Advances in Applied Mechanics, Volume 33

SOLID MECHANICS

Advances in Applied Mechanics Volume33

Editorial Board Y. C. FUNG AMES DEPARTMENT UNIVERSIW OF CALIFORNIA, SAN DIEGO LA JOLLA, CALIFORNIA PAULGERMAIN ACADEMIE DES SCIENCES PARIS,FRANCE C.-S. YIH(Editor, 1971-1982)

Contributors to Volume 33 YAKOVBEN-HAIM N. A. FLECK

GUSTAVO GIOIA JOHNW. HUTCHINSON ORTIZ MICHAEL

2.s u o

SOLID MECHANICS

ADVANCES IN

APPLIED MECHANICS Edited by John W Hutchinson

Theodore Y.Wu

DIVISION O F APPLIED SCIENCES HARVARD UNIVERSITY CAMBRIDGE, MASSACHUSETTS

DIVISION OF ENGINEERING A N D APPLIED SCIENCE CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA

VOLUME 33

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9 8 7 6 5 4 3 2 1

Contents vii

CONTRIBUTORS

ix

PREFACE

Robust Reliability of Structures Yakou Ben-Haim I. Introduction 11. Convexity and Uncertainty 111. Truss with Uncertain Static Load IV. Geometric Imperfections: Axially Loaded Shell V. Dynamic System: Lifting Devices VI. Modal Reliability VII. Fatigue Failure and Reliability with Uncertain Loading VIII. Reliability of Mathematical Models IX. Summary References

1 4 6 10 15 21 25 34 39 40

Compresssive Failure of Fiber Composites N. A. Fleck I. 11. 111. IV. V. VI.

Introduction Competing Failure Mechanisms in Composites Compressive Strength of Unidirectional Composites Due to MicrobucMing Propagation of a Microbuckle in a Unidirectional Composite The Notched Strength of Multi-axial Composites Directions for Future Research Acknowledgments References

43 44 62 94 103 109 113 113

Delamination of Compressed Thin Films Gustauo Gioia and Michael 0rti.z 120 122 132 163

I. Introduction 11. Experimental Background

111. Folding Patterns as Energy Minimizers IV. Film Morphologies V

Contents

vi V. Conclusion Acknowledgments References

187 188 188

Motions of Microscopic Surfaces in Materials

z. suo I. Introduction 11. Interface Migration: Formulation 111. Interface Migration Driven by Surface Tension and Phase Difference

IV. Interface Migration in the Presence of Stress and Electric Fields V. Diffusion on: Interface Formulation VI. Shape Change Due to Surface Diffusion under Surface Tension VII. Diffusion on an Interface between Two Materials VIII. Surface Diffusion Driven by Surface- and Elastic-Energy Variation IX. Electromigration on Surfaces Acknowledgments References

194 196 209 222 235 245 26 1 261 219 289 289

Strain Gradient Plasticity N. A. Fleck and J. W Hutchinson I. Introduction 11. Survey of Strain Gradient Plasticity: Formulations and Phenomena 111. The Framework for Strain Gradient Theory

IV. FlowTheory V. Single-Crystal Plasticity Theory Appendk J , Deformation Theory and Associated Minimum Principles Acknowledgments References

AUTHOR INDEX SUBJECT INDEX

296 301 333 341 349 355 358 358

362 369

List of Contributors

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

YAKOVBEN-HAIM (11, Faculty of Mechanical Engineering, TechnionIsrael Institute of Technology, Haifa 320001 Israel N. A. FLECK(43, 2951, Engineering Department, Cambridge University, Trumpington, St., Cambridge, CB2 lPZ, UK

GUSTAVO GIOIA(1191, Division of Engineering, Brown University, Providence, Rhode Island 02912 JOHNW. HUTCHINSON (2951, Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138 MICHAELORTIZ(1191, Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California 91125 Z. Suo (1931, Mechanical and Environmental Engineering Department, Materials Department, University of California, Santa Barbara, California 93106

vii

This Page Intentionally Left Blank

Preface

Five articles on recent developments in solids and structures comprise Volume 33 of the Advances in Applied Mechanics. Each chapter is a mix of field survey and new work. The topics include structural reliability, failure modes of composites and thin films, the mechanics of micro-structural evolution, and strain gradient plasticity. Yakov Ben-Haim writes on a new formulation of robust reliability of structures. Quoting from the author: “The current standard theory of reliability is based on probability: the reliability of a system is measured by the probability of no-failure. ... In this paper we will describe a different formulation. We measure the reliability of a system by the amount of uncertainty consistent with no-failure. A reliable system will perform satisfactorily in the presence of great uncertainty. Such a system is robust with respect to uncertainty, and hence the name robust uncertainty.” N. A. Fleck surveys much of the recent research on the compressive failure modes of fiber-reinforced composites. Compressive failure is the Achilles heel of these materials, and a major research effort has been underway the past few years to characterize this behavior. This article makes it obvious that there has been considerable progress toward this goal. There are many interesting aspects to the subject, combining microand macro-scale phenomena. Fleck has made many contributions to the subject, both theoretical and experimental, and new work on the problem area by the author is contained in Chapter 2. Buckling delamination of thin films under biaxial compression is the subject of Chapter 3 by Gustavo Gioia and Michael Ortiz. This is a fascinating phenomenon of technological importance giving rise to intriguing delamination patterns, many of which are illustrated in the chapter. Buckling instability and interface fracture interact nonlinearly to produce the unusual patterns which spread from initially debonded sites. The authors have developed a mathematical approach which is elegant in the way it simplifies one of the most difficult nonlinear aspects of the problem. “Motions of Microscopic Surfaces in Materials” is the title of Chapter 4 by Z . Suo. This is a relatively new subject area for mechanics, and Suo ix

X

Preface

provides us with an unusually comprehensive treatment of the subject. Much in the article is published here for the first time. The author considers problems such as the change in shape and size of grains and voids due to diffusion and other mass transport mechanisms for materials subject to various driving forces as diverse as stress and electron wind. While qualitative approaches to some of these problems goes back many years, the author has now placed the entire subject area within a unified quantitative framework. Furthermore, he has shown how the framework lends itself to numerical analysis using finite element methods. For his second contribution to this volume, N. A. Fleck is joined by one of the Editors, John W. Hutchinson, in producing the article “Strain Gradient Plasticity” which comprises Chapter 5 , the last of the book. The first half of the chapter discusses applications of strain gradient plasticity to problems at small scales where size effects are observed which cannot be accounted for by conventional plasticity. Experimental data and new experiments needed to establish and calibrate the theory are reviewed. Existing data for metals indicates that a material-length scale on the order of several microns must be introduced into the constitutive law. The second half of the article lays out details of the theory, along with variational principles, and extends the theory to single crystals. John W. Hutchinson and Theodore Y. Wu

ADVANCES IN APPLIED MECHANICS, VOLUME 33

Robust Reliability of Structures YAKOV BEN-HAIM Faculty of Mechanical Engineering Technion-Israel Institute of Technology Haifa. Israel

...................................... 11. Convexity and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Truss with Uncertain Static Load. . . . . . . . . . . . . . . . . . . . . . . . . I. Introduction

............... .........................

IV. Geometric Imperfections: Axially Loaded Shell V. Dynamic System: Lifting Device

.................................. Fatigue Failure and Reliability with Uncertain Loading . . . . . . . . . . Reliability of Mathematical Models . . . . . . . . . . . . . . . . . . . . . . .

VI. Modal Reliability. VII. VIII.

IX. Summary

1

4 6 10

15 21

25 34

........................................

39

.......................................

40

References

I. Introduction

Though the Twentieth century began with the promulgation of the profoundest deterministic theory since the Seventeenth century, Einstein remained an anomaly; scientific thought for the past hundred years has been mainly influenced by concepts of uncertainty. Also, just as the early years of the century witnessed profound revision of the philosophical understanding of mathematical thought, so too has our understanding and treatment of uncertainty broadened and changed enormously. The technological applications of the mechanical sciences have followed suit and uncertainty thinking has won a place in the engineer’s practice. The major application is reliability: its assessment and attainment in mechanical design. 1 ADVANCES IN APPLIED MECHANICS, VOL. 33 Copyright 0 1997 by Academic Press. All rights of reproduction in any form reserved. 0065.2 165 / 97 $2 5 .OO

2

Yakov Ben-Haim

In accord with the diversification of uncertainty models that have emerged in recent decades, are a variety of reliability theories. In this paper we concentrate on one approach: robust reliability based on convex models of uncertainty. The comparison of alternative possibilities is avoided because this tends to be polemical rather than technological. The choice of an uncertainty model is, to some extent, a matter of taste. Also, some comparison is to be found elsewhere [5,6, 15, 161. Finally, it is too early for a far-reaching analysis of the merits of alternative theories of reliability because the monopoly of classical probabilistic reliability has only recently been challenged. Some eligible theories, such as reliability based on a nonstandard probability logic like fuzzy theory, have yet to be thoroughly developed for applied mechanics. To rely on something means to have confidence based on experience. This is a plain English word that has a classical etymology and it has carried this meaning long before engineers started thinking scientifically or scientists starting thinking probabilistically. Reliability rests on two more primitive concepts: performance and uncertainty. Still speaking lexically and not technically, we can rely on something when, despite uncertainties, its performance is acceptable. We now ask for a quantitative theory that reflects the intuitive idea of reliability. The current standard theory of reliability is based on probability: the reliability of a system is measured by the probability of no failure. This approach is exceedingly useful and has been developed in recent decades by many able authors. In this paper we describe a different formulation. We measure the reliability of a system by the amount of uncertainty consistent with no failure. A reliable system will preform satisfactorily in the presence of great uncertainty. Such a system is robust with respect to uncertainty, and hence the name robust reliability. On the other hand, a system has low reliability when small fluctuations can lead to failure. Such a system is fragile with respect to uncertainty. The relevance of noise-robustness to the quality of technological systems is widely recognized. Consider for example the comment by Taguchi et al. [18, p. 31: The broad purpose of the overall system is to produce a product that is robust [italics in the original] with respect to all noise factors. Robustness implies that the product’s functional characteristics are not sensitive to variation caused by noise factors.

Robust Reliability of Structures

3

In robust reliability, a system has high reliability when it is robust with respect to uncertainties. It has low reliability when even small amounts of uncertainty entail the possibility of failure. A similar idea is found in robust adaptive control, where one seeks a control strategy which will perform acceptably in the presence of considerable uncertainty. Robust control has been motivated in large measure by severely limited information about the uncertainties, usually system-model imperfections, which characterize complex-controlled systems such as aerospace vehicles, power, or chemical plants.’ Robustness-to-uncertainty is a useful guideline in formulating control laws in the absence of detailed probabilistic information [l, 131. Finally, self-learning systems are notable for their ability to adapt themselves to uncertain environments. This is important in on-line learning precisely because this adaptability enhances the reliability of the system despite the uncertainties accompanying its operation. Robustness-to-uncertainty is a natural starting point for a theory of reliability. It is not the only possibility starting point, but it is a fruitful one, well suited considering the limited information available about the uncertainties of mechanical structures and devices. We will identify three primary components in our analysis of the robust reliability of mechanical systems: (1) a model of the mechanics, (2) a model of the uncertainties, and (3) a criterion of failure. For items (1) and (3) we will employ standard mechanical and physical theories. For item (2) we will use convex models of uncertainty. The relation between convexity and uncertainty, and the theoretical and practical aspects of convex models, are developed extensively elsewhere [3, 91, but a brief description is included here in Section I1 for convenience. Following that discussion, we provide a sequence of examples of robust reliability analysis, beginning in Section I11 with a heuristic example of the reliability of a truss supporting a platform bearing uncertain static loads. In Section IV we analyze the reliability of a cylindrical shell with uncertain geometrical imperfections that is subject to a static axial load. The third example in Section V deals with a dynamic system: a lifting device subject to time-varying and spatially uncertain loadings. These three case studies demonstrate the method of robust reliability analysis. In Sec-

’

“Logic, observation, and frustration, in what proportion is not at all clear, have forced us successively from our citadel of determinism to the, by now, fairly civilized fields of stochastic processes, and finally to the jungle of adaptive processes” [2, p. 2071.

4

Yakov Ben-Haim

tion VI we discuss the topic from a more general point of view by developing the idea of modal reliability: the assessment of the relative reliability of dynamic degrees of freedom of a structure. Up to this point, all the examples have dealt with linear, or linearized, mechanical models. In Section VII we analyze the robust reliability of an inherently non-linear phenomenon: structural degradation and failure by material fatigue resulting from uncertain, time-varying loads far below the yield limit of the material. Finally, in Section VIII we modify our subject a bit, and discuss the reliability of mathematical models of structures, rather than of the structures themselves.

11. Convexity and Uncertainty

Uncertainty: events occur, one after the other, accumulating into clouds, clusters, and patterns. Fragmentary information about the uncertain phenomena can be used to characterize the clustering of uncertain events. Without specifying anything about the probabilistic frequency-ofrecurrence of events, a convex model is a set whose structure is derived from the underlying properties of the event clusters. A convex model is a set of functions or vectors. Each element of the set represents a possible realization of the uncertain phenomenon. The usual engineering approach to formulating a convex model is to start with what we know about the phenomenon and to define the set of all functions consistent with that information. An an example, consider transient excursions in the vector f ( t ) of loads applied to a structure. Suppose that the nominal load history is known, f ( t ) , and that the cumulative energy of deviation from this nominal load is bounded. The set of all functions consistent with this information is a cumulative energy-bound convex model, defined as:

We may not know the value of a , and in robust reliability analysis, we do not need to know it. a is the uncertaintyparameter whose value determines the “size” of the set, the latitude of variation of the uncertain phenomenon, or the information gap between the anticipated nominal load and the loads which may occur in practice. The convex model defines a set

Robust Reliability of Structures

5

of functions consistent with particular prior information. It defines a cluster of events. Because we do not know the value of a , we will often think of a convex model as a family of sets: %( a 1 for a 2 0. Thinking in this way, we realize that every ,!,*-integrable function f ( t ) belongs to the family for sufficiently large values of a. The convex model is not a black-and-white division of events into “completely allowed” and “strictly forbidden.” Rather, the family of convex models %(a),a 2 0, arranges the space of all events in a particular order. It is not an order in terms of frequency of occurrence as in probability, but an order in terms of clustering. In the convex model of eq. (1) the events cluster around the nominal load profile, l’at the overall reliability:

h

=

h i ( & ) = &#i)

(59)

This is typical of the relationship between these three reliability indices, which is developed in more general terms [6, 71.

VI. Modal Reliability In the previous section we considered a dynamic system driven by uncertain inputs and subject to uncertainty in the failure condition. Figure 7 illustrates the general structure of the situation. The uncertain input set, %(a,),results in an uncertain output or response set, M a , ) .Both sets are scaled by the uncertainty parameter, a , , whose value is not known and which expresses the degree of uncertainty in the input and response. The system fails if the response set intersects the failure set, f l a f ) , which is itself scaled by an uncertainty parameter, af. The condition for no failure is disjointness of the response and failure sets: N a , ) n f l a f >=

0

(60)

So, for example, the values of input and failure uncertainty in Figure 8 do not allow failure to occur. But in robust reliability, we are not interested in the disjointedness or intersection of response and failure sets for particular values of the uncertainty parameter a, and af.In fact, we never need to know engineering or practical values of these quantities. The concept of robust reliability is to measure the reliability in terms of the amount of uncertainty consistent with the condition of no failure: how large can a, and af be before failure becomes possible? We evaluate the input reliability by keeping af fixed and allowing 9 ( a , ) to expand with a, until S'(a,) and N a f )are tangent. The value of a, at which this occurs is the input reliability index, &,( af).The failure reliability

FIG.7. Input, response, and failure sets.

22

Yakov Ben -Haim

FIG.8. No-failure condition: disjointness of the response and failure sets.

index, h f ( a i )is , evaluated similarly by fixing aiand expanding Sa,) until the sets are tangent. The overall reliability index, &, is obtained by expanding the uncertainty parameters together as a single parameter until tangency is reached. The response and failure sets we are considering are convex models of uncertainty, so they are convex sets. If they are closed and at least one of them is also bounded (this is almost invariably the case), then they are disjoint if and only if a hyperplane separates them [14]. In Figure 9 we show tangent response and failure sets, with a separating hyperplane whose orientation is specified by the vector w. This vector has a particularly interesting and useful interpretation, which we will explain in the remainder of this section.

