i - - - P i
E
(2.46) O, i,
where ~.E is the temperature stress coefficients and pE is the pyroelectric
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Tong-YiZhanget al.
coefficients at constant strain and constant electric field strength. The virtual work for a crack-free piezoelectric solid in static equilibrium may be expressed as
fF(~
fll()~iEjo'j~tti J-~pE~O'j)dI-IfI-I~f*dI-I-O, (2.47)
fF(O'ijnj'ui-J-D;l'lj'~))dI-'f. (~.Eo, j'Ui--pEO,j,~))d[" I--f, ,,,,,_o.
(2.48)
fF (--Ui'~
f~ (ui)~E'o'J-dpp~,Oj)dFl-frI ,w*dFl--0,
+ Djnj6~)dI-' + fo (ui~ij ,j f.(--ui(~oijnj * *
(2.49)
j
--
f,,...-o, (2.50)
where f*, h*, w*, and g* are the pseudoisothermal thermodynamic functions, which have the same forms as Eqs. (2.10)-(2.13) except that the stresses and electric displacements are replaced by the corresponding pseudoisothermal ones. For a piezoelectric ceramic containing a crack, we may define pseudoisothermal potential energies, similar to the isothermal potential energies given by Eqs. (2.19)(2.22), as follows:
(2.51)
PH: fF (~
L)jnj~))dI"-f~ (~.EO,jUi--pEO, j~)dIl-f
~*.~ (2.52)
P~V--fF(--ui'i;'nJ-~D;nj)d~-JfFl('i)~Eo'J-~)PEO'j)dI-If.w..., (2.53)
P~= fr (--UiO'ijnj-~-[)~njdp)dF+fl-l(Ui)~Eo'Jqt-pyO, jdp)dl-I-fN g*dI-l. (2.54)
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Consequently, the energy release rate for crack propagation is calculated from
J =
OP~ OA
OPh OA
=
=
aP~ OA
=
aPa OA
.
(2.55)
In this case, the path-independent integrals for two-dimensional problems are given by
J=fv
( f * n l --
J =
(h*nl - - ~ i j n j u i ,
f
oijnjui, l -- D~Eini)dF Jr- s
().iEjO,jUi, 1 + qbpEO,jl)dl-[, (2.56)
*
l --]- D i Elni)dF +
*
(~.iEjOjUi,l + pj OjE1
s
~
)dn
,
(2.57)
J = f v (w*n 1 + oij,* lnjui - D~Eini)dF -flq (ui~.EOjl -ckp~O ,jl )dFl , (2.58)
J = fv (g*nl --~-oij,lnjui + D*Elni)dF - fl-I (ui~EO'jl - p~O jEl)dH. (2.59) Note that the surface area F is the surface area of the volume H of a closed domain. Kishimoto et al. (1980) and Prasad (1998) presented a path-independent J integral with thermal stresses for conventional solids that also included an area integral similar to Eqs. (2.56)-(2.59). In the pseudoisothermal approach there is an alternative way to derive the virtual work and the J integral. We still use Eqs. (2.10)-(2.13) as the definitions of the pseudoisothermal thermodynamic functions. The virtual work for a crack-free piezoelectric solid in static equilibrium can be expressed as
l"
fv (6ijnj~ui - ~6Djnj)dF - Jn 6f dFl
O,
(2.60) (2.61)
f
f
-
- O,
(-ai~aijnj + D j n j ~ ) d F -
-
s
~gdFl - 0 ,
(2.62) (2.63)
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160
where the prescribed quantities on the boundaries are correspondingly the same as those for Eqs. (2.19)-(2.21). Accordingly, the pseudoisothermal potential energies are
P~.-fv(6ijnjui-~Djnj)dF-frfdI-I,
(2.64)
P[_l-fv(Sijnjui+~jnj~)dF-filhdrl,
(2.65)
P~v-fv(-~icrijnj-~Djnj)dF-frlwdI-I,
(2.66)
P~ --
(2.67)
(-Si~ijnj + Djnjd~)dF-
gdl-I,
for a piezoelectric ceramic containing a crack. The energy release rate for crack propagation is calculated from Eq. (2.55), and the path-independent integrals for two-dimensional problems are given by J -- fv (f*nl -0-ijnjui.1 -- D 1 E i n i ) d I - ' ,
(2.68)
0-ijnjui,1 + DiElni)dF,
(2.69)
0-ij.lnjui - D1Eini)dF,
(2.70)
+ 0-ij, l n j u i + O i E l n i ) d I " .
(2.71)
J
--
iF (h*nl -
J - fv (w*nl + J --
fF (g'n,
In the second pseudoisothermal approach, the thermal stresses are implicitly involved in the pseudoisothermal thermodynamic energies. Gurtin (1979) derived a J integral similar to Eq. (2.69) for conventional solids. In Eq. (2.41), we can take into account the symmetry of piezoelectric crystals to reduce the number of independent coefficients in the linear constitutive equations. If the contracted notations are introduced, 0-1 ~
fill,
81 - - 8 1 1 ,
0"2 - - 0"22,
0"3 --" 0"33,
0"4 --" 0"23,
82 =
83 - - 8 3 3 ,
84 =
822,
2823,
0"5 =
0"13,
85 - - 2 e 1 3 ,
0"6 - - 0"12, 86 =
(2.72)
2812,
the constitutive equation [i.e., Eq. (2.41)] can be rewritten as
0"• = c•
7 - ek• Ek ,
Di = ei•215 + Kik Ek,
V, 0 = 1,2 . . . . . 6;
i,k=1,2,3,
(2.73)
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161
where the subscripts given in Greek notation are from 1 to 6 and the subscripts given in the Roman alphabet are from 1 to 3, and E
ekij
Cyrl - - C i j k l ,
-
e
ek•
(2.74)
Kik -- Kik.
The following rules 11~-+ 1, 2 2 ~ 2 ,
23 or 3 2 ~ 4 ,
33~3,
13 or 31+-~5,
12 or 2 1 ~ 6 (2.75)
are applied to replace ij by y. Nye (1972) presented a comprehensive review of the physical properties of crystals in tensor and matrix forms 9 Table I lists the nonzero matrix elements for the different symmetrical classes, including quartz (class 32), lithium niobate (LiNbO3, class 3 m), cadmium sulfide (CdS, class 6 mm), poled piezoelectric ceramics (class 6 mm), and gallium arsenide (GaAs, class 3~3 m). TABLE I MATERIALS CONSTANTS
Dielectric constants Piezoelectric constant matrix e~i
Elastic constant matrix c,~
Crystal of trigonal symmetry (classes 32, 3 m)
t
Cl I C12 C14
C12 C13 Cll
C13
C14
9
--C14
Class 3 m
Class 32
9
ell
"
--e22 e22 9 e51
9
-e,,
--C14
9
"
9
C44
9 c66 = (Cll --
9 "
"
C44 C14 C14 C66/
;i
41
c12)/2
--e41
e51
--ell
e22
(Kl ") Kll
K33
:!3,)
Crystal of hexagonal symmetry (class 6 mm) Cll c12 c13 c12 ell c13
e13'~ r / e33J
m a t r i x Kij
9
e13~
ci13 c13. c33. c44"
i!
e |
c44 9
Kll
e51
K3
C6
C66 -- (ell -- C12)/2
Crystal of cubic symmetry (class 3~3 m)
t
Cl 1 C12 C12 C12 e l l C12 C12 C12 Cll 9 9 9 C44 9 . c44 9
9
KI1 9
e l
KII
e41 c
9) KI1
e41
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Tong-Yi Zhang et al.
III. Two-Dimensional Electroelastic Problems and Stroh's Formalism
There are two commonly used coordinate systems in the study of fracture mechanics for poled piezoelectric ceramics. The first is the material coordinate system (~l, ~2, ~3), wherein the poling direction is often in the -~3 axis. The second is the sample coordinate system (Xl, x2, x3), wherein the xl axis is often chosen to be along the crack direction. The nonzero elements listed in Table I are expressed in the material coordinate system ( ~ , ~2, ~3). Note that these material constants may have different nonzero elements and values after they are transformed into the sample coordinate system (x l, x2, x3) following the tensor transformation rules. In two-dimensional deformation, mechanical and electrical fields are functions of Xl and x2 only, whereas the poling axis could be along any direction in the sample coordinate system. Of all formalisms for two-dimensional deformation of anisotropic elastic materials, the Lekhnitskii fornaalism (1950, 1957) and the Stroh formalism (Eshelby et al., 1953; Stroh, 1958; Ting, 1996) are the two most extensively used by the engineering community. As Ting (1996) stated, the Lekhnitskii formalism essentially generalizes the Muskhelishvili (1953) approach for solving two-dimensional deformations of isotropic elastic materials. The Lekhnitskii formalism begins with stresses and assumes that they depend on Xl and x2 only. Therefore, the Lekhnitskii formalism is stated in terms of the reduced elastic compliances. The Stroh formalism starts with displacements, and the resulting formalism is in terms of elastic constants. The general solution form of the Stroh formalism resembles that of the Lekhnitskii formalism. Thus, we introduce only the general solution of the Stroh formalism for piezoelectric solids.
A. GENERAL SOLUTION BASED ON STROH'S FORMALISM
Using the Stroh formalism, Barnett and Lothe (1975), Suo et al. (1992), Pak (1992a, 1992b), Park and Sun (1995b), and Chung and Ting (1996) established the general solution in terms of strains and electric field strengths for linear piezoelectric solids undergoing an isothermal process. The solution, in turn, is expressed in terms of displacements and electric potential. Following Barnett and Lothe's (1975) treatment, we extend the three-dimensional space to four dimensions, as follows: 1. Define a new operator 8 Ox4
-0
(3.1)
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2. Denote the electric potential 4~ as U 4 3. Define new strain components as Sij = Eij , Si4 - - S4i -
1
and
--~ Ei
i,j=
$44=0,
(3.2)
1,2,3
4. Define new stress components as ~aij = O'ij ,
and
~2i4 = ]~4i = Di
E44 = 0,
(3.3)
i, j = 1, 2, 3
5. Define new elastic constant components as E C i j k I m Cijkl,
e Cakl4 - - Cnk4l - - Ck441 = Ck414 - - --Kkl, C 4 j k l - - C j a k l = C k l a j - - Cklj4 = e j k l ,
i,
j, k, 1 = 1, 2,
(3.4) 3
C44kl = Ckl44 - - C j 4 4 4 = C4j44 - - C 4 4 j 4 = C444j = C4444 - ' - 0 .
Note that O/Ox3 = O / O x 4 - - 0 . The extended elastic constant tensor still retains the following symmetric properties i, j, k, l = 1, 2, 3, 4.
Cijkl - - Cklij - - C j i k l - - C i j l k ,
(3.5)
Consequently, the governing equations can be expressed in the four-dimensional system as 1
Sij - - -~(ui, j + u j , i ) ,
(3.6)
]~ij = Cijkl Skl,
E i j , j = O,
i, j, k, l = 1, 2, 3, 4.
Hereafter, the repetition of a subscript in a term denotes a summation with respect to that subscript from 1 to 4, unless otherwise specified. If u i are functions of Xl and x2 only, the general solution can be written in the form !
uj = Aj f (z)
or
u--A
!
f (z),
(3.7)
where Z = Xl + p X 2 .
(3.8)
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164
Combining Eqs. (3.6) and (3.7), we have [Q + (R + Rr)p + Tp2IA ' = 0,
(3.9)
where Oik = C i l k l ,
and
Rik = Cilk2
Tik =
(3.10)
Ci2k2.
A non-zero solution of A' requires that det[Q + (R + Rr)p + Tp 2] = 0.
(3.11)
Equation (3.11) has eight roots that cannot be real because of the positive definiteness of the strain energy and electric energy densities (Barnett and Lothe, 1975; Ting, 1996). The eight roots form four complex conjugate pairs. Without loss of generality, we define the four roots as p,~ for c~ - 1, 2, 3, 4, with Im(p~) > 0. Introducing the vector L' - (R r + pT)A' - _ I ( Q
p
+ pR)A',
(3.12)
we can rearrange Eq. (3.12) in the standard form of eigen-equation N~ = p~',
(3.13)
where
."i)
N1 = _ T - 1 R r ,
(::)
N2-T-
1 - N 2 r,
N3 -
RT-1R r -Q
- N~',
(3.14)
with ~" being an eigenvector and p being an eigenvalue. The eigenvectors are uniquely determined up to an arbitrary multiplicative constant. They are normalized to satisfy AA T + A A r = L L T + L L T = 0 , LA T + L A r = A L T + A L T = I ,
(3.15)
where I is the identity matrix, the overbar signifies a complex conjugate of the corresponding function, and A-(A'~
A~
A~
A~),
L-(L'~
L'2
L'3 L~)
(3.16)
are the eigenvector matrices, in which (Li) -- ~'7 is thejth eigenvector of Eq. (3.13), which is associated with the jth eigenvalue p j, j = 1, 2, 3, 4. J
Fracture of Piezoelectric Ceramics
165
The general solution for two-dimensional problems can be written in the form
where f -
U = Af + Af,
(3.17)
~p = Lf + Lf,
(3.18)
[fl(Zl) f2(z2) f3(z3) fa(Z4)] r, ~ is the extended stress potential, and
Z -- X l -k- pjx2, j = 1, 2, 3, 4. If we express the extended stresses in matrix form E1 "- (0-11
0-21
0-31
Dl) r,
E2 = (0-12 0-22 0-32 D2) r,
(3.19)
then the extended stress matrices, 0-33, and D3 become ]El = - - 0 , 2 ,
0"33 - - ]~33 - -
(3.20)
E 2 - - 1//,1,
1
-~C33ij(ui,j Jr- uj.i),
D3 = E34 - - Z43 - -
(3.21)
1
-~C34ij(ui.j + uj,i)
It can be shown (Lothe and Barnett, 1976; Suo et al., 1992) that the matrix B - i A L -~
(3.22)
is a Hermitian matrix (i.e., B - !~v) and can be partitioned as
The upper left block B1 is a 3 x 3 matrix, and the lower fight element B44 is a scalar. For stable materials, Lothe and Barnett (1976) showed that B1 is positive definite, B44 < 0,
(3.24)
and -T
B2 - B 3 .
(3.25)
For coordinate transformation, an in-plane coordinate rotation matrix may be introduced (Suo et al., 1992)
_
cos0
sin0
0
0
-sin0 0
cos0 0
0 1
00
0
0
0
1
'
(3.26)
166
Tong-Yi Zhang et al.
where 0 is the angle between the x l axis and the rotated x]~ axis, in which the superscript asterisk denotes the quantities associated with the rotated coordinates. Then, we have A* = ~A,
L* = .qtL,
B* : ~ B ~ T.
(3.27)
The pure plane-strain deformation requires Ul = UI(X1, X2),
U2 = U2(X1, X2),
//3 = A3f + A3f = 0,
~b "-" t~(Xl, X2),
(3.28) while the pure antiplane deformation requires Ul
--
A l f + A l f = 0,
U2 = A2f + A2f = 0,
U3 m= U3(Xl, X2),
t~ "-- ~(Xl, X2) (3.29)
where A j, j - 1, 2, 3, is thejth row of the eigenvector matrix A. For purely elastic materials, Ting (1996) expressed the condition for decoupling the antiplane deformation from the in-plane deformation in terms of zero components of elastic constants. Although it is difficult to express the decoupling condition for general piezoelectric ceramics explicitly, for poled ceramics, the in-plane deformation decouples from the antiplane deformation when the poling direction is perpendicular to the (Xl, x2) plane (Suo et al., 1992).
B. GENERAL SOLUTION FOR ANTIPLANE DEFORMATION For antiplane deformation, the electric potential ~b and the displacement U3 can be expressed in terms of two complex potentials (Pak, 1990a, 1990b; Zhang and Hack, 1992; Zhang and Tong, 1996) 4~ = Im[~(z)],
U3 = Im[U(z)],
(3.30)
where U and ~ are analytic functions of z = Xl + ix2 and Im denotes the imaginary part of an analytic function. Then, the electric fields and the shear strains can be written as E2 + iE1 = - ~ ' ( z ) ,
F3e + iy31 = U'(z),
(3.31)
where the prime denotes differentiation with respect to z and ~'3i = 2e3i, i = 1, 2, are the antiplane engineering shear strains. The constitutive equations become O'32 + i0-31 -- c44U'(z) + e l S ~ ' ( Z ) ,
D2 + i D1 = elsU'(z) -- Xll ~'(z).
(3.32)
Fracture o f Piezoelectric Ceramics
167
Gao et al. (1997) developed a strip saturation model using a simplified electroelastic constitutive equation for in-plane deformation. They followed the approach used by Rice et al. (1994) for three-dimensional dynamic crack propagation with a slightly wavy crack front. The simplification allows them to solve in-plane problems in the same manner as antiplane problems.
C. AN ELLIPTICAL CYLINDER CAVITY UNDER REMOTE LOADING
Consider an elliptical cylinder cavity (x2/a 2 + x 2 / b 2 -- 1) in an infinite plane under combined remote mechanical-electric loadings (Fig. 3a), where a and b are, respectively, the major and minor semi-axes of the ellipse. The electric loading is in-plane, whereas the mechanical loading can be in-plane tension and/or shear (mode I and/or mode II) and/or antiplane shear (mode III). The boundary conditions on the surface of the cavity are as follows: crijnj = 0,
i = 1, 2, 3
(traction-free),
Djnj -- D~nj (surface charge-free), q~ = qbc
(continuity of electric potential),
(3.33) (3.34) (3.35)
where the range of summation of the repeated index is from 1 to 3 (i.e., j --- 1, 2, 3) and the superscript c refers to the quantities in the cavity whereas the parameters associated with the piezoelectric medium are without superscript, and n is the unit outward normal vector of the cavity surface.
