Advances in Applied Mechanics Volume 38
Editorial Board Y. C. FUNG DEPARTMENT OF BIOENGINEERING UNIVERSITY OF CALIFOR...
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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
ACADEMIC PRESS A Division of Harcourt, Inc.
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Copyright 9 2002 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the Publisher. The appearance of the code at the bottom of the first page of a chapter in this book indicates the Publisher's consent that copies of the chapter may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 01923), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-2002 chapters are as shown on the title pages. If no fee code appears on the title page, the copy fee is the same as for current chapters. 0065-2165/02 $35.00 Explicit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press chapter in another scientific or research publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given.
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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
2
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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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Paolo Maria Mariano
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 _ 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)-
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 - 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)
72
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)~ ~ 12~ 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.
177
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 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 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 1
] + d*,
~ -
(7.19a)
r/a > 1
(7.19b) where d* = D ~
(X)
~
+ azz M5
for c~/fl --+ O,
(7.20a)
Fracture of Piezoelectric Ceramics
227
d* = D z + CrzzOOM5 for c~/3" -+ a nonzero finite value, 1 + ~/3"
(7.20b)
d* = 0
(7.20c)
for ~ / 3 " --+ oo,
in which
* - xClx3o ,eff
x3oeff= det [M(1)]/det [M (3)],
/~5 - det
[M(5']/det [M(3)]. (7.21a)
In Eq. (7.21a) M (1), M (3), and M (5) are 3 x 3 matrices with the elements
M(S) 2i --
M li(1) = si[c44(1 + kli) + elskzi],
zri -~-[c44(1 + kli) + el5k2i],
M(1) zri 3i = -~--[els(1 + k l i ) - K l l k 2 i ] ,
M(li3) =
/!,4(1) ~'~li '
M(5) li
Aar(1)
--*'*li
/IA(3) ~(1) ~"~2i = "'~2i ' 13'(5)= isik2i,
'
*"2i
(7.21b)
/i//(3) -isikzi, ~'~3i =
~r
*"3i
__ M(~) 3i '
i -- 1 ' 2 ' 3 "
Equation (7.21a) gives the effective dielectric constant for three-dimensional piezoelectric solids. The results indicate that the mechanical and electrical fields are strongly dependent on the ratio of ot/fl*. This is similar to the two-dimensional case, in which the solution depends strongly on the ratio of ot/fl*. The two limiting results given by Eqs. (7.20a) and (7.20c) correspond, respectively, to the electrically permeable and impermeable boundary conditions along the crack faces. Defining Mode I stress intensity factor K I and electric displacement intensity factor Ko as KI
=
lim v/2rr (r - a)~rzz,
r---+ a
K o = lim v/2:r(r - a)Dz, r-+
(7.22)
a
we have
KI
=
2~rzz ~ , a
KD -- 2 ( D ~ - d . ) V ~a"
(7.23)
These results show that, as in the two-dimensional problems discussed in Section III, the Mode I stress intensity factor is the same as that in purely elastic media and is independent of the applied electric displacement. The electric displacement intensity factor relates not only to the applied fields, but also to the material properties in terms of d*. For the two limiting cases, we have
7 #I ~0 ~ KD = -2O'zzO 0 Ms~/~-
(7.24a)
228
Tong-Yi Z h a n g et al.
for electrically permeable cracks and KD -- 2D~//-f-n"
(7.24b)
for electrically impermeable cracks. Using the electrically permeable or impermeable boundary conditions, Kogan et al. (1996) and Huang (1997) obtained the intensity factors for electrically permeable cracks, whereas Zhao et al. ( 1 9 9 7 b ) obtained the intensity factors for electrically impermeable cracks. For a conductive crack, the electric field inside the crack disappears (i.e., E C= 0). Then the traction-free and electric potential continuity boundary conditions require 3
Z
sjk2jAj -- 0
(continuity of electric potential),
j=l 3 Z sj[c44(1 + k l j ) + e l s k 2 j ] A j j=l
7ri
- 0
(traction-free in the r diection),
3
(traction-flee in the z direction).
[c44(1 + klj) -+- el5k2j]Aj = r
2
(7.25)
j=l
Equation (7.25) yields the constants Ai, i - 1, 2, 3. Once the electrical and mechanical fields are calculated from the potentials Ui, i -- 1, 2, 3 of Eq. (7.14) with Eqs. (7.7) and (7.8), then we can calculate the intensity factors, which become or ~//~_ K* -- 2a=z 7r'
KD = 2 F ~ %z~ ~ //rS7'
(7.26)
where F o _ det[FU]/det[Ft],
(7.27a)
in which F u and F t are 3 x 3 matrices with F1u. -- F[i -- s i k z i A i ,
F2]. -- F~i -- si[c44(1 + kli) + elskzi],
F3u -- els(1 + kli) - Kllk2i,
F~i -- c44(1 -+- kli) -+- elskzi,
(7.27b) i -
1, 2, 3.
Equation (7.26) shows that the mechanical load induces the intensity factor of electrical displacement Ko due to the piezoelectric effect.
229
Fracture o f P i e z o e l e c t r i c C e r a m i c s 2. A s y m m e t r i c L o a d i n g
If only 0"~3 and D ~ are applied at infinity, the corresponding non-zero displacement and electric potential at infinity are related by the constitutive equations u ~ = 2e~3r cos0,
4) ~ -
oc 2c44e~3 - e l 5 E ~ -- 0"13,
cos0 + ~b0,
-E~r
(7.28)
2ea4e~ ,3 + X l , E ~ -- D ~ .
The electric potential inside the electrically insulating cavity is cDc -
- E c r cos 0 + q~D.
(7.29)
The potential functions become (Kogan et al., 1996) i = 1, 2, 3,
A i H ( r , Zi)COSO,
Ui -
U4 -
-A4H(r,
Z4)
sin 0,
(7.30)
1
(7.31)
where 2ziC 2
H ( r , Zi) -- rzi[(~l(qi) -]- ~ 2 ( q i ) ] -k- ~ ,
O1 (qi) -- qi
[l+q In (q/1)] -~-
qi-k- 1
rqi ,
02(qi)
qi(q 2 - - 1 )
9
The dependent variables qi(r, zi) and constants Ci are defined by Eq. (7.16). Substituting Eq. (7.30) into Eqs. (7.5) and (7.7) and then into the boundary conditions of Eq. (7.11) yields the equations for the five constants Ai, i - 1, 2, 3, 4, and E c, ~o
-
~;,
3
Z
AiC3 -- A4C3 -- 0
(boundness condition at r - 0),
i=1 3
Z
sikzi[l~l(qi'~ + 3(~2(qi,o)]Ai - E ~ -- - E c
i=1
(continuity of electric potential), 3
Z
Si[Ca4(1 -Jr-kli) nt- elskzi][l~)l(qi.o) nt- 3~2(qi,o)]Ai + sac44[l~)l(q4,o)
(7.32)
i=1 + 3 O 2 ( q a , o ) ] A 4 + o'13
= 0
(traction-free in the r and 0 directions),
3
Z
si[c44(1 + kli) + el5kzi]|
- 0
(traction-free in the z direction),
i=1 3
y ~ si[el5(1 + kli) - Kllk2i][~l(qi,o) -- 3~2(qi.o)]Ai i=1
+ saels[Ol (q4,0) + 3O2(qa,o)]A4 -t- D ~ - x c E c -- 0
(surface-charge-free).
230
Tong-Yi Zhang et al.
The mechanical and electrical fields can be determined from the four potentials Ui as defined by Eq. (7.30) after the five constants Ai, i = 1, 2, 3, 4, and E c are determined. However, the general solution of Eq. (7.32) is complicated. For example, here we consider only the case of c~ --+ 0 and ot/fl* --+ O. In this case, the cavity is shrunk to a crack, and the boundary conditions of Eq. (7.32) are simplified as 3
Z
Ai
-
-
(boundness condition at r - 0),
A4 -- 0
i=1
3 3 -7ri Z s j k z j A j - E ~ - - E c 2 j=l
-7ri 2
j=l
(continuity of electric potential),
sj[c44(1-+-klj) + el5k2j]Aj -k- s4c44A4
(7.33)
-k- o13
(traction-free in the r and 0 directions), 3 ~
[c44(1 -!- kli) + elsk2i]Ai = 0
(traction-free in the z direction),
i=1 3 Z [els(1 -k- kli) i=1
- Kllk2i]Ai
= 0
(surface-charge-free).
The stress at 0 - 0, z = 0 is given by (Kogan et al., 1996) 0-13(F,
0 O) -- 20-~3 F
?2 _+_ 1 + Arc 7/" L 72 ~/72 __ 1
'
6s4 A4i c44 , ?2 ~,/72 _ 1
+
tan
V/? 2 -
1]
? - r / a > 1.
(7.34)
Defining the Mode II and Mode III stress intensity factors as Kll = lim V/27r(r
r-+a
-
a)0-rz,
Kill -- lim V/2zr(r -- a)0-zO,
r--~a
(7.35)
we have
KII--(40-13 ~ Km=
a
Im[A416s4c44~/-~)cosO,
-Im[Aa]6sacan~/na sin 0.
(7.36)
The results show that the electric displacement has no singularity near the crack tip. In other words, the electric loading D ~ at infinity does not contribute to the
Fracture of Piezoelectric Ceramics
231
singularity at the crack tip. Wang (1992a) and Chen and Shioya (2000) obtained the same result for electrically impermeable cracks.
C. AN IMPERMEABLE PLANAR CRACK OF ARBITRARY SHAPE UNDER ARBITRARY LOADING
Consider a planar crack of arbitrary shape in the isotropic (Xl, X2) plane. The upper and lower surfaces of the crack are denoted by s + and s-, respectively. The prescribed tractions, Pl, P2, and P3, along the respective x l, x2, and x3 axes and the prescribed electric displacement boundary value nr(x, y) are all equal but opposite in sign on the upper and lower faces of the crack; that is, PI(X1, x2, 0 +) = -PI(X1, x2, 0 - ) -- Pl(x, y), P2(x1, x2, 0 +) -- - P 2 ( x I , x2, 0 - ) = P2(x1, x2),
(7.37)
e3(xl, x2, 0 +) = - P 3 ( x I , x2, 0 - ) = P3(xI, x2), U)'(X1, X2, 0 +) -- --UY(XI, X2, 0 - ) = UT"(X1,X2).
The boundary integral method can be used to solve this type of crack problems (Cruse, 1996). Zhao et al. (1997a) established the Green functions for the loadings of concentrated electric potential discontinuity and concentrated displacement discontinuity in a three-dimensional piezoelectric medium. Using the fundamental solutions, Zhao et al. (1997b) derived the boundary integral equations for electrically impermeable cracks of arbitrary shapes f
+
[tz~l Ilu311 + tzl IlOll]~ds(~, 1 r/) = P3(xl, x2),
f+ [Lzz211u311 + Lz2llcblll-~ds(~, 1 s+{[Lzr3
O) = nr(xl, x2),
(7.38a) (7.38b)
cos 2 0 -k- Lz03 sin 2 0]llul II
+ [Lzr 3 -
s+{[Lzr3 -
Lz03]
Lzo3]
1
sin0 cosOllu2ll}-~ds(~, rl) - Pl(Xl, x2),
(7.38c)
sin0 cos011Ul II 1
-+- [Lzr 3 cos 2 0 -k- Lzo3 sin 20]llu211}-~ds(~, O) - P2(xl, x2),
(7.38d)
232
Tong-Yi Z h a n g et al.
where r 2 --- ( X l - - ~ ) 2 ..]_ (X2 -- 0) 2, the symbol II II denotes a jump of the quantity across the crack, and the material constants are given by 3
3
Lzzl = ~ - ~ a i m i , i=1
1
Lzr3
-
-
-
-
2
1
Lz~
~
~
1
3 i=1
3
H123 i=1
3
Lzz2--Zaiti,
i=1
- - ~H123
~
3
Lzl--Zbimi,
Lz2 -- ~
i=1
biti,
i=1
si Hi(c44 + c44kli -+-el5k2i) -+- c44s4
(7.39)
,
Si Hi(c44 + c44kli -1- el5k2i) - 2c44s4 ,
in which
ai --
--c13 +
(c33kli -~- e33k2i)s 2,
bi -
- e 3 1 --t- (e33kli
m l - - [k12k23 - k22k13 + k23 -
k22]/(47rSl A),
m2 - - [k13k21 - k l l k 2 3 -k- k21 -
kz3]/(47rszA),
m3 - - [ k l l k 2 2 - k12k21 d- k22 -
k21]/(47rs3A),
t2 - -
[el
~(k21kl3 c44
- k l l k 2 3 ) + kl3 - k l l
]/
- K33k2i)s2, (7.40a)
(47rs2A),
(7.40b)
t3-- Iel---~5(kllk22-k12k21)-t-kll -k12]/(47r,3/~)'c44 A - - k12k23 - kl3k22 + kl3k21 - k l l k 2 3 + k l l k 2 2 - kl2k21, H1 -
1,
H2
-- (a3bl -
H3 - - ( a l b 2 - a 2 b l ) / ( a 2 b 3
alb3)/(a2b3 -a3b2),
a3b2),
(7.40c)
H123 - - H1 d- H2 d- H3.
Equations (7.38a-d) are hypersingular boundary integral equations (HSBIE). Without loss of generality, the coordinate system can be chosen such that its origin is at the crack tip, and the x~ and x2 axes are along the normal and the tangential directions of the crack front, respectively. We define the intensity factors at the crack tip as
KI -- lim
xl-~0
Km -
lim
~/27rXlO'33(Xl, X2, 0),
xl--~0
~/27gXlO'32(Xl, X2, 0),
KII --
lim x/2rrXlCr31(Xl, X2, 0),
xl--+0
KD -- lim ~/27rxl D3(Xl, X2, 0). xl--+0
(7.41)
233
Fracture of Piezoelectric Ceramics
Then the intensity factors can be expressed in terms of the jumps of the displacement and the electric potential across the crack faces K o = - J r x,--.0 lim ~2~/xl [tzl Ilu311 + L~2114~11], KI--Jr
lim 2r x,-~o
[tzzlllu311 + tzz2114~ll],
(7.42) Kit -- -7r lim x l ---~o Kill-
--Tr
lim
xl~O
2~xx~[~L , r 3
+ 31 -~ Lz03
Lzr3 + -s
1 Ilulll,
Ilu211.