FIG.9. Tangent response and failure sets with a separating hyperplane.

Robust Reliability of Structures

23

We have explained that robust reliability is evaluated by expanding one or both sets until they are tangent. This expansion is in fact isometric: occurring similarly and simultaneously in all directions. Consider for instance the tangent two-dimensional sets in Figure 10, in which the separating hyperplane is parallel to the x,-axis, so w = (1,O). Let us suppose, as a “thought experiment,” that we could expand S?(ai)along the x2-axis alone, as suggested by the dashed lines. This expansion could go on indefinitely without encroaching on the failure set; without entailing the possibility of failure. We can provide a qualitative interpretation of this situation by saying that degree-of-freedom x 2 can “tolerate more uncertainty” than x1; that x2 is “much more reliable” than xl.The elements of the vector w = (1,O) of the separating hyperplane indicate, when inverted, the relative reliability of the two dofs. Now let us return to Figure 9, which shows a situation where the separating hyperplane is inclined at 45” to the two axes, so w = (1,l). In this case, the two dofs contributed equally to the reliability. In general, the response and failure sets are mutidimensional, rather than two-dimensional as in our illustrations, and the separating hyperplane is likewise of the same dimension. Nonetheless, the elements of the vector w,which defines the tangent hyperplane separating the response and failure sets, are inversely proportional to the “relative reliability” of the degrees of freedom of the corresponding axes. This is a qualitative statement, an interpretation, but it provides useful insight into the relationship between dynamics and reliability.

r

4 FIG.10. Tangent response and failure sets with a vertical separating hyperplane.

24

Yakov Ben-Haim

To illustrate the idea, suppose one has used modal coordinates in Figure 10. Then the vertical orientation of the separating hyperplane implies that the second mode is much more reliable than the first. In other words, failure will tend to occur in dynamic maneuvers dominated by the first mode rather than by the second. We have used the term modal reliability to refer to the general idea of the relative reliability of the dynamical degrees of freedom of the system. To demonstrate the relation between the relative reliabilities in two different coordinate systems is a straightforward matter. If the response sets are convex and at least one is closed and bounded, then they are disjoint if and only if a hyperplane separates them. We have used this fact in developing the idea of modal reliability. This phenomenon of hyperplane separation can be expressed algebraically. The disjointness relation (60) holds if and only if there is a vector w such that: max w'r

I

min w'f f € 9 7 a,)

rE9(a,)

(61)

This vector w is, as before, perpendicular to the separating hyperplane and determines its orientation. The elements of w express the modal reliabilities. Now consider an arbitrary orthogonal transformation applied to both the response and the failure set: p = HTr

and

4 = H'f

(62)

Let P(cri) and @(af)be the corresponding transformed sets. Also define a new vector u = H T w . Thus, u'p = wTr and u'$ = w'f. Consequently, eq. (61) is equivalent to:

So w separates 9 from .F if and only if u separates P from @. But this then means, that given the coordinate reliabilities w of one coordinate system, we can calculate the coordinate reliabilities of any other coordinate system which is related by an orthogonal transformation. For instance, given modal reliabilities we can calculate the reliabilities of the displacement coordinates using the modal matrix. Furthermore, we can always choose a transformation matrix H such that the new vector u has one unity element and all the rest are zeroes. In this new coordinate

Robust Reliability of Structures

25

system, one degree of freedom dominates the reliability. This is further developed elsewhere [6,7].

MI. Fatigue Failure and Reliability with Uncertain Loading The mechanical properties of a solid structure degrade slowly in time when subjected to cyclical loads at stress levels well below the yield stress of the material. This high-cycle fatigue can ultimately lead to failure of the material by the evolution of many small cracks or the eventual catastrophic growth of a single large crack. The classical laboratory assessment of damage evolution by fatigue is represented by the S-N curve, as in Figure 11, which expresses the amplitude, S , of the cyclic stress versus the number, N , of load cycles to failure. Such curves are measured in idealized laboratory conditions, where the load cycles are carefully controlled. In practice, of course, the load history of a real engineering system is uncertain and quite variable, unlike the perfect harmonic loading used in measuring an S-N curve. A bridge, for instance, experiences complex repeated loads due to traffic, as well as variables such as wind or waves. In this section we will develop a phenomenological model for damage evolution and failure by high-cycle fatigue, and we will represent uncertainty in the load history with a convex model. This will serve as the basis for our analysis of the reliability with respect to fatigue failure. We will stress the connection between the dynamics of large-dimensional

Load cycles to failure. N FIG.11. Schematic S-N curve for harmonic loading.

Yakov Ben-Haim

26

structures and a phenomenological model for the microscopic process of fatigue damage. A. DAMAGE EVOLUTION Consider an N-dimensional solid structure subjected to uncertain timevarying load and displaying small damped vibrations that are described by: MX(t)

+ C i ( t ) + K x ( t ) = Bf(t)

(64)

where M , C, and K are constant mass, damping, and stiffness matrices respectively, and x(t) is the deflection and f(t) is the load. The natural frequencies of the undamped system are w1 ,. . ., w N . We assume these are all greater than zero. The corresponding mode shape We vectors are the columns of the modal matrix @ = I + ] , . . . , assume the damping matrix is orthogonalized by the mode shapes, and that their normalization is: = p, Sij ,

+TK+j

=

w?p, Sij , +TC+j =

21i wi piSij , (65)

where Sij is the Kronecker delta function. For zero initial conditions and sub-critical damping in each mode, li2< 1, the displacement vector can be expressed as:

and where V ( t )is a diagonal matrix whose ith diagonal element is:

dl

The ith-mode damped natural frequency is wid = wi - Ci2. The damping and stiffness matrices, C and K, change slowly as damage accumulates in the fatigue process as a result of repeated loading. A duty cycle is a short duration T during which the system undergoes many vibrations but only a very small increment of damage accumulates. We will assume that C and K are constant over durations T, and we will evaluate their long-range damage evolution by evaluating the small damage increments and revising the model parameters at the end of each duty cycle. We are not assuming that the system evolution is described by linear

Robust Reliability of Structures

27

dynamic equations. We only assume that the oscillation-dynamics are linear during each short duration T . Numerous micromechanical models have been developed for understanding the evolution of load-related damage in structures. In particular, the evolution of fatigue damage has been fruitfully associated with vibroacoustical dissipation of energy [4, 10, 11, 171. We apply this idea to the vibration model of eq. (641, in which energy is introduced by the load, flows through the inertial degrees of freedom, and is dissipated. Some of this dissipated energy results in damage that leads to fatigue. The dissipative force acting on the system at any instant is Cx [N]. An infinitesimal displacement dx does i T C d x [J] of work on the system. This displacement occurs in a time interval dt, so the rate of work on the system is i T C d x / d f = i T C x [W]. So, the accumulated energy loss in a short-duty cycle of duration T , during which the physical parameters p = ( C , K ) of the system remain unchanged, is:

The associated damage is modelled here as: 6 ( p )= y[E(p)lY

(69)

where y and I, are positive constants. We need expressions for variation of the model parameter matrices C and K with the total accumulated damage. The dominant assumption which we will make is that, while the accumulated damage, A, depends on the load history, f(t), the matrices C and K depend only on the magnitude of the accumulated damage, With this and some additional assumptions [4], one can show that the damage evolution must be exponential: C(A)

=

e-pcAC, K ( A )

=

e-PkAK

(70)

The parameters p, and Pk are scalars that need to be determined either from empirical data or from fundamental physical considerations. Failure occurs when A reaches the critical value, A c r . B. UNCERTAIN LOADH~STORIES

The input which drives the system and generates the dynamical fatigue aging, is subject to uncertainty. We will model this uncertainty with a band-limited cumulative energy-bound convex model. Assume that the

Yakov Ben-Haim

28

input can be represented during a duty cycle, [O, TI, by a sum of harmonic functions: k,rt

c “f

f ( t >=

[ a , cos

7

(71)

1

,=

where k , is an integer wave number, and a, and b,, are the Fourier coefficient vectors of the nth temporal mode of the input. These coefficient vectors are uncertain. Concatenate them in a vector & * = (a:, , . . ,a:f, br,. . . , bz,). Eq. (71) can be compactly expressed as: f(t)

=

(72)

Q(t)5

where Q ( t ) is a known matrix of trigonometric functions. The known nominal input, f(ct), has Fourier coefficient vector 2, and the cumulative energy of deviation of the actual inputs from the nominal inputs is bounded in each duty cycle. The band-limited cumulative energybound convex model is defined as the following set of functions:

i

Z ( a > = f(t):f(t)

=

and

Q(t)5

2

iT[f(t)

-f(t>l*[f(t)

I

- f ( t ) l d t I a*

(73)

The available inputs are vector functions, f(t), constrained to the wave numbers, k , , and with amplitudes, (a,, b,,), which are uncertain but constrained by the requirement that the cumulative energy of deviation from the nominal input does not exceed the uncertainty parameter, a*.Of course, the value of a* need not be known. The integral inequality in eq. (73) can be reduced to an algebraic inequality by exploiting the orthogonality of the trigonometric functions, resulting in the following equivalent definition of the convex model: Z ( a ) = (f(t):f(t)

=

Q ( t I 5 , and (5- $)*(5-

5 ) 5 a’)

(74)

C. MAXIMUM INCREMENTAL DAMAGE We can now evaluate the maximum increment of damage that can accrue during a duty cycle, for any load-history allowed by the convex model, %(a 1. The maximum increment of damage will be a function of the

Robust Reliability of Structures

29

uncertainty parameter, a. This will be important in establishing the fatigue reliability. Substituting eq. (72) into eq. (66) and differentiating with respect to t , one finds that the energy loss in a duty cycle, eq. (681, becomes:

z

(76)

where the matrix Z is real and symmetric and depends on the current values of the damping and stiffness matrices, C and K . Thus, the increment of energy loss E during a duty cycle is quadratic in the Fourier coefficients of the uncertain load history. The Fourier coefficients of the load are constrained by the convex model, so we must seek the maximum of E ( p ) on the set %(a>:

Combining this with eq. (69) we have the greatest damage that can occur in a duration T :

The greatest energy loss in a duty cycle is found by solving the following optimization problem: max tTZ6 subject to

(6- z)T(6-

5 ) 5 cr2

(79)

It is readily shown that the maximum will occur on the boundary, so the constraint becomes an equality. With the change of variables 8 = 6 - 8, the optimization becomes: max(8

+ i ) T Z ( 8+ 8 )

subject to

6% = a2

(80)

Adjoining the constraint to the objective function with an unknown Lagrange multiplier, A, and differentiating with respect to 8, results in the following condition for an extremum:

Yakov Ben-Haim

30

Z is a real symmetric matrix and so it is diagonalized by an orthogonal matrix: Z = H r H T , where r is the diagonal matrix of eigenvalues of 2. Eq. (81) can be rearranged as:

[r - A I ] H T 6= - r H T 5

(82)

We first consider the special case that $, = 0, for which one finds that the maximum increment of damage in a duty cycle is proportional to the maximum eigenvalue of 2:

&p)

=

a* maxeig ~ ( p )

(83)

Let % p ) denote the maximum eigenvalue in eq. (831, which depends on the current values of the model parameter matrices, C and K. Now consider the solution of eq. (82) with 2 # 0. The Lagrange multiplier, A, must be chosen to satisfy the constraint in eq. (80). We will assume that this will occur for A different from an eigenvalue of 2. When h # ri,eq. (82) can be inverted as:

=

-(I-

iz)5 1

The constraint, a 2 = OT6, implies that A must be:

where we have defined y , = 5 T Z n5. Combining eqs. (80), (89, and (86) the greatest increment of dissipated energy in a single duty cycle becomes: Y3

E(p)= 7 A

(87)

where we choose the lesser of the two A values in eq. (86) to get the maximum energy dissipation. The extremal energy loss depends on the damping and stiffness parameters p = ( C , K ) in the current duty cycle. These appear implicitly on the right of eq. (87) in the matrix Z. Equivalently, the greatest increment of energy loss can be calculated given the current total damage level, A, using relations (70).

Robust Reliability of Structures

31

D. THELEAST-LIFETIME RECURSION Starting from a given amount of damage, A, our aim is to evaluate M A ) : the least number of duty cycles that result in failure, driven by a convex model of load histories. This will be a function of the uncertainty parameter, a. From this dependence on a, we will derive the robust reliability to fatigue. Failure can be variously defined, not necessarily as an ultimate catastrophe. For example, one may choose the transition between different regimes of damage evolution, such as the transition from a microscopic-crack growth to growth of a single, dominant crack. The amount of damage is defined implicitly with respect to a reference model or a reference state of the material. Our analysis is based on the following assumptions: 1. Failure occurs when the damage reaches a critical value, A,,

.

2. The damage increases continuously from A to A,,. That is, the increments of damage accumulating in each duty cycle are very small compared to the critical value, so that variation of the accumulated damage with age can be treated as a continuously increasing process. 3. Starting from a system whose accumulated damage is A, the least time to failure, M A ) , depends only on A.

These assumptions imply that M A ) can be calculated from a recursive relation [4], which we explain as follows. Suppose the number of duty cycles to failure, starting from damage level A,, - y , is n:

MA,,

-y

)

=

n

(88)

Then failure occurs after n + 1 cycles, starting from damage level ACTy - E , if the maximum damage increment in one cycle is sufficient to raise the level of damage to A,, - y. That is: MA,, - y - E )

=B

+1

if

A,, - y

- E

+ $(A,,

-y -

8)

=

A,, - y (89)

Defining A = A,, - y , these recursive relations for the least number of duty cycles to failure become: E =

N(A

-

E )

$(A -

=N(A)

E)

(90)

+1

(91)

32

Yakoo Ben-Haim

The recursion begins at M A , , ) = 0. N ( A ) can be evaluated numerically as a function of A using relations (90) and (91) and the expressions for s^ derived in Section C , eq. (78) together with either (83) or (87). E. FATIGUERELIABILITY WITH COMPLEX UNCERTAIN LOADS

In the absence of load uncertainty, the fatigue-related damage level will reach its critical value after No duty cycles. When the load history is uncertain, the critical damage level may be reached earlier. For the purpose of evaluating the robust reliability, we will consider the system to fail if the number of duty cycles to critical damage is less than a critical threshold value:

N 5 N,,

(92)

The robust reliability is the upper bound of the load uncertainty that the system can tolerate without violating condition (92). N,, can be any value up to No, the number of duty cycles required to accumulate the critical level of damage in the absence of load uncertainty. When N,, < N,), it is considered as “acceptable,” with critical damage-accumulation in less than No cycles. Now we consider the fatigue reliability with complex uncertain loads, represented by the convex model of eq. (74), with one simplification. We let 5 = 0, which means that the nominal or “typical” load is zero. Nevertheless, we can still choose one of the wave numbers, k , , to be zero, implying that during each duty cycle the load vector varies around a constant value possibly different from zero. This constant value is uncertain and free to vary from one duty cycle to another. The greatest possible increment of energy dissipated in a single duty cycle is expressed by eq. (83). Combining this with eq. (78), the greatest increment of damage in a duty cycle is: 8(p)=

ykY2,

(93)

Let primes denote differentiation with respect to A, and approximate N(A - E ) = M A ) - N ’ ( A ) E . Now combining eqs. (901, (911, and (93) leads to:

Robust Reliability of Structures

33

i,the maximum eigenvalue in eq. (83), depends on the damage level, A. However, we can circumvent this in the following manner. Let NJA) be the least number of duty cycles to failure at a reference level of uncertainty, a o . That is, N o ( A ) is evaluated by solving recursion relations (90) and (911, for a given value, a o , of the uncertainty parameter. Using eq. (94), the derivative of No can be expressed:

Combining eqs. (94) and (95) leads to:

An integration from A,, to A leads to:

This is the least number of cycles to failure, as a function of the loaduncertainty parameter, a. As a increases (that is, as the uncertainty increases), the fatigue process can reach the critical damage level earlier, as expressed by the fact that N ( A ) decreases. Equating the right-hand side of this relation to the critical number of cycles, N,, , leads to the following expression for the robust reliability, the greatest tolerable uncertainty:

& is the greatest tolerable uncertainty while requiring that the critical damage level not be reached in fewer than N,, duty cycles. Recall that I, is the exponent in eq. (69), relating the energy dissipation to the damage increment. Eq. (98) shows how the reliability varies with the critical number of duty cycles, N,, ,and how it scales with the damage exponent, I,. When the load is pure harmonic excitation, the damage exponent is related to the slope of the S-N curve [4], which can be measured in the laboratory.