1. Solution Inside the Cavity
Because only an electric field exists in the cavity, the governing and constitutive equations, respectively, are reduced to (3.36)
vzt~ c = 0,
Ocbc
Djc = K c Ejc = _ x c , Oxj
j-
1,2,
(3.37)
where x c is the dielectric constant in the cavity. The cavity is assumed to be electrically isotropic. The dielectric constant has a value ranging from 8.85 x 10 -12 F/m (Farad/meter) for a vacuum cavity to infinity for an electrically conductive cavity. The general solution of Eq. (3.36) is r
= im[~C(z)],
(3.38)
Tong-YiZhang et
168
al.
o'i2 or e~2
L.
L. L.
(a) z- plane
E7
L. O"il
~
a
(~or
b
D;+
e2
L.
t
T
E 2 or D 2 c,o
L.
O'i2 or e~2
&
g.
(b) w- plane
g.
or -----~~
D~
or oo
.-?
g.
1
T
Eil
T
E~' or D ; FIG. 3. An elliptical cylinder cavity in an infinite piezoelectric medium under combined remote loadings in (a) the z plane and (b) the w plane, where the ellipse is mapped into a unit circle.
169
Fracture of Piezoelectric Ceramics
where ~ C ( z ) is a complex analytic function of z with z = Xl + ix2. The complex electric field E c and electric displacement D c, respectively, are given by Ec
_
~ _ E 2+iEcl-
d~ c dz
,
D c_
c c= D2 + i D l
d~ c
-x
c~. dz
(3.39)
In order to determine ~c, we introduce the transformation z = R(w
(3.40)
+ m),
which maps the ellipse in the z plane to a unit circle in the w plane, where w = vl + iv2, R = (a + b ) / 2 , and m = (a - b ) / ( a + b). The inverse mapping of Eq. (3.40) is w =
Z -'1- ~/Z 2 - - C 2
2R
.
(3.41)
The line segment ( - c , c), where c 2 = a 2 - b 2, o n the Xl axis in the z plane is mapped to a circle of radius ~ in the w plane (Fig. 3b). The electric field must be single-valued along the line segment in the z plane, which requires
~c(~/--mei~ ~c(w/me-i~
(3.42)
where 0 is the polar angle in the w plane. Under remote loading, the complex potential, ~c, takes the following form ~c _ RC
( ml w + -w
-
Cz,
(3.43)
where C is a constant to be determined. In other words, the electric field inside the cavity is constant.
2. S o l u t i o n in the P i e z o e l e c t r i c M e d i u m
The general solution for a piezoelectric material is given by Eqs. (3.17)-(3.18). Consider the four planes defined by z~ - xl + p ~ x 2 , c~ - 1, 2, 3, 4, where p~ is the eigenvalues of Eq. (3.13). In each of the four planes, the elliptical cavity is distorted. The transformation z,~ - R~ w~ +
(a not summed)
(3.44)
maps the distorted ellipse on the z~ plane to a unit circle on the w~ plane, where a - ip~b
R~ = ~
,
2
m~ =
a + ip~b a - ip~b
(a not summed).
(3.45)
Tong- Yi Zhang et al.
170
The inverse mapping function is +
wo" --
,
2Ro"
c~ -- 1, 2, 3, 4,
(c~ not summed)
(3.46)
where 2 -- a 2 + po" 2b2 . co,
(3.47)
To construct the general solution for remote loading, we assume the components of f of Eqs. (3.17) and (3.18) in the form
2Ro'ao'2 fo" -- Ro" ao'l too" nt- ao'2 ) _ ao'l (Zo" -Jr-v/Z 2 -- C2) 2 wo" 2 + zo" + V/z 2 - c 2 c~ - 1, 2, 3, 4
(c~ not summed)
(3.48)
with ao'l and ao'2 being constants to be determined by the boundary conditions. Using Eqs. (3.17)-(3.18), (3.20), and the mapping functions, one can rewrite the boundary conditions (3.33-3.35) on the unit circle in the wo" plane in the form
Ljka~l + Lji, a[,2 - - 0 ,
j = 1, 2, 3
Lnka~l + Lal~a~2 -- --Kc(c + m C ) / 2 Aaka~l + Anka~2 -- (C - m C ) / 2 i where Ajk and Ljk tively, and
,
aj -
(traction-free),
(3.49)
(surface charge-free),
(3.50)
(continuity of electric potential),
(3.51)
are components of the eigenvector matrices A and L, respec(R1
R2 R3 R4) T ---~alj ---~a2j ---~a3j ---~a4j ,
j-
1,2.
We can solve for C and a 2 in terms of I11 C
~2(B 4 +
4B--?)m(1 + x c B44)La~ + (1 - x c B44)La~ m2(1 + x c B44) 2 - (1 - x c B44) 2
a~ - L-ld- L-1La~',
,
(3.52) (3.53)
where
B4 - (B41 B42 B43 B44), d-
(0 0 0 d) r,
d - -xc(c
aj--(alj
+ mC)/2,
a2j a3j a4j
(3.54)
)r ,
j -- 1,2,
Fracture of Piezoelectric Ceramics
171
in which Bij are defined in Eq. (3.22). Note that B44 is a negative real parameter in the unit of the reciprocal dielectric constant (Lothe and Barnett, 1976; Suo et al., 1992). The constant vector a 1 is to be determined from the remote loading conditions. The remote electric loading is normally the electric field strengths Ei or the electric displacements Di, whereas the remote mechanical loadings are the stresses tTij or the strains 6ij, a s shown in Fig. 3a. The remote extended stresses are given by lim E2 = Lal + Lal = E ~ , z--,o~ lim E 1
Z----~ OO
- L (pu)al
(3.55)
L(p--~)a-i-= E ~176
where the superscript infinity (o~) refers to the remotely applied extended loads and ( ) denotes a diagonal matrix. There are only seven independent equations in Eq. (3.55). An additional supplementary equation is needed to determine al, a complex vector of four components. We may take the zero rotation around the outplane axis at infinity as the supplementary condition (Sokolnikoff, 1956), which leads to the equation A2al - (A(p~))lal + A2al - (A(p~))lal -- 0,
(3.56)
with A2 --- (A21 A22 A23 A24) and (A(p,~))l -- (AllPl A12P2 A13P3 A14P4). From the constitutive equation, we have lim $ -- C -l lim E = S ~176 Z---+ O~
(3.57)
Z---~ OO
where S is the extended strain tensor (or matrix) and C -1 is the inverse of the extended elastic constant tensor (or matrix), called the extended compliance tensor (or the extended compliance matrix). We can use Eq. (3.56) and (3.57) to determine al in terms of the applied remote strains. The extended displacements and electric potential in the material can be calculated from Eqs. (3.48) and (3.17), and the extended stresses can be calculated from Eq. (3.20), which can be written in the form 2~2 - -
Lf, l + Lf, l,
where f,j = (fl,j f2,j f3,j f4,j)r,
f~,l =
(
4 R~a~2
a~l - (z~ + v/Z 2 - c~) 2
fu,2 -- Potfa,1
E1 ---- - L f , 2 - Lf,2,
(3.58)
j = 1, 2 and
)
z~ +
-
2v/Z ~ - c 2
c~- 1,2,3,4 (ct not summed) (3.59)
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Tong-Yi Zhang et al.
The extended strains can be calculated from the stress-strain relation [Eq. (3.6)] or from the extended displacements [Eq. (3.17)].
3. Electric Field Inside the Cavity or Crack Equations (3.39) and (3.43) give a uniform electric field inside the cavitymthat is, E c - E~ + i E ~ -
-C.
From Eq. (3.52), the uniform electric field can be written as Ec
-
-
- ( B 4 4- B4) x
(1 - ct)(1 - / 3 ) L ( 1 - i p j u } a l + (1 + c~)(1 4-/3)L{1 - i p j u ) a l 2(c~ + fl)(1 + aft)
(3.60)
where c~ and fl are the dimensionless ratios defined as ot - b / a ,
(3.61)
(Zhang and Tong, 1996), and ,c elf is the effective dielectric constant defined as Keft-
-- 1/B44
(3.62)
(Zhang et al., 1998). Multiplying Eq. (3.60) by the dielectric constant of the cavity yields the electric displacement inside the cavity D c -- D~ + i DCl
__ (B4--]-B44~ (1 - o ~ ) ( 1 - / ~ ) L ( 1 - i p j o t ) a l + ( 1 + c~)(1 + j 6 ) L ( 1 - i p j o t ) a l 2(1 + ct/fi)(1 +cq~)
\
/
(3.63)
For fi --+ 0 (i.e., the dielectric constant of the cavity is much smaller than that of the piezoelectric material), the electric field inside the cavity still has a finite value Ec
~ ( B 4 + B4)
(1 - c~)L(1 - ipjot)al + (1 + c~)L(1 - ipjot)al. 2c~
(3.64)
Equation (3.64) indicates that the dimensionless parameter c~ has a great influence on the electric field inside the cavity. The smaller the or, the larger the electric field will be. The electric field approaches infinity as c~ approaches zero. Another
Fracture of Piezoelectric Ceramics
173
extreme of Eq. (3.60) is that the electric field inside the cavity tends to zero as the dielectric constant of the cavity grows large (i.e.,/3 ~ ~ ) . This is the result for a conductive cavity. When c~ --+ 0, the cavity is reduced to a crack. For simplicity, our present discussions focus on electrically insulating cracks only. Conductive cracks are examined in Section V. Readers may also refer to the work by Zhang et al. (1998), in which three limiting cases of a conductive crack were reported. An electrically insulating cavity or crack has a finite value of fl, which reduces Eq. (3.60) to
E c -- -(B4 -+- B--~)(1 - / ~ ) L a l + (1 + fl)Lal 2/3
(3.65)
Equation (3.65) shows that the smaller the value of/3 is, the larger the electric field inside the crack will be. An explicit result is given later for antiplane problems. As c~ --+ 0, Eq. (3.63) has three limits, which are, respectively,
Dc - (B4 + B4) [E~ + fl(Lal DC=(
D c = 0,
B n) 2+BB444
1 +E~c~//3'
forc~/fl--->0,
(3.66a)
for c~//3 ~ a finite nonzero value,
(3.66b)
for c~//3 --+ cx~.
(3.66c)
In Eq. (3.66b),/3 has the same order as c~, and in Eq. (3.66c),/3 is smaller than c~ by a few orders; therefore,/3 is negligible there in comparison with unity. Equations (3.66a-c) show that the electric displacement inside the electrically insulating crack approaches zero only if the ratio ~/fl approaches infinity. For the other two limits of ~/fl, the electric displacement does not equal zero. Equations (3.66a) and (3.66c) are related to the two most frequently used electric boundary conditions, which are discussed later.
4. Antiplane Solution To demonstrate explicitly the role of each physical property in affecting the fracture behavior of a piezoelectric ceramic, we examine the facture mechanics for mode III cracks. Equations (3.30)-(3.32) give the general solution for antiplane deformation. For simplicity, we consider a remote mechanical load cry2 and a remote electric field E ~ only. From the mapping function Eq. (3.40) and
174
Tong-Yi Zhang et al.
the boundary conditions Eqs. (3.33)-(3.35), we find the two complex potentials (Zhang and Tong, 1996) U'(z)
1 [ ( 0 ~ + e l s E ~ ) ( z + ~/z 2 - c 2) can
+
l
2
((ot--fl)el5 Oe + / ~
Ey
)
q- O'3C~
dp,(Z)___E~[Z'+'~/Z2--C2
2R2
]
1
~
Z + ~/Z 2 --C 2 ~/Z 2 --C 2'
ot--fl
2R 2
]
(3.67)
l
O~+ / ~ Z-~- ~/Z 2 --C 2 ~/Z 2 --C 2
Under the given loading condition, the electric field inside the cavity is uniform and the only nonzero component is in the x2 direction, which means l+c~ E c - E~ + iECl = ~ E~. c~+/~
(3.68)
For this case, the effective dielectric constant, x eff, of the piezoelectric material is given by
Keft --
KII +
e~5/r
(3.69)
For a given c~, if/3 -+ 0, meaning that the dielectric constant of the cavity is much smaller than that of the piezoelectric material, the electric field inside the cavity still has a finite value l+c~ U = ~E~. (3.70) O~
Similar to the general loading case shown by Eq. (3.64), Eq. (3.70) gives the explicit relationship between the electric field inside the cavity and the parameter c~, which indicates that the smaller the c~, the higher the E c is. However, for a conductive cavity,/3 --+ cx~ and the electric field strength in side the cavity becomes zero, no matter how large the finite value of c~ is. For c~ ~ 0, the electric field strength in the cavity is EC
1 x eft -- fi E y = x---7 E y .
(3.71)
For most of the widely used piezoelectric ceramics with vacuum or air-filled cavities, the ratio of xeff/x C is of the order of 1000. Therefore, the electric field strength inside a vacuum or air-filled crack could be three orders of magnitude higher than the applied field strength. The similar result is also true for the general loading conditions (Zhang et al., 1998). Multiplying Eq. (3.68) by the dielectric constant of the cavity yields the electric displacement inside the cavity. Similar to Eqs. (3.66a-c), we have the three limiting
Fracture of Piezoelectric Ceramics
175
cases for antiplane deformation
D c = D~ + i DCl = KeffE~, D c
KegE~ = 1 + oe//3'
D C = 0,
for ~//3 ~ 0,
(3.72a)
for ot/fl ~ a finite nonzero value,
(3.72b)
for c~/fl ~ oo.
(3.72c)
Equations (3.72a-c) explicitly indicate that the electric displacement inside an electrically insulating crack depends on the ratio of oe/fl. The electric displacement approaches zero only if the ratio of c~/fl approaches infinity, resulting in an electrically impermeable crack. When ot/fl -+ O, the electric displacement inside the crack is identical to that in the piezoelectric medium on the crack faces, and the crack becomes electrically permeable. Many researchers (e.g., Sosa, 1991; Sosa and Khutoryansky, 1996; Chung and Ting, 1996; Gao and Fan, 1998a, 1998b, 1999) studied the elliptical cavity problem for piezoelectric ceramics, and McMeeking (1989) discussed the appropriateness of the electrically impermeable boundary conditions for isotropic paraelectric materials by analyzing an elliptical flaw.
D. ELECTRIC BOUNDARY CONDITIONS ON ELECTRICALLY INSULATING CRACK FACES
In the study of the fracture mechanics of piezoelectric materials, a crack is usually treated as a mathematic slit. It is inconvenient to use the exact electric boundary conditions, Eqs. (3.33)-(3.35), because of the difficulty in calculating the electric field inside a slit if one does not treat the crack as the limit of an elliptical cavity. Therefore, simplifications are often used for the electric boundary conditions. Parton (1976) published a fundamental result on the fracture of piezoelectric materials. He assumed that the crack is traction-flee but electrically permeable. An electrically permeable crack requires that the electric potential and the electric displacement normal to the crack surface are continuous across the slit. Mathematically, the electric boundary conditions are O + -- 3 2,
(3.73)
~b+ = 4~-,
(3.74)
where the subscript n denotes the normal component of the crack face and the superscripts plus (+) and minus ( - ) mean the upper and lower crack faces, respectively.
Tong-Yi Zhang et al.
176
We can check the validity of Eq. (3.74) from the electric field inside the cavity. The electric potential drop across the crack faces equals the electric field strength multiplied by the cavity width. We have A~b = q~+ - q~- = ac~ sin 0 ( B 4 + B44) X
(1 - or)(1 - / 3 ) R e [ L ( 1 - ipjot)al] + (1 + or)(1 + fl)Re[L(1 - ipjot)al] (a + fl)(1 + aft) (3.75a)
from Eq. (3.60) for the general solution, where Re denotes the real part of a complex function, and A4~ = ~b+ -- qS- = --2ac~ sin 0
l+c~
c~+/~
E~,
(3.75b)
from Eq. (3.68) for antiplane deformation, where 0 is the elliptical parametric angle in the z plane (the polar angle in the w plane) ranging from 0 to zr. As ot approaches zero, Eqs. (3.75a) and (3.75b), respectively, reduce to AO = q5+ -- ~ - = a sin0(B4 + B 4 4 ) ~
(3.76a)
Aq0 -- 4~+ -- 4~- -- --2a sin0 ot/flE~ 1 + ~/r
(3.76b)
and
which indicate that AO5 -- 0 only if ot/fl --> O. The dielectric constant of a piezoelectric material can be three orders of magnitude higher than that of air or vacuum. Thus, ot/fl can be finite for a physical slit and the electric potentials at the two crack faces may be different. The permeable boundary condition Eq. (3.74) corresponds only to the case c~//3 --+ 0. Deeg (1980) analyzed dislocation, crack, and inclusion problems in piezoelectric solids. To simplify the analysis, Deeg set the normal component of the electric displacement to zero at the upper and lower crack faces D + -- D,~- -- 0.
(3.77)
This approximation is equivalent to treating a crack as an electrically impermeable slit by neglecting the electric field within the crack. Pak (1990a) gave a detailed argument for neglecting the electric displacement within the crack. From Eqs. (3.66c) and (3.72c), we see that the impermeable condition Eq. (3.77) corresponds to the case when oe//3 approaches infinity.
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Fracture o f Piezoelectric Ceramics
Hao and Shen (1994) introduced the following electric boundary conditions D + -- D~-, D+Au,
(3.78)
-- --KcA0,
(3.79)
on the crack surfaces, where AO is the potential drop across the crack and Aun is the crack opening. Equation (3.79) reduces to the permeable condition [Eq. (3.74)] if Au,, = 0, or to the impermeable condition [Eq. (3.77)] if K c -- O. We call the electric boundary conditions of Eqs. (3.78)-(3.79) the semi-permeable electric condition. When the crack lies in the xl axis, Au,, is Au2. This boundary condition is based on the crack profile after deformation and requires that the ratio of AO to Au2 be spatially independent. We may use the solutions of AO and Au2 derived from the exact boundary conditions to demonstrate the spatial independence. From Eqs. (3.17) and (3.83), we have the extended crack opening Au = ia[A(al - a2) -/il~(a-i-1- a22)] sin 0,
0 < 0 < Jr.