We can solve Eqs. (7.38a-d) to obtain the jumps of electric potential and displacements across the crack faces. Solutions for HSBIE problems for conventional materials are available (Ioakomidis, 1982a, 1982b; Tang and Qin, 1993). By analogy, we have the solutions of Eqs. (7.38a-d) for the piezoelectric solids, and then we can determine the intensity factors.
1. Symmetric Problems
Under symmetric loading, P1 = 0 and P2 = 0, the boundary integral equations involve only Eqs. (7.38a, b). Correspondingly, the HSBIE solution takes the form I1~11 4Jr(1# - v) fs+ ---~-ds(~, 11) - -t3(Xl, x2),
(7.43)
for isotropic elasticity, where II~ II is the jump in the normal displacement across the crack surfaces; # and v are, respectively, the shear modulus and Poisson's ratio; and t3 is the prescribed normal traction. One can show that the stress intensity factor is (Tang and Qin, 1993) # lim ~ 27r II~ II K ~ ( t 3 ) - 4(1 - v)x,--,0 x---7 "
(7.44)
The stress intensity factor is related to the loading through the displacement jump of Eq. (7.43). If the equations for piezoelectric solids have the similar form to
234
Tong-Yi Zhang et al.
that given by Eqs. (7.43) and (7.44), we are able to solve the piezoelectric HSBIE problems. To do so, we linearly combine Eqs. (7.38a, b) to yield f + [Lzlllu311 + Lzzllc~ll]-~ds(~, 1 17) = GE M P3(Xl,X2) -] GEEuff(XI,X2),
f+ [Lzzl Ilu311 + Lzz21lckll]-~ds(~, 1 rl) -
(7.45a)
G MMP3(X1, x2) + GME~o(XI, X2),
(7.45b) where GEM
_
L z 2 ( L z l
--
Lzz2)/AL,
2 GMM ._ ( L z z l L z 2 - Lzz2)/AL,
G EE
--
G Me-
(L zz I L z 2 - L2z l ) / A L , Lzzl(Lzz2- Lzl)/AL,
(7.46)
AL -- L z z l L z 2 - LzzzLzl.
It has been proved (Chen and Shioya, 1999; Zhao et al., 1997b) that G eE - G MM = 1,
G TM -- G ME -- O.
(7.47)
Equation (7.47) indicates that the electric and mechanical fields are decoupled in Eq. (7.45). Comparing Eqs. (7.42) and (7.45) with Eqs. (7.44) and (7.43), we can write the stress and the electric displacement intensity factors directly in the form KI -- KM(p3),
Ko-
Kzm(~r).
(7.48)
Equation (7.48) indicates that the stress and electric displacement intensity factors can be obtained from the stress intensity factor of the corresponding elastic problem by simply replacing the load t3 by P3 or ~ , provided that the corresponding solutions for the isotropic elastic materials are available. To demonstrate how to implement this method, we present the solution for a penny-shaped crack of radius a centered at the origin of the coordinate system. The crack is subjected to two point forces in opposite directions and at the same magnitude P at (rp, Op, 0 +) and two point charges Q at (ro, Oa, 0 +) on the upper and lower surfaces of the crack. From the stress intensity factor of the corresponding elastic problem (Cherepanov, 1974), we find the intensity factors at point (a, 0, 0) to be P
KI=
KD=--
(a 2 - r2) 1/2
7r ~'-~--d a 2 + r 2 - 2arp cos(O- Oe) Q (a 2 - r~)1/2 Jr ~ / ~
(7.49)
a 2 + r2a -- 2arQ cos(0 -- Oa) "
Using an extended potential function method (Fabrikant, 1989; Ding, Chen, and
Fracture of Piezoelectric Ceramics
235
Liang, 1997) for piezoelectricity, Chen and Shioya (1999) obtained the same results as those presented in Eq. (7.49). 2. Asymmetric Problems
In this case, P3 = ~ - 0. Introducing two new parameters Y and 8 to replace Lzr 3 and Lzo 3 through the relationship Y 8zr(1 - 8 )
1 - 28 Y 8~(1 - 8 2)
-- -Lzr3,
-- -Lz03,
(7.50)
we can rewrite the boundary integral equation (7.38b, c) as rl) f~+ {[(1 - 26) + 38cos 2 0]llu~ II + 36 sin0 cosOIlu21l}-~ds(~, 1
87r(1 --8 2)
Pl(Xl, x2),
(7.5 la)
rl) f + {38 sin0 cos011Ul II + [(1 - 28)+ 38cos 20]llu211l-~ds(~, 1
8zr(1 --8 2)
Pz(xl, x2). (7.5 lb) Y Equations (7.5 la, b) are in the same form as those of the HSBIEs for purely elastic problems. It can be shown that the stress intensity factors are = -
Y
.
r
KII(P1, P:) = 8(1 -- 82) J,l~m0 Z ]]ul ]l, i
KIll(P1, P:) =
r
(7.52)
Y
lim ./~2zr IIu 2 II. 8(1 + 8) x,--,o V xl
Solutions to Eqs. (7.51) and (7.52) can also be obtained easily from the corresponding elastic solutions (Tang and Qin, 1993) by replacing 2#(1 + v) and v with Y and 6, respectively. In summary, the solutions for impermeable planar cracks located in the isotropic plane of an infinite transversely isotropic piezoelectric medium can be obtained directly from the corresponding isotropic elasticity solutions. Analytic solutions have been established for the elliptical (Zhao et al., 1997b) and penny-shaped cracks (Chen and Shioya, 2000). Zhao et al. (1999a, b) also studied a pennyshaped crack in a half-space. Numeric methods such as the boundary element method or the finite element method are generally required for more complicated crack problems (Chen and Lin, 1993, 1995; Kumar and Singh, 1997a, b; Fang et al., 1998).
236
Tong-I'7 Z h a n g
et al.
D. A PERMEABLE PLANAR CRACK OF ARBITRARY SHAPE UNDER ARBITRARY LOADING
For a permeable planar crack without flee-charge loading on the crack faces, the electric boundary conditions on the crack faces are given by Eqs. (3.73) and (3.74). We repeat them as follows for clarity DJ- -- D 3,
4~+ -- 4~-.
(7.53)
In this case, if the applied mechanical loads on the crack faces are in the form of the first three equations of Eq. (7.37), the boundary integral Eqs. (7.38a, b) become
fs + [Lzzl fs +
Ilu3 II]~ds(~, 7) - P3(Xl, x2), 1
(7.54a)
-O3(x1, x2),
(7.54b)
[ g z z z llu311]-~ds(es , rl) -
whereas Eqs. (7.38c, d) remain the same. Then, the stress and electric displacement intensity factors are given by K I -- Kff(P3),
Lzz2
KD = L-~-z~K I .
(7.55)
In this case, it can be proved that the quantity 11)5 in Eq. (7.21a) equals to ~ I 5 -- - L zz_.___~2.
Lzzl
(7.56)
Equation (7.55) indicates that for electrically permeable cracks, the electric displacement intensity factor is proportional to the Mode I stress intensity factor due to the piezoelectric effect.
VIII. Nonlinear Approaches The linear electroelastic analysis of electrically insulating or conductive cavities and cracks is the first and most fundamental step toward understanding the fracture behavior of piezoelectric ceramics. As mentioned in Section I and shown in Fig. 2, both the polarization and the strain versus the electric field strength are nonlinear. Experimental observations, which are described in Section IX, show that the fracture behavior of piezoelectric ceramics is also nonlinear. Various models have been proposed to explain the nonlinear fracture behavior of piezoelectric ceramics. In this section, we briefly introduce four types of nonlinear approaches--namely,
Fracture of Piezoelectric Ceramics
237
the electrostriction, domain switching, domain wall kinetics, and polarization saturation at a crack tip.
A. ELECTROSTRICTION
If we apply an electric field to an electrically insulating material (dielectric material), we change its shape; however the strain remains unchanged if we reverse the direction of the applied electric field. This effect is called electrostriction, and the strain is proportional to the square of the applied electric field. The electrostrictive effect is induced by the Maxwell stress (Landau and Lifschitz, 1960), which is a tensor whose divergence equals the body force induced by the electric field in a dielectric material. Considering the Maxwell stress, Smith and Warren (1966, 1968) studied an elliptical inclusion in an infinite dielectric material, where the dielectric material has zero piezoelectric constants and its mechanical and electrical properties are isotropic. When the inclusion is a conductive crack, they found that the elastic stress field at the crack tip has a 1/r singularity, where r is the distance from the crack tip. McMeeking (1987) reanalyzed the problem of a dielectric solid containing a conductive elliptical flaw. He distinguished the material stress, which represents the force transmitted per unit area between neighboring elements of material, from the Maxwell stress field. At electric conductors adjacent to the boundary surface of the dielectric material, the normal components of the Maxwell stress cause the tractions, which induces electrostriction and material stresses (McMeeking, 1987). McMeeking (1987) believed that material stresses cause deformation, yielding, and cracking of the material. His results show that the material stresses have a l / x / 7 singularity at the crack tip. Due to the electrostriction, a remote electric field E ~ acting on a conductive crack with a length 2a located at ( - a , a) on the xl-axis generates a Mode I stress intensity factor, Kt ,~ v / [ x # : r a / ( l -
(8.1)
where x, #, and v are, respectively, the dielectric constant, shear modulus, and Poisson's ratio of the dielectric material. Using the relationship between the stress intensity factor and the energy release rate for plane strain, J -- K~(1 - v)/2#, and Eq. (8.1), we have 1 J - -~xrca(EC~) 2-
1 F~'K O ' . -~K
(8.2)
Equation (8.2) is a special case of Eq. (5.27), which indicates that in terms of the energy release rate, McMeeking's approach is equivalent to the linear approach introduced in Section V.
238
Tong-Yi Zhang et al.
Considering the electrostriction effect in an isotropic paraelectric material, Suo (1991) introduced the isothermal mechanical enthalpy per unit volume, which took the following form
w(D,
(Y ) = -- ~
O'j i O'ij -- ~1O+' m m PfYnn
-glammDnDn] -t-
--
0[(1 + 0)O'ji D i D j (8.3)
1 ~f ( A ) d A ,
where ~) is the electrostrictive coefficient, 0 is a dimensionless number, and D -(Dn Dn) 1/2. The function f gives the nonlinear dielectric response in the absence of stress, (8.4)
E = f(D).
The relationship between E and D may be simplified as three piecewise linear functions (Hao et al., 1996)
D-
xE, ]E] < Es, Ds, E > Es, -Ds, E __Kc,
(8.16)
the crack grows. Equation (8.16) is the commonly used failure criterion in the domain-switching model. Using the domain-switching model, Yang and Zhu (1998b) and Zhu and Yang (1999) provided a mechanistic explanation for the electric field-induced fatigue crack growth. They considered an electrically impermeable crack, which induced electric singularity at the crack tip. The high electric field drives domains to switch and, consequently, the domain switching generates an internal stress field. A cyclic electric load causes a domain switching sequence that generates a cyclic internal stress field, which mechanically fatigues ferroelectric ceramics. In this case, the
242
Tong-Yi Zhang et al.
term f2 in Eq. (8.13) is induced completely by the applied electric field, and the crack is driven solely by AK. Similarly, microcrack nucleation may also occur during repeated domain switching, which degrades the electric properties of ferroelectric ceramics (Jiang et al., 1994a, 1994b; Zhang and Jiang, 1995). In general, a domain-switching model requires the development of new constitutive equations and extensive numeric calculations. Recently, Huber et al. (1999) developed a nonlinear constitutive model for ferroelectric polycrystals under a combination of mechanical stress and electric field. They used a self-consistent analysis, as an extension of the self-consistent crystal plasticity scheme of Hill (1965a, 1965b) and Hutchinson (1970), to address ferroelectric switching and estimate the macroscopic response of tetragonal crystals under a variety of loading paths. In a qualitative way, this model captures several observed features of ferroelectrics, such as the shapes of the dielectric hysteresis and the butterfly loops, a Bauschinger effect under mechanical and electrical loads, and the depolarization of a polycrystal by compressive stresses (Huber et al., 1999).
C. DOMAIN WALL KINETICS MODEL
Polarization domain switching may be regarded as a result of domain wall motion. Studying domain wall kinetics (DWK) can bring insight to the mechanisms of failure processes in piezoelectric ceramics. In addition, the use of the DWK model can greatly simplify the mathematics involved in domain switching modeling. Under applied electric and mechanical fields, domains with low energy orientation grow and domains with high-energy orientation shrink as a result of domain wall motion. The displacement of 90 ~ domain walls causes a shear deformation. Thus, both domain wall displacement and volume deformation affect the material properties of ferroelectric ceramics (Herbiet et al., 1989). Arlt and Pertsev (1991) evaluated the force constant and effective mass of 90 ~ domain walls in ferroelectric ceramics. Domain size and grain size are involved in their model, which links the microstructures to the macroproperties of ferroelectric ceramics. The domain wall motion model explains the internal friction and the dielectric dispersion of ferroelectric ceramics (Pertsev and Arlt, 1993) quite well. Fu and Zhang (2000b) proposed a DWK model to explain the effects of temperature and electric field on the bending strength of PZT-841 ceramics. Either an applied stress or an electric field can drive domain walls to move. The equation for the motion of a domain wall under the electric and mechanical fields can be written as (Arlt et al., 1987) ~,ti + ~:u -- fM + fE,
(8.17)
Fracture of Piezoelectric Ceramics
243
where y zi and ku are, respectively, friction and clamping forces; u is the displacement; and fM and f e are the configuration forces exerted on the domain wall by mechanical and electrical fields, respectively. The inertial term has been neglected in Eq. (8.17) for static loads, monotonic loading at a low loading rate, or cyclic loading with a low frequency because of the light mass of the domain wall. For a cyclic load f = foe i~ the solution to Eq. (8.17) is
e
f o ~i Wt
u =
^
k + ioJy
,
ti=
. z" .,i wt iwJOe
,,
k + icoy
.