34

Yakoo Ben-Haim

VIII. Reliability of Mathematical Models The applied mechanician builds not only structures but also models, mathematical constructions, which he uses for design and safety analyses. These models underlie decisions. In this section we will discuss the reliability of these models in terms of robustness-to-uncertainty of the decisions. In the previous sections we measured the degree of reliability of a mechanical system in terms of the amount of uncertainty it can tolerate without failure. A system is reliable if it is robust to noise; it is unreliable if it is fragile to small, unknown variations. Applying this to mathematical models, we will say the model is reliable if the decisions that rely on the model are robust to uncertainties. The degree of reliability of the model is measured by the amount of uncertainty consistent with stability of the decision. The robust theory of mathematical models with multi-hypothesis decisions is developed elsewhere [7]. Here we will illustrate the basic idea with two examples. Mass and Stiffness of a One-Dimensional Mechanical Oscillator A small, mechanical device is subjected to a load that increases approximately linearly in time from zero. The load produces vibration in the device, and it is necessary that the acceleration of the device not exceed the critical value of ucr for a long duration, T . The vibration of the device is modelled as a one-dimensional undamped linear mass-spring system: mi(t)

+ Icr(t) = u ( t ) , d o ) = i ( 0 ) = 0

(99)

The equivalent mass and stiffness of the model are both uncertain, and we model their fractional uncertainties as:

where the chosen model values are m , and k , . The nominal slope of the ramp input is s, and the uncertain ramp input belongs to a slope-bound convex model:

For given uncertainty ai in the load, it is necessary to decide if the device will exceed its acceleration limit for a long duration, T. Is the mass-spring

Robust Reliability of Structures

35

model reliable based on the values (m,, k,), and is the decision robust with respect to uncertainties of the mass and stiffness, a,,,and f f k ? The degree of reliability of the model is expressed by the magnitude of the uncertainty parameters to which the decision is robust. That is, how large can a, and f f k be without altering the decision? The acceleration at time, t , in response to input, u , is: (102) where the displacement is:

and 0,” = k , / m , . The decision to be made is whether or not the acceleration exceeds the allowed value, ucr , throughout a duration, [O, TI:

Integrating eq. (103) by parts to replace u by u , and substituting this into eq. (102), results in: (105)

So, the maximum absolute value of the acceleration at time, t , for any allowed input occurs when ti switches between its extremal values as cos w(t - 7 ) changes sign. That is, the maximum acceleration occurs for:

+ ai, when

cos wo(r -

7)

2 0

(106)

u = s - a i , when

cos u,(t -

7 )

0 by comparison of (2.1) and (2.3). (ii) Fiber crushing occurs in preference to buckle delamination when the compressive strength associated with fiber crushing, uCcrush, is

Compressive Failure of Fiber Composites

59

less than the compressive strength given by (2.5) for buckle delamination. On writing q.crush = E c f c , where E is the longitudinal modulus of the composite and cC is the fiber-crushing strain, we deduce that fiber crushing occurs instead of buckle delamination Typically, EfC = 2% and so fiber crushwhen b / h < ~ / ( 2 ing dominates when b / h < 6.4. For most practical composites, the condition b / h < 6.4 is met and fiber crushing occurs in preference to buckle delamination. (iii) Fiber crushing occurs instead of plastic microbuckling when

6). p
6.4, the domain of fiber crushing is replaced by buckle delamination; the resulting map is of identical shape to that shown in Figure 9a but the axis c f c E / G is replaced by

The map shown in Figure 9b includes contour surfaces of constant ac/G.Each contour is composed of a set of orthogonal planes, as depicted in the figure. For example, in the region of plastic microbuckling contours

60

N. A. Fleck plastic micro buckling

f I ber crushing

buckle delamination

\

2.5,

L

G

elastic microb uckl ing

(a) plastic microbuckling

fiber Crushing

I

--__.-.---

I

buckling delominot ion

I

- - --

.

(b)

.C-

-\

G 1

FIG.9. (a) Three-dimensional failure map showing region of dominance of competing failure mechanisms; (b) Failure maps, with contours of strength added.

CompressiueFailure of Fiber Composites

61

of constant u,/C exist as planes normal to the $/y,-axis. The strength, u,/G, decreases with increasing $/y,, as given by relation (2.3). The maximum possible strength shown on the map is u,/G = 1; this is achieved by elastic microbuckling, but, as already discussed above, other failure mechanisms usually intervene and give a lower strength than this maximum value. A simpler two-dimensional version of the map is also instructive: it takes as axes the in-plane shear modulus of the composite G and the ratio of in-plane shear strength, T?, to fiber misalignment angle $, see Figure 10. In other words, we take as axes the long-wavelength limit of the Rosen formula (2.1) for the elastic microbuckling strength, and the Argon expression (2.2) for the plastic microbuckling strength. Contours of compressive strength are straight lines with a plateau value given by the onset of fiber crushing. Material data for a number of carbon fiber composites are included in the figure: in all cases failure is by plastic microbuckling. It is clear that an increased compressive strength would be achieved by increasing T? and by decreasing $, and that significant improvements in compressive strength may yet be made without the intervention of fiber crushing. The demand for improved toughness of composites has caused a progressive decrease in the shear strength of the matrix, so the only strategy available is to reduce the fiber misalignment $ from a value of 2-3" to

5000-

I

'.F

0 T800/924 c X T800/92Lc part cured 0 ASL/PEEK n EXAS HSI/DX6002 0 ASk/PEEK(T=77%) 8 ASLIPEEK (r=i320c) Graphite/epoxy A Carbon/carbon

I

I

+

Plastic

- - - - - - - - 2000 MPa

microbuckling

- -+-I000

-----I

0

2000

4000

6000

8000

MPa

I

10000

G (MPo) FIG. 10. Two-dimensional failure map, with test data added. Commercial polymer matrix composites fail by plastic microbuckling, as indicated.

N. A. Fleck

62

much lower values. The use of pultrusion in composite manufacture (whereby the fibers and partially cured matrix are pulled through a die) has shown itself to be a practical processing route for highly aligned fiber composites.'

111. Compressive Strength of Unidirectional Composites Due to Microbuckling

In this section a number of theoretical and experimental studies are summarized in order to elucidate various aspects of the microbuckling phenomenon. In Section A, the simplest estimates of initiation strength for microbuckling in rate-independent composites are determined by kinking theory: the fibers are taken to be inextensional and fiber-bending resistance is neglected. The kinking theory is then used to estimate conditions for microbuckling for a variety of composite constitutive laws: elastic-plastic, creep, visco-elastic, and low-cycle fatigue. In Section B, the effect of fiber-bending resistance on the infinite bandcollapse response is determined through the use of couple stress theory. A small strain theory is adequate to predict the initiation strength and the width of a microbuckle: the width is set by fiber fracture in bending. Finite strain calculations are used in order to estimate the post-collapse response, particularly when fiber fracture is not an issue. The calculations show that the post-collapse strength settles to a steady-state value, a,, associated with band broadening; this steady-state value can be calculated directly by a work calculation from kinking theory. The infinite bandbending theory is also used to estimate the compressive strength associated with random fiber waviness. This calculation uses digital signal techniques in order to calculate the probability density function of failure associated with a given power spectral density of fiber waviness. The compressive strength of the case of a finite region of initial fiber waviness is discussed in Section C. A two-dimensional finite element scheme is used, and couple stress theory is used to include the effects of finite fiber-bending resistance. Again, the fiber diameter sets the internal length scale of the microstructure. The main finding is that the compres-

'

Neptco Inc. have recently introduced a carbon fiber-epoxy composite GraphliteTM with a fiber misalignment of less than 0.5". The measured compressive strength is about 2.0 GPa (Neptco, 1994).

Compressive Failure of Fiber Composites

63

sive strength is given to a good approximation by kinking theory unless the region of initial waviness is small. The role of multi-axial loading in knocking down the axial compressive strength is addressed: the knockdown factors for a small region of fiber waviness are similar to those for an infinite band of initial waviness. 1. Assumed Collapse Response from an Initial Impe$ection in the Form of an Infinite Band

We begin by summarizing the qualitative details of rate-independent microbuckling for a composite with a pre-existing infinite band of imperfection in the form of fiber misalignment. Quantitative details are covered in subsequent sections. Assume that the fibers are uniformly misaligned by a constant angle 5 within the band in the stress-free initial configuration as shown in Figure 3. The normal to the band is taken to be at an angle p to the fiber direction. The fibers are assumed to possess a finite-bending resistance so that the additional fiber rotation 4 under load is continuous. Consider the collapse response for the geometry given in Figure 3, under an axial stress urn. The collapse response is presented in the form of remote axial stress versus maximum additional fiber rotation, &, in Figure 11. The deformation mode within the band is a combination of in-plane shear parallel to the fiber direction and direct-straining transverse to the fibers. Fibers within the band attain large rotations (of up to about 60" for the case p = 30") and so a non-linear constitutive response is appropriate for the smeared-out behavior of the composite within the band. Initially, fiber rotation occurs under an increasing remote stress, urn, as indicated by a typical point A of Figure 11. Note that the fibers within the band rotate by a greater amount than material outside the band. The additional rotation leads to geometric sofiening, which more than offsets the strain hardening within the band. With continued fiber rotation the load goes through a maximum (point B) and then decreases to a lower steady-state value as shown in Figure 11. The maximum is attained after only a few degrees of fiber rotation. Beyond the maximum load point, fiber rotation continues and the band continues to broaden, see point C of Figure 11. Eventually, the matrix strain hardens sufficiently within the band for the material to "lock-up" and continued end shortening of the structure is due to broadening of the band in the axial direction at a constant value of remote stress, up,point D. It is found experimentally that the lock-up state within the band is associated with a state of zero

N. A. Fleck

64

-

0.L

c

c

0.3 (300 -

G

0.2 -

0.1

.. A

0

I

10'

I

20"

I

I

30'

LOo

I

50'

I

60"

I

70'

@Ill

FIG. 11. Typical infinite band collapse response, with the role of fiber bending included. 4,,, is the maximum value of fiber rotation in the composite, for any given remote stress uT.

volumetric strain within the band: as the fibers rotate they first lead to dilation within the band, followed by compaction until lock-up occurs at vanishing volumetric strain (Chaplin, 1977; Evans and Adler, 1978; Sivashanker et al., 1995). The overall buckling response is akin to tensile drawing of a polymer, whereby a neck forms by geometric softening; subsequently, orientation hardening occurs within the neck and leads to steady-state neck propagation known as drawing. The total end shortening AT of the composite of finite length h is the sum of elastic shortening A E of the fibers and the end shortening A associated with fiber rotation; see Figures 12a and b. To an excellent approximation, the urnversus A collapse response may be considered to be the universal collapse curve, and the elastic end-shortening is given by A E = u m h / E L ,where EL is the longitudinal elastic modulus of the composite. The urnversus A, collapse response is sketched in Figure 12b for two different lengths h of composite: note that an increasingly strong snap-back behavior is predicted with increasing h . (Kyriakides et al., 1995) have confirmed this using detailed finite element calculations where they treat the fibers and intervening matrix as a series of discrete layers.) The

Compressive Failure of Fiber Composites

'

65

0.4 0

0.1

-

A

0.5

I

I

I

I

1

I

-

0.4 -

B'

-

--D'

0

2

L

I

I

6

0

D" 0 -

10

AT/d

(b) FIG. 12. Collapse response, presented in form of (a) grn versus end shortening A associated with fiber rotation, and (b) urn versus total end shortening A, associated with both fiber rotation and elastic axial compliance of the composite of height h.

66

N. A. Fleck

four representative stages of deformation A-D given in Figure 11 are also shown in Figures 12a and b. The description above of the collapse process represents the consensus of opinion from the work of Shih and co-workers (Liu et al., 1995; Moran et al., 1995), Kyriakides and co-workers (Kyriakides et al., 1995) and Fleck and co-workers (Sutcliffe and Fleck, 1994; Sivashanker et al., 1995). Liu et al. (1995) were the first to observe steady-state band broadening by testing a composite of sufficiently high-failure strain for fiber fracture not to intervene (IM7 fibers in APC-2 PEEK matrix). Experimental observations (Sutcliffe and Fleck, 1994; Sivashanker et al., 1995; Fleck et al., 1996) have been made recently of band broadening in fiber composites which exhibit fiber fracture. It is observed that fiber fracture is intermittent along the length of a microbuckle band, and that the average traction carried by a microbuckle band is almost identical to that carried by a microbuckle band displaying no fiber fracture: the infinite band collapse response is hardly changed by the occurrence of fiber fracture.

A. KINKINGTHEORY As reviewed in Section II.B, the theoretical studies by Argon (1972) and Budiansky and Fleck (1993) have shown that microbuckling in polymer matrix composites is associated with a non-linear plastic response of the matrix. The analysis of Budiansky and Fleck (1993) for plastic microbuckling considers the effects of initial imperfections, plastic strain hardening, and combined remote shear stress and axial compression. Slaughter et al. (1992) extended the Budiansky and Fleck (1993) analysis to general multi-axial loading. We begin by summarizing the treatment of Slaughter et al. (1992). The following kinking theory is an infinite band calculation in the spirit of one-dimensional shear localization analysis. A uniform imperfection in the form of a finite fiber misalignment angle 3 is assumed within a band (see Figure 31, and the evolution of fiber rotation within the band is deduced from algebraic equations for continuity of traction and displacement at the band boundary. The kinking analysis allows for the determination of analytical formulae for the critical stress for microbuckling. As in most treatments of compressive kinking, the fibers in these calculations are assumed to be inextensional. This has the effect of shielding the matrix

Compressive Failure of Fiber Composites

67

from axial stress, and therefore the actual effect of this stress on matrix plasticity is not taken into account. Consider the collapse of a kink band inclined at an angle p to the main fiber direction, as shown in Figure 13. (It is observed experimentally that kink bands are inclined typically at p = 20"-30".) It is assumed that the fibers are inextensible and that uniform straining within the kink band is associated with a fiber rotation, 4. It is further assumed that initial fiber misalignment within the composite is represented by the angle, 8. Two Cartesian coordinate systems, ( e l , e,) and ( E ~ E,), , are defined such that el and e2 are parallel and normal to the fiber axes outside the kink band, and E~ and E~ are parallel and normal to the fiber axes inside the kink band. These two coordinate systems are related by el e,

+ 4 ) - E, sin($ + 4 ) = E , sin(& + 4 ) + E~ co d 8 + 4 ) =

cod8

(3.1)

The stress components outside the kink band are defined by urn = -urelel

+ uFe2e2+ .r"(e,e, + e2e,)

(3.2)

and those within the kink band are defined by

FIG. 13. Definition of coordinates within kink band. The fibers are taken to be inextensional.

N, A. Fleck

68

Continuity of Tractions Continuity of tractions across the kink band interface can be expressed as

n-a"=n.a

(3.4)

+

where n = el cos p e, sin p is the unit normal to the kink band interface. Equations (3.1)-(3.3) lead to the two scalar equations for continuity of tractions in the s1 and E, directions, respectively,

Because the fibers are assumed to be inextensible, the axial stress in the kink band, a,, is of no interest in the analysis to follow and eq. (3.5) need not be considered further. It is assumed that the initial misalignment, $, is small. Furthermore, it is anticipated that consideration of small deformation angles, will be sufficient to examine the critical events associated with microbuckling. For small $ + linearization of eq. (3.6) provides

+,

+,

Equation (3.7) can be further approximated, when (7r/2) - p >> 0, by dropping the term ( T - T ~ ) ( $+ +)sin p from the right-hand side, to give

This form of the approximation is chosen so that, when the composite behaves elastically, a proper account of the terms involving the remote stresses is maintained.