(3.80)
Because AU4 - - A ~ , Eq. (3.80) yields A~b Au2
=
[A(al
-
a2) -
A(~-
a22)]4
[A(al -- a2) - A ( a ] ( - a2)]2
,
(3.8~)
which is a constant. Combining Eq. (3.79) with Eq. (3.81) shows that the normal component of the electric displacement along the crack face is also a constant that depends on the material properties and the loading conditions. Because the profile of the crack opening is elliptical, even when the crack lies on the x l axis, D2 may not be exactly perpendicular to the deformed crack surfaces. However, if the crack opening is small, D2 may be treated as approximately D,,. Thus, in practice, one may first use the electric boundary condition D + = O~- -- D o
(3.82)
to solve the boundary problems in terms of a constant D o and then use Eq. (3.81) to determine the value of D ~ It is more convenient to evaluate the electrical boundary conditions on undeformed crack surfaces. However, under combined mechanical and electrical loads, the crack opening is very sensitive to the electric field inside the opened crack, and the electric field, in turn, is affected by the profile of the opened crack. Zhang et al. (1998) conducted a self-consistent calculation of the crack profile using the exact boundary conditions that showed the effects of the boundary condition approximation on the energy release rate. Interested readers may refer to the work presented by Zhang et al. for the self-consistent calculation, which is a geometrically nonlinear
Tong-Yi Zhang et al.
178
electroelastic analysis. Dunn (1994b) and Shindo et al. (1996) also investigated the electric conditions along crack faces. In summary: We examined four sets of electrical boundary conditions on electrically insulating crack faces for studying the electroelastic fracture of piezoelectric ceramics. The four electric boundary conditions are, respectively, the exact [Eqs. (3.34) and (3.35)], the electrically permeable [Eqs. (3.73) and (3.74)], the semipermeable [Eqs. (3.78) and (3.79)], and the impermeable [Eq. (3.77)] conditions. In later sections, we examine the consequences of the different approximations in the electric boundary conditions.
E. INTENSITY FACTORS AND ENERGY RELEASE RATES
1. Electric and Mechanical Fields of a Slit Crack a. General Case When the cavity reduces to a slit, the extended displacements and stresses are given by Eqs. (3.17), (3.18), and (3.58), with f,~, f,~.l, and f~,2 reducing to
f~=
a~, (z,~ + v/Z2 - a 2) 2
+
a2a~2 2(zc~ + v/Z~ - a2) '
a2aa2 f~,l --
ac~l -
) Za + ~//Z2 --a 2
(zot + v/z 2 - a 2 ) 2
2 v / z 2 - a2
(3.83)
c~ = 1, 2, 3, 4 (c~ not summed)
fa,2 -- Pa f~,l An intensity factor vector can be defined as K* -
lim L (v/2Zr(z,~ - a ) ) [ , l
(3.84)
Z~ "--+O
for the fight crack tip. A substitution of Eqs. (3.83) into Eq. (3.84) yields K* = v/-Y-~L(a, - a2) 2
~/-7ra ( E l - d). 2
(3.85)
Recall that d - (0 0 0 d) T in Eq. (3.54). Using Eqs. (3.52) and (3.54), we find
d_(B4+B4)(1-~176176176 2B44
(1 + or)[ 1 + c~2 + c~//3 + or/3]
~ ' (3.86)
Fracture of Piezoelectric Ceramics
179
for the cavity. Again, we consider here only an electrically insulating crack that has a finite value of/3. Then, letting ct ~ 0 leads to three limiting results: m
d = (B4 + B 4 ) E ~ , 2B44 A
d =
(B4 -+- B4) ~ 2B44
1 1 + ct//~'
d - 0,
for ct/fl ~ 0,
(3.87a)
for ct//~ --+ a finite nonzero value,
(3.87b)
for ct//3 ~ c~.
(3.87c)
The three limiting results in d correspond respectively to the three limiting cases, shown by Eqs. (3.66a-c), of the electric displacement inside the crack. Zhang et al. (1998) showed that K* is complex for a conductive crack. For electrically insulating cracks, Eqs. (3.87a-c) indicate that d is real; consequently, the intensity factor vector K* is also real. The mode II, I, and III stress intensity factors and the electric displacement intensity factor are defined as twice the real part of K* K-
K* + K* -
K -
(Ktl
Kt
-d],
Kin
(3.88)
Ko) r.
Because d - (0 0 0 d) r, Eq. (3.88) may be written as ( g ll
g l
glll
g D ) -- x / - ~ ( t7 ~2
t72~
0"3~
O ~Z - d ) 9
(3.89)
Equation (3.89) shows that the stress intensity factors of modes I, II, and III are the same as those in elastic media and independent of the remote electric load, whereas the intensity factor of electric displacement can be a function of the remote mechanical loads through the piezoelectric effect. It can be seen in Eq. (3.87c) that d = 0 only ifct//~ ~ c~. Thus, for electrically impermeable cracks, the electric displacement intensity factor is simply (Suo et al., 1992)
Ko -- x/-~--aD~.
(3.90)
For most PZT ceramics containing a vacuum flaw (which has the smallest dielectric constant among all media), the ratio of the dielectric constant inside the crack to the effective dielectric constant of the material is of the order of 10 -3. To satisfy the condition d -~ 0 requires that ot = b/a > 0.01. Thus, caution is necessary in using fracture mechanics in treating a real physical cavity. Zhang (1994a) discussed the crack width effect. However, for electrically permeable cracks (ot/fl--+ 0) with d given by Eq. (3.87a), the electric displacement intensity factor induced by the piezoelectric
180
Tong-Yi Zhang et al.
effect is If"
KD =
~ / / ~ L(B41 -[- 841)0-12 -[- (842 -[- 842)0"22 -[- (843 -'[- 843)0"32 ]. 2B44
(3.91)
Equation (3.91) shows that the intensity factor of electric displacement is completely induced by the piezoelectric effect rather than by the applied electric loads. If c~/fl is finite, depending on the value of c~/fl, the applied electric displacement contributes to the intensity factor of electric displacement--that is,
KD = - 2B~(1 + ot/fl)
-[- (843 -[" 843)0"3T -
2844(ol/fl)O~].
(3.92)
Moving the origin of the coordinate system to the fight crack tip and introducing new variables z~ = z~ - a, we can express the electrical and mechanical fields near the right crack tip in terms of the complex stress intensity factor:
Z~-L ,/2~z: ~l
--
-L
/ / / / P'~
L-1K*
/-/ / +/
L
-
~
P'~
~_.
,
0.93) L-1K*.
(3.94)
It can be shown that the extended crack opening near the crack tip is given by
AU = -~-[B
+
I],IK,
(3.95)
where r is the distance from the crack tip. Hereafter, the phrase near a crack tip means that r << a.
b. Mode III
For antiplane deformation, Eq. (3.67) gives the two complex potentials for the elliptical cavity. Consequently, the mechanical and electrical fields in the material
Fracture of Piezoelectric Ceramics
181
are expressed by 1 F (0-~ + e , s E ~ ) ( z + ~/z 2 - 0 2)
V = V32 + iy31 =
+(~/~
c44
/
2
- 1
1 + ot/fl e l S E ~ + E=
Z -'1- ~ / Z 2 - - C 2
~ / Z 2 - - C2
(3.96)
E2+iE1 C2
_ wy[ z + ,/z ~
I
0"
0.32
--- 0"32
-5
+
ot/fl - 1 1 +Ot/j~
l+c~[
Xl
]
Z ' J - 4 Z 2 - - C2
+ i0"31 = C44V -- elsE,
As z approaches a along the
2R 2 D
= D2
1
J 4 Z 2 - - C2 '
-+- i D I
elsv +
=
Kll
E.
axis, the values of the four functions are
~
)"32 =
~
~32 - -
0"32 =
C44V32 - - e l 5 E 2 ,
elsflE~
]
(~ + t3)c44
,
l+c~
E2 -- ~ E ~ ,
c~ +
(3.97)
D 2 = e 1 5 v 3 2 -~- K l l E 2 .
Equation (3.97) shows that the shear strain approaches infinity as c~ approaches zero for finite/3. Singular strain produces singular mechanical stress; it also causes singularity in the electric displacement through piezoelectricity. However, as long as the dielectric constant of the cavity is not zero, and then I3 has a finite non-zero value and the electric field strength remains finite no matter how small c~ is. When the cavity is reduced into a crack, Eq. (3.96) is reduced to V =
Y32 -k- iV31
= ~
1 [
C44
z
~
0"32 ~ / Z 2 -
c~/fl z --elsE~ a2 -t- 1 + ot/fl ~ / Z 2 - - a 2
ot/~ E-
0-
Ez + i E 1 --- 0"32
-~- i0"31
E~ =
z
l + o~/fl ~/z 2 - a 2
C44Y - - elsE,
+
1 + ~ e l 5 1 + ot/fl
E~], (3.98)
, ] 1 + ot/fl
'
D = De + i Dl = elsv +
Kll
E.
As expected, the mechanical and electrical fields are related to the ratio of or~ft. Let us define the real strain, stress, electric field strength and electric displacement intensity factors at the right crack tip as Ko32 III _ lim v/27r(z - a)0-32 z~a E2 KIn -- lim V/27r(z - a)E2, Z-.--> a
KII>'~2 -- lim v/2rr(z - a)v32, z~a D, K i d -- lim v/2n'(z - a)D2. z.--~ a
(3.99)
182
Tong-Yi Zhang et al.
The near crack tip strain, stress and electric displacement can be expressed in terms of the strain, stress, and electric displacement intensity factors as K0-32 0 HI 0-32 - - ~ COS ~ ,
~32 - - ~
E2-
KY32 III
0
D2 KIH
0
~cos~,
2~~
/(,-Y32 "'~ HI
0 COS ~ ,
E2 Ki11
k" ~ " 9IH
0-31 =
}'31 -- -- ~
0
sin ~, (3.100)
E2 Ki11
El =
0
2v/~_7 s i n ~ , D2 KIII
.-.
cos D2 -- ~ - ~ r 2'
0
sin 2'
ul -
~
0
sin - . 2
If c~ is treated as zero first (i.e., ot/fl ~ 0), we find the mechanical and electrical fields in the material for a permeable crack. In this case, we have E -- E ~ from Eq. (3.98), indicating that the permeable crack has no influence on the electric field strength. Then, using Eq. (3.99) results in /,f,"~32
"'III
=
0-32x/~a C44
E2
'
oo / / ~
,
(3.10~) D2
K II1 -- O,
k,. 0-32
""111 n 0-32
Kin =
e150-32 C44
The strain and stress intensity factors are the same as those in elastic solids. There is no intensity factor of electric field strength and the intensity factor of electric displacement is induced by the piezoelectricity. If the dielectric constant of the cavity is treated as zero first (ot/fl --+ e~), the crack becomes electrically impermeable. Then, the various intensity factors take the following form: K•
III
_ ~0-~2
KIII - ~ - a E ~ ~
+ elsE~ C44
o32 ~ ~ 0 - ~ 2 , K I11
Klli -~/-~--d
(3.102) C44
+
1+
C44 K 11
Equation (3.102) shows that the electric field strength intensity factor is no longer zero for electrically impermeable cracks. As a result of the piezoelectric effect, both the applied mechanical and electrical loads induce singularity in the elastic strain and electric displacement.
Practure of Piezoelectric Ceramics
183
However, if the ratio u/fl has a finite value for u ---> 0, the intensity factors are "'~///
=
K cr3z __ 111
Dz
Kill
C44
0"32
VCffa 0-~2
-- ~
'
+ 1 + ~/flelsE~
e2 _ ,r
KIll
,
E~
[ el5 oc ( ~0-32 + gll E~ 1 + c44
ot/ fl 1 + ~ / fl '
c44xll
(3.103)
1 + c~//~
Comparing Eq. (3.103) with Eqs. (3.101) and (3.102) indicates that, for a finite non-zero value of ot/fl, the intensity factors, except for the stress intensity factor, have values between the corresponding values for impermeable and permeable cracks.
2. Energy Release Rates for Crack and Cavity Propagation In facture mechanics, the characteristic of the energy release rate is extremely important because it provides, from a thermodynamic point of view, a clear physical picture of crack growth. Many researchers have specifically studied the problem. For example, McMeeking (1999), Suo et al. (1992), and Park and Sun (1995b) evaluated the energy release rate of piezoelectric cracks. To demonstrate the relationship between the energy release rate and other parameters, particularly the intensity factors, next we look at the general cracks and mode III cracks. Explicit formulas are available for mode III cracks.
a. General Case As described in Section II, the energy release rate can be evaluated from each of the four thermodynamic functions: the free energy, the electric enthalpy, the mechanical enthalpy, and the full Gibbs energy, or from each of the four associated potentials. Using these functions, we can derive the energy release rate for the growth of an elliptical cavity, as shown in Fig. 3a, under the constraint that the ratio of the minor to the major semi-axis (i.e., the dimensionless parameter ~ = b/a) remains fixed. When the elliptical cavity reduces to a crack, the energy release rate automatically becomes that of crack extension. Because only two-dimensional problems are treated here, the crack area A(=I x a = a) and all properties are based on unit thickness. Thus, the energy release rate is given by
2 J - O P t 4 - f r-( O a
ti--~aOUi-Jr-Di Odpni) drOa
0aa
fn harI
(3.104)
184
Tong-Yi Zhang et al.
A factor of 2 is added here due to the presence of two-crack tips in this system (i.e., a is the length of the major semi-axis, or the half-length of the crack). Using the solutions given in the previous sections and after some tedious algebraic manipulation, we find J
~_
7ra T 4 {(E~) (B+I])E~-(1
+u)(E~)r(Bd+l]d)
+ ~ 2 ( E ~ ) r ( B + I~)E~ - i u ( 1 + u ) ( E ~ ) r ( B d - l~d)
(3.105)
- [i(1 + c~)(d - d) - 2orES] r (Aa~ + Aa~) - u [ ( 1 + u)(d + d) - 2 E ~ ] r [ A ( p j ) a l + ,~(p--j-)a]-i]}. Equation (3.105) shows that the energy release rate for an elliptical cavity depends on mechanical and electrical loads, the dimensionless parameter ot - b/a, as well as the value of d. For an electrically insulating crack, the dielectric constant of the crack medium x c has a finite value (for example, x c = 8.85 x lO-12F/m for a vacuum crack) and d is real. When the cavity reduces to a slit (i.e., ot = b / a ~ 0), the energy release rate for crack extension reduces to ara T J -- 4 [ ( E ~ ) (B + I~)(E~ - d)].
(3.106)
In terms of the intensity factors, J can be written as 1
J - ~ [K:r(B + I])K + 7/ad(B4 + ]~4)(X~ - d)].
(3.107)
Substituting the expression of d [Eq. (3.86)] into Eq. (3.107) leads to J --
Kr(B + I~)K 4
+
rca(ul3) [(B4
4
-+-
B4)X~z] 2
2B44(1 +
Ot//~) 2 "
(3.108)
For a finite value of o~/3, the intensity factor for electric displacement KD is given in Eq. (3.92). For both electrically impermeable cracks (o~/3 --+ cx~) and electrically permeable cracks (or~3 --+ 0), Eq. (3.108) becomes J =
K :r (B + I])K 4
,
(3.109)
in which the intensity factor of electric displacement is given by Eq. (3.90) for electrically impermeable cracks and Eq. (3.91) for electrically permeable cracks. For electrically permeable cracks (oe/fl --+ 0), the intensity factor of electric displacement is determined by the remote mechanical loads only and is independent
Fracture of Piezoelectric Ceramics
185
of the remote electric load. Substituting Eq. (3.9 l) into Eq. (3.109), we can further reduce the energy release rate for the electrically permeable cracks to
4J = (Kll K,
_
Bll + Bll
B13 -q- BI3~
Bl2 -k- B12
]~ "- /B21 -k- B21 B22 q- B22 ! \B31 + B31 B32 + B32
(3.110)
B23 -q- B23
_
1
(",4 + B14~
B33 + B33
|B24"+'-ff24|(B41+B41
2B44/ / \ B+ 3B-~34,/ 4
B42 + B42
B43 + B43),
where Bij, i, j = 1, 2, 3, 4, is defined in Eq. (3.22). Equation (3.110) indicates that the energy release rate is independent of the applied electric field. In other words, the applied electric field does not drive or impede the propagation of electrically permeable cracks.
c. Antiplane Deformation The energy release rate for the elliptical cavity propagation under a fixed c~ with antiplane deformation can be written as Jm = -g--(1 zrca +or) I cr3~ ( Y32~ -
(l+~176
9
\l+~/fl
(3.111) When the cavity reduces to a crack (i.e., a --+ 0), Eq. (3.111) becomes
Jill =
7ra [
-~
a3~
(y~ _ 32
e,sE~ ) ( c~/fl )1 ot/fl)c44 -E~Dy 1 + ot/-----fl "
(1 +
(3112)
"
We may express the energy release rate in terms of a 32 ~ and E ~ as 7ra[ (cr~2)2 Jill-- T c44
(1+
e~5 )xll(E~C)2( ot/fl
/~11c44
l _ql..0///~
)].
(3.113)
For electrically permeable cracks (c~/fl ~ 0), Eq. (3.113) is further reduced to Jill --
IH 2c44
""
(3.1.14)
As expected, the electric field contributes nothing to the energy release rate of
186
Tong-Yi Zhang et al.
electrically permeable cracks. If the crack is electrically impermeable (i.e., ct/ fl --~ c~), Eq. (3.113) becomes 1 [ t'k'~r32 2 ( ~,"~,II ) 1+
Jill = ~
C44
e~5 ) KllC44
E22] 9
K,I (KII 1 )
(3.115)
The contribution of applied electrical loads to Jm as given in the second term on the fight side of Eq. (3.115) is always negative. Thus, the electric field impedes the mode III crack propagation for electrically impermeable cracks. If the ratio of ~//~ has a finite value, the energy release rate is given by Eq. (3.113) and the intensity factors are calculated from Eq. (3.103). It should be pointed out that the energy release rate calculated from Eq. (3.104) depends on the value of d b / d a , while the J-integral for a slit crack is uniquely determined because the crack tip fields are obtained after the limiting process of b --, 0. Very recently, Zhang and Zhao (2001) derived the condition for a mode III crack, under which the engergy release rate calculated from Eq. (3.104) is identical to the J-integral. The condition states that the area of the ellipse should be remained unchanged during the calculation of the energy release rate.