(8.18)
The corresponding damping factor is _ wy
(8.19)
For a static load, after a period of time, At, the displacement and the average velocity of the domain wall become
u=--fo k -
2
, (8.20)
2
li---~tf~
2u ~
A----~'
where 2 - 1 - e x p ( - A t / f ) , f -- V/~:, and u ~ is the equilibrium displacement. It takes only several f for a domain wall to complete its movement from its initial position to its final position under a static load. When the time period is larger than a few times f, 2 ~ 1 and u ,~ u ~.
1. Temperature Dependence of Compliance We shall consider the motion of a 90 ~ domain wall. Because a 90 ~ domain wall is actually a twin boundary, a local shear deformation, etocat = esu/d, is associated with the domain wall displacement u (Arlt, 1990), where d is the distance between domain walls and es is the spontaneous strain. This local deformation is accommodated by the neighboring medium, resulting in a global deformation UCX~
eD -- ~es d '
(8.21)
where/~ is a proportionality factor linking the local deformation to the global strain. By definition, eD -- s Dcr, where SD is the contribution of the domain wall displacement to the compliance. We then have uCX~
~es --~ -- SDCr.
(8.22)
244
Tong-Yi Zhang et al.
Equation (8.22) links the domain wall displacement to the macroscopically mechanical behavior. Under a static mechanical load, combining Eqs. (8.20) and (8.22) leads to the force exerted on a domain wall by the applied load o fM-
~:dso
~es a.
(8.23)
If we know how u and d change with temperature, we can predict the temperature dependence of so from Eq. (8.22). There is an alternative way to determine so. Letting u = ~ao, where ao denotes the representative lattice constant and ~ is a dimensionless factor, we rewrite Eq. (8.22) as ~ e s a o / d - socrc with ac - o/~.
(8.24)
The variations of the lattice constant ao and the domain spacing d with temperature are about the same, such that the ratio of ao/d is independent of temperature. Equation (8.24) indicates that if the applied stress, as a function of temperature, required the domain wall to move the distance ao is available, we can determine the temperature dependence of so. It is widely accepted that lattice exhibits a frictional resistance, known as the Peierls-Nabarro stress, against dislocation motion (Peierls, 1940; Nabarro, 1947). Thus, we assume that the lattice also exerts a friction force against the domain wall motion, resulting in the friction term ?,ti in Eq. (8.17). When the applied load is small and the domain wall displacement u ~, is less than aomthat is, ~ < l m t h e domain wall moves elastically around its equilibrium position. As the applied load reaches the critical value at which u ~ = ao, or ~ -- 1, the domain wall begins to sweep over the lattice sites and induces damping. Comparing Eq. (8.24) with Eq. (8.22), we conclude that Crc defined in Eq. (8.24) is simply the critical stress for triggering damping. It should be possible to determine Crc from the damping behavior of the material. As observed in the experiment on PZT-841 ceramics (Fu and Zhang, 2000b), for an applied cyclic stress, there is a threshold temperature Ti at which the damping factor starts from zero and increases rapidly with temperature. The three threshold temperatures are around 41 ~ 91 ~ and 139 ~ C, respectively, for the peak stresses of 30, 15, and 7.5 MPa. This result implies that for a given temperature, there is a critical level of applied cyclic stress at which damping rises. The plot of the log of the threshold stresses against the corresponding temperatures shows a linear relationship ~c = ~o e x p ( - T/ To),
(8.25)
where ~o = 53.8 MPa and To = 70.7 ~ C are parameters determined by the linear
245
Fracture of Piezoelectric Ceramics
regression. Equation (8.26) gives the threshold stress for triggering damping as a function of T. Combining Eqs. (8.24) and (8.25) yields (8.26)
SD(T) = 6~es(T) e x p ( T / To),
where c~ = ~ao/(o'od). The domain wall motion and the volume deformation inside the domains contribute to the total elastic compliance--that is, (8.27)
s(T) = sv + SD = Sv + 6tes(T)exp(T/To),
where sv represents the contribution of the volume deformation to the compliance. Furthermore, we assume s v to be independent of temperature within the temperature range of this study. Fu and Zhang (2000b) measured the spontaneous strain as a function of temperature for PZT-841 ceramics and found es(T)=0.062
1-~c c
+ 0.170 1 - Tc
-0.142
1-
- 0.077
1-
,
(8.28)
where Tc(~272 ~ C) is the Curie temperature. Fu and Zhang (2000b) measured the total elastic compliance directly and found that the measured data fit Eq. (8.27) perfectly (Fig. 8a). Equations (8.27)-(8.28) predict the existence of a maximum compliance, which occurs at Tm= 227 ~ C. The directly measured maximums of the elastic compliance for the three applied stress levels all occur at about 225 ~ C, in good agreement with the prediction.
2. Effect of Temperature on Bending Strength
The fracture of ceramics with well-polished surfaces can be considered to be a crack nucleation-control process. This is because ceramics are very brittle, and their fracture is triggered once a crack is initiated; microcracks may be initiated at pores or grain boundaries. Rice and Freiman (1981) defined the criterion for the crack initiation at grain boundaries as ( I f + o.in - -
v/9Y?'8/g - o.8,
(8.29)
where o.f is the critical applied stress, o.in is the internal tensile stress perpendicular to the plane of the initiated microcrack, Y is the Young's modulus, ?'8 is the grain boundary fracture energy, g is the grain size, and o8 is the corresponding grain boundary strength. Equation (8.29) shows that internal stress plays an important role in initiating microcracks. Moving a domain wall embedded in a grain produces
Tong-Yi Zhang et al.
246
18 .~
17
0
0
16 O n
~ 14 O
r,.) 13
DWK model
12 50
100
(a)
150
200
Temperature
250 Tc
(~
100
~
9o
m
8o
~
7o
DWK model .
.
.
.
|
.
.
.
.
50
(b)
i
.
.
100
.
.
|
.
.
.
.
150
i
.
.
200
.
.
i
.
250
.
.
.
300
Temperature (~
FIG. 8. Temperature dependence of (a) the bending strength and (b) the elastic compliance for PZT-841. The solid curves are based on the domain wall kinetics (DWK) model. a serration at the grain boundary. The representative local tensile strain in the serration is given by (Arlt, 1990) e-
Ss u ~ d "
(8.30)
This corresponds to an internal stress of 8s u ~ O'in
svd '
(8.31)
Fracture of Piezoelectric Ceramics
247
where the volume elastic compliance, s v, is used because the serration is simply a localized volume deformation. Combining Eqs. (8.22) and (8.31), we have the following relationship trin = ~ S D t r ,
(8.32)
where X - 1/(~sv). Equation (8.32) indicates that the internal stress due to 90 ~ domain wall motion is proportional to the applied stress. Combining Eqs. (8.29) and (8.32) at the critical condition of tr = try yields
t r f ~"
trB 1 + Xso'
(8.33)
which relates the bending strength to the compliance contributed by 90 ~ domain wall motion. Equation (8.33) clearly shows that the minimum bending strength corresponds exactly to the maximum elastic compliance, indicating that the anomalous minimum of the fracture strength is a result of elastic softening due to domain wall motion. Substituting Eq. (8.26) into Eq. (8.33) yields tr8 try -- 1 + ~es(T)exp(T/To)'
(8.34)
where ~ - 6 t / ( ~ S v ) = ao/(dsvtro). Thus, the bending strength is a function of temperature. Within the temperature range of 30-272 ~ C, trB may be assumed to be independent of temperature. Thus, the bending strength varies with temperature through the term of es(T) exp(T/To). Using Eq. (8.34), Fu and Zhang (2000b) fit experimental data of bending strength for various temperatures by the least square procedure (Fig. 8b), yielding tr8 = 99.3 MPa and t? = 1.94. They estimated the values of sv and & for the three applied stress levels with the parameters/~ and d from the fitting results. Using tro = 80.7 x 106 N / m 2, t~ = 1.94, t~ -~ 23 x 10 -12 m2/N, and sv .~ 13 x 10 -12 mZ/N, together with the relations t~ = ~ao/(dtro) and t~ 6t/(~sv), Fu and Zhang (2000b) obtained/~ ~ 0.9 and ao -~ 2 • 10-3d. Taking ao ~ 4 • 10 -4 /zm gives d ~ 0.2 #m, which is in approximate agreement with the results evaluated from Fig. 1b.
3. Effect of an Electric Field on Bending Strength We determine the influence of an applied electric field on the bending strength. Miller and Savage (1959) demonstrated that the velocity of the 180 ~ domain wall
248
Tong-Yi Zhang et al.
motion in BaTiO3 single crystals depends on the electric field and is given by the empirical equation {~ = Vo exp(1 - Eo/IEI),
(8.35)
where Eo is the activation field strength and Vo is the domain wall velocity under Eo. We assume that Eq. (8.35) also holds for 90 ~ domain walls. We may express Vo by V o - Uo2/ZXt, where Uo is the equilibrium displacement of a domain wall under the activation field. If only an electric field is applied, combining Eqs. (8.20) and (8.35) gives f e -- kuo exp(1 - Eo/IEI),
(8.36)
which is the force exerted on a domain wall by an applied electric field E. Substituting Eq. (8.24) and Eq. (8.36) into Eq. (8.20) with f/ = fM + fE, and )? ~, 1 and u ~ u ~, we have u ~ - SDdo-/(fles) + Uo exp(1 - Eo/IEI).
(8.37)
Substituting Eq. (8.37) into Eq. (8.31) leads to O-in -- S D O - / ( ~ S v )
"[- O'E,, exp(1 - Eo/IEI),
(8.38)
where O-eo -- esUo/(Svd) is the equivalent internal stress induced by the activation field of Eo. Substituting Eq. (8.38) into Eq. (8.29) yields O-f =
O-8 - O-e,,exp(1 - Eo/IEI)
1 +)~SD
.
(8.39)
Equation (8.39) gives the fracture strength as a function of the applied electric field. For simplicity, we assume )~so to be independent of the applied electric field. Taking O-8 = 99.3 MPa, Fu and Zhang (2000b) fit the experimental data (see Fig. 17) and found that O-eo - 35.7 MPa and Eo = 14.2 kV/cm for the positive fields, and O-eo = 22.8 MPa and Eo ---=14.0 kV/cm for the negative fields. It is interesting to note that the activation field under positive electric loading is almost the same as that under negative electric loading. Equations (8.27) and (8.39) give the elastic compliance and the bending strength as a function of temperature and electric field. Equation (8.39) shows that the anomalous minimum of the fracture strength is directly related to the inelastic relaxation of the 90 ~ domain wall motion. In this model, the internal stress induced by the domain wall motion is the dominant mechanism causing the strength degradation observed in the experiments. An internal stress field develops when
Fracture of Piezoelectric Ceramics
249
a 90 ~ domain wall is moved by mechanical and/or electrical loads. This internal stress field, in turn, assists the applied load/electric field to fracture the sample. Because the internal stress field varies with temperature and electric field, the bending strength also depends on temperature and electric field. Using Eqs. (8.27) and (8.39), one can estimate the parameters used in the model by regression of the experimental data. The parameters obtained by Fu and Zhang (2000b) given in the previous paragraph seem to be reasonable. The DWK model is essentially the same as the domain-switching model. Both models consider the internal stress field induced by applied mechanical and electrical loads. Depending on its nature, the internal stress field may assist or resist the applied loads to fracture the samples. For bending smooth samples, the failure initiates at locations of high total tensile stress. Grain boundaries, domain boundaries, and other defects are the potential locations for the initiation of cracks.
D. POLARIZATION SATURATION MODEL Gao et al. (1997) proposed a strip polarization saturation model to examine the electrical yielding effect on the fracture behavior of electrically insulating cracks in piezoelectric ceramics under combined electrical and mechanical loading. In the strip polarization saturation model, piezoelectric ceramics are treated as mechanically brittle and electrically ductile materials. This saturation model is analogous to the classic Dugdale model (1960). The crack tip is completely shielded electrically by a polarization saturation zone in the saturation model. As a result, the local energy release rate is only mechanical in nature (Gao et al., 1997). Applying the Griffith theory yields the fracture criterion, jl > j1_
2y,
(8.40)
where J is the J integral, the superscript l denotes local, and y is the specific surface energy. To emphasize the physical insight, Gao et al. (1997) considered a simplified piezoelectric material to make the derivation process straightforward. Subsequently, electrically impermeable and conductive cracks with the constitutive equations given by Eq. (2.41) have been investigated (Gao and Barnett, 1996; Fulton and Gao, 1997, 1999; Ru, 1999; Ru and Mao, 1999; Mao et al., 2000; Wang, 2000). We briefly discuss the analysis of an electrically impermeable crack. Figure 9 shows a semi-infinite crack lying on the negative xl axis in an infinite piezoelectric medium. The applied stresses and electric displacements are expressed in terms
Tong-Yi Zhang et al.
250
a (~) Km
t
--~ K,]
1' K~~
X2
D2=Ds c
-x, I
Saturation s t r ~
FIG. 9. Schematic depiction of the strip saturation model for a semi-infinite crack.
of the intensity factors K a - (K']I K'] K']II K~) r. For the electrically impermeable crack, the extended traction vanishes on the crack facesmthat is, 2~2 - - O,
for xl < O.