69

Compressive Failure of Fiber Composites

Kinematic Relations Kinematic conditions for kink-band deformation are now examined. Consider a material point P within the kink band, as shown in Figure 13. The position vector r to point P is c

+ rZsz= y ( - e ,

= ( 1 ~ 1

where the scalar lengths tl, 13 and are related by

tan p

t 2 ,x , and y

+ e,) + X

E ~

(3.9)

are defined as shown in Figure

The velocity of point P is v

=y j m e l

+ yt:e, + X

~

E

~

(3.11)

where i," and e y are, respectively, the shear strain rate and transverse strain rate outside the kink band, and f(t) = df(t)/dt. The strain rate tensor within the kink band is related to the velocity field by 1 T (3.12) & = -[Vu (VU) 1 2

+

where the superscript T denotes the transpose, and the gradient operator V is (3.13)

With the strain-rate components within the kink band defined by i: = t

1 2

T ~ +Z- ~ ~ , 2 ( E ~ E +~E , E ~ )

(3.14)

eq. (3.121, along with eqs. (3.1) and (3.9-3.10, gives the kinematic relations

tT= 4 tan( p - 3 - 4) + [ t; cos(3 + 4) - i," sin($

+ +)]cos p sec( p - $ - 4 ) i, = 4 + [i," cos(3 + 4) + 67 sin($ + +)]cos p sec( p - 3 - 4)

1

(3.15)

70

N. A. Fleck

Differentiating eq. (3.91, and noting that r follows from eq. (3.11) that

j

=

-(i; sin p

+

= u , y = yi;,

cos p)cos p

and

El = $ E ~ it ,

(3.16)

For $ + 4, em,, and ym small, eqs. (3.15) and (3.16) reduce to the approximate kinematic equations

(3.17a)

where po is the kink-band angle associated with zero remote straining. In the limiting case p + 0, the analysis reduces to the case of pure shear deformation within the kind band, as outlined by Wisnom (1990). In the case of vanishing remote transverse strain and shear strain, the rate equations (3.15) can be integrated to give

Y=4 eT = log

cod p - $ - 4 ) cod p -

3)

1]

(3.17b)

Note that (3.1%) implies that the volumetric strain in the band vanishes when the fibers have rotated to the point when 4 = 2( p - $1. Constitutive Relations: Deformation and Flow Theory Versions of Plasticity If the composite deforms elastically, then the stress components ( u T7, )in the band are related to the strain components ( e , , y ) in the band via UT = ETeT

(3.18a)

r=Cy

(3.18b)

where E , and C are the transverse and shear elastic moduli for the composite. Similarly, the components of remote stress (a;, 7-1 are related to the remote strain state (em,,y m ) via CT; = ETeF and rm = Gym.The state of stress within the band may be related to the rotation 4 within the band via (3.171, to give

"i

(3.19)

71

Compressive Failure of Fiber Composites

The exact equation for continuity of tractions, eq. (3.6), combined with this result and then linearized gives the approximate elastic kink-band response 0:

+ a;

-

27@tan p = [G

+ ET tan2 p ]

+

7

+++

(3.20)

An examination of eq. (3.81, the approximate equation for continuity of

tractions, shows that it also reduces to the correct result for elastic kink-band response, eq. (3.20).

Flow-Theory Version The following constitutive equations for a flowtheory version of plasticity have been derived by Budiansky and Fleck (1993) and the derivation is only outlined here. A similar approach has been adopted by Sun and Chen (19891, Sun and Yoon (19911, and Schapery (1995). The elastic component of the strain tensor is given by (3.18). Assume that the composite is characterized by the quadratic yield condition

(;)2=(;)2+(q 2

(3.21)

OTY

where T~ and uT,,are the plane strain yield stresses in pure shear and pure transverse tension in the case of perfect plasticity (when T, = T ~ a, constant). The effective stress, T,, which can be rewritten as

is used as a plastic potential for the plastic strain rates, y P and if. The parameter R = u ~ ~ /defines T ~ the eccentricity of the yield ellipse, which expands homogeneously with increasing T, due to strain hardening. Discounting the possibility of elastic unloading, the associated flow theory relations for plastic-strain rates, based on T, as a plastic potential, can be written as

(3.23)

N. A. Fleck

72

where &‘(re)is a measure of the rate of strain hardening. A workequivalent effective plastic strain rate, ,y: is defined by

79’

+ U ~ k f= T,y:

(3.24)

and it follows that

Thus, we interpret F(TJ as the inverse of the tangent modulus of the versus y P response in pure shear.

T

Deformation Theory Version Next, we derive a deformation theory version which coincides with the above flow-theory version for the special case of proportional loading. Substituting eq. (3.24) into eq. (3.22) and assuming proportional loading leads to

(3.26)

Note that the functional dependence of y,P on T, is taken to be the same as that of y p on r for pure shear. Equations (3.26) have the form of a deformation theory of plasticity. The linear elastic portion of the strain state is related to the stress state via (3.18). It is unclear whether deformation or flow theory is the more appropriate constitutive law for addressing fiber microbuckling. We shall see below that the maximum compressive stress is carried by the kink band at rather small values of fiber rotation (a few degrees). In this regime, relation (3.17a) indicates that straining is proportional within the kink band for the case of uniaxial compression: the distinction vanishes between deformation and flow theories. In the post-collapse regime of large fiber rotations within the kink band, straining is non-proportional and the flow theory prediction is stronger than the deformation theory prediction. There are strong theoretical arguments and convincing experimental evidence that deformation theory is more accurate in the prediction of plastic buckling loads in metallic structures; see for example the review by Hutchinson

Compressive Failure of Fiber Composites

73

(1974). Fleck and Jelf (1995) have performed non-proportional loading tests on hoop-wound carbon fiber-epoxy tubes; they found that the deformation theory description (3.26) is more accurate than the flow theory version (3.23). We shall continue with the deformation theory description and now derive the compressive strength of a kink band for a rigid-perfectly plastic solid, and then for a Ramberg-Osgood strainhardening solid.

2 . Rigid-Pe$ectly Plastic Solid under Multi-AxialLoading Slaughter et al. (1992) have derived an algebraic expression for the axial collapse strength, a,,under multi-axial loading for the limiting case of a rigid-perfectly plastic solid of shear yield strength ry,

where a 5 dl + R2 tan’ p . This relation indicates a large knockdown effect of both in-plane shear stress and transverse stress on the axial compressive strength. Jelf and Fleck (1994) have confirmed the accuracy of (3.27) for the case of combined axial and shear loading by performing compression-torsion tests on unidirectional carbon fiber-reinforced epoxy tubes. 3. Effect of Strain Hardening on Compressive Strength

The in-plane shear response of polymer matrix composites at small levels of strain may be adequately described by the Ramberg-Osgood description (3.28)

in terms of the three-parameter fit ( 7 , , , y y , n ) . The shear modulus G follows as G = rU/yr. For polymer matrix composites, r y is in the range 40-60 MPa, y y equals approximately 1%, and the strain-hardening exponent n is in the range 3-10, as collated by Fleck and Jelf (1995).

N. A. Fleck

74

For the case of an inclined kink band, the strain state within the band is composed of both shear and transverse straining. The Ramberg-Osgood description (3.28) generalizes to Ye

'e

(3.29)

where 7, is given by (3.22) and the effective strain ye is composed of an elastic part

and a plastic part Y,"

=

T 3YY (

$)n.

For the deformation theory solid, y,P is given by (3.26), and for the flow theory solid, y,P is defined by (3.25). Budiansky and Fleck (1993) have developed the following analytical expression for the uniaxial compressive strength for the deformation theory solid, (3.30)

where G* = [ l + R2 tan2 PIG and y: = y y / d l + R2 tan2 p . The critical stress a, is achieved in the regime of proportional straining at small and so the above result also remains valid for values of fiber rotation flow theory. Figure 14 shows how ac varies with $/ y; for n = 3 , 5 , 9 and w: we note that strain hardening has little effect on the strength and so the elastic-perfectly plastic estimates of Section 1I.B remain valid. Typically, the measured compressive strength of polymer matrix composites is a,/G = 0.2 and the corresponding estimated value for fiber waviness is $/ y;t: = 4 from Figure 13. With y: taken as 1% this suggests a fiber misalignment angle of about 2.3".

+,

4.

Time-Dependent Kinking

Many fiber composites are known to exhibit time-dependent deformation behavior, or creep. These include polymer matrix composites (Horoschenkoff et al., 1988; Ha et al., 1991) and woods (Dinwoodie, 1981).

Compressive Failure of Fiber Composites

75

FIG.14. Effect of fiber misalignment angle 5 upon the collapse strength mc for a range of values of strain-hardening exponent n. The results are taken from the kinking analysis of Budiansky and Fleck (1993).

The literature on visco-elastic buckling is extensive. Early analyses (e.g., Biot, 1957) performed a bifurcation analysis to predict the critical load for a perfect structure. More recently, the significance of initial imperfection and material non-linearity have been appreciated in governing timedependent microbuckling (Schapery, 1993; Slaughter et d.,1992; Slaughter and Fleck, 1994a). In each case, the above equilibrium and kinematic relations hold for the kink band and the lifetime is calculated by timeintegrating the rate of fiber rotation in the band, for a range of assumed constitutive laws. Unfortunately, there remains a lack of experimental data on visco-elastic microbuckling, and the critical event dictating the failure life, tr, has not been resolved. There are several possibilities which remain to be explored: (i) a critical value of fiber rotation & can be assigned, corresponding to tensile fracture at the fiber-matrix interface. (ii) failure occurs by static plastic kinking when 4 + 4 reaches a value of misalignment that, together with (T = a,, satisfies the static criterion (3.30); or (iii) we simply say that an upper bound to tr corresponds to = m.

At elevated temperatures, metal matrix and ceramic matrix composites undergo creep. Slaughter et al. (1993) have performed a theoretical analy-

76

N. A. Fleck

sis of creep microbuckling, based on power-law viscous behavior within the kink band. The composite is assumed to creep under in-plane shear by a shear-strain rate .i, related to a shear stress T via m

Y

(3.31)

where ( T ~ , + ~and ) the creep exponent m are material constants. The amount of fiber rotation 4 as a function of time t for kinking under a fixed-axial stress u is deduced by substituting the constitutive law (3.31) into the equilibrium and kinematic statements given above to obtain (Slaughter et al., 19921, 1

1

1

(3.32)

Here, the inclination p of the kink band is taken into account through the definitions

where (see eq. (3.22)) R can be regarded as a parameter equal to the ratio of transverse creep strength to shear creep strength. The creep lifetime depends upon the particular criterion chosen to define failure. If we adopt the assumption (iii) given above that the creep lifetime, r f , is set by the time for C#I + w, then (3.32) reduces to (3.34) It is of some interest to estimate whether creep kinking might be an issue in ceramic fiber/metal matrix composites, under conditions of moderately elevated temperature and sustained high load. Assuming the plausible values T~ = 100 MPa, To = lo7 s - ' , u = 1500 MPa, 3 = 3", and m = 5 , gives tf = 120 hours, which suggests that creep kinking may indeed have to be considered in the design of metal-matrix composites. 5 . Kinking Fatigue

Slaughter and Fleck (1992) have analyzed fatigue kinking from two viewpoints: (i) fatigue failure by low-cycle fatigue of the matrix within the kink band, and (ii) failure by cyclic ratchetting of the material within the

Compressive Failure of Fiber Composites

77

kink band until the plastic strain accumulation is sufficient to trigger the plastic microbuckling instability. Little experimental data are available on compressive fatigue failure of fiber composites. Soutis et al. (1991a) observed fatigue kink growth from a circular hole in a carbon fiber epoxy composite, and Huang and Wang (1989) measured the stress-life fatigue curve for un-notched specimens made from alumina fibers in an aluminum alloy matrix. Slaughter and Fleck (1992) found that the predictions of their ratchetting fatigue model were in better agreement with the experimental results of Huang and Wang than the predictions of the low-cycle fatigue model. Further work is clearly required in order to elucidate the fatigue failure mechanisms as a function of material composition. B. THE ROLE OF FIBERBENDING: INFINITE-BAND ANALYSIS The infinite-band analyses described above suffer from two main limitations:

(i) they are unable to predict the width of the microbuckle band, as the constitutive law contains no length scale, and (ii) they assume that the initial imperfection exists as an infinite band with an assumed orientation j3 rather than as a finite region. In the current section the first assumption is relaxed by including the role of fiber bending in an infinite-band analysis. In the following Section, III.C, the compressive strength is calculated for a two-dimensional initial region of waviness, relaxing the second assumption. Commonly, the fiber-bending strength is sufficiently small for fullydeveloped kink bands to be bounded by fiber breaks. The onset of fiber fracture during collapse sets the kink-band width, defined as the fiber length w within the kink band. A n early analysis (Budiansky, 19831, based on the simplifying assumptions of perfectly aligned fibers and rigid-ideally plastic behavior of the composite in shear and transverse tension, together with incorporation of the effects of couple stresses provided by fiber bending, gave

(3.35)

78

N. A. Fleck

for the ratio of the final kink width to the fiber diameter d , in terms of the longitudinal composite modulus EL and the p-modified shear yield strength T; = ( ~ 7This ~ . formula was based on the additional assumption that the fibers were perfectly brittle in tension. Measurements of kink-band widths by Jelf and Fleck (1992) were in good agreement with (3.35) over a wide range of parameters. 1. Summary of the Couple Stress Analysis of Fleck et al. (19956)

In this analysis, the individual responses of the fibers and matrix are smeared out, and the composite is considered to behave as a homogeneous, anisotropic solid. A couple-stress formulation is used to take fiberbending resistance into account. Descriptions of kinematics and equilibrium are now outlined, followed by constitutive laws and a criterion for fiber fracture. Kinematics In the initial stress-free configuration, the fibers are assumed to possess a small, initial angular misalignment, &, which is perfectly correlated along a direction inclined at an angle p to the transverse direction. Thus, 3 depends on the single variable x + y tan p, where the Cartesian coordinates (x,y ) are parallel and transverse to the ideal fiber direction, respectively, as shown in Figure 15a. For y = 0, it is assumed that 3 is an even function of x. Equilibrium Consider equilibrium of a representative material element in the deformed configuration (see Figure 15b). The element is subjected to a longitudinal compressive stress (T aligned with the fiber direction, a sliding shear stress T ~ a,transverse shear stress T ~ and , a transverse tensile stress, (T~ The . fibers embedded in the material offer bending resistance; thus, the representative material element carries a bending moment per unit area, or couple stress, m . Moment equilibrium gives

(3.36)

The presence of couple stresses makes the stress tensor unsymmetric, with rS # T ~ .

Compressive Failure of Fiber Composites

\\

79

t'

FIG.15. (a) Deformed shape of infinite band. Under load, the fibers bend and rotate from an initial inclination $(XI to a deformed inclination $ ( X I+ +(n). (b) Stresses on a representative element of solid. The fibers and matrix have been smeared out to a continuum.