IV. Piezoelectric Dislocation and Green's Function
Piezoelectric dislocation, an important crystalline defect that can adversely influence the performance of piezoelectric materials, has attracted a great amount of research interest (Faivre and Saada, 1972; Barnett and Lothe, 1975; Farvacque et al., 1977; Deeg, 1980; Pak, 1990b; Chung and Ting, 1995a, 1995b; Liu et al., 1997b; Meguid and Deng, 1998; Deng and Meguid, 1999). A piezoelectric dislocation is defined as a mechanical displacement jump across a slip plane, identified as the Burgers vector b, and an electric potential jump across an electric dipole layer. The dislocation core can be subjected to a line force and/or a line charge. Clearly, the solution for a piezoelectric dislocation is the same as a Green function. The characteristics of various Green's functions have been studied extensively by many researchers (Lee and Jiang, 1994; Chung and Ting, 1995a, 1995b; Liang and Hwu, 1996; Liu et al., 1997a; Gao and Fan, 1998b; Lu and Williams, 1998; Qin and Mai, 1998). Usually, one uses the Laurent or power series to analyze elliptical inclusions and the interaction of a piezoelectric dislocation with an elliptical cavity (Gao and Fan, 1998a, 1998b; Meguid and Deng, 1998). To demonstrate the physical picture, Eshelby (1979) adopted an image dislocation approach to solve the boundary value problems of dislocation-cavity interaction. We use the Laurent series to examine the interaction of a general piezoelectric dislocation with an elliptical cylinder
Fracture of Piezoelectric Ceramics
187
cavity, and use the image dislocation approach to investigate the interaction involving screw piezoelectric dislocations. The series solution from the image dislocation approach converges much faster than the Laurent or power series.
A. INTERACTION OF A PIEZOELECTRIC SCREW DISLOCATION WITH AN ELLIPTICAL CAVITY
If a piezoelectric screw dislocation with its line being along the poling direction that is chosen to be in the x3 axis, the deformation associated with the screw dislocation is antiplane. The general solution for antiplane deformation can be expressed in terms of two complex potentials [see Eqs. (3.30)-(3.32)]. For a piezoelectric screw dislocation located at Zd in a homogeneous infinite medium, the mechanical and electric complex potentials are simply (Pak, 1990b) U - ,~ ln(z -- l~ ln(z --
Zd), Zd),
(4.1)
where ,~ -- ,41 + i,4e and l~ - / 1 1 + i/3e are named the mechanical and electrical complex constants, respectively, representing the characteristics of the dislocation. The real parts of,~ and l~ are related to the Burgers vector b3 and the jump in electric potential A~b, respectively. The imaginary parts of ,~ and !~ are related to the line force and the line charge. For any Burgers circuit enclosing the dislocation only, we have
frOU3dl=b3'fr O d p d l - A c ] ) ' O31l
(4.2)
The force and charge balance require the following equations"
~ aijnjdl -- -Fi,
~ Djnjdl = q,
(4.3)
where Fi is the line force per unit thickness or per unit dislocation length in the xi-direction and q is the line charge per unit thickness. For anti-plane deformation, F1 -- F2 -- 0. Using Eqs. (4.1)-(4.3), we find b3
lil = 2Zr'
--xll F3 + elsq /~2 --
2rr (C44Kll -}- e25) '
/~1 -/32 --
ar 27r
,
(4.4)
e15F3 -+- c44q
- 2rr (c44x,, + e125) "
(4.5)
Figure 4a shows schematically a piezoelectric screw dislocation near an elliptical hole in the z plane. Figure 4b shows the corresponding screw dislocation near the circular hole in the w plane and the image dislocations used to construct the solution for the material. For antiplane deformation, the boundary conditions
Tong-Yi Zhang et al.
188
X2 (a) Z- plane _1_ z~
b
..~a
h.
x,
V2 1
(b) w- plane
"~'~ N" . ; . ~ 1
./
/ .L,r
/.'w)m Wd
....\.\ :" : m w, i I "'"....! i . . /
. vl
m/w; --";:L.
FIG. 4. A piezoelectric screw dislocation near an elliptical cylinder cavity in (a) the z plane and (b) the w plane with image dislocations for construction of the solution in the material. Eqs. (3.33)-(3.35) at the cavity surface can be written in terms of the three complex potentials as
c44(U + O) + e15(~ + ~ ) -- O, e15(U + U) - x11(~ + ~ ) -- - x c ( ~ c + ~c),
(4.6)
~-~--~c-~c. From Eq. (4.6), we find the following relationships between the complex electric potentials in the piezoelectric material and in the cavity:
xeff(O + ~)) -- xC(OC + Oc)'
(4.7)
189
Fracture of Piezoelectric Ceramics
where x eff, defined in Eq. (3.69), is the effective dielectric constant of the material under antiplane deformation (Zhang and Hack, 1992; Zhang and Tong, 1996). We use the image method to solve the problem. First, consider a circular cavity of unit radius, where a = b -- 1 and m = (a - b ) / ( a + b) = 0. In this case, the single-valued condition [Eq. (3.42)] is satisfied automatically, and the w plane is identical to the z plane. Consider a piezoelectric screw dislocation located at W d ( W d > 1) with a given electrical constant i~ in the w plane. To satisfy Eq. (4.7), we added an image dislocation of electrical constant f21~ at 1/~--d and an image of electrical constant -f21] at the center. It can be shown that a dislocation of electrical constant AI~ located at Wd generates the electric field inside the circular hole, where K c -- K eff
fl -- 1
x ~ + x~ff
fl + 1'
2 K eff
2
K c _Jr_K ef f
1~ + 1
m
A
(4.8)
Thus, the solution to Eq. (4.7) is given by r
1~ln(w
Wd) -{- ~21~In ( 1
~
fl3
9 ~ -- AI~ ln(w -
W d
)
(4.9)
,
(4.10)
Wd).
Consequently, from Eq. (4.6), we calculate the mechanical complex potential as --
--
Wd
-U9
-- ~ ( f 2 + 1)!~ In C44
Wd
9
t/)
(4.11) For a general elliptical cavity, m - r 0 and the single-valued condition of Eq. (3.42) must be satisfied. However, the complex potential given by Eq. (4.10) does not meet the single-valued condition. We use image dislocations to satisfy the single-valued condition Eq. (3.42) and the boundary conditions in Eq. (4.7) alternatively. The approach is thus called the i m a g e d i s l o c a t i o n m e t h o d . To make Eq. (4.10) satisfy the single-valued condition, we add to the complex potential ~c an image dislocation of electric constant AI~ at m / W d and another image of - A l l at the center. The two images generate a potential AI~ l n ( m / w - Wd), which together with the dislocation of Eq. (4.10) meet the single-valued condition inside the cavity. Unfortunately, the new ~c violates the boundary conditions of Eq. (4.7). To enforce the boundary condition again, we add to the complex potential 9 an image at Wdd/m associated with an electric constant of f2B and use the two images ^
Tong-Yi Zhang et al.
190
with each having an electric constant I~ inside the hole and, respectively, at m/wa and at the center, as shown in Fig. 4b. These images produce additional potentials 1~ln(m/w - Wd) + f2[I ln(m w -- Wd) in the material. At this stage, we complete the first round of the image dislocation approach. Now the solution satisfies the boundary conditions and the single-valued condition except generating a new dislocation at ~-~/m associated with ~1~. It is interesting to note that the new dislocation is 1/ m (> 1) times farther from the cavity, and its electric constant is f2 (< 1) times weaker than the original one. To cancel this new dislocation, we introduce a dislocation at ~dd/m associated with -E2B to ~, and thus start the second round of the image dislocation approach. Finally, we have the sequent solution ^
= 1~ln(w -- Wd) + ~'-2]~In ( 1
tV
+
(--1)1~1+11~ In (mw
Wd
)
wa) +"ln( mw Wd) -Jr"
(-- 1)1 ~'21]~In
(m2 1/)
nt-(-- 1)2~22+11~In ( m2w
Wd) +
+ (--1)l~21l~ In
(--1)2f221~ln ( mgw w d ) + ' " ,
(4.12)
[(mw---~-j) (m2 tV
}
+(-1)2~2"ln[(mZw-wcl)(; ~
,413,
The associate mechanical potential is then given by
U ~i'ln(w -- wd) -- ~l'ln ( w c44e15{ (f2 + 1)1]In ( lw + (-1" f2'+'l~ In ( m W
+(--1)2f22+lflln
( m2 W
Wd) We) + I~ In (mw
We)
lV d ) + (--1)l~ll~ln ( m2 lV
)
Wd
+ (--1)2f22]~ln
wd)
(4.14)
( m3 W
Application of the mapping function to Eqs. (4.12)-(4.14) yields the potentials in the z plane.
P'racture of Piezoelectric Ceramics
191
B. INTERACTION OF A G E N E R A L PIEZOELECTRIC DISLOCATION WITH AN ELLIPTICAL CAVITY
For a general piezoelectric dislocation, we use Stroh's formalism with the general solution given in Eqs. (3.17)-(3.20). For a dislocation at w~d in an infinite body in the w plane, the solution takes the form of f0 = (In ( w ~ - w~a))q.
(4.15)
The parameter vector q is related to the extended Burgers vector, b = ( b l b2 b3 A4~)r, of the dislocation and the extended force, F = (F1 F2 F3 q)r, acting at w~a through (Ting, 1996) 1
q -- ~-~-/(ArF + Lrb).
(4.16)
First we consider an electrically impermeable elliptical cavity. The extended traction is zero along the cavity surface, which is satisfied if all the components of the extended stress potential 1/ti = 0,
i = 1, 2, 3, 4, (traction-free)
(4.17)
at w~ = e iO in the w plane. The complex vector f in Eqs. (3.17)-(3.18) can be written in the form f = f0 + re,
(4.18)
where f0 is given by Eq. (4.15) and fe is induced by the cavity and called the image vectorto be determined. Using Eqs. (3.18), (4.15), and (4.16), we can write Eq. (4.17) as
Lfo(e iO) + Lfe(e iO) -+-Lfo(e -iO) -+-Lfe(e -iO) = O.
(4.19)
Equation (4.19) is satisfied if fe(w~) = Zfl,
(4.20)
,
(4.21)
with
zij . .L .l ln(1 .
tO i
i, j -- 1, 2, 3, 4 (i, j not summed)
The solution of Eq. (4.20) is given in the w plane; then, using the mapping function [i.e., Eq. (3.46)], we obtain the solution in the z plane.
Tong- Yi Zhang et al.
192
For a general electrically insulating elliptical cavity, we use the Faber series to express the complex potential in the w plane for the electric field inside the cavity oo
c~C(w) -
a,,~
Z
-Jr- m n w -n
),
(4.22)
n=l
where a no is a constant to be determined by the boundary conditions and m =
(a - b)/(a + b). The Faber series satisfies the single-valued condition Eq. (3.42). Similarly, we construct (x)
( f j)e --
Cj,nIIOj
j = 1, 2, 3, 4 (j not summed)
,
(4.23)
n:O
for fe in the material, which satisfies the remote stress free condition. The boundary conditions [Eqs. (3.33)-(3.35)] on the cavity surface can be expressed (Lf0 + Lfe + Lf0 + Lfe)j = 0,
j = 1, 2, 3 (traction-free),
(Lfe + Lfe + Lfo + Lfo)4 =--x
' [s c
2
( a o. + a -L--d .m
n ) einO
+ ( a . om
n
+ a . o) e -i'~
n=l
]
(4.24)
(surfacecharge-free), (4.25)
(Afe + Afe + Afo + Afo)4 --2i Z
( a~ _ anm n
einO
- (a.m~
n + an)e-inO-2-5
(continuity of electric potential).
n=l
(4.26) Substituting Eq. (4.23) into Eqs. (4.24)-(4.26) and using the series expansion In (e iO - w j )d-
oo
In ( - w e ) - n ~ !
el'~
j = 1, 2, 3, 4, (j not summed) (4.27)
for f0(e/~ we determine the constants
cj,o -- - q j In ( - w J ) ,
j -- l, 2, 3, 4, (j not summed)
c--~--- L - I L (
g,=(O
0
1 ) n(wJ) n q
0
g,)r,
1 c gn-----~X ( a o. + a %-"5 . m ) ,n
(4.28a)
+ L-lgn,
(4.28b)
Fracture of Piezoelectric Ceramics
(B4L o
an--2
193
,)
(1)
iA4) n(~j)~ ftmn(1 -- fl) 4- (B4L 4- iA4)I n(wi),, q(1 +/3) m2n(1 -- fl)2 _ (1 4- fl)2
(4.28c) Equations (4.15), (4.18), (4.22), (4.23), and (4.28) provide the solution for a piezoelectric dislocation near an electrically insulating elliptical cavity in the w plane. Again, the mapping should be applied to have the solution in the z plane.
C. EXTENDED LINE FORCE ON THE ELLIPTICAL CAVITY SURFACE
Consider an extended line force per unit thickness (or per unit dislocation length) applied to the surface of the elliptical cylinder cavity. In this case, the extended Burgers vector becomes zero and Eqs. (4.15) and (4.16) take the following forms f0(wa) -- (ln[w~ - eiC"])qF,
(4.29)
ArF ~. qF -- 2:ri
(4.30)
The solution for the extended line force can be obtained easily by replacing q with qF and w~d with e ir in Eqs. (4.20) and (4.28). For electrically impermeable cavities, the image vector has the same form as that of Eq. (4.20) f e ( w ~ ) - - L - l [ , f o ( w , ~ ) - ZilF,
(4.31)
with
Zkl -- -LkjlLj----71n( ~1- e
--i~d),
wk
k, 1 =, 12,3,4.
(4.32)
For electrically insulating cavities, these constants are
cj,o -- --(qF)j ln(--eiO"), e-in~Jd Cn = ~ L - 1 L q F ?/
g,,--(0
0
0
j -- 1, 2, 3, 4,
(4.33a)
+ L-lgn,
gn) T,
(4.33b)
g, -- --~l Kc (a,~ + a~ '') o_ 2 an
(B4L - iA4)flFmnein~/d(1 -- fl) + (B--44L+ iAa)qFe-inr n(m2n(1 _ fl)2 _ (1 4- fl)2)
fl) (4.33c)
Again, the mapping function must be applied to obtain the solutions in the z plane.
Tong-Yi Zhang et al.
194 D.
SOLUTIONS
FOR CRACKS
As discussed previously, if we first treat the dielectric constant of the electrically insulating cavity as zero and then let the cavity shrink to a slit, we have the solution for an electrically impermeable crack. However, if we reduce the cavity to a crack while keeping the dielectric constant of the cavity unchanged, we obtain the solution for an electrically permeable crack. In the following subsections, we manipulate the general cavity solution in these two ways to derive the solutions for electrically permeable and impermeable cracks.
1. Electrically Permeable Mode III Cracks When the cavity is shrunk to a slit, we have m ---> 1. For electrically permeable mode III cracks, 13 has a finite value, and the solution to antiplane deformation [Eqs. (4.12) and (4.13)] consequently reduces to
= I] ln(w -
w a ) + [I
In ( 1
wd),
(4.34a)
W
(pc
--
(1 + fl)l]ln [( w 2/~
__
wd)( 1 w
//3 d
)1 - ~ f - l B l n x 2f
[ ( w - ~--j) ( 1 ) wd ] w (4.34b)
If a constant 1~ln[--a/(2Wa)] is added to Eq. (4.34a), we have = 1~ln(z - Zd).
(4.34C)
Because a constant in the potential does not affect the solution, Eqs. (4.34c) and (4.34a) are equivalent. Equation (4.34c) indicates that a permeable crack does not affect the electric field of the material. For electrically permeable mode III cracks, m --+ 1 and the mechanical potential is reduced to
Wd) e,5{.ln(1 ) ( 1 )}
U = ,~ ln(w
-
l/3d) --
~ ' l n ( lto
+ I] In
C44
//3
wd
W
.
(4.35)
9
195
Fracture o f Piezoelectric Ceramics
Using the potentials and the mapping function, we calculate the mechanical and electrical fields of the piezoelectric medium, which are
/~(Z "if-~/Z 2 - - a 2)
y=
Z - Zd + ~/Z 2 -- a 2 -- ,u/ Z ] -- a 2
-+-
~
a 2 - (z + ~/z 2 - a2)(Z-d"+ V/~2d 2 - - a 2)
(Be15 / c44 )a 2 a 2 -(z
E
(~k nt- ~el5/C44)a 2
+
/ _
]
+ ~/z--~2 - a--~2i(Z--da; ( z 2 -- a 2)
1
a~/z2 ' - - - --~
'
(4.36)
Z -- Zd
D
cr = c44Y - e l s E ,
=
el5Y -t--t C l l E .
Using the definition for the real strain, stress, electric field strength, and electric displacement intensity factors at the right crack tip [i.e., Eq. (3.99)], we calculate the intensity factors at the right crack tip, which are given by
{ ~, + f3els/cn4
Ky3•
III = -
KltI E2
Zd -- a + ~/Z2 - a 2
--0,
o32
K Ill
~
.
{
Ac44 +
.
.
.
I~e15
+ ~e15/c44
+ Z--d
192
{
a + V/--~d - - a 2
(4.37) ~c44 + l~e15
+ z~-
Ki11 = --el5
] ~/-~a
A + [le15/c44 Zd -- a + ~ Z 2 - a 2
a + V/--~d -- a 2
/ ,ff~a,
I
+ i + 1 /c44 I ,/-YS. Z---d-- a + ~-~d -- a 2
I
As expected, there is no the intensity factor of electric field strength for electrically permeable cracks. In this case, the energy release rate for crack propagation is the same as that given by Eq. (3.114).