(8.41)
In the strip electric saturation zone 0 < x~ < c, the boundary conditions are D2 -- D s ,
(8.42)
f o r 0 ~ Xl ~ c ,
where Ds is the saturation electric displacement. The Green function for an electrically impermeable crack of a finite length 2a is given by Eqs. (4.42a-b). We rewrite the solution in the form below 1 fl
V/Z2
=
(
Z~ + v/Z2~ - a 2
a 2 Z~+v/Z 2 - a 2 - z ~a - V/ (z~) 2
1 Lik 1Lkj I _ k=l ./Z 2 __ a 2 V"
__
a 2
)>tq+ Z, li[l,
4
Zij,1 = Z
X
(
02
)
m
a2-
(Zi + v / z 2 - a 2 ) ( z J
+ v/(-Z~j)2-a 2)
(8.43a)
f,2 --
(( z2 a2
4
Fracture of Piezoelectric Ceramics
z + z2a2/
z~ + , / z 2 . 02.
~
a2
q + Z,2(I,
Pi
Zij,2 -- Y~ L~ 1Lkj /
.,
(8.43b)
( X
z~ . . V(z~) 2
)>
251
02
)
~
a Z - (zi + V/z~-aZ)(z~ + v/(-~j) 2
m
a 2)
where i, j = 1, 2, 3, 4, and i and j are not summed; and the vector q is given by Eq. (4.16) for a general problem. For the current problem, the extended force, F = 0 , and the extended Burgers vector, b = (0 0 0 A4~)r, we have
q _ ~/A~b(g41
L42 L43 L44) r -
2rriA4~L4r
(8.44)
and L4i, i = l, 2, 3, 4, are the components of the eigenvector matrix L defined by Eq. (3.16). Consequently, we shift the origin of the coordinate system to the right crack tip and then let the crack length approach infinity. The solution for the interaction of a dislocation with a semi-infinite impermeable crack becomes
(
,
>
r~2 - Lfl + E l l - L 2~/g2-~(V~-2~_v/~d) q + L Z l r t -+-L(
Zij,1 -- - Z
4
k=l
1
> (t + LZ~q,
1
L~l Lkj
2v~i(V/~_7 + c-~_~/zj)'
~[~1 -- - - E l 2 - E l 2 -- - L
2~~(~//~-
~d)
~z/~d) ( t - LZzq,
4 Zij,2 -- - Z k=,
(8.45a)
L~I
Lkj
Pi
2v/-~7(v/~-7 + ~-v/z~)'
where i, j = 1, 2, 3, 4 and i and j are not summed.
q -- LZ'21~
(8.45b)
Tong-Yi Zhang et al.
252
If the electric dislocation is located on the xz axis, the electric displacement along the Xl axis calculated from Eq. (8.45a) has the compact form
D2 = ~f~(x, -- x~) (L4q + L4q).
(8.46)
Substituting Eq. (8.44) into Eq. (8.46) yields
~l a A~b (L4L4r _ L4L4r) D2 - ~ ( X l - xla) 2:ri
(8.47)
From Eq. (3.15), we have L4 LT - - L 4 L T. Hence, Eq. (8.47) can be further reduced to D2
--
~la A~bL4LT. ~ - 7 ( X l - Xla) 7ri
(8.48)
The intensity factors produced by the electric dislocation located anywhere are defined as
(K d
K]
Kdl
KdD)T = lim(L(v/2n'z,,)fl+L(v/2n'~--d}~,).
(8.49)
Zu---*0
Substituting Eq. (8.45a) into Eq. (8.49) yields
(K d K d KdI KdD)T = --ff~-~ L ~ d
q-t-L
.
(8.50)
If the dislocation is located on the Xl axis, Eq. (8.50) reduces to
Kd KdI Kd)T = (Kd Kd KdI Kd)T = (K d
a [Lq + L(t].
(8.51)
Substituting Eq. (8.44) into Eq. (8.51) leads to
Kai Kffll Kao)r _ i A4) [LL4r _ LLr] _ i
A ,/5 LL4r.
(8.52)
As given by Eq. (3.93) and discussed in Section III, the mechanical and electrical fields near a crack tip can be expressed in terms of the intensity factor vector. For
Fracture of Piezoelectric Ceramics
253
an electrically impermeable crack, Eq. (3.93) can be simplified as
E2 = 5
L v/2zrz,~ I'-I + 1" v/2ZrT~
(8.53)
Thus, the electric displacement on the xl axis under the applied loads is dependent only on the applied intensity factor of electric displacement and is given by
(8.54)
D2 -- ~ .
~/2zrxl
Let f(x' l) be the distribution function of the electric dislocations such that the number of dislocations located in the interval dx'1 at x'~ is f(x' 1)dx'1. The boundary condition in which the electric displacement D2 equals the saturation value Ds in the polarization zone is described by
K---a--P~~ - f 0 c ~/27rXl ~- B
J
f(xl)~/~l l / dx S(x,
-
x'l)
'1 --
Ds
(8.55)
,
where B -- AqOL4L~/(7ri) is a constant related to the Burgers vector. The uniqueness of a distribution f(x' l ) with zero value at x'l = 0 and c requires that
Kao -- 2 V/-2Ds v/-~/ ~/-~.
(8.56)
Thus, the solution to Eq. (8.55) is
f (x l) -- - ~
In
l
(8.57)
"
The electrical dislocations in the saturation zone produce stress intensity factors and intensity factor of electric displacement at the crack tip, which are given by
(K~
K]
K~t
T fo c f(x'l)dx'= K d o ) T - - i xAq~X~LL4 /~ ~1
LL~ L4L].K~.
(8.58)
The vector of the local intensity factors is the sum of the intensity factor vectors induced by the applied field and the dislocations--that is, K t = K a + K d.
(8.59)
254
Tong-Yi Zhang et al.
Combining Eq. (8.58) with Eq. (8.59) yields LlL4r ( K~t - ~rL4L K ~4
g~
K~III
Klall
L2L4v
a
L4LTKD
(8.60)
L3L4T a L4L T KD 0
2
where Lj is the jth row of L. Equation (8.60) indicates that the electric displacement at the crack tip is completely shielded by the saturation zone. Ru (1999) and Wang (2000) obtained results equivalent to Eq. (8.60) from a different approach. In their results, the ratio L/L4r/(L4L4T) is replaced by a ratio [(B + 1~)-1.]i4/[(B-+- ]~)-1144, i = 1, 2, 3. Using the relation given by Eq. (3.15), we can prove that the two ratios are the same. As stated in Eq. (3.109), the local energy release rate can be expressed in terms of the intensity factors as jl
=
(Kt) r (B + 4
I~)KI.
(8.61)
Substituting Eq. (8.60) into Eq. (8.61) yields
jl
:
K~
-
L1L4r a L4L----~K D
a
g l
L2L~ L4L~ K~9
L3L~ ) K~I -- L4L----~K~9
L4LTK~9 m
(B1 + B1)
g~
L2L4T a
L4LTKD
(8.62)
L3L4y K a
k g ~ l l --
t4t-----~ Dj
where Bl is the 3 x 3 upper left block in B, as introduced in Eq. (3.22). When the local energy release rate is adopted as a failure criterion, it yields a linear relationship between the applied mechanical load and the electric field (Gao and Barnett, 1996). In this case, one may also use the local intensity factor as a failure criterion, which predicts the linear relationship between the applied load and electric field as well (Wang, 2000).
Fracture of Piezoelectric Ceramics
255
IX. Experimental Observations and Failure Criteria In the literature, there are a number of experimental studies concerning the fracture behavior of piezoelectric ceramics under purely mechanical loading. However, experimental reports on the subject of combined electrical and mechanical loading are limited. In this section, we provide an overview of the experimental observations of the fracture behavior of piezoelectric ceramics. Along with the experimental results, we introduce failure criteria. A. EXPERIMENTAL OBSERVATIONS
1. Effects of Microstructure and Temperature In 1988, Pohanka and Smith presented an overview of the fracture and strength of piezoelectric ceramics under purely mechanical loading. Typical values of Ktc for commercially available piezoelectric and dielectric ceramics, such as barium titanate and lead zirconate titanate (PZT), range from 0.8 to 1.7 MPa 4'-~. The fracture strength depends on temperature and the grain size and composition of the material. For barium titanate, the fracture energy at room temperature is a function of grain size and varies from about 3 to 12 J~ m 2, whereas the fracture energy at 150 ~ C (above the Curie temperature of 130 ~ C) is almost independent of the grain size with a value of about 3 J / m 2. The effect of the microstructure on the enhancement of the fracture energy is attributed to twinning toughening and microcracking toughening. The maximum fracture energy occurs when the grain size is around 40 #m, at which point the contribution to toughening due to twinning and microcracking is balanced by a linkup of microcracks (Rice and Freiman, 1981). In PZT, Kzc depends also on the Zr/Ti ratio. The minimum K IC occurs at the morphotropic boundaries between phases of different crystal structures due to the reduction of microcracking toughening, where the piezoelectric coefficients are at the maximum (Freiman et al., 1986). Both the fracture strength and the fracture toughness of piezoelectric ceramics are sensitive to temperature. Cook et al. (1983) carried out controlled flaw tests on unpoled BaTiO3 ceramics of nominal grain size (7 #m). The flaws were introduced at room temperature by indentation with a load of 30 N, which produced cracks well in excess of the grain size. Then, four-point bending tests were conducted in a heated oil bath at temperatures ranging from room temperature to above the Curie point. The results show that the fracture toughness at each temperature during heating is the same as that during cooling. The fracture toughness decreases with
256
Tong-Yi Zhang et al.
increasing temperature between room temperature and the Curie point, indicating the existence of intrinsic thermal effects in the toughness parameter. Mehta and Virkar (1990) reported similar results for unpoled PZT samples, in which the fracture toughness decreased from a maximum 1.4 MPa4Fm at room temperature to about 1.0 MPav/-~ at 500 ~ C, far above the Curie point of 350 ~ C. They examined domain switching under electrical and mechanical loading using x-ray diffraction and attributed the observed temperature dependence to the toughening effect of 90 ~ domain switching. If one measures the fracture toughness at temperatures near the Curie point, one may find the minimum fracture toughness at a temperature just below the Curie point. Zhang et al. (1993) observed that the minimum bending strength and the minimum fracture toughness of both poled and unpoled PZT-4 ceramics are near the Curie point. The similar phenomenon was also observed for PBZT and PZTNV- 1 ceramics (Kramarov and Rez, 1991 ). The reasons that Cook et al. (1983) and Metha and Virkar (1990) did not find this behavior may be a result of the fact that, in their experiments, the increment in temperature near the Curie point was too large. Fu and Zhang (2000b) conducted three-point bending tests on PZT-841 ceramics at 25, 122, 220, and 268 ~ C to measure the temperature dependence of the bending strength. Twenty samples were tested at each temperature. In the tests, the poling direction was parallel with the jig surface, in which configuration the maximum tensile stress induced by bending was perpendicular to the poling direction. The bending strength exhibits valley-shaped floors at a temperature below the Curie point, as shown in Fig. 8b. The bending strength decreases from 97.8 MPa at room temperature to 85.9 MPa at 122 ~ C, and further decreases to 74.8 MPa at 220 ~ C, then increases to 90.5 MPa at 268 ~ C. Figure 8a shows that the elastic softening, which accompanies the bending strength reduction, has a peak at the same temperature, where the bending strength is at its minimum. Note that the smooth curves in Figs. 8a and Figs. 8b are plotted using Eq. (8.18) and (8.25), respectively, indicating that the experimental results were explained well by the domain wall kinetics model.
2. Effects of an Alternating Electric Field
The effects of an alternating electric field on crack initiation and growth have long been a focus of interest in the study of the reliability of piezoelectric ceramics. Four major mechanisms for electric cycling damage have been identified experimentally: aging (Robels and Arlt, 1993; Warren et al., 1996), ferroelectric fatigue (Duiker et al., 1990; Warren et al., 1995), cracking (Winzer et al., 1989; Uchino, 1997) and dielectric breakdown (Desu and Yoo, 1993; Chen et al. 1994). Here,
Fracture of Piezoelectric Ceramics
257
we focus on cracking and the effect of pores, flaws, and cracking on ferroelectric fatigue. Ferroelectric fatigue is characterized by the loss of switchable polarization with repeated polarization reversals. Although no cracking is involved in the definition of ferroelectric fatigue, pores and flaws may play important roles. McHenry and Koepke (1983) observed that in unpoled and poled Navy-III-type PZT ceramics subjected to deadweight loading in a double torsion mode, crack propagation was enhanced by the application of a static or an alternating electric field perpendicular to existing cracks. In the poled PZT, the applied electric field always "turned" the crack in the direction opposite to the poling direction. Furuta and Uchino (1993) observed crack initiation and propagation near the internal electrode tip in multilayer piezoelectric actuators during cyclic loading. Aburatani et al. (1994) studied the failure mechanisms in ceramic multiplayer actuators by three simultaneous observations: (1) visual observation with a charge coupled-device microscope; (2) fieldinduced strain measurement; and (3) acoustic emission measurement. They found that during cyclic electric loading, cracks initiate from the edges of internal electrodes and propagate obliquely to other electrodes in piezoelectric samples. Jiang and Cross (1993) showed that ferroelectric fatigue failure occurred in low-density (93-97%) lanthanum-doped PZT (PLZT) ceramics after 104 switching cycles, whereas the high-density (> 99%) PLZT specimens of the same composition did not fail after 10 9 switching cycles. For low-density ceramics, the ferroelectric fatigue rate, defined as the loss rate of switchable polarization with switching cycles, was also much higher than that for high-density ceramics. This indicates that the porosity is one of the key factors affecting ferroelectric fatigue behavior. Microcracks may be a reason for the reduction of remnant polarization (Carl, 1975; Kim and Jiang, 1996). Jiang et al. (1994b) conducted ferroelectric fatigue tests on hot-pressed finegrain PLZT. All the specimens with conventionally cleaned surfaces showed significant fatigue after 105 switching cycles, but specimens cleaned with a new cleaning procedure did not show fatigue, even after more than 108 switching cycles. This type of fatigue was found to be due to microcracking generated at the ceramicelectrode surfaces. White et al. (1994) also observed microcracks induced by an alternating electric field at the resonant frequency of the material in PZT specimens precracked by indentation. At temperatures higher than 150 ~ C, the microcracks were dispersed in small clusters, whereas at temperatures below 86 ~ C, microcracks were generated in a densely populated region near the indentation site. Jiang et al. (1994a) investigated the effect of composition and temperature on ferroelectric fatigue of PLZT ceramics. Their results show that at temperatures higher than the temperature where the dielectric constant is the maximum, no
258
Tong-Yi Zhang et al.