As explained by Fleck et al., (1995b), considerations of equilibrium and the kinematics of Figure 3 provides the governing differential equation of equilibrium (3.37) In order I proceed, we need a constitutive law in order -J relate the generalized stress components ( m , T ~oT) , to the fiber rotation, 4. Constitutive Law The fibers are treated as linear elastic beams undergoing inextensible bending, and the matrix contributions to couple stresses are neglected. Simple beam theory for circular fibers of diameter d,

N, A . Fleck

80

Young’s modulus Ef, and volume fraction c gives the relation between the couple stress m on the composite and the associated curvature d 4 / d x as (3.38)

The shear and transverse responses of the composite are taken to be those of a non-linear deformation theory solid, as suggested by (3.181, (3.261, and (3.28). Fiber Fracture Criterion It is found experimentally (see for example Soutis and Fleck, 1990) that the width of the kink band is set by fiber fracture in tension due to local fiber bending. The strain in the fibers is the sum of the bending strain and the compressive strain associated with the axial stress u. We equate the maximum tensile strain in each fiber with the tensile fracture strain of the fiber E, to obtain the fracture criterion (3.39)

where Id4/dulm,, is the maximum absolute value of curvature along each fiber, and E is the longitudinal Young’s modulus of the composite. This fracture condition will be satisfied at two locations, x = fx,, and the width of the kink band is defined as the distance 21x,1 along the fibers between the points of fiber fracture. It is noted that the assumption of inextensible fibers was made in the kinematics and equilibrium relations, but axial straining of the fibers is implicit here in the fiber fracture condition. Budiansky and Fleck (1993) have included fiber extensibility in the kinematic and equilibrium relations of a particular version of their kinking analysis, and found that for typical polymer-matrix composites, fiber extensibility has little effect on the collapse response. Solution Method The equilibrium equation (3.37) may be reduced to a differential equation in 4 by eliminating m, T ~ and , uT via (3.26) and (3.38) to give Ed2 d2+

-16

dx2

+ u(4 + $1 = TeJ1

+Ptan2p

(3.40)

where the composite modulus E has been used as an approximation for cEf. The effective stress T~ is related to the effective strain ‘ye by the

Compressive Failure of Fiber Composites

81

non-linear Ramberg-Osgood relation (3.28), and in turn, ye may be written in terms of 4 as ye =

Cp~iTiFGqi

(3.41)

We treat (3.40) as the governing non-linear differential equation for the rotation + ( x ) to be solved, together with (3.41) and the Ramberg-Osgood relation (3.28). The imperfection in fiber alignment $ ( x ) is assumed to take the form

P

&XI

=

(3.42)

0

where the magnitude of the fiber waviness q0 and the wavelength i7 characterize the imperfection. Using the above analysis, Fleck et al. (1995b) generalized the result (3.35) for an elastic-ideally plastic composite, with finite fiber failure strain s F ,and obtained the implicit equation for the kink width w as

For cF = y; = 0, this reduces to Budiansky's previous result (3.35). Typically, for polymer matrix composites, 7 ; / E is in the range 0.005-0.015, and w / d is in the range 10-20 for a wide range in value of cF = 0-2% and y ; = 0-2%. Fleck et al. (1995b) further showed that w/d is rather insensitive to the value of strain-hardening exponent, n, for the case of a composite which strain hardens in accordance with (3.28). The inclination of the kink band, p, and the width and magnitude of the initial waviness also have little effect on w / d . The typical collapse response is shown in Figure 11 for the strainhardening case (n = 3) with a small initial imperfection ($/ y;". = 4, width of imperfection E/d = 20). With increasing fiber rotation, plastic deformation spreads along the fiber direction and the compressive stress attains a maximum value q . Fleck et al. (1995b) showed that the value of o, exceeds the kinking strength (3.30) by less than lo%, provided E / d exceeds about 20. In other words, kinking theory suffices in order to predict compressive strength unless the physical size of the imperfection is

N. A. Fleck

02

less than about 20 fiber diameters. During collapse the fibers rotate within the microbuckle band, leading to an end shortening of the composite, as shown in Figure 12a. Eventually, at a fiber rotation of 4m = 2 ( p - $>, volumetric lock-up occurs and continued end shortening of the composite is associated with broadening of the locked-up central region, as depicted in Figures 11 and 12. A steady state is achieved, with the remote stress equal to the band-broadening stress, u b . 2. Calculation of the Band-Broadening Stress, crb The band-broadening stress can be calculated by a work calculation, along similar lines to that given by Moran et al. (1995). In steady state, remote material outside the microbuckle band is convected into a state of simple shear within the locked-up band; the stored elastic-bending energy within the band of locked-up material vanishes. Finite strain-kinking theory is adequate in order to calculate the associated energy change, provided we assume the composite behaves as a deformation theory solid. Consider the steady-state limit of broadening of a band of locked-up fibers under a constant remote stress, . We neglect the effect of fiber fracture and examine the work done when a strip of width Sw is convected into the locked-up state 4 = 2 p within the microbuckle band from a remote state of uniaxial compression. The end shortening is Sw(1 - cos(2p)) and the external work done W E is given by

The internal work done W' in rotating fibers within a band of fiber length Sw from 4 = 0 to 4 = 2 p is given by W'=Swj

+2p

q5= 0

[

cod p - 4 ) cos p

1

7"Gd 4 dye

(3.45)

by making use of the kinematic relations (3.17b) and the constitutive law (3.26). The band-propagation stress is calculated by equating the internal work and the external work; typical results are shown in Figure 16 for a Ramberg-Osgood strain-hardening curve, as specified by (3.29). In order to interpret Figure 16, we assume that the propagation direction p of the microbuckle is set by the details of the propagation process (elucidated in Section IV), and is therefore an assumed parameter rather than a prediction from the band-broadening analysis. Typically, p is in the range

Compressiue Failure of Fiber Composites

83

0.15 -

'b G 0.1

-

0.05

-

0

\ \

10'

20'

30'

LOo

50'

E lo

P FIG.16. The steady state band-broadening stress, ub,for a Ramberg-Osgood deformation theory composite, under uniaxial stress. The magnitude of ub depends upon the assumed inclination p of the microbuckle hand.

20-30", giving a value for a b / G in the range 0.03-0.12, depending on the assumed value for the strain-hardening exponent n. We note that the band-broadening stress is significantly less than the typical collapse strength of q / G = 0.2: initiation of microbuckling from the initial imperfection determines the load-carrying capacity of the structure. In contrast to the initiation strength a,, which is relatively insensitive to the strain-hardening exponent n, the band-broadening stress a, increases substantially with increasing n. The band-broadening stress will be used later in Section IV as one ingredient in a crack model of microbuckle propagation. 3.

Knockdown in Strength under Multi-Axial Loading

The knockdown in compressive strength due to the presence of in-plane shear and transverse stress has already been discussed in Section III.A.6, for the case of kinking theory. For the rigid-ideally plastic solid, the knockdown in compressive strength follows from (3.27) as (3.46)

84

N. A. Fleck

dl

where a,, is the uniaxial compressive strength and a = + R 2 tan2 p . Slaughter et al. (1992) and Shu and Fleck (1996) have shown that (3.46) remains reasonably accurate when strain-hardening and fiber-bending resistance are taken into account. 4. Infinite-Band Strength in Case of Random Waviness In practice, the fiber misalignment angle 4 varies throughout the composite in a three-dimensional random manner. The underlying relationship between fiber waviness and processing conditions remains unexplored. Although Yurgatis (1987) has shown that the distribution in misalignment angle is roughly Gaussian in nature, little information is available on the statistical nature of random fiber waviness. Significant progress has been made on this difficult experimental task: Clarke et al. (1996) have measured the power spectral density of a glass fiber-reinforced epoxy by confocal laser-scanning microscopy. In a related study, Slaughter and Fleck (1994b) determined the relation between the power spectral density of fiber waviness and the observed Weibull distribution of compressive strength. Briefly, they assumed a flat-power spectral density for the misalignment angle as a function of distance along the fiber direction. The fiber shape is taken to be invariant along lines inclined at an angle ( ~ / 2- p ) to the fiber direction. Slaughter and Fleck further assumed the power spectral density was characterized by a given value for the meansquare spectral slope, with a lower cut-off for the spectral wavelength. Monte-Carlo realizations for the fiber waviness were generated, and the compressive strength for each realization was determined by the couplestress formulation of Fleck et al. (1995b) in order to take fiber bending resistance into account. The ensemble of results was then used to compute a probability density for compressive strength, and a Weibull fit was conducted to extract the Weibull parameters. Good agreement was observed between the predicted Weibull parameters and the values measured independently by Jelf and Fleck (1992). Thus, the underlying relationship between the statistics of fiber waviness and the resulting distribution of compression strength was determined. Further work is required in order to measure imperfection spectra and the corresponding distribution of compressive strength for a range of composites. Additional micromechanical calculations are needed in order to establish the relationship between the two.

Compressive Failure of Fiber Composites

c.

INITIATION STRENGTH FOR A

85

FINITEIMPERFECTION

So far, we have idealized the initial fiber misalignment as an infinite band, so that the response can be calculated in a one-dimensional framework. Recently, this assumption has been relaxed and the compressive strength has been estimated for a two-dimensional distribution of initial fiber misalignment. Two alternative strategies have been adopted:

1. The composite is treated as distinct, perfectly-bonded layers of fibers and matrix. 2. The effect of the individual fibers is “smeared-out’’ by treating the composite as a Cosserat continuum capable of bearing couple stresses. Kyriakides et al. (1995) used the first strategy to study the early stages of microbuckling from a small region of waviness. In similar fashion, Sutcliffe et al. (1996) used this method to calculate microbuckle initiation and early growth from a sharp, open notch under remote compressive loading. This approach is useful when the initial region of fiber waviness extends over only a small number of fibers, but becomes prohibitively expensive in computer time when a large number of fibers are considered. Fleck and Shu (1995) and Shu and Fleck (1996) have adopted the alternative strategy of smearing out the effects of each individual fiber and developed a finite strain, finite element code based on couple-stress theory. Thus, it is conceived that each element contains many embedded fibers. Here, we summarize the two-dimensional theory and then collect the main results found to date. In order to obtain the constitutive law, the fibers are assumed to behave as elastic Timoshenko beams embedded within a non-linear dilitant plastic matrix. A virtual work expression is obtained for a two-dimensional unit cell consisting of a fiber of volume fraction c adhered to matrix of volume fraction (1 - c ) . Macroscopic stress and strain quantities are thereby derived for the smeared-out homogeneous composite. It is found that the governing equations are identical to those of Cosserat couple stress theory (Cosserat and Cosserat, 1909). The significance of the unit cell analysis is that the independent micro rotation angle 8 in the general couple stress theory is shown to be the independent rotation angle Or of the fiber cross section. The bending resistance of the fibers is set by the fiber diameter, d , and so the constitutive law involves the fiber diameter as the internal length scale. Deformation and flow theory versions of a dilitant plasticity law for the composite are proposed along the lines of (3.21-3.29).

N. A. Fleck

86

The finite element procedure is based upon a Lagrangian formulation of the general finite deformation of the composite, and can deal with both geometrical and material non-linearities (see Fleck and Shu, 1995, for details). A version of the modified Riks algorithm (Crisfield, 1991) is adopted to handle snap-back behavior associated with the microbuckling response. Imperfections in the form of fiber waviness are included in the formulation. The finite element code has been used to determine the effect of a finite region of initial waviness upon the compressive strength of the composite. The main results to date are summarized below. 1. Effect of Impe$ection Size on Compressive Strength

Shu and Fleck (1996) have explored the effect of a finite region of initial fiber misalignment on the collapse response of the composite. Again, a uniform remote compressive stress urnis applied in the x,-direction, as shown in Figure 17. Consider the case where the initial fiber misalignment is confined to an ellipse of length f and width w in the ( x , ,x , ) plane, as

.( ax*

i.yfiber

FIG. 17. Sketch of initial imperfection. Fiber misalignment length k' and width w.

3 is confined to an ellipse of

Compressive Failure of Fiber Composites

87

shown in Figure 17. The axes of the ellipse, Cx; , x i ) , are rotated through an angle, p, about the x 3 axis with respect to ( x , ,x2) axes, such that x',

=x,

cos p

+ x2 sin p

and

x i = -xl sin

p

+ x 2 cos p.

(3.47)

In the region outside the ellipse, the fibers are straight and perfectly aligned in the x,-direction. The fiber misalignment follows a cosine distribution within the elliptical region, as specified by

(3.48)

Where (3.49)

As the band length / + M, the imperfection tends to an infinite band as described in the previous section. At the other limit of / + 0, the fiber misalignment vanishes and the compressive strength q approaches the Rosen value of a, = G, where G is the in-plane shear modulus of the composite. These limits provide a useful check to our finite element calculations. A finite element mesh of the unidirectional composite was constructed of six-noded triangular elements with three degrees of freedom at each node (two in-plane displacements and one rotation along the normal to the plane of deformation). Full details can be found in Fleck and Shu (1995). The mesh was loaded parallel to the fiber direction by applying uniform end displacements u:, and the finite element calculation gives the corresponding remote stress urn. A typical plot of the average remote stress urnversus the end shortening uy is given in Figure 18 for the inclination p = 0" (n = 3, q,,/yy = 4, w / d = 20, / / d = 50). The response is almost linear with a sharp snapback behavior at maximum load. Since we focus our attention on the initiation and early propagation of a microbuckle, the calculation of the post-buckling response was stopped when the load dropped to about 75%

N. A. Fleck

88 50 -

I

1

1

I

I

I

I

I

-

-

-

-

-

30 -

-

LO

-

-Om TY

-

20 -

-

I

30

&

I

LO

of the maximum load. The severe snap-back response of Figure 18 is due to the fact that the mesh is long in the fiber direction (4000d). The snap back is more severe than in the infinite band case, as the fibers surrounding the finite imperfection remain almost straight at maximum load. Numerical experimentation showed that the weakest orientation is p = 0", as found previously for the infinite band limit by Budiansky and Fleck (1993) employing kinking theory, and by Fleck et al. (1995b) employing couple stress theory. The progressive nature of the collapse is exhibited in Figure 19 in the form of contours of total fiber rotation 4t,for the three stages of loading A-C shown in Figure 18. State A is the initial unloaded configuration with 4 = 0 and c$t = state B is immediately post-maximum load (99.7% of maximum load); and state C is at 78% of maximum load. We note that state B, at just past maximum load, displays:

4;

(i) a relatively small maximum value of fiber rotation 4 = c#+ - $. The maximum fiber rotation is 4 = 4.4" for the case p = 0".

Compressive Failure of Fiber Composites

89

a+0.1 b-0.5 c = 1.0 d r 1.5

e = 2.0

VI

2

a = -0.5 b = 0.5 c = 2.5 d: L . 5 e z

6.5

0

FIG. 19. Contours of total fiber rotation dl (in degrees) at the three stages marked A, B and C in Fig. 18 for p = 0". Remote stresses at stages B and C are respectively 99.7% and 78.2% of the maximum load. The shape of the deformed fibers at state C is included at the bottom of the figure.

90

N, A. Fleck

(ii) a spatially small region of fiber rotation. At maximum load the region over which the total fiber rotation exceeds 0.5” is only 70d, i.e. 2.3% of the width of the mesh. Two versions of the finite element code were written: in version I the phenomenon of “fiber lock-up” was neglected and large compressive transverse strains were allowed to accumulate within the microbuckle band. Experimental observations of microbuckle bands suggests that volumetric lock-up occurs such that the transverse strain does not become strongly negative. (A useful indicator of the magnitude of the transverse strain is given by (3.17a), from the infinite band calculation.) In version I1 of the finite element program, fiber lock-up is included, and the constitutive response is taken to be elastic when the transverse strain becomes negative. Both versions of the finite element code revealed that the microbuckle grew initially at an orientation /3 = 0”: the microbuckle band initially propagates in the transverse x 2 direction. In version I of the code, large, compressive transverse strains (of order 5%) accompanied transverse propagation of the microbuckle. In version 11, fiber lock-up occurred and the direction of propagation of the microbuckle increased to realistic values of order 20”. Typical results for the orientation of a growing microbuckle from an initial small-defect oriented one at p = 0” is shown in Figure 20, by using the “fiber lock-up” version I1 of the finite element code. In parallel studies, Kyriakides et al. (1995) and Sutcliffe and Fleck (1996) modelled the tip region of a propagating microbuckle by alternating layers of fibers and matrix. The microbuckle was observed to propagate in a similar manner to that of an inclined mode I1 crack at an inclined angle p = 5-30”, depending upon the strain-hardening exponent n and the shear-yield strain yr of the composite. The effect upon the collapse strength u, of the initial length /‘ and orientation p of the imperfection, is shown in Figure 21 (page 92). As the length L increases from zero to infinity, the collapse strength decreases from the elastic bifurcation strength a, = G given by Rosen (1965) to the infinite band result given by Fleck et al. (1995b). The collapse strength is mid-way between the elastic bifurcation value and the infinite band value at a “transition length” L/d = 20. For L> 0, the strength decreases with increasing magnitude of initial misalignment q0 and with increasing strainhardening index n.

Compressive Failure of Fiber Composites

I c

91

1"" 1

Initial imperfection

@ = L O

contour

FIG. 20. Evolution of an in-plane microbuckle from an initial circular imperfection of diameter 10d, and C#J~ = 5". The effect of volumetric lock-up is included in the constitutive law from the composite. Note that the microbuckle grows to a steady-state orientation of about 17".

2. Effect of Impefection Shape on Compressive Strength

The effect of the shape of the region of fiber waviness on compressive strength has been explored by Shu and Fleck (1996). They considered three types of initial imperfection, as shown in the insert of Figure 22: 1. Infinite band of width L inclined at p

=

0.

2. Circle of diameter L.

3. Ellipse of width 20d and length L, oriented at p

=

0.

All three shapes of imperfection are described by (3.47)-(3.49), with = 4 (we take y y = 1%, giving 6, = 2.3").The compressive strength as a function of imperfection size of each of the three shapes is shown in Figure 22, for n = 3. We note that the infinite band prediction is signifi-

+JyY

N. A. Fleck

92

0.8 -

5

0.6 -

G

0.4 -

0.2-

0'

I

'

,

1 1 1 1 '

I 1 1 1 1 1

I

, , , 1 l l l l

I

lo2

10'

,

I

lo3

4 /d FIG.21. Compressive strength as a function of the length k' of the elliptical region of fiber misalignment. w / d = 20. The infinite band results are taken from Fleck, Deng, and Budiansky (1995). I

-c

1

/n-.