2. Electrically P e r m e a b l e General Cracks
For a general piezoelectric dislocation, we have Rj --+ a / 2 andmj --+ 1, j -- 1, 2, 3, 4, when the ellipse is reduced into a crack. For an electrically permeable crack,
Tong-Yi Zhang et al.
196
the solution is still given by Eq. (4.18) with Eqs. (4.15), (4.23), and (4.28), except that the constants gn and an0 are reduced to the following form: 1
g,, = - ~ x C(a ~ + a~
1)
(4.38a)
(1)
n(-~)n (1(1 -- f l ) + (B4L + iA4) n(wd)n q(1 + 13) 0
B
a n
(4.38b)
3. Electrically Impermeable Mode III Cracks For electrically impermeable cracks under antiplane deformation, the two parameters in Eq. (4.8) have the value of f2 = - 1 and A = 2, and the solution is simplified as -- 1~ ln(w -- Wd) -- I] In ( 1
//9
Wd),
(4.39a)
~c = 0, -
(4.39b)
)
-
Wd 9 (4.39c) w As expected, there is no electric field within an impermeable mode III crack. The mechanical and electrical fields in the piezoelectric medium are calculated from the potentials and take the following forms in the z plane a 2)
/~k(Z + x/Z 2 -
V--
/_ Z -- Zd
a t- X / Z 2 - - a
2 --
,/Z~ -- a 2 u ~,
-
]
,~a 2
+
a~ - ( z + ,/z ~ - a ~ ) ( ~ + V / ~ - a~) E -- - [
1 ,/z 2
a 2'
l~(Z + X/Z 2 - - a 2)
(4.40)
z - Zd + x/Z 2 - - a 2 -- v/z 2 - a 2 l~a 2 a~ - (~ + ,&~ - a ~ ( ~ O" =
C44 V - -
el5E,
D = el5?' +
] + V / ~ - a~) K'll
E.
1 "/z~ - a~
197
Fracture o f P i e z o e l e c t r i c C e r a m i c s
Consequently, we have the following real intensity factors at the right crack tip
f K Y32 __ --/
Ill
KIll -
k,.o-32
"~m
__
I Zd --
{
/~ a +
+ z~
--
z. - a + v/z 2 - a 2
-
{
Z---d-- a + V/--~d -- a 2
a 2
} ,/-~a,
+ z~-
Ac44 --I- Bel5
-
/ el5 "ll
a2
_
}
Ac44 + Bels
+
z d -- a + V/ z ~ -- a 2
Km - -
a + V/~-
(4.41) ~/-~a,
z---~_ a + v/--~d _ a 2
+
Z d -- a + V/ Z2 -- a 2
e15
}
~ffa.
z---~_ a q_ v/--~d _ a 2
The energy release rate takes the same form as that presented in Eq. (3.115).
4. E l e c t r i c a l l y I m p e r m e a b l e G e n e r a l C r a c k s
For a general piezoelectric dislocation, the solution is the same as that given by Eqs. (4.15), (4.18), (4.20), and (4.21). In mapping the solution into the z plane, one should take into account that Rj ~ a / 2 and mj --+ 1, j = 1, 2, 3, 4. The extended stresses can be calculated from Eqs. (3.18) and (3.20) with
fO, 1 -~- [e, 1 =
v/z2
Zij, 1 = E
a2
z~ + , / z ~
- 02 -
~
-
V//( z ~ ) ~ - 02
4 1 ( L ~ I Lkj
k=l
q+Z,l(l,
)
02
v/Z 2 - a 2
a 2-
(4.42a)
(Zi q- v / Z f - a 2) (Z--d q'- V/(7) 2_ a 2)
f0,2 -+- re,2
gij,2
,/~2_ a2
4
= ZL~I-~kJv/z2
k=l
zo + , / z 2 - a 2 _ z~ -
,i( --
a 2
V/( z ~ )
2 - a2
02
a2--(Zi-}-
q+Z2~,
-_ v/Z 2 - - a 2 ) ( Z J -}- V/(~fd.) 2 a 2)
(4.42b)
) !
198
Tong-Yi Zhang et al.
The intensity factor vector at the right crack tip induced by the dislocation at z~d is K* -
lim L(v/2rr(z~ - a))(fo,1 + re, l)
Zot----~a
1
-
z d - - a 4- ~(Zd)2 -- a 2
) ( q+L
1
z d - - a 4- ~(~d~d)2 -- a2
)1 fl
9
(4.43) Equation (4.43) shows K* is real. Thus, the intensity factor vector, K, for the mode II, I, and III stress intensity factors and electric displacement intensity factor are twice that of K* (i.e., K = 2K*). Consequently, the energy release rate can be calculated by Eq. (3.109).
E.
FORCE
ON A PIEZOELECTRIC
DISLOCATION
Pak (1990b) calculated the force on a piezoelectric dislocation under external mechanical and electrical loads. The force acting on a piezoelectric dislocation is a configuration force, which relates the change in energy when the dislocation moves an infinitesimal distance. Thus, from a thermodynamics point of view, we may add the changes in isothermal thermodynamic functions associated with the dislocation movement into each of Eqs. (2.27)-(2.30), d E = P d A + V d Q - J . dl,
(4.44a)
= P d A - Q d V - J . dl,
(4.44b)
dn
dW = -AdP
4- V d Q - J . dl,
(4.44c)
dG = -AdP
- Q d V - J . di,
(4.44d)
where l is the dislocation displacement vector and J is the force acting on the dislocation, which is defined as -
_
A,Q
_
445,
p, Q
A,V
-~i
P, V
The force acting on the dislocation can be evaluated from the each of the isothermal potential energies defined in Eqs. (2.19)-(2.22) such that Ji
--
0 PF Oli
--
0 PH 31i
--
0 Pw Oli
=
0 PG 31i
.
(4.46)
199
Fracture of Piezoelectric Ceramics
For a straight dislocation line along the x3-axis, the generalized Peach-Koehler forces are J, = a~2bl + o'2a2b2+ o'~2b3 + D~ Aq~ + u~, 1F, +
J2 - - - ~
2,1F2 +
Ua
b,
a 3,1F3 + Elq
Ua
- - o'~2b 2 - O~l
,
b3 - D? A~b +
(4.47) U a1,2
F,
.+_ua2,2F2 .+_U3,2 a F3 "+- E2q a .
V. Conductive Cracks
Internal electrodes in electronic and electromechanical devices made of piezoelectric ceramics may act as conductive cracks or notches, causing the devices to fail under electric and mechanical loads. It is therefore of practical importance to study the fracture mechanics and failure behavior of conductive cracks in piezoelectric ceramics. Many researchers have worked on this topic (McMeeking, 1987; Furuta and Uchino, 1993; Suo, 1993; Zickgraf et al., 1994; Lynch et al., 1995; Chung and Ting, 1996; Ru and Mao, 1999). Using compact tension samples with conductive notches, Fu, Qian, and Zhang (2000) performed extensive fracture tests on lead zirconate titanate (PZT) ceramics under purely electrical or mechanical loading. The experimental results indicate that both purely electric and mechanical fields can propagate conductive cracks (notches) and fracture the samples. Under purely electric loading, there is a critical energy release rate at fracture called the electric fracture toughness. The electric fracture toughness is about 25 times larger than the mechanical fracture toughness, the critical energy release rate at fracture under purely mechanical loading. Like the mechanical fracture toughness, the electric fracture toughness is a material property, which is defined as the resistance of a material against fracture or as the energy per unit area absorbed by the material as the crack propagates. Following the approach used by Orowan (1952) and Irwin (1956, 1958), Fu, Qian, and Zhang (2000) attributed the high electric fracture toughness to electrical plastic deformation. In this section, we study conductive cracks within the framework of linear fracture mechanics. As in the preceding sections, we establish the solution for an elliptical cylinder cavity first, and then reduce the cavity to a crack. For simplicity, we consider here only the case that no net free charge exists on the conductive cavity or crack (i.e., f DinidF -- 0 for any integral path enclosing only the cavity or crack).
Tong-Yi Zhang et al.
200
A. UNIFORM REMOTE LOADING For a conductive cavity under uniform remote loads, the components of the vector f of Eqs. (3.17)-(3.18) have the same form as those of Eq. (3.48). Using the mapping functions Eqs. (3.44)-(3.46), we write the boundary conditions on the conductive cavity surface in the w plane along the unit circle in the following form:
Ljiail* -Jr-Ljiai2* -- 0 ,
j -- 1 2, 3 (traction-free),
A4iai* 1 -Jr-A4iai*2 -- dpo (continuity of electric potential),
(5.1) (5.2)
where 4~0 is a reference potential. For simplicity, we take 4~0- 0 and rewrite Eqs. (5.1) and (5.2) in the compact form of
Qa~ + Qa~ -
O,
(5.3a)
where
Q
Lll L21 L31 A41
L12 Lee L32 A42
L13 Le3 L33 A43
-7 a2 -
_Q-1Qa~.
L14'~ L24 / L341 A44] |
(5.3b)
o
|
Equation (5.3) gives (5.4)
Recall that
a; -
RI ---~alj
R2 ---~a2j
R3 ---~a3j
R4 --ff a4j
)T
,
j -- 1,2.
From the remote loading conditions,
J -Qa, +Qa,,/ 3vJ
-- -Q
(5.5)
we can determine al. Similar to electrically insulating cracks, there are only seven independent remote input loads in Eq. (5.5). We need a supplementary condition to determine a unique al. As before, we take the null rotation around the x3 axis at infinity as the supplementary condition (Sokolnikoff, 1956) by requiring A2al -- (A(pa))lal + A2al -- (A(p,~))lat -- 0.
(5.6)
Fracture of Piezoelectric Ceramics
201
When the conductive cavity is reduced to a conductive crack, from Eq. (3.48) we have
f,~ =
aoel (zoe + ~/-z~ -- a e)
a2aoee
+
2
2(zoe + v/Z 2 - a 2 ) '
a2aoe2 foe, 1 = foe,2 - -
) zoe + v/z 2 - a 2
aoel - (zoe + v/Z~ Poe
foe,1,
2v/z~ -- a 2
a2) 2
(5.7)
O~ -- 1, 2, 3, 4 (a not summed).
Equation (5.7) together with Eqs. (3.17)-(3.18) and (3.20) give the mechanical and electrical fields for a conductive crack in an infinite piezoelectric medium under remote loads. We can define an intensity factor vector at the right crack tip as K* -
lim Q(v/27r(zoe - a))f,1
(5.8)
Z ct ---+ a
Equation (5.8) gives
x/2~rz~ )Q-IK*
f, 1 - -
for
Iz~l << a
(5.9)
where z~ = zoe - a and Iz~l is the absolute value of the complex variable z~*. For Iz~l << a, we consequently have f--
2
* ~/Q-IK
/
*
f2--( Poe ) 1
v/2n.z * Q - K*
(5.10)
(5.11)
A substitution of Eq. (5.7) into Eq. (5.8) yields
( ~r~2 / a2~
K*- ~ -
2
I ~73~
\-E 7
( KII 1 ] KI
~
~ I Kill
(5.12)
\-KE,
which shows that the intensity factor vector K* is real and its components equal half of the stress intensity factors of modes II, I, and III and the intensity factor of
202
Tong-Yi Zhang et al.
electric field strength. Let
p __
All
A12
AI3
A14
A21 A31 L41
A22 A32 L42
A23 A33 L43
A24 A34 L44
(5.13)
The near crack tip fields can be written as
uUi 1
O-12 ~ / -~ / -El/
l
o-11 0"21 ~ E2
\-Ke,
/
/( K//
2,~ Q~Q-'+0
~
i,,,, '
(5.14)
\-KE, { K//
_
2v/~- Q ~
Q_
p~
~
KI \-KE,
The extended crack opening near the crack tip is ul A
u2 u3
KII
_
lP4
i~2-7 ~ [pQ-l _ p Q - 1 ]
KI glll
(5.15)
--KE,
The energy release rate can be calculated by the methods introduced in Section II. Using the solution for conductive cracks, we find the energy release rate { K// J -- (Kit
KI
Kill
_ K e , ) i [ p Q - 1 _ pQ-1] / KI 4 | KIII \--KE,
(5.16)
B. ANTIPLANEDEFORMATION
To illustrate the dependence of the energy release rate on the material constants, we examine the antiplane deformation of a conductive crack. The general solution
Fracture of Piezoelectric Ceramics
203
of Eqs. (3.30)-(3.32) is valid for the antiplane deformation of conductive cracks. From the traction-free and constant electric potential boundary conditions,
o31nl -+- tY32n2 = Im[(c44U' + elsdP')e iO] = 0, (traction-flee)
(5.17a) (5.17b)
Im[~] = 0, (continuity of electric potential)
we find the two complex potentials for a conductive elliptical cavity
U(z) = H1
~(Z)--
H3
Z "-[- ~/Z 2 - - C 2
2 Z -'l- "~/Z2 --C 2
2
-+-
-+-
2R2H2 z
-+- x / Z 2 -
C2
(5.~8)
2R2H4 z + x/z2 - c 2
where CX)
9
el___55(E~ + i E ~ ) ,
H i __ 0"32 "1"-/O'31 + C44
H2
--
-
C44
cry2 - i o ' ~ C44
el5 + --(E~
(5.19)
-iE~~
C44
Ha -- - ( E ~ + i E T ) ,
Ha -- - ( E ~ a - i E ~ ) .
Then, using Eqs. (3.31) and (3.32) yields the mechanical and electrical fields
Z + ~/Z 2 - - C 2 Y = 1"32 nt- iy31 - -
2
H1
2R2H2
]
1
z + ~/z 2 - c 2 ~/z 2 - c 2
E= E2+iE1 F,(E~a + i E ~ ~ z +
L
I
X/Z 2 - - C 2
2
(7 - - 0"32 -I-- it731 = C44(}/32 -+-
2R2(E~a - i E ~ ) ] Z + ~ / Z 2 - - C2
iv31) - els(E2 +
i E1),
D = D2 -k- iD1 = els(Y32 -+- iv31) + Xll(E2 + iE1).
1 ~ / Z 2 - - C2
(5.20)
204
Tong-Yi Zhang et al.
When the cavity is shrunken to a crack, Eq. (5.20) reduces to e 15
~
C44
C44
Y =Y32+iy31 = ~ E ~ + i
~
+
C44
e 15
+i~E
C44
z N/Z 2
0 2'
iE~z E = E2 + i E1 = E ~ + ~/z 2 - a 2 ,
(5.21)
OG
(7" --" 0"32 + i0-31 - - 10"31 " c~ +
D= "--
Dz+iD1
(
KI1
2
O'32 Z ~ / Z 2 __ a 2
)
+ e----~5 E ~ + iel5 C44
O'31
+
C44
I
e150-~xz 2 C44
+i
x11+~
C44 ,,/
E~
%/CZ2 - - a 2
We define the strain, stress, electric field strength, and electric displacement intensity factors at the right crack tip as K~I = lim V/2Zr(z - a)y,
KIWI1 -- Z----~ lima v/2rr(z - a)0-,
KEI = lim v/2rr(z - aiE,
KIID -- Z---+ lima ~/2rr(z - a)D.
z---~ a
Z-.-+ o
(5.22)
Then, the strain, stress, electric field strength, and electric displacement near the crack tip may be expressed in terms of their corresponding intensity factors as K lall
KYII
0- = x/27r z* ' E =
Kfll
Y -- ~/2rr z* KIID
~/2jrz ~,
D-
~/27r z* '
(5.23) z* - - z - a .
Substituting Eqs. (5.19) and (5.21) into Eq. (5.22) yields K ~I1 =
~0"32 + i ~el5 E ~ C44
KEI_.iE~,-~--d,
x/~ a ,
oc/-~a K lall "-- 0"32 ,,
,
C44
KIID- [e150-~2 ic44 +
(5.24)
( Kll+ e~--~5)E~l~/-~-d'c44
Equation (5.24) indicates that the strain intensity factor and the intensity factor of electric displacement are complex, and the intensity factor of electric field strength is purely imaginary. Both 0-3~ and E ~ have no contribution to the intensity factors. In this case, the energy release rate for the mode III conductive crack is given by 1 Y cr )/ cr E D Jill -- ~ ( KHI KIH ~- KHI Kill - KIll KIll - KEI KID) 9
(5.25)
205
Fracture o f Piezoelectric Ceramics If we define the real intensity factors as • _ lim x/2n'(z - a)Y32, K HI
Ko32 lit = lim ~/2zr(z - a ) 0 3 2
Ki1Elt
Di K m = lim ~/2zr(z - a)D1
Z--~ a
--
lim x/2zr(z - a ) E l ,
z..-+ a
z--~ a
(5.26)
Z---~ a
the energy release rate takes the following form"
1 t' t,,-•
k"O'32
El
Di
(5.27)
Jzn = -~ t " m "tit + Kilt K m )
C . INTERACTION OF A DISLOCATION W I T H A C O N D U C T I V E CRACK
1. General Piezoelectric Dislocation The complex solution vector of a dislocation at w~a in an infinite piezoelectric body is given by Eq. (4.15). In the presence of a conductive crack, the solution has to satisfy the traction-free and the constant electric potential boundary conditions, which take the following form in the w plane: Q(f0 +
fo
fe)
+ Q(f0 + fe) = 0,
(5.28)
= (ln(e i~ - w~))q
The solution that satisfies Eq. (5.28) is
fe -- Yet,
4 Q~ l~-~kj In (1- - - ~/d.)
(5.29)
Yij ------ Z
k=l
//)i
where i, j = 1, 2, 3, 4 and i and j are not summed. The same notations are used in the following equations of Eqs. (5.30)-(5.32). Mapping back to the z plane gives
fo, l
+
fe,1
( ( 2 1
,/z~ - c~ 4
Yij, 1-- ~
z~ + , / z ~
Z~ + ~/Z~-C~
)lq + Y,l el,
~ ~ - z~ - v/ (z~) ~ - c~
1
Q~I Qkj ~/z2 - c 2
x(
1-I=, +
(5.30a)
1
)
Tong-Yi Zhang et al.