fatigue effect is detected. They also found that compositions of rhombohedral symmetry exhibit little or no sign of fatigue in comparison to compositions of tetragonal and orthorhombic symmetry. Compositions close to the phase boundaries display significant fatigue behavior. Electric fatigue arises from the pinning of domains or from microcracking. Hill et al. (1996) studied the effect of mechanical cycling (four-point bending) and electrical cycling on the degradation of the mechanical properties of PZT-8 bars. Microcracks were found to originate from second-phase material located at triple junctions. High intergranular microcrack density was observed in the mechanically cycled samples and in samples electrically cycled at temperatures of 80 ~ or lower. Samples electrically cycled at 180 ~ showed much lower microcrack density. Hill et al. (1996) believed that elevated temperatures (,~,180 ~ are necessary to cause depolarization in PZT-8 under resonance-induced stresses. With precracks introduced by indentation, steady crack growth was observed perpendicular to the applied field in both PZT and PLZT ceramics under alternating electric fields larger or smaller than the coercive field (Cao and Evans, 1994; Lynch et al., 1995; Zhu and Yang, 1998; Tajima et al., 2000). Cao and Evans (1994) and Lynch et al. (1995) found that electric fatigue was characterized by step-by-step cleavage. Tobin and Pak (1993) showed that fatigue crack growth took place even at field amplitudes as low as 5% of the poling field (e.g., ~0.83 kV/cm). Jiang and Sun (1999b) investigated the fatigue behavior of PZT-4 ceramics using prenotched compact tension specimens and two types of loading. In the first type of loading, the voltage was kept constant while a tension-tension cycling mechanical load was applied. In the second type, the specimen was under a constant tensile load while a time-varying electric field was applied. The results illustrate that the magnitude and direction of the electric field influence the crack growth rate significantly. Jiang and Sun proposed that the mechanical and electrical loads can be combined into a single parameter, and that the mechanical strain energy release rate alone should be used to characterize fatigue crack growth. They fit the fatigue data with a law similar to the Paris law, wherein the mechanical strain energy release rate replaced the stress intensity factor. Xu et al. (2000) reported an in situ transmission electron microscopy (TEM) study of the effect of cyclic electric field on microcracking in a single crystal 0.66Pb(Mgl/3Nb2/3)O3--0.34PbTiO3 ferroelectric ceramic. They observed microcracks initiated from a fine pore under an applied alternating electric field. The microcracks laid on { 110} planes, which are common domain boundaries in ferroelectric materials. The pore was also distorted by the alternating electric field. Tan et al. (2000) confirmed that the electrically induced crack growth in the -oriented piezoelectric Pb(Mgl/3Nb2/3)O3--PbTiO3 single crystal was along
Fracture of Piezoelectric Ceramics
259
the 90 ~ domain boundary. Under an alternating field with a peak value of 6.5 kV/cm at a frequency of 0.3 Hz, a crack extension of 2.1/zm was produced after 20 electric cycles, yielding an average crack growth rate of 10 .7 m/cycle. Tan et al. (2000) observed that cracks can grow under a static electric field of 10 kV/cm. It seems that the sample holder in the TEM for in situ observations (Tan et al., 2000; Xu et al., 2000) might have constrained the mechanical displacement of the samples. Thus, in addition to other mechanisms that are still under investigation, the constraint could have induced a stress field through the piezoelectric effect to drive the crack propagation. Nuffer et al. (2000) studied the damage evolution in commercial bulk PZT ceramics induced by bipolar cycling. Polarization, strain hysteresis loops, and acoustic emission were monitored in the experiments. They found that higher cycling fields yield stronger ferroelectric fatigue and higher acoustic emission energy, and that the threshold for the onset of acoustic emission events is lower at high cycle numbers. They suggested that the bipolar cycling led to the agglomeration of point defects, which clamped domain walls, therefore reducing the number of switchable domains. Under electric cycling, fewer and fewer domains can be switched due to the coalescence of point defects; thus, the remnant polarization decreases, resulting in ferroelectric fatigue. Clearly, all the experimental results demonstrate that an alternating electric field can damage piezoelectric ceramics, and the damage mechanisms are still under investigation. 3. Effects of a Static Electric Field
The fracture behaviors of piezoelectric ceramics under combined static mechanical and electrical loads have been studied using indentation induced fracture, three- or four-point bending, and fracture of prenotched compact-tension (CT) specimens. Tobin and Pak (1993) conducted indentation tests on PZT-8 ceramics in a static electric field. Cracks normal and parallel to the electric field (the poling direction) were induced, and crack lengths were measured from the comer of the impression to the crack tip using either optical micrographs or a calibrated eyepiece. Figure 10 shows the average crack length and the associated standard deviation of 10-15 separate indents under a load of 4.9 N. Analysis of the standard deviations of the experimental data suggests that the variation in the crack growth due to the electric field application is statistically significant. For example, the average crack length was 21.50 #m under a negative electric field o f - 4 . 7 kV/cm and the associated standard deviation was 4.82 #m, leading to a relative error of 22.4%. For cracks normal to the applied electric field, Tobin and Pak (1993) found
Tong-Yi Zhang et al.
260 60
9 O
For cracks perpendicular to poling For cracks parallel to poling
50 ::t.
40 30 20 10 |
-6
-4
.
i
-2
.
,
0
.
|
.
2
|
4
6
Applied Field (kV/cm) FIG. 10. Effects of static electric fields on the crack length of PZT-8 under an indentation load of 4.9 N (with the experimental data by Tobin and Pak, 1993). that cracks under a positive electric field (same as the poling direction) were longer than those under a negative field (opposite to the poling direction). This indicates that a positive field assists the applied mechanical load in propagating the crack, whereas the negative field retards the crack propagation. The positive electric field of 4.7 kV/cm produces a crack normal to the poling direction almost 70% longer than the crack for zero applied electric field. In the mean time, the crack under a negative electric field of - 4 . 7 kV/cm is about 30% shorter than the crack under zero applied electric field. The electric field does not appear to have much effect on the propagation of cracks parallel to the poling direction. Wang and Singh (1997) observed the opposite trend in the indentation fracture tests of PZT EC-65 ceramics under load ranging from 4.9 to 11.76 N. They found that for all indentation loads, a positive field of 5 kV/cm generated shorter cracks than the negative field o f - 5 kV/cm in both the perpendicular and parallel directions. For both directions, the difference in crack lengths under positive and negative electric fields is larger at smaller indentation loads. The difference diminishes as the indentation load increases. Other data also show that the fracture behavior of piezoelectric ceramics tested by the indentation fracture technique is load dependent but follows different trends. Figure 11 plots the indentation results on PZT-4 ceramics under the indentation loads of 4.45 and 22.24 N (Sun and Park, 1995). At 4.45 N, the results follow the same trend as those in Fig. 10 (i.e., a positive applied electric field enhances crack propagation normal to the electric field, whereas a negative field retards such crack propagation). However, if the indentation load is 22.24 N, both negative
Fracture of Piezoelectric Ceramics
261
130 (a) 4.45 N Load 120
110 :1. e~o
100 ,
9
,
9
,
.
,
9
,
.
|
.
,
.
,
,
,
.
|
.
360 (b) 22.24 N Load
r,.)
340
320
300 9
-8
,
.
-6
|
-4
.
,
-2
.
!
0
,
2
4
6
8
Applied Field (kV/cm) FIG. 11. Effects of static electric fields on the crack length of PZT-4 under an indentation load of (a) 4.45 N and (b) 22.24 N (with the experimental data by Sun and Park, 1995).
and positive fields facilitate crack propagation. Jiang and Sun (1999a) proposed a wedge model to explain the load-dependent behavior. Without considering the applied electric field, they modeled the tensile stress acting on the indentation crack front as a plastic wedge induced by indentation. As a result of the piezoelectric effect, the wedge elongates at a positive electric field and produces a wedge force. In contrast, if the field is negative, 180 ~ domain switching takes place because the voltage on the crack surface is high and the length of the wedge is small. As a result, the wedge effect takes place under a positive as well as a negative electric field. Jiang and Sun (1999a) introduced a reduction factor to link the wedge force with the applied electric field. Allowing the reduction factor to change with applied mechanical load, the wedge model predicts the load-dependent phenomenon well. Lynch (1998) conducted indentation fracture tests with a 39.2 N load on PLZT ceramics. His results show that a positive electric field enhances the growth of cracks perpendicular to the poling direction in PLZT ceramics. This effect seemed
Tong-Yi Zhang et al.
262
to be saturated at 500 kV/cm, about 1.2 times the coercive field. When the ceramics were indented under high electric field, a series of microcracks developed in the stress field of the indentation at a small distance away from the impression. The microcracks were not extensions of the radial or lateral crack systems. Lynch did not report the effect of a negative field. Fu and Zhang (2000a) conducted indentation fracture tests on PZT-841 ceramics with a load of 49.0 N. Under each of the electric fields of 4-4 kV/cm, about 10 indentations were made. The fracture toughness was determined from the following equation (Anstis et al., 1981):
(Y33) 1/2P K1c - 0.016 - ~ c3/2,
(9.1)
where P and c are, respectively, the applied mechanical load and the crack length perpendicular to the poling direction; H is the hardness; and 1133is Young's modulus along the poling direction. Figure 12 shows the variation of Kzc with the applied electric field. Under pure mechanical loading, the averaged K ic was 1.01 4-0.06 MPaVr~. The mean K1c was reduced by both a positive and a negative applied electric field. The reductions were 0.21 MPav/-m and 0.10 MPaWrm, respectively, for the negative and the positive field of 4 kV/cm. This result seems to indicate that a negative field has a stronger influence than a positive field on the averaged KI c. Using the indentation fracture technique with a 40 N load, Schneider and Heyer (1999) studied the effect of a static electric field on the fracture behavior of
1.1 -9
0 0 0
0
1.0
0 0
0
0.9 0
;
0
0.8
o 0
0.7
!
-6
-4
o
0
o ,
,
-2
mean i
0
,
|
i
2
4
6
Applied Field (kV/cm) FIG. 12. Effects of static electric fields on the fracture toughness of PZT-841 under an indentation load of 49 N.
263
Fracture of Piezoelectric Ceramics
ferroelectric barium titanate. Their results indicate that the measured crack length versus the applied electric field shows hysteresis similar to the strain hysteresis (i.e., the butterfly curve introduced in Section I). Curves for cracks parallel and perpendicular to the electric field direction are symmetric to each other. The lengths of cracks perpendicular to the poling direction under either positive or negative electric field are longer than the length of cracks without the applied electric field. This last result is consistent with that of Fu and Zhang (2000a). Park and Sun (1995a) and Fu and Zhang (2000a) conducted fracture tests on prenotched CT samples of PZT-4 and PZT-841 ceramics, respectively. About 10 samples of PZT-841 ceramics were tested at each level of the electric fields, except that 33 samples were tested at the electric field of 15 kV/cm to study the scattering of fracture toughness. The apparent fracture toughness Ktc was calculated based on the critical mechanical load at fracture only. Figure 13 shows the variation of the normalized K ic with the applied electric field for PZT-4 and PZT-841 ceramics.
1.4
o
(a) PZT-4
8
1.2
8 ra e ~
0
1.0
0
O O
0.8 ,
o ~D N
9,-,
|
.
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15
Applied Field (kV/cm) FIG. 13. Effects of static electric fields on the fracture toughness measured from prenotched compact tension samples of (a) PZT-4 ceramics (with the experimental data by Park and Sun, 1995a) and (b) PZT-841 ceramics. The fracture toughness is normalized by its values under purely mechanical loading.
264
Tong-Yi Zhang et al.
The results for PZT-4 ceramics reveal a nearly linear effect of the electric field on the fracture load; the results for PZT-841 ceramics show that the applied electric field increases the scattering of the measured apparent fracture toughness. The applied electric field, either positive or negative, reduces the mean of the apparent fracture toughness. Under purely mechanical loading, the average K IC for PZT841 is 1.12 -4- 0.05 MPa~/~, which is almost the same as that obtained from the indentation fracture tests. A negative field of 7.5 kV/cm reduces the averaged KIC by 0.25 MPav/-~, whereas a positive field of the same strength reduces the average KIC by 0.10 MPa~/-m. Applying a positive electric field of 15 kV/cm reduces further the averaged KIC to 0.92 + 0.14 MPax/-~, a relative reduction of about 18%. Applying an electric field generally increases the scattering in the measured fracture toughness considerably. The largest scattering in fracture toughness is at the electric field of 15 kV/cm. The distribution of K1c for 33 samples is shown in Fig. 14. The ratio of the standard deviation of 0.14 MPax/~ to the associated mean of 0.92 MPa~/-~ leads to a relative error of about 15%. As shown below, the relative error for the bending strength at an electric field of 10 kV/cm is about 33% (Fu and Zhang, 1998), which is more than double that observed in the CT tests. Using CT specimens, Kolleck et al. (2000) measured the fracture resistance curves (R curves) of BaTiO3 and commercial PZT-PIC 151 ceramics under an applied electric field parallel to the crack front. They found an increase in the fracture toughness with growing the electric field, and proposed a domain-switching model to explain the observed phenomenon.
FIG. 14. Distribution density of the fracture toughness under a static electric field of 15 kV/cm for CT tests on prenotched PZT-841 samples.