1

T y

P -

I

-

L

1

1

0

I

, , , I

10

I

1

I

I

1

1

I

I

l

100

30

1

I

I

300

L /d FIG. 22. Effect of imperfection size and shape upon compressive strength. n qJo/Yu = 4.

-

=

3,

93

Compressive Failure of Fiber Composites

cantly weaker than the other shapes for the L / d values considered. In the limit of large L / d (greater than about 30 for the infinite band, and greater than about 300 for the circle and t h e ellipse) the strengths converge to the asymptote given by the kinking solution (3.30). It is instructive to compare the strengths for the ellipse and for the circle at L / d > 20. Then, the circle circumscribes the ellipse; the circular patch has the larger physical size but gives less of a stress-concentrating effect than the ellipse. These two factors compete, and result in the circle being slightly stronger than the ellipse, for the same value of L / d . The main practical conclusion to draw from Figure 22 is that compressive strength is significantly influenced by both the shape of the imperfection, and by the size in relation to the fiber diameter d . 3. Effect of Multi-Axial Loading upon the Knockdown in Strength for a Finite Imperfection The knockdown in compressive strength due to in-plane shear and transverse tension has been calculated by Shu and Fleck (1996) for the case of a circular patch of waviness, of diameter L = 20d. Contours of compressive strength are plotted in Figure 23 for the case n = 100. The contours are approximately straight lines of constant spacing, suggesting that the knockdown in strength can be given by the following analytical formula: 0; ~

rm - 0.1 4 1 - 0.8 7-Y

U C0

(3.50)

TY

where a,, is the compressive strength in the absence of in-plane shear stress and transverse stress. The knockdown in strength for an infinite band is given by (3.46). For the case p = 20" and R = 2 we have (Y = 1.24 and (3.46) becomes

a,

-= 1

-

roc

4

0.8 - - 0.3 -

(3.51)

We conclude that the knockdown effect of shear stress on compressive strength is the same for the infinite-band case and for the case of a circular imperfection. The presence of transverse stress causes a greater reduction in strength for the infinite-band imperfection than for the circular imperfection.

94

N. A. Fleck

tw t Y

FIG.23. Contour plot of normalized compressive strength of a circular imperfection region of diameter 20d, under general in-plane loading.

IV. Propagation of a MicrobucMe in a Unidirectional Composite So far, we have been concerned primarily with the compressive strength associated with the initiation of a microbuckle from a region of preexisting fiber waviness. In this section we examine the growth of a microbuckle in unidirectional material. First, recent experimental findings are reported on the stable propagation of a microbuckle, and then a mode I crack propagation model is described in order to model the propagation response. A. EXPERIMENTAL OBSERVATIONS The investigation of kink-band propagation in fibrous polymer composites is difficult, since unstable propagation usually occurs as soon as kinking has initiated. Notched unidirectional carbon-fiber epoxy composites typically split at the notch ends when loaded along the fiber direction. Fleck and co-workers (Sivashanker et af., 1995; Fleck et al., 1996) overcame this problem for edge-notched plates of unidirectional composite by first nucleating a microbuckle at the root of a starter notch by an

Compressive Failure of Fiber Composites

95

indentation technique. The specimen was then loaded in axial compression and stable microbuckle growth was observed in a consistent, repeatable manner for microbuckle lengths limited only by the ligament width of the specimen (35 mm for the geometry employed). The specimen geometry is shown in Figure 24. Typical plots of remote axial stress versus microbuckle extension are shown in Figure 25 for three representative carbon fiber composites: (i) Toray T800 medium strength fibers in a toughened epoxy matrix, Ciba Geigy 924c. (ii) Medium strength AS4 fibers in a thermoplastic PEEK matrix, ICI APC-2, and (iii) High strength Hercules IM8 fibers in a PEEK matrix.

It is clear from Figure 25 (page 97) that there is little effect of composition upon the collapse response. In each case, out-of-plane microbuckle propagation occurred, as sketched in Figure 26 (page 98). Examination of the side face of the specimens after the microbuckle had grown about 15 mm revealed that the microbuckle grew in a crack-like manner: the width of the microbuckle increased roughly as the square root of distance back from the microbuckle tip, see Figure 27 (page 99). A typical view of the microbuckle tip in the T800-924c material is given in Figure 28 (page 99): the SEM micrograph shows progressive broadening of the flanks of the microbuckle with increasing distance back from the microbuckle tip, If the microbuckle growth were to occur in a dislocationlike fashion, then one would expect the width of the microbuckle band to be constant. For the case of IM8-PEEK, the fibers are sufficiently strong for no fiber fracture to accompany microbuckling, Moran et al. (1995) and Fleck et al. (1996). The T800 and AS4 fibers are weaker and undergo fiber fracture within the microbuckle band. Figure 28 shows that the number of microbuckles increases with increasing distance back from the tip of the microbuckle, but the average width w of each individual kink band is constant at 15-25d. The phenomenon of multiple kinking in the wake of a growing microbuckle is the same phenomenon as steady-state band broadening for an infinite microbuckle band. Strain gauges were placed along the trajectory of microbuckle propagation in the edge-notch tests and were used to measure the compressive stress across the flanks of the microbuckle. The results are summarized in Figure 29 (page 100). For all three materials, band broadening is observed to occur at a constant value of bridging stress of about 100 MPa across the flanks of the microbuckle.

N. A. Fleck

96

,

Aluminium end -tabs

50 rnm

f

-

Strain gauges

\

mas LSmm

FIG. 24. 3 mm thick edge-notched unidirectional specimens for measurement of microbuckle propagation.The 15 mm notch is indented in order to nucleate a microbuckle of length about 2 mm. Anti-buckling guides, lubricated with PTFE spray, prevent Euler macrobuckling of the specimen.

Compressive Failure of Fiber Composites 300

250

97

3

t

IM8 1 PEEK

200 5

Remote 150 stress cc (MPa)

-

AS4 / PEEK

50

0

0

5

10

Microbuckle

15

Extension

20

25

(mm)

FIG. 25. Typical plots of remote axial stress versus microbuckle extension for unidirectional composites.

Note that the magnitude of the steady-state bridging stress in the wake of the microbuckle is approximately equal to twice the shear yield strength of the materials (7y= 60 MPa), in agreement with the estimate for the band-broadening stress at high n and p = 20 - 30°, as shown in Figure 16.

B. THEORETICAL PREDICTIONS The observation that a microbuckle propagates in a crack-like manner rather than in a dislocation-like manner suggests that a crack-bridging line model can be used to estimate the relation between applied stress and microbuckle length. The out-of-plane microbuckle development in the thin composite plate is reminiscent of the cracking behavior of thin metallic plates under mode I tension: an inclined crack forms with out-of-plane displacements close to the crack tip and a mode I displacement field forms farther from the crack tip. A pragmatic approach for the microbuckling problem is to treat the microbuckle as an overlapping mode I crack. The infinite band response of remote stress urnversus shortening A can be used to provide the crack-traction versus crack-overlap displacement law in

N. A. Fleck

98

L

p =20°

thickness direct ion

I

FIG.26. Sketch showing out-of-plane microbuckle growth in unidirectional carbon fiber composite specimens (cf Fig. 24) under axial loading.

a mode I cohesive zone at the crack tip. Outside the cohesive zone, the cracked structure is treated as a linear, elastic orthotropic solid. A typical plot of urnversus A is repeated in Figure 30 (page 101): the stress peaks at the Rosen value urn= G for the case of vanishing initial imperfection, and urn rapidly falls to the steady-state band-broadening stress ub,as discussed in Section 111. We use the infinite band urnversus A response as the non-linear spring law for the cohesive zone at the crack tip. The analysis is simplified considerably by partitioning the area under the urn versus A curve into two parts: (i) the area below the line uTa= ub,and (ii) a finite remainder, termed Gti,,. This partitioning allows us to treat the propagating microbuckle as a crack carrying a constant bridging stress u,, along its flanks, with a mode I tip toughness C f i pas , depicted in the insert of Figure 31 (page 102). (A similar strategy has been adopted by Palmer and Rice (1973) in the study of mode I1 shear faults in soils and rocks). The tip toughness G f i , has been calculated using the finite strain couple-stress code of Fleck and Shu (19951, which has already been outlined in Section 1II.C. Numerical experimentation shows that the non-

Compressive Failure of Fiber Composites

99

too0

A ASL/PEEK

800

I

E

-32

600

-6 Y

u

3

2 LOO

.-u

E

Lc

0

5 200

E

3

0 0

2

L 6 8 10 12 Distance behind microbuckle tip I m m )

1L

16

FIG. 27. The crack-like nature of microbuckle growth in carbon fiber composites. The width of the microbuckle band increases with increasing distance back from the microbuckle tip.

FIG.28. Side view of microbuckle tip region of T800-924c composite. Near the tip of the microbuckle, two planes of fiber fracture are evident and a single kink band is formed. With increasing distance back from the microbuckle tip, the number of parallel fracture planes increases: multiple kink bands are formed.

N A. Fleck

100

00

6-

0 0 0

-

0

5C

vi

:L -

amm Qxxxx)oaDo

-

0

0

n Y

G

Y

3-

m

-

m c o

0

r

0

0'

I

2

0

I

L

I

6

I

8

I

10

I

I

12

14

Distance behind microbuckle t i p ( m m l FIG. 29. The number of individual kink bands within the overall microbuckle band as a function of distance from the tip of the microbuckle. Unidirectional composite made from TSOO carbon fibers in a 924c epoxy matrix.

dimensional group G,i,/d \/.yEL is only weakly dependent on the value of the strain-hardening exponent n and to the ratio of longitudinal modulus EL to shear yield strength T ~ Typical . results are shown in Figure 32 (page 103) for p = 30". For rough design calculations we may state that G,,,/d 0.3 - 0.5.

d x

1. Comparison of Mode I Model with Experimental Data

Sivashanker et al. (1995) and Fleck et al. (1996) have implemented the above fracture mechanics model in order to compare the predicted response of microbuckle length in terms of applied stress with the observed behavior for their edge-notched panels, described in Section 1V.A above. They made use of existing calibrations for the mode I stress intensity factor and crack-opening profile for an orthotropic edge-cracked strip as laid down by Bao et al. (1992) and Wu and Carlsson (1991). Representative results for the ASCPEEK material are presented in Figure 33a, page 104. The value of the applied stress at the onset of microbuckle growth was

Compressive Failure of Fiber Composites

101

Bridging stress ( MPa)

-924 C A S 4 -PEEK

I I

I

-10 Distance

I

I

0

I

10

g ( m m ) from

I

20

microbuckle tip

FIG. 30. Estimate of the bridging stresses across a propagating microbuckle in three carbon fiber composites. Wire resistance strain gauges were placed about 1.5 mm from the plane of trajectory of the microbuckle, and the strain was recorded with increasing microbuckle extension past the gauge. The gauge stress was estimated by multiplying the axial strain detected by the gauge by the longitudinal modulus of the composite (thereby neglecting the small Poisson ratio effect).

used to estimate the tip toughness Grip= 20.0 kJm-* (corresponding to a compressive mode I stress intensity of 30.3 M P a 6 ) . Strain gauge measurements of the bridging stress across the flanks of the microbuckle indicated a, = 100 MPa. With these assumed values for C l i pand for a,, the crack model gave good agreement with the observed dependence of microbuckle length upon applied stress; see Figure 33a. A n additional comparison can be made between model and experiment, by comparing the predicted width of microbuckle band with the observed profile. Kinking theory suggests that the crack overlap displacement S due to fiber rotation to lock-up of 4 = 2 p within a band of width w and inclination /3 is given bY 6

=

w(1 - c o s 2 p )

(4.1)

Thus, the width of the microbuckle band may be estimated from the predicted mode I displacement profile 6 behind the crack tip of the crack model, and by converting these values to a profile in w via (4.1). The comparison is shown in Figure 33b. Again, good agreement is observed in firm support of the simple mode I crack model of microbuckle propagation in preference to a dislocation model. Similar agreement is observed for the

N. A. Fleck

102

0.

0

2

L

6

U/d FIG.31. Sketch of the infinite band collapse response, assuming zero initial imperfection. The collapse response is calculated by couple stress theory. The area under the curve is split into a rectangular region under the line a m = a, and a remaining area designated G,,,. This suggests a mode I crack model, with a constant bridging stress of ob and a tip toughness G,,p,as shown in the insert of the figure.

other two materials tested by Fleck et al. (1996): T800-924c and IM8-PEEK unidirectional composites. Fleck et al. (1996) and Sutcliffe and Fleck (1996) have measured C l i pfor the T800-924c, AS4-Peek, and IM8-PEEK unidirectional composites using the experimental procedure described above. They find that G,,,/d , / - lies in the range 0.5-1.1, which exceeds the predicted values by a factor of about two. The discrepancy between the predictions of the elastic crack-line model and experimental data has been discussed by Sutcliffe and Fleck (1996). In brief, the microbuckle tip is surrounded by a plastic zone and the coupling of shear tractions to normal tractions on the microbuckle is significantly different from that predicted by the crack model. We conclude that the cohesive zone approach, based on the infinite band response, provides a useful but approximate estimate of the manner by which a microbuckle propagates in unidirectional material. A two-dimensional analysis of microbuckle propagation is more satisfactory but more cumbersome (Sutcliffe and Fleck, 1996).

Compressive Failure of Fiber Composites

0.3l

0

I

0.1

I

0.2

103

I

0.3

I/ n FIG.32. Calculated tip toughness G,,, from the infinite-band couple stress theory. fi

=

30".

V. The Notched Strength of Multi-Axial Composites

Microbuckle initiation and growth in multi-directional composites is much less understood than microbuckle development in unidirectional material. Failure is dominated by microbuckle initiation and growth in the load bearing 0" plies. To a first approximation, the compressive failure strain of unnotched multi-directional laminates is the same as for unidirectional material, and so laminate plate theory can be used to estimate the compressive strength (Soutis et al., 1993). This is consistent with the observation that the compressive strength is set by microbuckle initiation from a local region of fiber misalignment in the 0" plies. Further work is required to explore experimentally and theoretically the effect of fiber lay-up on the initiation strength. A more difficult but highly practical problem is the prediction of compressive strength for multi-directional panels containing notches, such as holes. Composite panels contain holes, either by design (holes for fasteners, inspection holes, etc.) or by accident (service or battle damage). The effect of transverse impact on a composite plate is to induce a region of extensive damage in the form of fiber fracture, microbuckling, and

N. A. Fleck

104 300

-

I

I

1

.

crb = 200MPa

u b=100MPa ub=50MPa 100

-

-

a, = 0

-Experiment Bridging analysis I

I

5

10

I 15

I

Microbuckle extension ( m m )

(a)

-E 3a

1000

800

d C 0

L1

-w

600

Y

u 3

n

g

.-

LOO

E u-

0

5 200 '0 3 0

0

2

6

L

8

10

12

Distance behind microbuckle tip [ mm)

(b) FIG. 33. Comparison of measured and predicted microbuckle growth in AS4-PEEK material. (a) Remote stress versus microbuckle length. (b) Measured width of microbuckle band for a microbuckle which has grown 14 mm from the notch root.