206
fo~ + fe~ :
4z~
c~
z~ + 4 z ~ - c~ ~ - z~ - V / (z~) ~ - c~~
4
Yij,2 -- Z Q~l--~kJ k=l
•
Pi
v/Z2 _ c2
(5.30b) l
1--(Zi + ~/Z2 --C2)(Z--~j-'~~/(~jd) 2 - --~j) / 4 Ri-~j )
When the cavity is reduced to a crack, Eqs. (5.30a-b) reduce to
f0,1 + f e , , -
( ~/z21_ ae (
z'~+v/zZ-a2 zo + , / z ~
4
1
~=l
v/z/~-a~
Yij,1 -- ~ Q~I Qkj x(
))q + Y ,(1,
- a~ - z~ - v / ( z ~ ) ~ - a~
(5.31a) a2
)
aZ- (zi + v/zZi -aZ)(-~j + V/(-~j)Z-a2 ) f0,2 + fe,2 --
4Z 2 __ a2
4
V/ z~ + 4 z ~ - a~ - z~ -
q + Y,2(I,
(z~) ~ - a~
Pi
(5.31b)
aZ- (zi + ~z~ -aZ)(z~ + ~(-~j)Z -a2) For a semi-infinite crack with the origin of the coordinate system at the crack tip, we have
f0,1 + fe, l --
( 2~/_~(~_~_ l x/~a) )q + Y,ls (5.32a)
4
1
k:,
2~/~-(~7 + v/z j)
Yij, l -- -- ~ O~l Okj
/-..:_
Fracture of Piezoelectric Ceramics 2~/~(~/~-~-
fo,2 + fe,2 =
~/~d)q + Y , 2 q ,
4
r'ij,2 = - ~
207
(5.32b)
Pi
Q;I Qkj
2~-7(~&-7 + V~jd.)
k=l
near the crack tip. The intensity factor vector at the right crack tip induced by the dislocation at z~a is K*
=
lim Q(v/2zr(z,~
-
a))(fo,1
+
fe, l)
Z ot ----~ a
= Eo(
z~ - a + r
1 -
e- a 2
)q + O (
z--~- a + r
1
-
2- a 2
)1q
(5.33) for a finite conductive crack and K * = lim Q(v/2Zrz~)(f0,1 + fe,1)Za---~0
2
Q ~ z ~d q +
0(5)0] (5.34)
for a semi-infinite conductive crack. Equations (5.33) and (5.34) indicate that the intensity factors induced by a dislocation are real, and therefore Eq. (5.12) holds. As a consequence, the energy release rate is also given by Eq. (5.16).
2. Piezoelectric Screw Dislocation Now consider a piezoelectric screw dislocation near a conductive elliptical cavity. In this case, the two parameters introduced in Eq. (4.8) have the values of g2 -- 1 and A = 0 because/3 --+ cx). We do not need to consider the single-valued condition inside the cavity because of the zero electric field inside the conductive cavity. Consequently, the complex potentials in the w plane are derived in a similar manner as those for deriving Eqs. (4.9) and (4.11) and take the following form: U
,~ln(w
m
Wd) _ ( t ] - + 2 e l S l ] ) l n ( c44 /
-- I~ ln(w - Wd) + !~ In ( lw
1 w
l/3d ) ,
(5.35)
Wd)
where .~ and !~ are complex constants defined in Eqs. (4.4) and (4.5). Using the
Tong-Yi Zhang et al.
208
mapping function, we calculate the mechanical and electric fields in the z plane
~,(z + ~/z 2 - c 2)
y= z -
zd + ,/z 2 -
-t-
c2 -
V/z 2 -
c2
( i + 2~els/C44)4R 2
}
1
4R 2 - ( z + ~/z 2 - c2)(Td + V/~2d 2 - c 2) E = -{
~/z2 - c2' (5.36)
l](z -']- ~/Z 2 --C 2)
z - ~ + ,/~2 - ca - V/Z~ -
~2
l~4R 2
}
1
. .....
4R 2-
(Z q- X/Z 2 -
cr = c44Y - elsE,
-
C2)(Z--d if- V/~2d 2 --C 2)
~/z2
C2'
D = elsy + tcllE
When the cavity is shrunk to a crack, Eq. (5.36) reduces to 2) /_ Z - zd + ~/Z 2 - a 2 - v / z ~ / - a 2 ~k(Z + ~/Z 2 - - a
y=
nt-
(ilk -+- 2__~el__~5/c44)a2
]
a 2 - (z + ,,/z 2 - a2)(z--d" + V/Z-~d- a 2) E =-[
! Z-
I](z+x/z 2-a
1 x / z 2 - a2 (5.37)
2) /
Zd + ",/Z 2 - a 2 - v/z 2 - a 2 l~a 2
a 2 - (z + ~/z 2 - a2)(z--d + V / ~ d - a 2)
}
1
~ / z 2 - a2
D = el5g -t- Kll E
O" --" C44V -- elsE,
Using Eq. (5.22), we obtain the intensity factors at the right crack tip
/r
- -{
,i
+
Zd -- a -+- v/z 2 -- a 2
(~ + 2~e~5/c44) z---d- a + V/--~d - a 2
1~
K,~,- { Zd - a + v/z2 - a 2
K~II_ c44 K~I - e 15KIII, E
D KIII
},r
z~-a ~
+ V/~-
,. }vqa, a2
el5 KI.Y + Ir K~I
(5.38)
Fracture of Piezoelectric Ceramics
209
Moving the origin of the coordinate system to the fight crack tip and letting the crack length approach infinity, we obtain the mechanical and electrical fields of a semi-infinite conductive crack
(~ if- 2~e15/c44) } Y -- 2v/-~ ~/~ - v/T~E--2--- ~ o" =
C44F
+
v/-~_V/-~- + V ~ + v / ~ d --
elsE,
,
'
(5.39)
D = elsF + tell E
The intensity factors at the tip of a semi-infinite conductive crack induced by a piezoelectric screw dislocation at za are /G=-
,
IG-{ K~t -- c44K~III--
(5.40) el5K1EI ,
Kffl -- e,sK~l '
+ KllKIEI
Using the intensity factors and Eq. (5.25), we can calculate the energy release rate induced by a piezoelectric screw dislocation.
VI. Interface Cracks
In engineering applications, piezoelectric ceramics often take the form of multilayered structures that use the accumulative results of stacks to enhance the ceramic's efficiency and sensitivity. The integrity of interfaces between the stacks is of great concern because interfacial fracture is one of the major failure modes in multilayered structures. Interface cracking of piezoelectric ceramics is, therefore, of paramount importance and has drawn much attention (Kuo and Barnett, 1991; Suo et al., 1992; Beom and Atluri, 1996; Chen, Yu, and Karihaloo, 1997; Qin and Yu, 1997; Zhong and Meguid, 1997; Deng and Meguid, 1998; Ru et al., 1998; Shen and Kuang, 1998; Gao and Wang, 2000; Qin and Mai, 1999; Herrmann and Loboda, 2000; Ru, 2000). Because the analysis of an interface crack in piezoelectric media is enormously complex, we study only electrically impermeable piezoelectric interface cracks in detail and discuss the fracture mechanics for other types of piezoelectric interface cracks only briefly.
210
Tong-Yi Zhang et al. X2
Material 1 Xl
Material 2
FIG. 5. A semi-infinite interface crack.
A. I M P E R M E A B L E I N T E R F A C E C R A C K S
1. Semi-Infinite Interface Cracks Figure 5 shows a semi-infinite piezoelectric crack with the X 1 axis along the interface and the coordinate origin at the crack tip. For an electrically impermeable crack, the extended traction-free boundary conditions along the crack faces require Z~
--" ~-]2 - - 0,
Xl
<
0, X 2 --- 0
(6.1)
where the superscripts + and - indicate the limiting values taken from the upper and lower half-planes, respectively. On the bonded interface, the continuity of the extended traction and extended displacement is signified by
~ - = ~2
(6.2)
u + = u-
(6.3)
for Xl > 0, X 2 - - 0. Using Eqs. (3.17)-(3.20), we express the extended displacements and stress functions, respectively, in the form U(k) - A(k)f(k)(Z) + A(~)f(~), Y]2(k) - - lP'(k),l,
r
El(k)-
--1/](k),2,
k - 1, 2(k not summed),
(6.4)
-- L(k)f(k)(Z) + L(k)f(k)
where the subscript (k) denotes the upper half-plane when k - 1 or the lower
Fracture o f P i e z o e l e c t r i c C e r a m i c s
211
half-plane when k = 2. Following the one-complex-variable approach (Suo, 1990; Beom and Atluri, 1996), the vector function, f, is defined as f(z) = [fl(z) f2(z) f3(z) fa(z)] T
(6.5)
in which f represents analytic functions of z = X l + px2 ( I m p > 0). Once the solution of f(z) is obtained, a replacement of z by zi(i -- 1, 2, 3, 4) should be made for the component fi(z) of f(z). By analytic continuation, one finds from Eqs. (6.1) and (6.2) (Suo et al., 1992; Beom and Atluri, 1996) that L1F~l)(Z)- L2~'~2)(z) = L2F~2)(z) - Ll~'~l)(Z) = 2h(z)
(6.6)
where h(z) is analytic in the entire z plane and F
-
A,(2)~'(2)(Z)
--
A~e)F<e)(z) -
2~k(1)~"(1)(Z)
(6.7)
which holds in the entire z plane except on the crack line. Define F~)(xl) = F~l)(xl + 0•
(6.8)
Equation (6.1) can be reduced to a homogeneous Hilbert problem in the following form L
xl < 0.
(6.9)
To determine F
(6.10)
where I is the 4 x 4 unit matrix and
M~ : (D~,)- D,e))(D,I)+ D,2))-', 1
-1
Mt~ = (D~, + D,2))
-l
( W , , ) - W,2)),
D,k) = -[Im(A(k)L(-~l)] -l, W~k) : -[Re(A~k)L~-~I)],
(6.11)
k - 1, 2, (k not summed)
in which the two real matrices M~ and M~ are the generalized Dundurs parameters for piezoelectric bimaterials, D~) and D~2) are real and symmetric, whereas W~) and W~2) are real and asymmetric (Suo et al., 1992). If materials in the upper and lower half-planes are identical, M~ and Mt~ become zero.
Tong- Yi Zhang et al.
212
Substituting Eq. (6.10) into (6.9) and noting that ~"(1)(Xl -qt. 0 - p ) - F~I)(Xl), we obtain (I + iM~)L(l)F~l)(Xl) + (I - iMt~)L(l)F~)(Xl) - 2(1 + M,~)h(Xl),
forxl < 0 (6.12)
The homogeneous solution of Eq. (6.12) may be expressed in the form (Beom and Atluri, 1996) L(1)F(1)(z) - z- 89
(6.13)
where I is a characteristic length and v is a constant vector. The constant vector v is determined by the eigenvalue equation (M~ + iXI)v = 0,
X = tanh(rr ~)
(6.14)
in which X is an eigenvalue. The four eigenvalues of Eq. (6.14) are (Suo et al., 1992), ~l :
17,
~.2 :
--17,
Z3
= -i0),
~4 = i0)
(6.15)
where / / _ {I(~tr(M~) 1 2)
2
__
det(Mt~)]1/2 _ 1~tr (M~)}1/2,
0)--{ [(~tr(M~)) 2- det(M/~)l 1/2-+- ~tr(Mg)}1/2
(6.16) (6.17)
Then, the homogeneous solution of Eq. (6.12) can be further written as L(l)F(l)(Z)- 2 2 ~1~ (i + i M ~ ) y E ( ; ) ie (;)Klg(z),
(6.18)
where g(z) is an arbitrary analytic function over the entire plane, and
1 e -- --1 tanh-l //, x----tan 7/"
2"/"
,2
Y(~' ~') -- 2 //2 ..}_0)2 (~ + ~:-1) + //2 +0)2 (r
1[
-1
(6.19)
0)
1
_+_~.-1) I
i0)2
+ ~ ~(~2 + ~o2)(~ - ~ - l ) - (-0(//2~2 + 092)(r
'[_
+2
1[
,
l
//2 +0)2 (~ -+-~" 1)+ //2 +092 (r + r
i
1
__ ~.-1) 1 M~
_,]
) M~
-1]
2 7(//2 + 092) (~ -- ~-1) + 09(//2 + 0921(~. __ ~. ) M~,
(6.20)
Fracture of Piezoelectric Ceramics
213
in which ~ = ( z / l ) ie and ~" = (z/l) K. The matrix Y(~, ~') satisfies (Beom and Atluri, 1996) Y(1, 1) = I,
Y(s~l~'2,~'1~'2) - -
(6.21)
~'l)Y(s~2, ~'2)
Y(~l,
The general solution of Eq. (6.12) is the sum of the homogeneous solution plus a particular solution
1
-1
F(1)(z) -- 2 ~ ~ z L ~ l ) ( I + iM~)Y
[(ztie ( Z)X ] g(z) + L(~)(I -1 + 7 ' 7
M,~)h(z) (6.22)
A substitution of Eq. (6.22) into Eq. (6.10) gives h(z) = -h(z),
(6.23)
g(z) = g(z)
a. Stress and Electric Displacement Intensity Factors Substituting Eq. (6.22) into Eq. (6.6) yields F(2)(z) - 2~_~zL(2)(I - iMt~)Y
~-
Z ~-
,
- 1) (I - M , ~ ) h ( z ) g(z) + L(2
(6.24) Using Eqs. (6.4), (6.22), and (6.24), we obtain the singular extended stress vector along the bonded interface near the crack tip ~2(xl)-
1
~/2zrxl
Y
,
xl --/-
x l / l << 1
g(xl),
(6.25)
As in elasticity, the solution is oscillatory unless e = 0. From Eqs. (6.16) and (6.19), we know that e - 0 if M~ -- 0, and there is no oscillation. The stress and electric displacement intensity factors can be defined as
K = [Kit Kt
Kilt KD]T -- x,--,olimv/2zrxlY
[(/)-ie ( )-x1 ,
--/-Xl
~2(xl) (6.26)
This definition is an extension of the definition for anisotropic elasticity (Wu, 1990; Qu and Li, 1991) to piezoelectricity. Because both Y[(xl/l) -i~, (Xl/1) -K] and Ez(xl) are real, K is real. Note that the intensity factors defined in Eq. (6.26) have the same physical dimensions as those for homogeneous piezoelectric materials [see Eq. (3.88)] and can be reduced to those intensity factors when the two materials become the same.
214
Tong- Yi Zhang et al.
U s i n g ~]2 -- LF + LF, we find the analytic functions F~k)(z), k -- 1, 2, near the
crack tip as 1 - ' ( I - k - i M / ~ ) Y [ ( ~ ) ie F(l)(z) = 2 2~--~L(1) (6.27) F(z)(z) : 2 2v,,~-~L(2)
}-
For an interface crack, one may define the intensity factors in slightly different ways. Suo et al. (1992) proposed the definitions of one complex intensity factor and two real intensity factors based on the components of the traction vector in the principal (eigenvector) coordinate system. The intensity factors defined by Eq. (6.26) depend on the characteristic length. However, one may define the intensity factors without using a characteristic length, as Beom and Atluri (1996) did. Rice (1988) reexamined the elastic fracture mechanics concept for conventional interface cracks and suggested the use of a characteristic length in the definition for intensity factors. Zhang and Lee (1993) discussed this issue in detail for conventional interface cracks.
b. J Integral Based on the solution to Eq. (2.38). Choosing J and substituting Eqs. intensity factors (Beom
given previously, we can calculate the J integral according a small circle around the crack tip as the integral path for (6.27) into (2.38), we find the J integral in terms of the and Atluri, 1996)
1
J -- - K T U - I K , 4
(6.28)
where, from Eq. (6.11), U -1 = (D~I + D~I)(I + M ~ ) -
-1)(i + iMt~). (I + iMt~) r (D~)l + D(2 )
(6.29)
2. Several Important Cases Using the solutions described previously and the mutual integral, Boem and Atluri (1996) studied several important cases of interface cracks. Here, we briefly present their results of the intensity factors for semi-infinite and finite cracks subjected to distributed loads on the crack faces and the results of the interaction between an interface crack and a piezoelectric dislocation.
Fracture of Piezoelectric Ceramics
215
a. A Semi-Infinite Crack with Crack Face Loading Consider a semi-infinite interface crack subjected to prescribed tractions t + (X l) and t-(xl)(Xl < 0) on the upper and the lower crack faces, respectively. The intensity factor can be calculated from K = ~-
1
CO
Ia - ~ Y[(21 )-it, (.~)-x 1 ](I + i M , ) -
x [(I - M~)t+(xl) - (I + M ~ ) t - ( x l ) ]
1
dx------L-1
(6.30)
where 21 -xleiJr/I. If the tractions are equal but opposite in sign on the upper and lower faces, -
t+(xl) -- - - t - ( x l ) -- t(xl)
(6.31)
Eq. (6.30) reduces to K-
~
fO___ Y[(3;1) -ie, oo
(21)-x](I + i M ~ ) - I t ( x l ) ~ dxl ~/-Xl
(6.32)
For the homogeneous case with material 1 being identical to material 2, we have e -- x = 0, thus Eq. (6.32) is further simplified to K -
~
ffJ~
dxl t(xl) ~_----------~1
(6.33)
b. Interaction of a Semi-Infinite Crack with a Piezoelectric Dislocation If a piezoelectric dislocation with an extended Burgers vector b (bl b2 b3 A4~)r - (bl b2 b3 b4)T and an extended concentrated force F = (F1 F2 F3 q)r _ (Fl F2 F3 F4)r is located at z d in the lower half-plane, the intensity factors are
Km---~Re{~~kZ
Y[(zf)-ie,(Zf)-X]m nH~l1g-1 (2)kl(L(2)jkbj-k- A(z)jk Fj )] ,
k=l
m -- 1, 2, 3, 4
(6.34)
where g l - - gll, K2 _ KI, K3 _ KIll, K4 -- KD, and z^dk _ Zd / l The same notation is used hereafter unless specifically noted, and H -l - (I + i M ~ ) U
(6.35)
Tong-Yi Zhang et al.