Fracture of Piezoelectric Ceramics
265
30 25 r.t3
O9
20 15
d:Z r O9
O9
b(x), x > x f = b - l ( y ) ) ,
(on Sb,
lY[ < b(x))
y 3 W(Xl, Z'l)dxl , _ b 2 3xl
(b < lyl < bm),
(4.24)
310
Theodore Yaotsu Wu
wherexf = b - l (y) is the inverse function of y = b ( x f ) ~ b f forgiven y ~ (b(x) < lYl < bm) so that ( x f , y = b y ) locates the point on the trailing edge from which the vortex line starts to shed to reach a point of given (x, y)(b(x) < lYl < bm) on the vortex sheet. In the tail section (Xc < x < e), the trailing vortex sheets continue to have the z-component velocity transported along the characteristic line x - U t - - ~ , yielding Wv(y; x, t) -- -
fx xc
Y
~
O
i v/y 2 - b ~ o x l
W ( x l , rl)dXl ,
(bc < ]y]
Xc),
(4.25) y
= --W(Xm' Tm) v/Y 2 -
b2m
_ f x x' ~ y 3 (Xl, ~W m v/Y2 -- b~ OX1
"t'l ) d x l
(on So, JYl > bm),
(4.26)
This result shows that the upwash being transported with a retarded time "t"1 has a spanwise distribution dependent on y. Consequently, this y-variant correction to be imposed on the velocity field in the caudal fin section accordingly requires a generalization of the present simpler theory to a more general theory, which we address in Section IV.B.
3. The Caudal Fin Section (xc < x < g.)
When the vortex sheets shed from side fins enter into the caudal fin section, the interaction between the vortex sheet and caudal fin has been argued as being capable of augmenting the lift and thrust generation handsomely, especially when they become opposite in phase. However, the problem of evaluating this dynamic interaction is complicated as a result of several new issues, a primary one being that the upwash induced by the vortex sheets transported into this caudal fin section bears with it a conspicuous spanwise variation given by Eqs. (4.25) and (4.26). An earlier attempt by Wu (1971 c) contains a deficiency that was removed by Wu and Newman (1972), who took the body thickness effect into account, yielding a solution to the caudal fin problem in terms of an integral equation of the Abel type that can be evaluated numerically. A consistent slender-body theory was subsequently developed by Newman and Wu (1973) and Newman (1973) by taking into account both the kinematic and dynamic interactions and the effect of body thickness on trailing vortices. This problem is discussed in Section IV. B for the generalized case for a thin body-fin system moving with displacement h(x, y, t) varying with both x and y.
On Theoretical Modeling of Aquatic and Aerial Animal Locomotion
311
In closing this section, we note that for fishes known for their high performance using the so-called lunate, or crescent-moon shaped tails for propulsion, it calls for abandoning slender caudal fin analysis in favor of adopting high aspect-ratio oscillating wing theory, as commented on in Section II, and is discussed in more detail in Sections V and VI.
B. A GENERALIZEDLINEAR SLENDER-BODY THEORY OF FISH LOCOMOTION The case of body-fin movement that varies with both x and y, as that found in amiiform, gymnotiform, and other similar modes, has been investigated by Wu (1983). With this generalization, the simple and elegant physical concepts and simple calculations presented previously for the simpler case with rigid transverse body sections are no longer sufficient for attaining solutions. However, we may resort to more powerful mathematical methods for resolving such challenging biofluiddynamic problems to attain new results of interest. Next, we present a synopsis of the mathematical construction of the solution before we turn to consider the fully nonlinear, large-amplitude theory of aquatic and aerial animal locomotion. The general problem formulated at the beginning of Section IV.A is in a form of the Riemann-Hilbert problem in its simplest version. Our analysis to derive the required solution to more general problems can be greatly facilitated by using complex function analysis and application of Plemelj's formula (see, e.g., Muskhelishvili, 1953) G(~ ) -- ~
1
g(y~) dy~ Yl -
(4.27)
(~ q~ s
1 1 f c g(Y~) dy~ G+(y) - +-~g(y) + ~ yl - y
(y ~ s
(4.28)
hence g(y) = G+(y) - G_(y), and G(~') = ~
1 fz: G + ( y l ) - G - ( Y l ) d y ~ (~ q~ s yl - ~"
1 f~ G + ( y l ) - G_(Yl)dyl G+(y) + G_(y) = rci yl - y
(y ~ s
(4.29) (4.30)
where G(~') is a complex-valued function of ~ = y + i z, analytic in the entire ~" plane except for ~" 6 s s is an arbitrary smooth line (here taken to be the entire real axis of ~'), G+(y) and G_(y) are the limiting values of G(~') as ~" approaches from the left (upper) or right (lower) side of s as viewed along s respectively, to a point y on s these limits being assumed to be Hrlder continuous, and further, the integral in Eqs. (4.28) and (4.30) assumes Cauchy's principal value.
Theodore Yaotsu Wu
312
By classical analysis and using Plemelj's formula, the required solution can be obtained for different longitudinal sections Sx of the body as follows.
1. The Anterior Leading-Edge Section (0 < x < Xm, d b / d x = b'(x) > O) In this body section, the side edges along y = +b(x) are assumed to be either well rounded or moving well feathered to flow to avoid local flow separation as before, except that the body transverse motion can now have variations in both x and y. To find the solution to v, we first note that its boundary conditions have the symmetry v+ + v_ = - 2 i W for lyl < b; v+ - v_ = 0 for lyl > b; and we next seek its homogeneous counterpart [say, H(()], such that H+ + H_ = 0 for lyl < b; H+ - H_ = 0 for lyl > b. From this, it follows that G(() = v ( ( ) H ( ( ) can be determined immediately by applying Plemelj's formula (4.29) and (4.30), provided that the H ( ( ) so found makes v(() satisfy all the remaining boundary conditions. Thus we find the unique solution as
df ~--p--v-iw--
1 f ~ H+(yl ; b) W(yl" x, t) Yl - ( dyl
d(
7r ~_b H ( ( ; b ) H((;b)=v/(
2 - b2
( 0 < x <Xm),
(4.31)
(( = y + iz).
Here, the function H ( ( ; b) is again as defined in Eq. (4.12). From Eq. (4.31), the corresponding f a n d F can be readily deduced by integration. In particular, if W is independent of y, Eq. (4.31) reduces to the simple solution (4.12).
2. Trailing Side-Edge Section (Xm < X < Xc, d b / d x = b'(x) < O) Within this section, the elongated dorsal and ventral fins have edges slanted to trail behind the flow (with b' (x) < 0), and can move to vary with y, at which edge the Kutta condition is invoked. Under this condition, we note again the complete analogy between the two sets of boundary conditions I(i) on 4~z = R e ( d f / d ( ) = W and II(i) on ~z = R e ( d F / d ( ) = D W, other conditions being equal. Whence, by analogy with Eq. (4.31), we have (Wu, 1983)
d F _--
1 J_f~ H+(yl; b) DW(yl; x , t ) d y I
d(
Jr
b H((;b)
YI-(
(Xm < X
<Xc).
(4.32)
On integration, we obtain F as F ( ( ; x, t) --
l f_~ 7r
d(
cr H ( ( ; b )
b
H+(yl.b) DW(y , 9x,t)dy ' Yl --
which can be seen to satisfy also the Kutta condition and condition (v).
(4.33)
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This F field provides the pressure distribution within this trailing-side edge body section. The corresponding velocity potential field can be obtained, straightforwardly, by integrating Eq. (4.8) (i.e., D f = ft + Ufx = F), along the mathematical characteristic lines x - Ut = ~ = constant, starting from x = Xm [at which f is known from Eq. (4.31)], with F given by Eq. (4.33). Then the velocity field can be deduced from v = d f / d ( , thus yielding the ( + )-side limit of v = v - i w over Sb and Sw (for details, see Wu, 1983). The result of w+ = w(y, 4-0; x, t) =-Wv(y; x, t) gives the "sidewash" induced by the entire vortex system, which consists of the bound vorticity lying in the planar body surface Sb and the free vortex sheet over the wake surface Sw : (b(x) < [y[ < bm, Xm < X < Xc). This induced velocity field, when convected with the free vortex sheet into the caudal fin section, is capable of assuming an important role in interacting with the caudal fin for the possibility of enhancing its thrust production and propulsive efficiency.
3. The Caudal Fin Section (xc < x < ~)
In this body section, the caudal fin has new leading edges in the presence of the vortex sheets incident from upstream that in turn interact with the fin motion. In the framework of linear theory, the velocity field can be constructed by superposition of two components, one being vv, which is that induced by the entire vortex sheet convected (invariant along the characteristics) downstream into the caudal fin section (with velocity U, as if without caudal fin), and the other, vc, which is the velocity due to the motion of the caudal fin itself. This gives within the caudal fin section the velocity field as v ( ( ; x , t) = v c ( ( ; x , t) + v~,((; x, t)
(xc < x < ~),
(4.34)
where x = ~ marks the caudal fin trailing edge. Based on the known sidewash velocity, Wv, induced by the vortex sheets on the caudal fin surface (which is assumed to be slender like the entire body configuration), we therefore obtain for v~ the solution in integral form (Wu, 1983) as Vc(~;x, t) =
Jr
H + ( y l ; b ) W ( y l ; x , _..t)- W v ( y , x , t) dyl , b H(~'; b) Y l - - ~"
(4.35)
where W~(y; x, t) is the sidewash induced by the vortex sheets as described in Section IV.B.2. This solution exhibits several important features. First, this v~ reduces to Eq. (4.31) for the anterior section when W~ vanishes. When W~ --/: 0, the interaction between the caudal fin movement and the vortex sheet sidewash is seen as being possible to enhance greatly the thrust generation and propulsive efficiency,
314
Theodore Yaotsu Wu
especially when their own induced velocity fields maintain opposite in phase because the two W terms in Eq. (4.35) then totally augment each other. Typical cases of numerical examples have been investigated by Su and Yates (1983) and by Yates (1983), yielding results that lend quantitative support to our qualitative discussion of Eq. (4.35).
V. A Unified Approach to Nonlinear Theory of Flexible Lifting-Surface Locomotion In pursuing further improvement of slender-body theory for investigating fish swimming, Lighthill (1971) developed a large-amplitude elongated-body theory of eel-like fish locomotion with arbitrary amplitude. By taking the trajectory of a swimming slender fish lying in a fixed horizontal level and using a Lagrangian coordinate a to identify a point x(a, t), z(a, t) on the spinal column, Lighthill generalized his reactive force theory by distributing along the moving spinal column (or backbone) flow singularites that are relevant: here, the momentum carrying dipoles (source doublet), with which he evaluated the water momentum given by the virtual mass of the surrounding fluid in the transverse section perpendicular to the column per unit length times the velocity component of fish in that direction. Lighthill further proposes that propulsive thrust can be obtained by considering the rate of change of momentum within a volume V, which includes the fish and excludes the wake by introducing a geometric plane FI perpendicular to the caudal fin through its posterior end, which separates the wake from flow volume V. The momentum calculation then takes into account both the transfer of momentum across I-I and the action of pressure at the plane FI resulting from the swimming motion. The geometric FI plane, introduced to exclude the wake momentum calculation, is supposed to be thought of as swinging around with the caudal fin throughout the fin's waving motion. With respect to this new concept and the details of calculation, we refer the reader to the original paper (Lighthill, 1971). In the spirit of Lighthill, this present study pursues further improvement in several new aspects, including one attempt to make the trajectory and amplitude of fish body movement entirely arbitrary, another to relax the restriction to bodies being slender, and still another to admit more versatile compositions of the added mass in order to account for the effects due to the free vortex sheets shed from trailing edges of appended fins of arbitrary configuration. Consideration of these issues is believed necessary to determine the nonlinear effects generally involved in fish locomotion studies. With these extensions, the nonlinear unsteady theory of lifting-surface locomotion appears to be applicable in a
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unified manner not only to aquatic locomotion of fish and cetaceans, but also to the aerial locomotion of birds and insects. These objectives actually have had a strong appeal to Wu (2001) ever since his preparation for a lecture he was invited to deliver on 5 January 2000 at a special International Conference on Biofluiddynamics in Memory of Sir James Lighthill held at Technion and University of Haifa. That work, a forerunner of the present study, presented a three-dimensional, large-amplitude, flexible lifitingsurface theory for modeling aquatic and aerial animal locomotion at high Reynolds numbers, with the nonlinear effects fully taken into account, and is for the simulation of the propulsive movement of fish, bird, or insect with flexible lifting surface and with appended fins of negligible thickness, but otherwise of quite arbitrary configuration, moving with arbitrary amplitude and trajectory. For this general case, it is necessary to designate the displaced flow boundary of not only the lifting surface of the body [say, Sb(t)], which is prescribed, but also the vortex sheets shed from the lifting surface to trail as a free wake, denoted by S~(t), with its location unknown a priori (aside from the condition on the vortex sheets remaining attached to the trailing edges of the fins). The theoretical formulation is based on the assumption that the fluid is incompressible and inviscid, and the flow is irrotational, obeying the following principles: (i) The inherently nonlinear kinematic and dynamic boundary conditions are to be expressed in terms of the surface variables on the flow boundary and then applied exactly. (ii) Closure of the system of model equations is accomplished by expressing the solution for the velocity potential in terms of a surface distribution of dipoles as the relevant flow singularities for facilitating comp,Jtational work as well as for analytical studies. (iii) Hydrodynamic thrust and power expenditure in propulsion are to be derived from the pressure obtained from the solution of the exact model equations based on the momentum balance involving the entire flow field in reaction to the propulsive movement of the boundary. Following these principles, the solution for the velocity potential is expressed in terms of a distribution of surface dipoles over Sb(t) + Sw(t); the dipole strength is determined by the prescribed flow velocity normal to Sb(t) together with satisfying the dynamic condition of zero pressure jump across the trailing vortex sheet at Sw(t), including the trailing edge of Sb(t) for the Kutta condition. For the initial value formalism, all the flow disturbances are required to vanish at infinity. Along
316
Theodore Yaotsu Wu
this approach, the analysis leads to a set of differential-integral equations for calculation of the surface dipole distribution over the lifting surface and its vortex wake. In summary, Wu (2001) presented the basic principles and a method of solution to the general three-dimensional problem of flexible lifting-surface locomotion at high Reynolds numbers with the motion of body surface Sb(t) expressed in terms of z = h(x, y, t) for (x, y) lying in the projection of Sb(t) on the (x, y) plane. The final integral equation can be directly applied for computational purposes. Nevertheless, it is noted that the explicit expression of the lifting surface movement in terms of the class of displacement function z = h(x, y, t) is not versatile enough to describe body movement in its more general forms, such as with sharp sideor U-turns in fast maneuvering and impulsive large-amplitude control operations, such as figure-eight wing beats in hovering hummingbirds and highly curved turns in lunate-tail swimming. Such shortcomings have subsequently been overcome by adopting Lagrangian variables for describing body motion for the general purpose. In the following section, we shall consider the two-dimensional theory, and show that considerably more analysis can be carried out further in this manner to expose the intricacies of the challenging tasks involved in computation, and to streamline the complete procedure of calculating the final solution.