Compressive Failure of Fiber Composites

105

delamination. One approach is to neglect the load-carrying capacity of this damage region and to treat it as a hole in the panel. It is observed experimentally that failure from a hole in a multidirectional composite plate under compression is by microbuckle nucleation at the edge of the hole (Soutis et at., 1993). The local axial stress at the hole edge for microbuckle nucleation is found to be approximately equal to the compressive strength of an unnotched multi-directional specimen. This suggests that a cohesive zone model may be used to predict microbuckle initiation, with a peak value of bridging stress equal to the unnotched strength of the multi-directional material. With increasing remote load, the microbuckle grows from the hole edge, first in a stable manner and then unstable at peak load. A crack-bridging model has been developed as a useful engineering tool for prediction of notched compressive strength. The model is described in the following section, and representative predictions are summarized. The model has been incorporated in PC-driven software to predict notched strength for a wide range of geometries and fiber architectures (Xin ef al., 1995). A. LARGE-SCALE CRACK-BRIDGING MODEL

Soutis ef at. (1991b) have developed a crack-bridging model for the initiation and growth of compressive damage from the edge of a blunt notch such as a hole. The damage zone is simulated by a compressive mode I crack with a cohesive zone ahead of its tip. Consider compressive failure of a finite width, multi-directional composite panel, which contains a central circular hole. It is assumed that microbuckling initiates when the local compressive stress parallel to the 0" fibers at the hole edge equals the un-notched strength of the laminate a,, , that is

k,a"

=

o,,,

(4.2)

where k, is the stress concentration factor and am is the remote axial stress. Damage development by microbuckling of the 0" plies, delamination, and damage by plastic deformation in the off-axis plies is represented by a crack with a cohesive zone at its tip; see Figure 34. For the sake of simplicity, a linearly softening spring law is taken for the cohesive zone: the crack-bridging normal traction T assumed to decrease linearly with

N. A. Fleck

106

Damage zone (microbuckling delamination I

(bl

i FIG.34. Cohesive zone model for microbuckling of multi-directional composites.

increasing crack-face overlap 2 v from a maximum value (equal to the un-notched compressive strength ouLn of the composite) to zero at a critical crack-face overlap of 2vc. The cohesive zone is assumed to remove any singularity from the crack tip and stresses remain bounded everywhere. It is assumed that the length /' of the equivalent crack represents the length of the microbuckle. When the remote load is increased, the equivalent crack grows in length, thus representing microbuckle growth. The evolution of microbuckling is determined by requiring that the total stress intensity factor at the tip of the equivalent crack K,,, equals zero, K,,,

=

K"

+ K,

=

0,

(4.3)

where K" is the stress intensity factor due to the remote stress urn,and K , is the stress intensity factor due to the local bridging traction T across the faces of the equivalent crack. When this condition is satisfied, stresses remain finite everywhere. The equivalent crack length /' from the circular hole is deduced as a function of remote stress urn using the following algorithm. For an assumed length of equivalent crack L, we solve for om and for the crack-bridging tractions by matching the crack-opening profile from the crack-bridging law to the crack profile deduced from the elastic solution for a cracked body. The cracked body is subjected to a remote stress om

Compressive Failure of Fiber Composites

107

and crack-face tractions T. At a critical length of equivalent crack, f,,, the remote stress urnattains a maximum value, a,, and catastrophic failure occurs. 1. Input Parameters for the Model

The model contains two parameters which are measured independently from specimens made from the same material and same lay-up: the unnotched strength a,, and the area G, under the assumed linear tractioncrack displacement curve. a,, is measured from a compression test on the un-notched multi-directional laminate, and G, is measured from a compressive fracture toughness test. The concept of compressivefracture toughness may be explained as follows. Consider a finite specimen containing a single crack, with a cohesive zone at the crack tip. The cohesive zone is assumed to be much smaller than other in-plane dimensions. Then, stresses decay remotely with radius r from the crack tip as r-’I2, characterized by the remote mode I stress intensity factor, K. A cohesive zone exists at the crack tip such that the total stress intensity factor at the tip of the cohesive zone vanishes. Rice (1968) has shown that the work done to advance the crack by unit area G, equals the area under the crack traction versus crack-opening displacement curve, G,

=

2 I v r c ( v ) d u= cunvc,

(4.4)

0

where 2u, is the critical crack-closing displacement on the crack traction-crack displacement curve, as shown in Figure 34. For an orthotropic plate in plane stress, the fracture energy G, is related to K , by (Paris and Sih, 1969)

where E and C are the laminate in-plane extensional and shear moduli, respectively, and u is Poisson’s ratio in the reference system shown in Figure 34. This is analogous to the fracture mechanics relationship for an isotropic elastic plate in plane stress, G, = K , ~ / E . We assume that the toughness G, represents the total energy dissipated by fiber microbuckling, matrix plasticity in the off-axis plies, and by delamination. The compressive toughness G, of a laminate may be mea-

108

N. A. Fleck

sured by performing a compressive fracture toughness test to measure K , , and then by using (4.5). The compressive fracture toughness concept is meaningful provided the damage zone at the onset of crack advance is much smaller than other specimen dimensions. Also, the crack faces much not interfere at distances remote from the crack tip. 2. Application of the Model The approach has been applied to a wide range of specimen geometries (Sutcliffe and Fleck, 19931, and has been used to examine the effect of fiber architecture upon notched strength of carbon fiber laminates (Soutis et al., 1993). Additional tests have been performed on 2-D- and 3-D-woven composites, on woods, and on plywood. In each of these cases, the model adequately predicts the compressive strength and the corresponding microbuckle length at maximum load. As an example, results are presented for [( f45, O,),], panels made from T800-924C and ASCPEEK carbon fiber composites. The typical damage state immediately prior to failure is shown in Figure 35a for an ASCPEEK specimen. (The T800-924c failed in a qualitatively similar manner.) Microbuckles initiate from the edge of the hole in both the 0" and 45" plies; some splitting of the 0" plies and delamination between the 0" plies and the 45" plies are also apparent. The evolution of microbuckle length with remote, applied stress was monitored periodically by interrupting a test and X-raying the specimen: a typical response showing initial stable microbuckle development prior to catastrophic failure is given in Figure 35b. The predicted response by the Soutis et at. (1991b) model is included in the figure and is in good agreement with the observed initiation and growth of a microbuckle. The model slightly underestimates the stress for initiation of a microbuckle but accurately predicts both the maximum load and the associated microbuckle length. The notched strength a, of the T800-924C material is compared with that of the AS4-PEEK material in Figure 3%. Again, predictions of the

*FIG. 35. (a) Dye penetrant enhanced X-ray micrograph of [(*45, O2),Is ASCPEEK laminate, with 10 mm diameter central hole. (b) Comparison of the predictions of the Soutis er al. (1993) model with the observed growth of a microbuckle from a 5 mm central hole in a (145, O,),], AS4-PEEK laminate. (c) Comparison of the measured and predicted compressive strengths IT,,for [( *45, O,),], AS4-PEEK and TSOO-924c laminates. The panels are of width w = 50 mm wide, and the notch strength is plotted as a function of hole radius R. The theoretical predictions-are given by the Soutis er al. (1993) model.

Compressive Failure of Fiber Composites

FIG.35.

'

109

110

N. A. Fleck

Soutis et al. (1993) theory are included in the figure: excellent agreement is found between theory and experiment for both materials. The AS4-PEEK material has a higher compressive fracture toughness ( K , = 55 MPa than the T800-924C laminate ( K , = 46 MPa 6) and is less notch sensitive. The notched strengths of both materials lie between the limits of notch insensitivity (where the net-section failure stress equals the unnotched strength) and the perfectly brittle limit (where the local stress at the root of the notch equals the un-notched strength). The cohesive zone model of Soutis et al. (1991b) has proved to be a useful engineering approach to compressive failure of notched, multidirectional laminates. However, further work is required in order to relate the failure toughness of a multi-directional laminate to that of unidirectional material, as described in Section 1V.B.

VI. Directions for Future Research A number of important problems remain in order to complete our understanding of the compressive failure of fiber composites. Some of these topics are listed below. 1. Dynamic Microbuckling

Lankford (1989,1991,1994) has measured the compressive strength of a range of ceramic matrix and polymer matrix composites, and has observed a sharp increase in strength with increasing strain rate, at strain rates above about 300 s-'. Slaughter et al. (1996) have analyzed the response of a fiber composite to a suddenly applied stress pulse; they include the effects of material inertia and initial imperfection, and perform a onedimensional infinite band calculation using couple-stress theory. The more realistic calculation of a suddenly applied pulse in velocity to the end of the specimen, remains to be addressed. Dynamic kinking is an important topic as structural composites are commonly subjected to shock loading, particularly in military and sports applications. 2. Microbuckling from Random Waviness In practice, fiber misalignment exists as a random three-dimensional distribution throughout the composite. The compressive strength is determined by some combination of the magnitude of the fiber misalignment

Compressive Failure of Fiber Composites

111

and its physical size. Work remains to be done on the statistical characterization of waviness and its implications for the distribution in compressive strength for an ensemble of composite parts. A similar strategy of reliability analysis exists for the imperfection-sensitive buckling of shell structures, for the tensile failure of ceramics using Weibull statistics, and for the fracture of brittle solids using probabilistic fracture mechanics. It is envisaged that a defect assessment can be made of composite parts in terms of the measured distribution of imperfections such as fiber misalignment. 3. Microbuckling from Defects Such As Voids Compressive failure of fiber composites occurs from holes and voids, as well as from regions of fiber waviness. The presence of matrix voids is difficult to avoid in composite manufacture. A large cylindrical hole reduced the compressive strength by a factor of about three, depending on the degree of orthotropy. In the other limit of vanishing hole size in relation to fiber diameter d , no reduction in strength occurs. Thus, a significant hole size effect is expected for small holes. Preliminary data showing the effect is presented in Figure 36: tests were performed on unidirectional specimens of T800-924c carbon fiber-epoxy composite containing holes of diameter in the range 0.3-2 mm. Microbuckling occurred from the hole and led to failure in some of the cases. In other, nominally identical specimens, splitting occurred from the edge of the hole, thereby reducing the stress-concentrating effect of the hole. In general, the specimens which displayed splitting were stronger than the specimens for which splitting is absent; see Figure 36. Further experimental and theoretical work is needed to determine the relation between hole and void size and compressive strength.

4. Tunnelling of Microbuckles in Multi-Directional Composites Recent observations of microbuckle propagation in multi-directional laminates (Fleck et al., 1996) suggested that microbuckles grow in the 0” plies of thick composites by a tunnelling mechanism, as sketched in Figure 37. When the microbuckle is sufficiently long, it advances under constant applied stress, termed the “tunnelling stress.” The phenomenon of crack tunnelling has been described in detail by Hutchinson and Suo (1992). For the case of tunnel microbuckling, the tunnelling process involves the combination of microbuckling of the 0” plies and mixed-mode

N, A. Fleck

112 2000

1

I

I

-0

X---

1

microbuckling

x splitting

25 1500 d 0,

c

m

2 1000

._ -

a a

Q

500

0

0

100

200

300

U nln A i n m n t n r I l Y l F Y I U I I I ~ I

LOO

500

ll

~Y I ,

fiber diameter. d

FIG.36. Effect of hole diameter D in unidirectional TSOO-924c carbon fiber-epoxy composite upon the measured compressive strength. A size effect is noted, whereby the strength is greater for holes of diminishing diameter. Failure is by microbuckling from the hole edge.

FIG. 37. Tunnelling of a microbuckle in a thick multi-directional composite, under a tunnelling stress u : .

Compressive Failure of Fiber Composites

113

delamination between 0” plies and adjacent off-axis plies. Calculations remain to be done on the magnitude of the tunnelling stress as a function of layer thickness and the delamination toughness.

Acknowledgments

The author is most grateful for the extensive collaboration with B. Budiansky and M. P. F. Sutcliffe, and for the research support given by C. Soutis, P. M. Jelf, J. H. Shu, S. Sivashanker, and X. J. Xin. Financial support is gratefully acknowledged from the EPSRC, from the Defence Procurement Executive, and from the U.S. Office of Naval Research (contract number 0014-91-5-1916,directed by Y. D. S. Rajapakse).

References Argon, A. S. (1972). Fracture of composites. Treatise Mater. Sci. Technol. 1, 79-114. Ashby, M. F., and Hallam, S. D. (1986). Acta Metall. 34(3), 497-510. Bao, G., Ho, S., SUO,Z., and Fan, B. (1992). The role of material orthotropy in fracture specimens for composites. Int. J . Solids Struct. 29(9), 1105-1116. Barker, A. J., and Balasundaram, V. (1987). Compression testing of carbon fibre-reinforced plastic exposed to humid environments. Composites 18(3), 217-226. Biot, M. A. (1957). Folding instability of a layered viscoelastic medium under compression. PIVC. R. SOC. London A242, 444-454. Biot, M. A. (1967). Rheological stability with couple stresses and its application to geological folding. Proc. R. SOC.London A298,402-423. Budiansky, B. (1979). Remarks on kink formation in axially compressed fibre bundles. I n Preliminary Reports, Memoranda and Technical Notes on the Materials Research Council Summer Conference, La Jolla, California, July 1979, Sponsored by DARPA. Budiansky, B. (1983). Micromechanics. Comput. Struct. 16,3-12. Budiansky, B., and Fleck, N. A. (1993). Compressive failure of fiber composites. J . Mechanics Phys. Solids 41(1), 183-211. Chaplin, C. R. (1977). Compressive fracture in unidirectional glass-reinforced plastics, J. Mater. Sci. 12, 347-352. Chattejee, S. N., and McLaughlin, P. V. (1979). Inelastic shear instability in composite materials under compression. Proc. 3rd Engineering Mechanics Div. Specialty Conf., Sept. 17-19, 1979, University of Texas at Austin, ASCE, pp. 649-652. Clarke, A. R., and Archenhold, G., Davidson, N., Slaughter, W. S., and Fleck, N. A. (1996). Determining the power spectral density of the waviness of unidirectional glass fibres in polymer composites. Applied Composite Materials 2, 233-243. Cosserat, E., and Cosserat, F. (1909). “Theorie des Corps Deformables.” Herman et fils, Paris. Cox, B. N., Carter, W. C., and Fleck, N. A. (1994). A binary model of textile composites. I. Formulation. Acta Metall. Mater. 42(10), 3463-3479.

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Crisfield, M. A. (1991). “Non-Linear Finite Element Analysis of Solids and Structures,” Vol. 1, Chapter 9, Wiley, London. Curtis, P. T. (1986). An investigation of the mechanical properties of improved carbon fibre composite materials. RAE Tech. Rep. 86021 DRA Farnborough, Hants., England. deTeresa, S. J., Porter, R. S., and Farris, R. J. (1988). Experimental verification of a microbuckling model for the axial compressive failure of high performance polymer fibres. J. Muter. Sci. 1886-1894. Dinwoodie, J. M. (1981). “Timber, Its Nature and Behaviour,” Van Nostrand-Reinhold, Berkshire, England. Evans, A. G., and Adler, W. F. (1978). Kinking as a mode of structural degradation in carbon fiber composites. Actu Metull. 26, 725-738. Ewins, P. D., and Potter, R. T. (1980). Some observations on the nature of fibre reinforced plastics and the implications for structural design. Philos. Truns. R . Soc. London A294, 507-517. Fleck, N. A., and Jelf, P. M. (1995). Deformation and failure of a carbon fibre composite under combined shear and transverse loading. Actu. Metall. Muter. 43(8), 3001-3007. Fleck, N. A., and Shu, J. Y. (1995). Microbuckle initiation in fibre composites: A finite element study. J. Mech. Phys. Solids 43(12), 1887-1918. Fleck, N. A., Jelf, M., and Curtis, P. T. (1995a). Compressive failure of laminated and woven composites. J. Compos.: Technol. Res. 212-220, July 1995. Fleck, N. A., Deng, L., and Budiansky, B. (1995b). Prediction of kink width in compressed fiber composites. J. Appl. Mech. 62, 329-337. Fleck, N. A., Sivashanker, S., and Sutcliffe, M. P. F. (1996). The propagation of a microbuckle in unidirectional and multidirectional composites. Acru Metall. Muter. submitted. Fried, N. (1963). The compressive strength of parallel filament reinforced plastics-the role of the resin. Proc. 18th annual meeting of the Reinforced Plastics Division, Society of Plastics Industry, Section 9-A, 1-10. Gibson, L. J., and Ashby, M. F. (1988). “Cellular Solids: Structure and Properties.” Pergamon, New York. Greszczuk, L. B. (1972). Failure mechanisms of composites subjected to compressive loading. AFML-TR-72107,US. Air Force. Greszczuk, L. B. (1975). Microbuckling failure of circular fibre-reinforced composites. A M J . 13(10), 1311-1318. Grossman, P. U. A., and Wold, M. B. (1971). Compressive fracture of wood parallel to the grain. Wood Sci. Technol. 5, 147-156. Gupta, V., Anand, K., and Kryska, M. (1994). Failure mechanisms of laminated carbon-carbon composites-I. Under uniaxial compression, Actu Metull. Muter. 42(3), 781-795. Ha, S. K., Wang, Q., and Chang, F. (1991). Modelling the viscoplastic behavior of fiberreinforced thermoplastic matrix composites at elevated temperatures. J. Compos. Muter. 25, 335-373. Hahn, H. T., and Williams, J. G. (1986). Compression failure mechanisms in unidirectional composites, “Composite Materials: Testing, and Design (Seventh Conference), ASTM STF’ 893,” (J. M. Whitney, ed.), pp. 115-139. American Society for Testing and Materials, Philadelphia, Pennsylvania. Hahn, H. T., Sohi, M., and Moon, S. (1986). Compression failure mechanisms of composite structures. NASA CR 3988. Harte, A.-M., and Fleck, N. A. (1996). Unpublished research. Horoschenkoff, A., Brandt, J., Warnecke, J., and Bruller, 0. S. (1988). Creep behaviour of carbon fibre reinforced polyetheretherketone and epoxy resin. “New Generation Materials and Processes” (F. Saporiti, W. Merati, and L. Peroni, eds.), pp. 339-349. Milan.