216
x2 Material 1
Xl -a Material 2
FIG. 6. A finite interface crack.
For a homogeneous medium, A ( l ) - A ( 2 ) - A, L ( 1 ) - L ( 2 ) - L, Eq. (6.34) reduces to
Im ~k=l ~ k
g m -- -
AjkFj),
m -- 1, 2, 3, 4
(6.36)
c. A Finite Interface Crack under Crack Face Loading
Consider a finite crack located from - a to a on the Xl axis (i.e., Lc = [ - a , a]), as shown in Fig. 6. The crack is subjected to the prescribed tractions t +(xl) and t - ( x l ) ( - a < Xl < a) on the upper and the lower crack faces, respectively. The intensity factor at the right crack tip is given by
K
m
1 f~a Y[(21 )-iE , (21)-~1(I +
2C%-S
-(I
+ M~)t-(xl)]
i M , ) - I [ ( I - M~)t+(xl)
a + Xl dxl - ~ y [ ( f i ) - i E ,
a --Xl
2v/-~
• (I + M~) -1 [M~(I - M ~ ) + (I - M~)S/l)]tc
(fi)-x] (6.37)
where a n
2a l '
tc --
F a
~ ( - - x l e i~r -- a) Ycl - - a ( - X l eizr + a ) '
[t+(xl) + t - ( x l ) ] d x l ,
S(l) -- W(1)D(l)
(6.38)
Fracture of Piezoelectric Ceramics
217
If the materials are homogeneous and the prescribed tractions are uniform and in opposite directions t + ( x l ) -- - t - (x l) = Z ~
(6.39)
then Eq. (6.37) is reduced to the intensity factor as that given by Eq. (3.88). If the characteristic length l is chosen to be 2a, we have Y[(5) -iE, (5) -x ] - I.
d. Interaction of a Finite Crack with a Piezoelectric Dislocation Similar to the case of a semi-infinite interface crack, the piezoelectric dislocation located at z d in the lower half plane generates the following intensity factor
=
-- -
gk -Jf-ay((~d -- a
Re
, (~d))mn
k=l
X H~ll 1L(2)k - l I (L(2)jkbj -+- A(2)jk Fj) ] -+- 1 Y(({t) -ie , ({t)-X)mnUnjbj -1 __Y((cl) -is, (cl)-X)mnUnl(n(-ll S~l , - M~D(,) )tjFj, ~/7r5a
(6.40)
m-1,2,3,4 at the right crack tip, where d
(6.41)
zkd + a For a homogeneous medium, Eq. (6.40) reduces to
Km -" - ~ I m
k--1
Vz~ - a
1
Lmk(Ljkbj + AjkFj) 1
+ 2 r--(Dmjbj~/Tra - Sjm Fj),
m -- 1, 2, 3, 4
(6.42)
Tong (1984) obtained a similar result for purely elastic media. Using the relations of D(k) -- -2iL(k)L(k7"),
S(k) -- i (2A(k)L/~) - I) ,
one can show that Eq. (6.42) is equivalent to 2K* in Eq. (4.43).
(6.43)
Tong- Yi Zhang et al.
218 B.
OTHER
RELEVANT
WORK
1. Curve-Shaped Interface Cracks
Zhong and Meguid (1997) developed a mathematically rigorous method to solve partially debonded circular inhomogeneity problems under antiplane shear and inplane electric fields with impermeable crack boundary conditions. The energy release rate can be derived and given by a closed-form solution. The solution for a straight crack is a special case wherein the radius of the circular inhomogeneity tends to infinity.
2. Permeable Interface Cracks
Gao and Wang (2000) studied the electrically permeable collinear interfacial cracks. They reduced the problem to a Hilbert problem and obtained explicit, closed-form solutions for the electric and mechanical fields in both piezoelectric media.
3. Conductive Cracks and Conductive Inclusions
Thin internal electrodes are widely used in devices and structures made of piezoelectric ceramics. The electrodes act like conductive inclusions. For a slit conductive inclusion in the (xl, x3) plane, the boundary conditions along the inclusion faces, x2 = 0, are the continuity of traction and displacements (Ru, 2000)
cr~j -- cr2j , -
+
-
uj -- u j ,
j -- 1, 2, 3,
E-~ -- E 1 -- O, f
llD2(Xl)lldxl = 47rq,
Xl
~ Lc,
xl ~ Lc, X2 =
X2 =
0,
0,
(6.44) (6.45)
IID2(xl)ll = D ] ( x l , O) -- D 2 ( x l , 0)
(6.46)
where q is the total electric charge of the electrode. If the electrode layer is much more compliant than the surrounding piezoelectric ceramics, the inclusion can be approximately treated as a crack with traction-free faces and Eq. (6.44) can be replaced by O'~j - -
0"2~ - -
O,
j -- 1, 2, 3,
Xl 6 Lc,
X2 - -
0
(6.47)
If the electrode layer is much stiffer than the surrounding piezoelectric ceramics, the electrode can be approximately treated as a rigid and conductive slit (Deng and
Fracture of Piezoelectric Ceramics
219
Meguid, 1998). In such a case, the displacement boundary conditions are
+-uj
Uj
-
--
UjO,
j=
1, 2, 3,
xl 6 Lc,
X2 = 0
(6.48)
w h e r e UjO are rigid body displacements. Note that the net force and net moment on the rigid and conductive slit should both equal zero, which are supplementary conditions to determine the rigid body displacements. Ru (2000) studied conductive interface inclusions subjected to the conditions given by Eqs. (6.44)-(6.46). He discussed the case in detail that the two piezoelectric half-planes are of the same material but have opposite poling directions. The poling directions of the upper and lower half-planes are along the positive and negative x2 directions, respectively. He found that the electroelastic field exhibits the inverse square root singularity without the normal interface oscillatory feature. Ru (2000) expressed the stress and electric fields in the vicinity of the crack tip in the form
B23 ] r 2q (crzl cr22 D2) r, = + 0 :q: B22 1 ~/a 2 - x 2
x E Lc
(6.49)
where B22 and B23 are two elements of the matrix B defined in Eq. (3.22), and + and - denote the upper and the lower edges of the electrodes, respectively. Equation (6.49) shows that the electrical and mechanical fields disappear if q -- 0 and that debonding is likely to occur at the interface between the electrode and piezoelectric ceramic blocks when the electrode carries an electric charge.
4. Contact Zone Model Many of the solutions for interfacial cracks in piezoelasticity have oscillatory singularities, as in the case of elasticity (Williams, 1959; England, 1965; Erdogan, 1965), which causes the overlapping of crack faces, a physically unreasonable phenomenon. To correct this shortcoming, Comninou (1977) developed a crackface contact model for conventional solids. Qin and Mai (1999) developed this model further for electrically impermeable interface cracks in linear thermopiezoelectric media. Assuming the contact zone to be friction-free, Qin and Mai (1999) obtained a set of singular integral equations supplemented by the condition that o"22 equals zero at the ends of the contact zone for the determination of the contact zone size. The stress intensity factors and the size of the contact zone can be calculated numerically by solving the integral equations. Herrmann and Loboda (2000) studied electrically permeable interface cracks by assuming that an artificial contact zone existed. Thus, they were able to reduce the problem for transversely isotropic piezoceramics to a combined Dirichlet-Riemann boundary value problem, which
220
Tong-Yi Zhang et al.
can be solved analytically. Numeric results for electrically permeable interfacial cracks of CTS-19/PZT-4 bimaterials show that the contact zone is four orders smaller than the crack length when the applied loads satisfy the condition
O'1~176 2
< 20,
D~~ = 0.
(6.50)
If the applied shear load is dominant, the contact length becomes significant. For example, when 0"12/0"22 oo oo - - 100 and D~~ = 0, the contact length is about oneseventh of the crack length (Herrmann and Loboda, 2000).
VII. Three-Dimensional Electroelastic Problems
Three-dimensional problems are of great importance in the fracture of piezoelectric ceramics and in practical applications. As early as in 1980, Deeg solved the electro-elastic problem for a three-dimensional piezoelectric inclusion embedded in an infinite piezoelectric medium. In the 1990s, many important threedimensional results were obtained for transversely isotropic piezoelectric media (Sosa and Pak, 1990; Wang, 1992a, 1992b; Dunn, 1994a; Huang and Yu, 1994; Wang, 1994; Wang and Huang, 1995a, 1995b; Wang and Zheng, 1995a; Kogan et al., 1996; Dunn and Wienecke, 1997; Huang, 1997; Zhao et al., 1997a, 1997b, 1999a-c; Chen and Shioya, 1999, 2000; Chen, Shioya and Ding, 1999). In practice, most poled piezoelectric ceramics possess 6 mm symmetry, as discussed in Section II. Thus, we limit the discussion to three-dimensional problems in transversely isotropic media.
A. GENERALSOLUTIONS The governing equations are the same as those given in Section III. Let the (xl, xe) plane of the Cartesian coordinates coincide with the isotropic plane, and let the poling direction be along the x3 axis. By virtue of the constitutive equations and the strain-displacement relations, the governing equations can be written in terms of the displacements, U l, U2, U3, and the electric potential, 4~
OZul 02//1 02Ul 1 02U2 Cl,-~Xl2 + C66 OX 2 -']-C44-~X23 -'1- -~ ( Cl , + C12)OXlOX2
02//3 02~ +(C13 + C44) OXIOX3 -k- (el5 + e31) OXlOX-----~-- O,
Fracture of Piezoelectric Ceramics 02U2
02U2
c66- Xl
+ 44
02U2
1
221
02Ul
+ (Cl, +Cl )0 10
02U3 02(]) + (C13 Jr- C44) OX20X3 nt- (el5 -~- e31) OXzOX3 = O, 02U2
C441k OX~ Jr- OX2 / nt- C33-~X32 + (C13 -~- C44) 0X;~X3 t OX20X3
+el, 0Xl + 02U2
el5\ Ox 2 -Jr- OX2 ~] + eaa--~x32 + (el5 '1- e31) Ox15x3 t OX20X3 --Kll
~
) (7.1)
)
+ OX2 / -- K33-~X32 -- 0,
where cij with C66 -- (Cll --
C 1 2 ) / 2 ; and eij and Kij are the elastic, the piezoelectric, and the dielectric constants, respectively. For three-dimensional deformation, the displacements and the electric potential may be expressed in terms of potentials U and X (Wang and Zheng, 1995b; Ding, Chen and Liang, 1996; Kogan et al., 1996; Zhao et al., 1997a)
OU Ul "-- OXl
OX OX2 ,
OU OX UZ -- ~X2 + -~Xl'
OU U3 -- k,-~x3'
OU 4) -- k20x3
(7.2)
where kl and k2 are constants to be determined. After substituting Eq. (7.2) into Eq. (7.1), we have
02Ui 02Ui 02Ui = 0, x--~l-~O x--~2 O nt- ~,i x---~3 O OZx Ox 2
i = l, 2, 3(i not summed) and
02X
02X
(7.3a) (7.3b)
where ~.4and ~i(i
-
2c44 Cll
(7.4a)
-- C12
l, 2, 3) are the roots of the characteristic equation
C1 ~.3 --~ C2 ~.2 Jr- C3 ~ -+- C4 - 0,
(7.4b)
222
Tong-Yi Zhang et al.
in which (?l, C2, C3, and C4 are materially related constants defined as follows: C;1 -- Cll(e25 + C44Kll), C2 -- 2el5(el5cl3 -- e33c11) -1--2c13(e15e31 + C44Kll)
-c44(e~l + cllK33) + Kll(C~3 -- CllC33),
(7.4c)
C3 -- e33(e33Cll --I- 2e15c44) -- 2e33(e15 -t- e31)(C13 + c44) -- K33(C23 -k- 2C13C44 -- C11C33) -k- K11C33C44 -I- c33(e15 -q- e31
C4 --
--c44(e23 +
)2
,
C33K33) 9
Equation (7.4b) is derived from the elimination of kl and k2 to obtain Eq. (7.3a). In the following discussion, we consider only the case that the first three roots ~i(i = 1, 2, 3) are distinct. In many cases, it is more convenient to use the polar cylindrical coordinates (r, 0, z), where the z axis coincides with the x3 axis. Letting U4 = X, we can express the displacements and the electric potential in a more compact form in terms of the four potentials Ui. Equation (7.3) can then be written as 02Ui 1 OUi 1 oZui -----5Or + -r -~r -~ r2 O0 2
oZui
where zi = si z and si -- 1 / ~ i . Ur-
i -- 1, 2, 3, 4 (i not summed),
~- OZ2 = O,
3 OUi Z O; i=1
(7.5)
The results of Eq. (7.5) are
1 OU4 r OO '
Uo-
3 OU i Uz -- Z k l i ~ , i=1 Oz
~
-
3 1 OUi OU4 Y~ r O0 ~ ~Or , i=l
(7.6a)
3 OUi Z k2i Oz ' i=1
where kli and k2i(i -- 1, 2, 3) are constants related to ~-i by C44 + (C13 + c44)kli + (el5 + e31)k2i
c33kli + e33k2i
Cll
C13 "]- C44 + c44kli -'}- elsk2i
=
e33kli - K33k2i el5 -+- e31 + el5kli - Kllk2i
(7.6b) = ~.i.
Consequently, the stresses are given by 3 F 02Ui 10Ui 1 02Ui - - 002 ~ -~- (Cl3kli d- e31k2i)--~z 2 tTrr -- i")~9 - LCll--~r2 + C12 r 0r + Cl2r2
1 02U4) + (Cll -- C12)
r2 00
r O00r
'
223
Fracture o f Piezoelectric Ceramics
3 E
02Ui
C12---~-F2 + Cll
~176176 -- /El "__
10Ui
1 02Ui -~- C1 l -r2- ~ 002 -Jr-(cl3kli Jr- e31k2i) Oz 2
r 0r
02U4) r OOOr ' 1
-- (Cll -- C12) r 2 O0
3 02Ui tTzz -- Z ~.i(C44 -Jr-Cl44kli + el5kzi) Oz 2 , i=l
3[
tYzO -- i 9~ 1 (C44 "+-c44kli-1t- elsk2i) 102Ui r OzO0
(7.7)
]
0204
OrOz
+c44~
3 [ O2Ui 1 02U4 tTzr -- i ~ (c44 -~- c44kli -1I- elsk2i) OrOz - c44 -
rS-b
3 [ tTr~ ~---/El'= (cll -c12) 1
-~- ~(Cll --C12)
(1 r
02Ui
0rO0
(02U4
r 2 00
10U4 r Or
0r 2
1
02U4)
r 2 002
'
and the electric displacements are given by
3 [ Dr -- E
i=l 3[
Do --/~l'= Dz --
O2Ui 1 02U4 (el5 d- el5kli - gllk2i) OrOz -- el5-
r OOOz
(el5 q- el5kli -
3
~.i(e15-~- el5kli E i=1
1 02Ui]
02U4
Kllk2i)-r aOaz + e 1 5OrOz ~
(7.8)
O2Ui - Kllk2i) Oz 2 .
B. AN ELLIPSOIDAL CAVITY UNDER REMOTE LOADING For a transversely isotropic piezoelectric medium, Fig. 7 shows an ellipsoidal cavity in an infinite body under remote loading. The center of the cavity is located at the origin of the coordinate system, and the ellipsoidal surface is described by
r2
Z2
a2 q- ~-~
1,
(7.9a)
224
Tong-Yi Z h a n g et al.
az~, D7
X3
X2
,,o ~
( ~ rr
Xl
Fro. 7. An ellipsoidal cavity under combined remote loadings.
which can be rewritten as r2
z2
q2----~ + q---~ -- C2'
(7.9b)
where C 2 - b 2 - a 2 and q0 = b/Co. Then,
n
--
{nr,
O, nz}
(7.10a)
is an outwardly normal unit of the ellipsoidal surface with r nr =
a2N'
n, -"
z b 2N
,
N-
/r 2
22
~/ --a-~ + _-7.
(7.10b)
q'b
The boundary conditions on the cavity surface are the same as those given by Eqs. (3.33)-(3.35) and take the following form in the cylindrical coordinate system r - - 4~c
(continuity of electric potential),
~ r r n r q- ~Trzrtz - - O, Crrznr + O'zzn z - - 0
O'Orrlr + croznz -- O,
(traction-free),
D r n r + Dznz -- DCn~ + D~nz,
where the superscript c denotes the cavity.
(surface-charge-free),
(7.11)
225
Fracture of Piezoelectric Ceramics 1. A x i s y m m e t r i c L o a d i n g
Under the remote loads of {rrr and {r=~ and the remote electric displacement D z , the corresponding displacements and electric potential are (X)
{30
OO
Ur -- errr,
{90
{~)OO
u z -- ezzz,
-
OO
- E z z + Cko,
(7.12a)
where 4}o is a reference electric potential, and co
oo
O'rr ~ -- (Cll + Cl2)er~r + Cl3e=z -- e31E z , ~ -- e33 E .,~ , {rzz~ -- 2Cl3Srr + C33e,z. Dz
(7.12b)
~ 9 -- 2el3err~ + e33ezz~ + x 3 3 E =
Under remote loading, the electric field strength E c inside the cavity is uniform with the electric potential ~c _ _ E ~ z + 4}~,
(7.13)
where 4}~)is another reference electric potential and E c is to be determined from the boundary conditions. In this case, the four potentials take the following forms (Kogan et al., 1996): Ui -- A i H ( r ,
U4 - - 0 ,
i = 1, 2, 3,
Zi),
(7.14)
where l[z~qgl- (qi) 4- r 2 q92(qi) - C2i990(qi)], n ( r , zi) -- -~ l(qi-t-1) 990(qi) = ~ In qi - 1
1
991(qi) = ~ In
(qi+l) qi-
'
(7.15)
1 1
qi
1 ln(qi+ 1) 1 qi 992(qi) -- --~ qi -- 1 -~ 2 q 2 - - - ~ ' with constants, A i , i = 1, 2, 3, and qi(r, zi) defined implicitly by r2 z2 - C 2 (i not summed) with C 2 = _b2 _ q 2 _ 1 +- q2 ki
_ 0
2
"
(7.16)
On the ellipsoidal surface, using Eq. (7.9a) and z 2 - z2/)~i yields q2 _ q2i,O ~ b 2/ ( b2 -- ~.ia2) 9
(7.17)
226
T o n g - Y i Z h a n g et al.