VI. A Unified Nonlinear Theory of Two-Dimensional Flexible Lifting-Surface Locomotion The theory of two-dimensional lifting-surface locomotion is of fundamental importance in several aspects. First, it provides a valuable limiting case for asymptotic evaluation of lifting surfaces of large aspect-ratio as found in various modes of aquatic and aerial animal locomotion. Furthermore, limiting to two-dimensional configurations may afford the simplicity needed to develop efficient and even elegant methods of solution that can enable generalizations to three-dimensional and more general cases more effectively with improved experience and understanding. Thus we consider the irrotational flow of an incompressible and inviscid fluid generated by a two-dimensional lifting surface Sb(t) of negligible thickness, moving through the fluid in arbitrary manner, even with a good degree of flexibility, as found among various animals in nature. Its motion can be described parametrically by using a Lagrangian coordinate ~ to identify a point X(~, t), Y(~, t) on Sb(t) varying with time t as
Sb(t):x=X(~,t):(X(~,t),Y(~,t))
(-l 0),
(6.7)
t), Y(~m, t)] of the starting vortex
~m(t) at time t after being shed at t - 0 when motion began, whereas
X(~, t) of Sw(t) is to be determined for (1 < ~ < ~m, t > 0) as a part of the solution. To describe the flow in a neighborhood of Sw(t), we adopt the same intrinsic coordinates (~, r/) such that 0 - 0 coincides with S~(t) for 1 < ~ < ~m, t > 0. Physically, it is of theoretical and practical interest to note that the wake S~(t) is continually being opened just beyond the trailing edge to create a new stretch 3X(1, t) - U(1, t ) r t in a small time interval 6t tangentially to the trajectory traversed by the trailing edge at the rate U (1, t). So, in time interval 0 < t' < t since the motion started, the past trajectory of the trailing edge (X(1, t'), 0 < t' < t) is in fact the "birthplace" of the wake vortex sheet, from which the free vortices would have been convected by their local induced velocity field that may be weak for wellfeathered motions of Sb(t) but can be very significant for impulsive body motions. By analytic continuation, the intrinsic coordinates (~, 0) form a complex reference plane ( = ~ + i ~ such that the ( plane and the z -- x + i y plane are related by a conformal transformation, which generally exists Vt > 0 and is denoted by z - z((, t) and has the following general properties. First, for sufficiently smooth bodily movement [X(~, t), Y(~, t)], d z / d ( # 0 or c~ on Sb(t), even including its
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319
end points because Sb(t) is smoothly curved and both Sb(t) and its image plate have zero thickness. In fact, d z / d ( at an end point of Sb has its modulus equal to the distance of the end point to the origin and has its argument equal to the slope angle of Sb(t) at that end, in accordance to the general theory. Furthermore, the point of z = c~ is generally not mapped onto ( = cx~ (unless Sb remains to be a flat plate) but to some point (say, (0) in the finite ( plane, which tends to ( = c~ as Sb reduces its curvature everywhere to 0. Otherwise, z -- z((, t) is analytic in the entire ( plane except at (0 and also possibly at ( = c~. It is unnecessary to determine the actual time-dependent conformal mapping z -- z((, t), a time-consuming and very tedious task, because we need use only the analytic relationship of z -- z((, t) in a small neighborhood of Sb(t) + S~(t). For further analysis, it is convenient to introduce a transformation between the Eulerian field description with (x, y, t) and the Lagrangian material description with (~, 17, t') in terms of: x--s
Y=Y(e,O,t'),
and
t=t',
(6.8)
where the overhead-symbol indicates a functional relation. As understood, the time derivatives with respect to t and t' mean that for a differentiable function
(6.9)
F(x(~, O, t'), y(~, O, t'), t) - P(~, 17, t'), we have, by the chain rule,
OF aP I _ OF OF OF dF at' = Ot7 ~,~ -at + U-~x + V-oy - -dr
(6.10)
signifying that these equivalent operators all mean material time differentiation by following instantaneously the fluid particle identified by (~, 0) at time instant t' - t, and the subindices (~, 0) indicating that they are held fixed in differentiation. In the absence of viscous effect, Sb(t) and Sw(t) are singular surfaces across which certain flow variables may have jump discontinuities. For an arbitrary flow variable F(~, r/, t'), we denote its value on the 4- side of Sb(t) and Sw(t) (located at r / = 4-0) by P(~, r/ -- -+-0, t ' ) - - P + ( ~ , t ' )
(-l 0),
(6.17)
on Sw(t)"
Un+(~,t') -----Un(~, t t) -- Un(~, t')
(l < ~ < ~m, t' > 0).
(6.18)
Physically, this signifies that the flow velocity normal to the surface is continuous at the surface and equal to the normal velocity of the (moving) body surface, but the tangential velocity at the boundary surface generally has a jump in magnitude across the surface. With such discontinuities, the surface Sb(t) becomes a bound vortex sheet, whereas Sw(t) forms a free vortex sheet being convected away with the fluid. In viscous fluid, they form a thin boundary layer in which the vorticity is
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321
distributed within the layer, whose integral across the layer gives the velocity jump in the inviscid region just outside the thin layer (which reduces to zero thickness with the viscous diffusion effects neglected). In this sense, we have the distinction that Un(~, t') is prescribed on Sb(t), but is unknown a priori on S~(t). The latter is to be determined by also imposing a dynamic condition invoking that the pressure be continuous across S~(t), namely, p+(~, t') = p_(~, t')
(1 < ~ < ~m, t > 0),
(6.19)
signifying that free vortex sheet cannot sustain any pressure jumps, including the trailing edge at ~ = 1 for the Kutta condition, where the pressure p is given by the Bernoulli equation P + aq~ + 1 (u 2
p
--~
-p -q a ~
2
-~ s + U , ) - - P
at'
1 (u 2 + u,) 2 -- 0,
2 s
(6.20)
in which the latter form follows from using Eq. (6.10), so that, under conditions (6.17) and (6.18), ~(p-
- p+) -
-g
+
~
_ + ~ ( ~ s + + , s ~ ( , s + - Us)
(z 6 s = Sb + Sw) a
= ~,(,+ -,_~-
(6.21)
1
~(u~ + ~;)(U+s - Us) (z 6 s = Sb + S w ) .
(6.22)
To construct the unique solution to this problem, we now follow Principle (ii) to adopt the integral representation of a surface distribution of mass dipole (or mass doublet) of strength # per unit length streamwise (and unit depth transversewise) over the boundary surface Sb(t) + S~(t), giving the complex potential f = 4~ + i~p as
t)dz, f (z, t) -- ~ 1 fL #(~" z'-------~z As z --+ Z(~, t) 6 E from the r / = •
f+(z, t) - • 1
( z ' = z(~', t'), z ~/2 = Sb + S~).
(6.23)
side, we have, by Plemelj's formula (4.28),
t) + ~il fc #(~"z'----~zt)dz'
(z = Z(~ t) ~ s
(6.24)
with the integral assuming Cauchy's principal value. We therefore have the dipole strength related to the jump in 4~ as /z(~, t) = ~b+(~, t) - 4~-(~, t)
(dipole strength/length)
(6.25)
Theodore Yaotsu Wu
322
while the normal derivative of 4~is continuous across/2 - Sb + S~, [see, Eq. (6.29)]. From this, we derive the complex velocity from w(z, t) - d f / d z to obtain 1
w(z)-
~i
d
#(~', t)
_
1
-
2re i
1
dz'=
dz' z ' - z
~/(~', t ) d ~ ,
y -
2rri ~ / z ( ~ ' , t ) [
~
d~
Ose ' z ' - z
z r
z' - z
-~'
by integration by parts, where t) -
0# o~
t) -
0 o~
(ep+
-
ep_) -
(U+s -
Us)
(6.26)
is the vorticity per unit length along 12 -- Sb + Sw or along the ~ axis, which is equal to the jump in tangential velocity across the boundary/2. By applying Plemelj's formula once again, we find that
1 /dz 1 ~y(~',t) ~z-7 d ~2-' z w+(z, t) = 4--~ y(~, t) -d~ + ~
(z = Z(~,t) 6 12);
(6.27)
and hence, from Eqs. (6.16) and (6.27), there results
1 1 dz ~ y(~', t)d~,. Us~ - i Un -- -+--~)/ ( ~ , t) + 2 rr----~d--~ z '----~-Z
(6.28)
From this equation (6.17) and (6.18) are automatically recovered and further, for both z, z' ~ s
s zdfg b+&. Y(~" 't) d~' z } u+(~, t) - u-~(~, t) - -~1 Re { d-:-7
U sm ~
l(u++u_)_
-2
s
1 2re Im
{dzf -~
Z t
(6.29)
--
y(~',t) } b+&, z ' - z d~'
(6.30)
of which the first equation shows the continuity of normal velocity u,, = Och/On across 12 and the latter gives the algebraic mean of tangential velocity Us across the boundary/2. Moreover, the fact that the us and u,, velocity components arise in Eq. (6.28) so naturally lends simplicity to the subsequent analysis as well as to related computational work. Summing up, we have obtained Eqs. (6.29) and (6.30) that characterize the kinematic boundary conditions as follows: on Sb
:
on Sw :
Un(~, t) = U,(~, t) -- is prescribed,
y(~, t) - is unknown.
y(~, t) - is material invariant,
un(~, t) - is unknown.
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323
To this end, we point out that by the material invariance of free vorticity, it is meant that according to Helmholtz's theorem, vorticity in irrotational flow of incompressible inviscid fluid cannot be generated in any interior bulk of the fluid, but only at a boundary surface; and once generated, it leaves the surface as a conserved property of the local fluid material. In viscous fluid, the boundary layer generated at such high Reynolds numbers is very thin, but distributed with all the vorticities of large values on both sides of Sb(t) that finally merge to be convected smoothly downstream from the trailing edge. With the viscous effect neglected in the ideal fluid (without diffusion), this layer reduces to a singular surface of vortex sheet Sw(t), with the vorticity frozen with the local fluid as a conserved property. It therefore implies that Oyw at
+Usm
Oy~
a~
=0
(1 < ~ O ) , -
(6.31)
-
where Usm is given in Eq. (6.30). Under either condition (6.19) or (6.31) serving as the Kutta condition, the trajectory X(~, t') of the free vortex sheet Sw(t) can be determined either analytically, by means of an integral equation, or numerically, by a time-marching procedure, to be delineated later. To account for the total net vorticity, we evaluate Kelvin's circulation, F(t), given by the line integral of the velocity component along the contour (taken clockwise) from point P_(~, r / = 0 - ) to point P+(~, r / = 0+) without cutting the boundary surface/2 = Sb(t) + S~(t), r ( ~ , t) -
f
u. dx =
d ~ - 4~+(~, t) - 4~-(~, t) -
.(~,
(-l 0).
(6.33)
This also results from Stokes's theorem, which states that the line integral of F is equal to the surface integral of all the vorticity V x u (pointed into the surface) spanning the integral contour. Thus, the circulation around any contour E completely circumventing s must be conserved, Fz(t) --
f0
y(~, t)d~ = F(0),
(6.34)
a constant equal to the circulation of a stationary initial state, which is zero for a rest state. This is known as Kelvin's circulation theorem. Of particular interest
324
T h e o d o r e Yaotsu Wu
is that by Eq. (6.34), the circulation Fb(t) around lifting surface Sb(t) and the circulation around wake S ~ ( t ) are related as Fb(t) =
u . dx -
?,(~, t)des - -
?,(es , t)des -
-Fw(t).
(6.35)
B
Now consider the variation 31-'b = Fb(t + 6t) - Fb(t) during a smooth motion at a small time 6t apart, this 6Fb, by Kelvin's circulation theorem, must be equal and opposite to the new vorticity being shed from the trailing edge into the new gap 6~ created by the forward moving trailing edge with velocity Us plus the fluid motion with tangential mean velocity Usm; that is, 6Fb(t) =
6r~(t) = - y ( 1 , t)6~ = - y ( 1 ,
t)[Us(1, t) + Usm(1, t)]6t,
(6.36)
where U s ( l , t) is the tangential velocity of the forward moving trailing edge. This vorticity balance does not involve the vorticity sheet that has previously been shed because the latter moves on invariant while being convected with the local fluid bulk. With the vortex shedding rate determined, we can set up a primarily numerical method for computation of the initial-value problem as follows.
A. METHOD I--COMPUTATIONAL TIME-MARCHING METHOD
For computation of solution by discrete mathematics, we take a time sequence to = 0, tl, t2 . . . . . (tk+l -- tk = At, k = 0, 1,2 . . . . ), with At sufficiently small. At to, the body surface Sb(0) is taken at the stretched-straight position in an unbounded flow field, which is taken at rest (or at a stationary state). At tl, Sb(tl) assumes its new position given by X(~, tl) of Eq. (6.1) for ( - 1 < ~ < 1), while S w ( t l ) has its first onset grid opened to length A~ as a smooth extension of Sb(t~) beyond the trailing edge to receive the first element of a starting vortex that is being shed to form the new wake, with A~ and the shed vortex strength y(1, tl) given by Eq. (6.36). The unknown vorticity y(~, tl) over ( - 1 < ~ < 1) on Sb(tl) is determined from the integral Eq. (6.29) under condition (6.17) by applying some efficient numeric method of high precision. This way of evaluating the discretized unknown y(~, t) for ~ only over the Sb(tl) stretch of s is considered to be necessary and sufficient because the new vortex sheet Fw(tl ) = --Fb(tl ) involves neither any additional new unknowns because of Eq. (6.36), nor are there any additional kinematic conditions required on S~(tl). After y(~, tl) is so determined over ( - 1 < ~ < 1 + A~), we can deduce u~ from Eq. (6.28) and un on A~ of S ~ ( t l ) from Eq. (6.29), with which we can revise the data on S~(t~) by iteration for any improvement on the length and direction of A~. We can next update all the boundary variables by applying
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325
the evolution Eq. (6.15) for the displacement of s and Eq. (6.31) for the wake vorticity, which are needed for the next time step at t2. At t2, Sb(t2) assumes its assigned position from X(~, t2) of Eq. (6.1) whereas the starting vortex 6F1 -- 3Fw(1, tl) has moved to its new position by convection with the local known flow velocity while maintaining its magnitude invariant henceforth, leaving a new gap A~(t2) to receive new shed vortex sheet of strength y(1, t2) given by Eq. (6.36), so the same calculation can be performed for y(~, t2) as done for y(~, tl) to obtain the solution at t2. Thus, with the solution accomplished for time step at tk, the computation can be carried forward to obtain the solution at tk+l. In this manner, this time-marching procedure can, in principle, be extended forward indefinitely. The problem is thus reduced to computation of the unknown vortex element just leaving the trailing edge and entering the wake at each time step. This is a point of paramount importance to forming a physical concept, conducting mathematical analysis, and making numeric algorithms and codes for the computation of a solution. Concerning the algorithm, quadratures, and numerical filters to be used for resolving the integral Eq. (6.29) under condition (6.17), reference may be made to the existing numerical methods for evaluating unsteady airfoil dynamics (e.g., Smith and Hess 1967; Giesing et al., 1971; Ashley and Rodden, 1972; Katz and Weihs, 1978; McCune et al., 1990; some being purely numerically aimed). Relative to these existing numerical methods, the present method is a further extension to the general case of a flexible lifting surface moving with arbitrary form along arbitrary trajectory, taking into account the nonlinear effects involved. However, it should be noted that even in the simpler case of rigid airfoils, satisfaction of the Kutta condition would generally involve some sort of artifices to various degrees of discrete mathematics, making each numerical method seem to carry a "fingerprint" characteristic of its own, rendering critical comparisons between theories and between theory and experiment not entirely straightforward. This is particularly true at the beginning of a sudden start or right after a sudden stop, whenever the airfoil undergoes an impulsive or discontinuous movement. To pursue further development and improvement, we explore ways of reducing some of the shortcomings of Method I by investigating another method.
B. METHOD II - - GENERALIZED WAGNER--VON KARMa, N-SEARS METHOD
In the history of pioneering development of linear unsteady airfoil theory, Herbert Wagner (1925) was the first to have generated an integral equation for calculating the wake vorticity shed from the wing. This approach was subsequently further developed by Theodore von K~irm~in and William R. Sears (1938), who
Theodore Yaotsu Wu
326
made a lasting contribution by providing an alternative and innovative formulation of the basic theory, and elucidating the underlying physical significances of the mathematical analysis advanced, and the results they accomplished. The improved derivation is based on an ingenious decomposition of the bound vorticity Fb on the wing into two parts, one for its "quasi-stationary wakeless" flow, F0, and the other, F~, due to wing's reaction to the trailing vortex sheet, with both F0 and F~ lucidly analyzed using linear approximations. In addition, there are other pioneers who have made valuable contributions. Theodorsen (1935) found a fundamental solution for the simple harmonic oscillatory wing motion in heaving and pitching. Ktissner (1936) obtained excellent experimental results and his theoretical explanations for these results. Robert Jones (1940) converted the unsteady-lift functions for wings of infinite span to that for elliptic wings of large aspect-ratio. Nearly all these latter theories are based on using orthogonal function expansions, integral transforms, or other operational methods that are basically for linear analysis, and hence not practical for extension to nonlinear problems. However, of these linear theories, the Wagner-von K~irm~in-Sears method is unique in affording a direct generalization to fully nonlinear theory, as has been accomplished by McCune and co-workers (1990, 1993) for investigating unsteady two-dimensional airfoil problems. Here, we make further generalization to two-dimensional flexible lifting surface performing arbitrary movements for modeling aquatic and aerial animal locomotion. In the spirit of Wagner, von K~irm~in,and Sears, we adopt for t > 0 the following vorticity distribution: on Sb(t) :
T(~, t) = Yo(~, t) + Yl(~, t)
(--1 < ~ < 1),
on Sw(t) :
T(~',t) = Yw(~', t)
(1
< ~
(6.60)
If only the starting motion is of interest, 0 < U t - s 0), (6.62)
which is Wagner's famed result, stating that an impulsively started flat-plate airfoil generates instantly half the final lift, which it eventually achieves, 1 - ~ ( U t ) ~ 1 as t --> o~ (i.e., after a time delay), which is known as the Wagner effect. In addition, von K~irm~in and Sears (1938) showed that the function 9 can be used on linear theory to calculate the lift acting on an airfoil that is subjected to an arbitrary transient variations of quasi-steady circulation F0. Here, the point is that in application of this fundamental solution for constructing solutions to general problems, it is vital to notice that the small-time behavior of its flow field exhibits various singular features, including the bound vortex Yl having square-root singularities at both the leading and trailing edges of the airfoil
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(with the starting vortex sitting right at the trailing edge) at t = 0 [as is clearly indicated by Eq. (6.61) adjacent to the trailing edge], together with the pressure possessing a Dirac delta function 3(Ut) over the whole plate (Nakamura, 1992). Involvement of these generalized functions may be ascribed at least in part to the difficulties in attaining consistent results between various primarily numerical methods when assessed on the basis of equal accuracy. For this purpose, Wagner's fundamental solution stands as a critical test. Having obtained the solution to the entire vortex distribution, we can proceed, heeding Principle (iii), to calculate the net force and moment of force acting on the lifting surface Sb(t) by integrating the pressure distribution of Eq. (6.22) over Sb(t). With respect to the solution at hand, a few remarks are in order. First, we note that to have body Sb(t) possess flexibility and move with time-dependent velocity U(t) is necessary for evaluating such maneuvering operations that are marked with strong nonlinear effects as starting and turning, bending and twisting of the lifting surface, stretching and shortening of wing span, operating duration and loading variations in the pronation-supination strokes, and other operations that are commonly displayed in flapping flight of birds and some insects as well as skates and rays in water. We further note that determination of the wake vortex y~ by using the generalized Wagner equation depends only on wing's quasi-stationary circulation F0 rather than the detailed bound vorticity distribution Y0(~, t), an interesting result that points to the brilliant decomposition of Fb = F0 + F~ introduced by von Kfirm~in and Sears (1938). In addition, the leading term of the solution to the bound vorticity [Y00(~, t) given by Eq. (6.42)] shows that the contribution of the local normal velocity U, of Sb to the differential lift has a weighting function proportional to {~/1 + ~/~/1 - ~ }, which clearly favors greater Un input near the trailing edge than near the leading edge. The feature of this relationship ought to help explain why birds in free flight have their wings more bent down near the trailing edge during the downward power stroke than near the leading edge.
VII. On Experimental Differentiation between Thrust and Drag in Fish Locomotion In Gray's (1936) pioneering study on aquatic animal locomotion referred to earlier, he found evidences of incompatibility between the mechanically implied frictional drag on high-performing aquatic animals and the biologically expected muscle power required of the observed high performance as compared with the known specific muscle power (per unit muscle weight) of warm-blooded animals,
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with the discrepancy found beyond reconciliation by a factor of severalfold. This perplexing conclusion stimulated extensive investigations over several decades with interests from both biological and mechanical fields. There are a great variety of viewpoints held for hypothetic resolution, including such ideas as seeking for low-drag possibilities (e.g., streamlined body shape; compliant skin; mucus secretion; active boundary-layer control, such as expelling water jets from fish gills) on the one hand, and on the other hand, seeking for improved ways of measuring the drag on animal (e.g., in accelerating mode or decelerating in quiet coasting of life specimens). However, the collected data on drag coefficient Co of fish measured under various conditions show such a wide scatter (by a few to tens of times the Co of a fiat plate at comparable high Reynolds numbers; see, e.g., Gray, 1968, and Webb, 1975, for survey) that no sound conclusion seems feasible. In addition, there is also the enlightening assumption that the drag on fish during swimming is different from the drag on the same specimen (or its ideal scale model) that is not self-swimming, such as being held with fixation, being towed, or even in quiet coasting. We refer the reader to the extensive references in the literature (e.g., Lang, 1975; Nagai, 2000). On this issue of differentiating between the thrust and drag involved in animal swimming, we propose that the foremost criterion is to determine whether the net force (including all the forces) acting on the specimen in question is either zero (said to be in State LMO) or nonzero (in State LM1), and, when body rotation is involved, whether the net torque acting on the specimen is zero (in State AMO) or nonzero (in State AM1). It is then obvious that is State LM0 of the linear momentum, the thrust and drag are equal and opposite to each other, but not in State LM1; similarly we can conclude in regard to the propulsive and resistive torques concerning the angular momentum when body rotation is present. Based on this criterion, there is of course no validity in taking the drag measured in State LM1 (of a specimen that physically is an open system) to imply the same drag in State LM0 (of a self-propelling specimen that momentumwise is a closed system). The reason for this criterion is simple: It is because the pressure distributions over the body surface in these two states are fundamentally different, so therefore are all the boundary-layer properties. Furthermore, it is essential to realize that the validity of this criterion is based on Newton's first law (for which this is a sophisticated applicationmnamely, to animated bodies) and is true regardless of what the Reynolds number of the swimming motion in question may be. Let us explore the physical significance of this issue for the case of low Reynolds number regime because the distinction between the two dynamic states of microorganism locomotion is even more dramatic. Here, the two photographs in Fig. 3, taken by Keller and Wu (1977) for comparative studies by visualization, vividly exhibit this distinction between two drastically different types of streamlines, one
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FIG. 3. Streak photograph of streamlines produced by (A) a freely swimming Paramecium caudatum, at left, and (B) a dead specimen of Paramecium caudatum sedimenting under gravity at right. (Adapted from Keller and Wu, 1977.)
for a freely swimming microorganism, Paramecium caudatum (on the left) and the other for an inert impermeable specimen (immobilized, slightly denser than water) falling in water under the influence of gravity, which is an external force (on the fight). On theoretic argument, the inert specimen sedimenting in fluid exerts on the fluid a total force equal to its weight in the fluid. Applying the low-Reynolds number (in this case, Re ~ 10 -3) fluid mechanics, this body action on the fluid can be represented by a distribution of stokeslet (standing for a point force on fluid) and its higher-order poles over the body, with the net stokeslet strength equal to the total force. As the stokeslet is a long-range singularity, with its induced velocity falling off inversely proportional to the distance from it, this singular force moves the fluid from far behind to far ahead, as clearly displayed by the streamlines streaking past the body. In contrast, the freely swimming specimen would still need a (different) stokeslet distribution to represent the ciliary forcing strokes, but of zero net strength, as required by Newton's first law. Accordingly, the effects of the stokeslets appear to stop right at the enveloping edge of the ciliary layer, leaving the exterior flow invariably irrotational at all times, a flow field that can always be represented by a distribution of mass dipoles and its higher-order poles. Nevertheless, the stokeslets, even with zero net, are playing a vital role in generating at the edge of the ciliary layer a differential velocity distribution that is required just to match exactly the exterior irrotational flow so that the flow velocity is continuous everywhere (in and out the ciliary layer) and can be said to satisfy the
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(. FIG. 4. The streamlines in the laboratory frame past a prolate-spheroid of eccentricity 0.9, with the body in (A) self-propelling motion at left, and (B) inert translating under gravity, at right. (Adapted from Keller and Wu, 1977.)
no-slip condition for the exterior irrotational flow at the beating ciliary envelope. It is, therefore, remarkable to observe irrotational flows to manifest at such a small scale (about 50-150/zm in length for microorganisms) with such a marvelous fit with the strongly viscous flow inside the ciliary layer. It is no less important to note that although irrotational flows are also solutions of the Navier-Stokes equations for viscous fluids, and therefore possess viscous stresses, but these stresses acting over a (closed) body surface of arbitrary shape always result in zero net force, as can be readily shown to hold for incompressible viscous fluid with constant viscosity. With the preceding observation, Keller and Wu (1977) pursued in parallel a theoretical study using a prolate ellipsoid to simulate the paramecium body shape. The corresponding results are shown in Fig. 4, one for the streamlines around the body in incompressible irrotational flow for modeling a self-propelling body in the laboratory frame (on the left), and the other for the streamlines around the body in Stokes flow (on the right) for comparison with an inert body translating under an external force (viz. gravity). The comparison between theory and experiment is strikingly similar, notwithstanding the discrepancies that the boundary conditions used in the theory are at variance with the experiment due to the proximity of glass plates and sealing gel used for making microscopic observations (for more details of the experiments, see Keller and Wu, 1977). In regard to energetics, a topic we explore next in Section VIII, we note that despite the remarkable achievement by ciliates in manufacturing a perpetual zero net force environment while sustaining impressive swimming speed (easily from tens
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to scores of body lengths per second), their cilia do have to expend nonvanishing work on the viscous fluid outside at the rate of the local flow velocity at the ciliary layer edge (in addition to the work on the fluid within the layer). For instance, for a spherical ciliate of radius R swimming at velocity U in water of viscosity #, its rate of working can be found to be 127r# U2R. With reference to the intrinsic force fo -- # v (pertaining to the fluid, # = p v being the dynamic viscosity coefficient of the fluid of density p) working at the same rate, their ratio gives a specific energy cost for the ciliate specific energy cost-- 12zrlzUZR/foU = 12zrUR/v = 12toRe,
(7.1)
which is very small since the Reynolds number Re