A substitution of Eqs. (7.14)-(7.17) into the boundary conditions of Eq. (7.11) gives the following equations for the constants Ai and E c as follows: ~0
r
-
Ez
-
3[
/~l
93i~l ~k2i i g)l (qi ,o)Ai
Ec -
(continuity of electric potential),
-
c44(1 + kli) nu el5k2i ] ).i qgl (qi,o) Ai nt- ff r~r -- 0
(C12 -- Cll)gO2(qi,o) --
".I
(traction-free in the r direction),
3 2 Z[c44(1 -t- kli) "k- elsk2i]q92(qi,o)Ai - Ozz -- 0 i=1 (traction-free in the z direction), 3 2 Z [el5(1 q- eli) - Kllk2i]g)2(qi,o)Ai - D z - trCE c (surface-charge-free), i=1 (7.18) where x c is the electric permeability of the cavity. Note that the traction-free condition associated with ~Or and CrOz is satisfied automatically due to the axial symmetry. Once the coefficients A i , i = 1, 2, 3, and E c are determined from Eq. (7.18), the stress and electric displacement fields can be calculated from Eqs. (7.14), (7.7) and (7.8). As discussed in Section III for a two-dimensional cavity, caution should be used when shrinking the cavity to a crack because the results depend on the sequence of how the limits are reached. Letting b approach zero (i.e., oe = b / a ~ 0) and using Eqs. (7.7), (7.8), and (7.18), we obtain the stress and the electric displacement on the crack plane (z = 0) =
D z ( r , O) =
1 715
+ A r c t a n V/~ 2 -
%/~2 __ 1
tOz
,
Jr
%/~2
1
1
]
+ A r c tan v/? 2 - 1
? -- r / a > 1
] + d*,
~ -
(7.19a)
r/a > 1
(7.19b) where d* = D ~
(X)
~
+ azz M5
for c~/fl --+ O,
(7.20a)
Fracture of Piezoelectric Ceramics
227
d* = D z + CrzzOOM5 for c~/3" -+ a nonzero finite value, 1 + ~/3"
(7.20b)
d* = 0
(7.20c)
for ~ / 3 " --+ oo,
in which
* - xClx3o ,eff
x3oeff= det [M(1)]/det [M (3)],
/~5 - det
[M(5']/det [M(3)]. (7.21a)
In Eq. (7.21a) M (1), M (3), and M (5) are 3 x 3 matrices with the elements
M(S) 2i --
M li(1) = si[c44(1 + kli) + elskzi],
zri -~-[c44(1 + kli) + el5k2i],
M(1) zri 3i = -~--[els(1 + k l i ) - K l l k 2 i ] ,
M(li3) =
/!,4(1) ~'~li '
M(5) li
Aar(1)
--*'*li
/IA(3) ~(1) ~"~2i = "'~2i ' 13'(5)= isik2i,
'
*"2i
(7.21b)
/i//(3) -isikzi, ~'~3i =
~r
*"3i
__ M(~) 3i '
i -- 1 ' 2 ' 3 "
Equation (7.21a) gives the effective dielectric constant for three-dimensional piezoelectric solids. The results indicate that the mechanical and electrical fields are strongly dependent on the ratio of ot/fl*. This is similar to the two-dimensional case, in which the solution depends strongly on the ratio of ot/fl*. The two limiting results given by Eqs. (7.20a) and (7.20c) correspond, respectively, to the electrically permeable and impermeable boundary conditions along the crack faces. Defining Mode I stress intensity factor K I and electric displacement intensity factor Ko as KI
=
lim v/2rr (r - a)~rzz,
r---+ a
K o = lim v/2:r(r - a)Dz, r-+
(7.22)
a
we have
KI
=
2~rzz ~ , a
KD -- 2 ( D ~ - d . ) V ~a"
(7.23)
These results show that, as in the two-dimensional problems discussed in Section III, the Mode I stress intensity factor is the same as that in purely elastic media and is independent of the applied electric displacement. The electric displacement intensity factor relates not only to the applied fields, but also to the material properties in terms of d*. For the two limiting cases, we have
7 #I ~0 ~ KD = -2O'zzO 0 Ms~/~-
(7.24a)
228
Tong-Yi Z h a n g et al.
for electrically permeable cracks and KD -- 2D~//-f-n"
(7.24b)
for electrically impermeable cracks. Using the electrically permeable or impermeable boundary conditions, Kogan et al. (1996) and Huang (1997) obtained the intensity factors for electrically permeable cracks, whereas Zhao et al. ( 1 9 9 7 b ) obtained the intensity factors for electrically impermeable cracks. For a conductive crack, the electric field inside the crack disappears (i.e., E C= 0). Then the traction-free and electric potential continuity boundary conditions require 3
Z
sjk2jAj -- 0
(continuity of electric potential),
j=l 3 Z sj[c44(1 + k l j ) + e l s k 2 j ] A j j=l
7ri
- 0
(traction-free in the r diection),
3
(traction-flee in the z direction).
[c44(1 + klj) -+- el5k2j]Aj = r
2
(7.25)
j=l
Equation (7.25) yields the constants Ai, i - 1, 2, 3. Once the electrical and mechanical fields are calculated from the potentials Ui, i -- 1, 2, 3 of Eq. (7.14) with Eqs. (7.7) and (7.8), then we can calculate the intensity factors, which become or ~//~_ K* -- 2a=z 7r'
KD = 2 F ~ %z~ ~ //rS7'
(7.26)
where F o _ det[FU]/det[Ft],
(7.27a)
in which F u and F t are 3 x 3 matrices with F1u. -- F[i -- s i k z i A i ,
F2]. -- F~i -- si[c44(1 + kli) + elskzi],
F3u -- els(1 + kli) - Kllk2i,
F~i -- c44(1 -+- kli) -+- elskzi,
(7.27b) i -
1, 2, 3.
Equation (7.26) shows that the mechanical load induces the intensity factor of electrical displacement Ko due to the piezoelectric effect.
229
Fracture o f P i e z o e l e c t r i c C e r a m i c s 2. A s y m m e t r i c L o a d i n g
If only 0"~3 and D ~ are applied at infinity, the corresponding non-zero displacement and electric potential at infinity are related by the constitutive equations u ~ = 2e~3r cos0,
4) ~ -
oc 2c44e~3 - e l 5 E ~ -- 0"13,
cos0 + ~b0,
-E~r
(7.28)
2ea4e~ ,3 + X l , E ~ -- D ~ .
The electric potential inside the electrically insulating cavity is cDc -
- E c r cos 0 + q~D.
(7.29)
The potential functions become (Kogan et al., 1996) i = 1, 2, 3,
A i H ( r , Zi)COSO,
Ui -
U4 -
-A4H(r,
Z4)
sin 0,
(7.30)
1
(7.31)
where 2ziC 2
H ( r , Zi) -- rzi[(~l(qi) -]- ~ 2 ( q i ) ] -k- ~ ,
O1 (qi) -- qi
[l+q In (q/1)] -~-
qi-k- 1
rqi ,
02(qi)
qi(q 2 - - 1 )
9
The dependent variables qi(r, zi) and constants Ci are defined by Eq. (7.16). Substituting Eq. (7.30) into Eqs. (7.5) and (7.7) and then into the boundary conditions of Eq. (7.11) yields the equations for the five constants Ai, i - 1, 2, 3, 4, and E c, ~o
-
~;,
3
Z
AiC3 -- A4C3 -- 0
(boundness condition at r - 0),
i=1 3
Z
sikzi[l~l(qi'~ + 3(~2(qi,o)]Ai - E ~ -- - E c
i=1
(continuity of electric potential), 3
Z
Si[Ca4(1 -Jr-kli) nt- elskzi][l~)l(qi.o) nt- 3~2(qi,o)]Ai + sac44[l~)l(q4,o)
(7.32)
i=1 + 3 O 2 ( q a , o ) ] A 4 + o'13
= 0
(traction-free in the r and 0 directions),
3
Z
si[c44(1 + kli) + el5kzi]|
- 0
(traction-free in the z direction),
i=1 3
y ~ si[el5(1 + kli) - Kllk2i][~l(qi,o) -- 3~2(qi.o)]Ai i=1
+ saels[Ol (q4,0) + 3O2(qa,o)]A4 -t- D ~ - x c E c -- 0
(surface-charge-free).
230
Tong-Yi Zhang et al.
The mechanical and electrical fields can be determined from the four potentials Ui as defined by Eq. (7.30) after the five constants Ai, i = 1, 2, 3, 4, and E c are determined. However, the general solution of Eq. (7.32) is complicated. For example, here we consider only the case of c~ --+ 0 and ot/fl* --+ O. In this case, the cavity is shrunk to a crack, and the boundary conditions of Eq. (7.32) are simplified as 3
Z
Ai
-
-
(boundness condition at r - 0),
A4 -- 0
i=1
3 3 -7ri Z s j k z j A j - E ~ - - E c 2 j=l
-7ri 2
j=l
(continuity of electric potential),
sj[c44(1-+-klj) + el5k2j]Aj -k- s4c44A4
(7.33)
-k- o13
(traction-free in the r and 0 directions), 3 ~
[c44(1 -!- kli) + elsk2i]Ai = 0
(traction-free in the z direction),
i=1 3 Z [els(1 -k- kli) i=1
- Kllk2i]Ai
= 0
(surface-charge-free).
The stress at 0 - 0, z = 0 is given by (Kogan et al., 1996) 0-13(F,
0 O) -- 20-~3 F
?2 _+_ 1 + Arc 7/" L 72 ~/72 __ 1
'
6s4 A4i c44 , ?2 ~,/72 _ 1
+
tan
V/? 2 -
1]
? - r / a > 1.
(7.34)
Defining the Mode II and Mode III stress intensity factors as Kll = lim V/27r(r
r-+a
-
a)0-rz,
Kill -- lim V/2zr(r -- a)0-zO,
r--~a
(7.35)
we have
KII--(40-13 ~ Km=
a
Im[A416s4c44~/-~)cosO,
-Im[Aa]6sacan~/na sin 0.
(7.36)
The results show that the electric displacement has no singularity near the crack tip. In other words, the electric loading D ~ at infinity does not contribute to the
Fracture of Piezoelectric Ceramics
231
singularity at the crack tip. Wang (1992a) and Chen and Shioya (2000) obtained the same result for electrically impermeable cracks.
C. AN IMPERMEABLE PLANAR CRACK OF ARBITRARY SHAPE UNDER ARBITRARY LOADING
Consider a planar crack of arbitrary shape in the isotropic (Xl, X2) plane. The upper and lower surfaces of the crack are denoted by s + and s-, respectively. The prescribed tractions, Pl, P2, and P3, along the respective x l, x2, and x3 axes and the prescribed electric displacement boundary value nr(x, y) are all equal but opposite in sign on the upper and lower faces of the crack; that is, PI(X1, x2, 0 +) = -PI(X1, x2, 0 - ) -- Pl(x, y), P2(x1, x2, 0 +) -- - P 2 ( x I , x2, 0 - ) = P2(x1, x2),
(7.37)
e3(xl, x2, 0 +) = - P 3 ( x I , x2, 0 - ) = P3(xI, x2), U)'(X1, X2, 0 +) -- --UY(XI, X2, 0 - ) = UT"(X1,X2).
The boundary integral method can be used to solve this type of crack problems (Cruse, 1996). Zhao et al. (1997a) established the Green functions for the loadings of concentrated electric potential discontinuity and concentrated displacement discontinuity in a three-dimensional piezoelectric medium. Using the fundamental solutions, Zhao et al. (1997b) derived the boundary integral equations for electrically impermeable cracks of arbitrary shapes f
+
[tz~l Ilu311 + tzl IlOll]~ds(~, 1 r/) = P3(xl, x2),
f+ [Lzz211u311 + Lz2llcblll-~ds(~, 1 s+{[Lzr3
O) = nr(xl, x2),
(7.38a) (7.38b)
cos 2 0 -k- Lz03 sin 2 0]llul II
+ [Lzr 3 -
s+{[Lzr3 -
Lz03]
Lzo3]
1
sin0 cosOllu2ll}-~ds(~, rl) - Pl(Xl, x2),
(7.38c)
sin0 cos011Ul II 1
-+- [Lzr 3 cos 2 0 -k- Lzo3 sin 20]llu211}-~ds(~, O) - P2(xl, x2),
(7.38d)
232
Tong-Yi Z h a n g et al.
where r 2 --- ( X l - - ~ ) 2 ..]_ (X2 -- 0) 2, the symbol II II denotes a jump of the quantity across the crack, and the material constants are given by 3
3
Lzzl = ~ - ~ a i m i , i=1
1
Lzr3
-
-
-
-
2
1
Lz~
~
~
1
3 i=1
3
H123 i=1
3
Lzz2--Zaiti,
i=1
- - ~H123
~
3
Lzl--Zbimi,
Lz2 -- ~
i=1
biti,
i=1
si Hi(c44 + c44kli -+-el5k2i) -+- c44s4
(7.39)
,
Si Hi(c44 + c44kli -1- el5k2i) - 2c44s4 ,
in which
ai --
--c13 +
(c33kli -~- e33k2i)s 2,
bi -
- e 3 1 --t- (e33kli
m l - - [k12k23 - k22k13 + k23 -
k22]/(47rSl A),
m2 - - [k13k21 - k l l k 2 3 -k- k21 -
kz3]/(47rszA),
m3 - - [ k l l k 2 2 - k12k21 d- k22 -
k21]/(47rs3A),
t2 - -
[el
~(k21kl3 c44
- k l l k 2 3 ) + kl3 - k l l
]/
- K33k2i)s2, (7.40a)
(47rs2A),
(7.40b)
t3-- Iel---~5(kllk22-k12k21)-t-kll -k12]/(47r,3/~)'c44 A - - k12k23 - kl3k22 + kl3k21 - k l l k 2 3 + k l l k 2 2 - kl2k21, H1 -
1,
H2
-- (a3bl -
H3 - - ( a l b 2 - a 2 b l ) / ( a 2 b 3
alb3)/(a2b3 -a3b2),
a3b2),
(7.40c)
H123 - - H1 d- H2 d- H3.
Equations (7.38a-d) are hypersingular boundary integral equations (HSBIE). Without loss of generality, the coordinate system can be chosen such that its origin is at the crack tip, and the x~ and x2 axes are along the normal and the tangential directions of the crack front, respectively. We define the intensity factors at the crack tip as
KI -- lim
xl-~0
Km -
lim
~/27rXlO'33(Xl, X2, 0),
xl--~0
~/27gXlO'32(Xl, X2, 0),
KII --
lim x/2rrXlCr31(Xl, X2, 0),
xl--+0
KD -- lim ~/27rxl D3(Xl, X2, 0). xl--+0
(7.41)
233
Fracture of Piezoelectric Ceramics
Then the intensity factors can be expressed in terms of the jumps of the displacement and the electric potential across the crack faces K o = - J r x,--.0 lim ~2~/xl [tzl Ilu311 + L~2114~11], KI--Jr
lim 2r x,-~o
[tzzlllu311 + tzz2114~ll],
(7.42) Kit -- -7r lim x l ---~o Kill-
--Tr
lim
xl~O
2~xx~[~L , r 3
+ 31 -~ Lz03
Lzr3 + -s
1 Ilulll,
Ilu211.
We can solve Eqs. (7.38a-d) to obtain the jumps of electric potential and displacements across the crack faces. Solutions for HSBIE problems for conventional materials are available (Ioakomidis, 1982a, 1982b; Tang and Qin, 1993). By analogy, we have the solutions of Eqs. (7.38a-d) for the piezoelectric solids, and then we can determine the intensity factors.
1. Symmetric Problems
Under symmetric loading, P1 = 0 and P2 = 0, the boundary integral equations involve only Eqs. (7.38a, b). Correspondingly, the HSBIE solution takes the form I1~11 4Jr(1# - v) fs+ ---~-ds(~, 11) - -t3(Xl, x2),
(7.43)
for isotropic elasticity, where II~ II is the jump in the normal displacement across the crack surfaces; # and v are, respectively, the shear modulus and Poisson's ratio; and t3 is the prescribed normal traction. One can show that the stress intensity factor is (Tang and Qin, 1993) # lim ~ 27r II~ II K ~ ( t 3 ) - 4(1 - v)x,--,0 x---7 "
(7.44)
The stress intensity factor is related to the loading through the displacement jump of Eq. (7.43). If the equations for piezoelectric solids have the similar form to
234
Tong-Yi Zhang et al.
that given by Eqs. (7.43) and (7.44), we are able to solve the piezoelectric HSBIE problems. To do so, we linearly combine Eqs. (7.38a, b) to yield f + [Lzlllu311 + Lzzllc~ll]-~ds(~, 1 17) = GE M P3(Xl,X2) -] GEEuff(XI,X2),
f+ [Lzzl Ilu311 + Lzz21lckll]-~ds(~, 1 rl) -
(7.45a)
G MMP3(X1, x2) + GME~o(XI, X2),
(7.45b) where GEM
_
L z 2 (