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Huang, Y. H., and Wang. S. S. (1989). Compressive fatigue damage and associated property degradation of aluminium-matrix composite. Proc. 4th Japan-U.S. Conf. on Composite Materials, 27-29 June, 1988, Washington, D.C., pp. 602-632. Technomic, Westport, Connecticut. Hutchinson, J. W. (1974). Plastic buckling. Adu. Appl. Mechanics 14, 67-144. Hutchinson, J. W., and SUO,Z. (1991). Mixed mode cracking in layered materials. Adu. Appt. Mechanics 29, 63-191. Jelf, P. M., and Fleck, N. A. (1992). Compression failure mechanisms in unidirectional composites. J . Compos. Muter. 26(18), 2706-2726. Jelf, P. M., and Fleck, N. A. (1994). The failure of composite tubes due to combined compression and torsion. J. Muter. Sci. 29, 3080-3084. Jelf, P. M., Soutis, C., and Fleck, N. A. (1990). Notched compression failure of carbon fibre PEEK laminates. Presented at 3rd. Int. Symp. on Composites, Patras, Greece, Sept. 1990. Johnson, A. M. and Ellen, S. D. (1974). A theory of concentric, kink, and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers. I. Introduction. Tecionophysics 21, 301-339. Johnson, A. M., and Ellen, S. D. (1975a). A theory of concentric kink, and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers. 11. Initial stress and nonlinear equations of equilibrium. Tectonophysics 25, 261-280. Johnson, A. M., and Ellen, S. D. (1975b). A theory of concentric kink, and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers. 111. Transition from sinusoidal to concentric-like chevron folds. Tecronophysics 27, 1-38. Johnson, A. M., and Ellen, S. D. (1976). A theory of concentric, kink, and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers. IV. Development of sinusoidal and kink folds in multilayers confined by rigid boundaries. Tectonophysics 30, 197-239. Johnson, N. J., and Hergenrother, P. M. (1987). High temperature thermoplastics: A review of neat resin and composite properties. 32nd Intl. SAMPE Symposium, April 6-9, 1987. Kaute, D., Ashby, M. F., and Fleck, N. A. (1996). Compressive failure in ceramic matrix composites. Manuscript in preparation. Kumar, S., Adams, W. W., and Helminiak, T. E. (1988). Uniaxial compressive strength of high modulus fibers for composites. J . Reinf. Plust. Compos. 7 (March 19881, 108-119. Kyriakides, S., Arseculeratne, R., Perry, E. J., and Liechti, K. M. (1995). On the compressive failure of fiber reinforced composites. Int. J. Solids Strucr. 32(6/7), 689-738. Lager, J. B., and June, R. R. (1969). Compressive strength of boron/epoxy composites. J. Compos. Muter. 3(1), 48-56. Lankford, J. (1989). Dynamic compressive fracture in fiber reinforced ceramic matrix composites. Muter. Sci. Eng. A107, 261-268. Lankford, J. (1991). Compressive damage and failure at high loading rates in graphite fiber-reinforced polymeric matrix composites. Cerum. Trans. 19, pp. 553-563. Lankford, J. (1994). Shear versus dilatational damage mechanisms in the compressive failure of fiber-reinforced composites, to appear in Composites. Liu, X. H., Moran, P. M., and Shih, C. F. (1995). The mechanics of compressive kinking in ductile matrix fiber composites. To appear in a special issue of Composites Engineering on Thick Composites. Moncunill de Ferran E., and Harris, B. (1970). Compression of polyester resin reinforced with steel wires. J . Compos. Muter. 4, 62-72. Moran, P. M., Liu, X. H., and Shih, C. F. (1995). Kink band formation and band broadening in fiber composites under compressive loading. Acru Meiull. Muter. 43(8), 2943-2958. Nemat-Nasser, S., and Horii, H. (1982). J . Geophys. Res. 87(B8), 6805-6821.

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Neptco Inc. (1994). Product information sheet on Graphlite. Box 2323, 30 Hamlet St., Pawtucket, Rhode Island, 02861-0323. Palmer, A. C., and Rice, J. R. (1973). The growth of slip surfaces in the progressive failure of over-consolidated clay. Proc. R. SOC.London, A332, 527-548. Paris, P. C., and Sih, G. C. (1969). Stress analysis of cracks. Fracture Toughness and Applications ASTM STP 381, 30-83. Parry, T. V., and Wronski, A. S. (1981). Kinking and tensile, compressive and interlaminar shear failure in carbon fibre reinforced plastic beams tested in flexure. J. Mater. Sci. 16, 439-450. Piggott, M. R. (1981). A theoretical framework for the compressive failure of aligned fibre composites. J . Mater. Sci. 16, 2837-2845. Piggott, M. R., and Harris, 8. (1980). Compression strength of carbon, glass and Kevlar-49 fibre reinforced polyester resins. J. Mater. Sci. 15, 2523-2538. Piggott, M. R., and Wilde, P. (1980). Compressive strength of aligned steel reinforced epoxy resin. J. Mater. Sci. 15,2811-2815. Prandy, J. M., and Hahn, H. T. (1990). Compressive strength of carbon fibers. Proc. 35th International SAMPE Symposium, April 2-5 1990, pp. 1657- 1670. Rhodes, M. D., Mikulas, M. M., and McGowan, P. E. (1984). Effects of orthotropy and width on the compression strength of graphite-epoxy panels with holes. AIAA J . 22(9), 1283-1292. Rice, J. R. (1968). Mathematical analysis in the mechanics of fracture. In “Fracture” (H. Liebowitz, ed.), Vol 2, Chapter 3. Academic Press, New York. Rosen, B. W. (1965). Mechanics of composite strengthening. “Fiber Composite Materials.” pp. 37-75. Chapter 3, American Society of Metals. Sammis, C. B., and Ashby, M. F. (1986). Acta Metall 34(3), 511-526. Schapery, R. A. (1993). Nonlinear viscoelastic effects in the compressive behavior of fiber composites. “Mechanics of Thick Composites, AMD-162.” (Y.D. S. Rajapakse, ed.), pp. 81-90. American Society of Mechanical Engineers. Schapery, R. A. (1995). Prediction of compressive strength and kink bands in composites using a work potential. Int. J . Solids Struct. 32(6/7), 739-765. Schulte, K., and Minoshima, K. (1991). Mechanisms of fracture and failure in metal matrix composites. 12th Rim Int. Symp. on Materials Science: Metal Matrix Composites Processing, Microstructure and Properties (N. Hansen et al., eds.), pp. 123-147. Shu, J. Y., and Fleck, N. A. (1996). Microbuckle initiation in fibre composites under multiaxial loading. Proc. R. SOC. London, to appear. Sivashanker, S., Fleck, N. A., and Sutcliffe, M. P. F. (1995). Microbuckle propagation in a unidirectional carbon fibre-epoxy matrix composite. To appear in Acta Metall. Mater. Slaughter, W. S., and Fleck, N. A. (1993). Compressive fatigue of fibre composites. J. Mech. Phys. Solids 41(8), 1265-1284. Slaughter, W. S., and Fleck, N. A. (1994a). Viscoelastic microbuckling of fibre composites. J. Appl. Mech. 60(4), 802-806. Slaughter, W. S., and Fleck, N. A. (1994b). Microbuckling of fiber composites with random initial fiber waviness. J . Mech. Phys. Solids 42(11), 1743-1766. Slaughter, W. S., Fleck, N. A., and Budiansky, B. (1992). Microbuckling of fibre composites: The roles of multi-axial loading and creep. J. Eng. Mater. Technol. 115(3), 308-313. Slaughter, W. S., Fan. J., and Fleck, N. A. (1996). Dynamic compressive failure of fiber composites. J. Mech. Phys. Solids 44(11), 1867-1890. Soutis, C., and Fleck, N. A. (1990). Static compression failure of carbon fibre T800/924C composite plate with single hole. J . Compos. Mater. 24, 536-558. Soutis, C., Fleck, N. A., and Smith, P. A. (1991a). Compression fatigue behaviour of notched carbon fibre-epoxy laminates. Znt. J . Fatigue 13(4), 303-312.

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Soutis, C., Fleck, N. A., and Smith, P. A. (1991b). Failure prediction technique for compression loaded carbon fibre-epoxy laminate with open holes. J. Compos. Mater. 25(11), 1476-1498. Soutis, C., Curtis, P. T., and Fleck, N. A. (1993). Compressive failure of notched carbon fibre composites. Proc. R. SOC.London 440, 241-256. Sun, C. T., and Chen, J. L. (1989). A simple flow rule for characterizing nonlinear behavior of fiber composites. J. Composite Mater. 23, 1009-1020. Sun, C. T., and Yoon, K. J. (1991). Characterization of elastic-plastic behavior of AS4/PEEK thermoplastic composite for temperature variation. J. Compos. Mater 25, 1297- 1313. Sutcliffe, M. P. F., and Fleck, N. A. (1993). Effect of geometry upon compressive failure of notched composites. Int. J. Fract. 59, 115-132. Sutcliffe, M. P. F., and Fleck, N. A. (1994). Microbuckle propagation in carbon fibre-epoxy composites. Acta Metall. Mater. 42(7), 2219-2231. Sutcliffe, M. P. F., and Fleck, N. A. (1996). Micobuckle propagation in fibre composites. Acta Metall. Mater. to appear. Sutcliffe, M. P. F., Fleck, N. A., and Xin, X. J. (1995). Prediction of compressive R-curve for long fibre composites. To appear in Proc. R. SOC.London. US. Polymeric (1990). Data sheets on properties of carbon fibre epoxy composites, 700E Dyer Rd., Santa Ana, California, 92707. Whitcomb, J. D. (1986). Parametric analytical study of instability-related delamination growth. Composite Science and Technology 25, 19-48. Wilkinson, E., Parry, T. V., and Wronski, A. S. (1986). Compressive failure in two types of carbon fibre-epoxide laminates. Composite Science and Technology 26, 17-29. Wisnom, M. R. (1990). The effect of fibre misalignment on the compressive strength of unidirectional carbon fibre/epoxy. Composites 21(5), September 1990, 403-408. Wu, X. R. and Carlsson, A. J. (1991). “Weight Functions and Stress Intensity Factor Solutions.” Pergamon, New York. Xin, X. J., Sutcliffe, M. P. F., Fleck, N. A., and Curtis, P. T. (1995). “Cambridge Composite Designer: A User Manual.” CUED/C-MATS/TR.226, August 1995. Young, R. J., and Young, R. (1990). Strain measurement in fibres and composites using Raman spectroscopy. Proc. Fourth European Conference on Composite Materials, September 25-28 Stuttgart, pp. 685-690. Elsevier, Amsterdam. Yurgatis, S. W. (1987). Measurement of small angle misalignments in continuous fibre composites. Composites Science and Technology 30, 279-293.

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ADVANCES IN APPLIED MECHANICS. VOLUME 33

Delamination of Compressed Thin Films GUSTAVO GIOIA' Division of Engineering Brown Uniuersity Prouidence. Rhode Island

MICHAEL ORTIZ Graduate Aeronautical Laboratories California Institute of Technology Pasadena. California

...................................... I1 . Experimental Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Sources of Residual Stresses in Thin Films . . . . . . . . . . . . . . . . B. Observations of Delamination in Compressed Films . . . . . . . . . . I . Introduction

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122 122 124

C . Experimental Observations of Blister Morphologies and Folding Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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111. Folding Patterns as Energy Minimizers . . . . . . . . . . . . . . . . . . . . .

132

A . Energy of Delaminated Films . . . . . . . . . . . . . . . . . . . . . . . . . B. Thin-Film Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . A Two-Variant Construction . . . . . . . . . . . . . . . . . . . . . . . . . D . General Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E . Comparisons with Experiments . . . . . . . . . . . . . . . . . . . . . . . . F. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IV . Film h8orphologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Energetics of Blister Growth . . . . . . . . . . . . . . . . . . . . . . . . . B . Boundary-Layer Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Equilibrium Morphologies: The Telephone Cord . . . . . . . . . . . . D . Comparisons with Experiments . . . . . . . . . . . . . . . . . . . . . . . . E. Growth of Telephone-Cord Blisters . . . . . . . . . . . . . . . . . . . . .

....................................... Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V . Conclusion

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'

Present address: Department of Aerospace Engineering and Mechanics. 107 Merman Hall. University of Minnesota. Minneapolis. MN 55455. USA. 119

ADVANCES IN APPLIED MECHANICS. VOL . 33 Copyright 0 1997 by Academic Press. All rights of reproduction in any form reserved . 0065-2165/97 $25.00

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Gustavo Gioia and Michael 0rti.z

I. Introduction

Thin-solid films deposited over substrates are presently used in strikingly diverse technological applications. For instance, wear-resistant coatings in metal-cutting tools have attained considerable technological importance and, as such, have been the subject of extensive research. Since its appearance in 1969, the coating of cutting tools has afforded an order-ofmagnitude improvement in performance and productivity in the machine tool industry (Quinto, 1988). This is a remarkable feat because the volume of the coating barely amounts to about 1% of the volume of the tool. The coatings are thin films of single elements (e.g., C, B); binary compounds, including metallic (e.g., Ti, Zr, Hf, Ta, and V nitrites, carbides, and borides), covalent (e.g., Sic, B,C), and ionic compounds (e.g., Al,O,, Ti, Th, Zr, and Hf bioxides, Be and Mg oxides); and ternary compounds (e.g., nitrites, in particular Ti,-,Zr,N and Ti,_,Al,N) (Musil et al., 1993; Randhawa et al., 1988). Usual coating thicknesses range from 5-10 pm. Most commercial coatings comprise three layers, e.g., TiC/TiCN/TiN or TiC/Al,O,/TiN, albeit as many as lo3 nanometric layers may be utilized. A more recent development concerns the utilization of “gradient coatings,” wherein the composition of the coating varies continuously across its thickness, leading to improved mechanical performance (Holleck et al., 1985; Quinto, 1988; Musil et al., 1993). The most common substrate materials used in metal-cutting tools are steels and silicon. Some of the films that find application in cutting tools are also of importance in the manufacture of integrated circuits. A case in point is that of TiN (also TaN, Ta, W, V, Cr, Nb) films used as diffusion barriers between silicon substrates and metallic (Al, Au, Ag) conductive layers (Leusink et al., 1993). These barriers are inserted to prevent the diffusion of substrate atoms into the conductive layers, and the attendant degradation of the electrical performance of the latter (Granneman, 1993). A second example involving thin anodic oxide films is provided by the microelectronics industry, where such films are employed as insulating and protective coatings (Stark et al., 1993). Additional examples of film/substrate systems are afforded by the automotive and optical industries, where glass has been largely replaced by optical polymers. These materials are advantageous from the standpoint of weight and formability. However, optical polymers suffer from a number of shortcomings, including low surface hardness, loss of transparency and mechanical properties when exposed to UV radiation, and severe sensitiv-

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ity to certain chemicals, including hydrocarbons. The introduction of transparent silicon oxide and silicon nitrite coatings has been instrumental in overcoming these problems. A further example of application of thin films in the manufacture of optical devices is the deposition of oxidic coatings (silicon, tantalum, tungsten, and nickel oxides) on glass. These coatings are employed as high- or low-refractive layers in lenses, filters, and mirrors (Pulker, 1987). In yet another optical application, thin multilayer films made of heavy metals (Zr, W) are used in combination with carbon substrates in reflective components for soft X-ray regions (Aouadi et al., 1992). Ceramic thermal barrier coatings are extensively used in the aircraft and automobile industries. These coatings increase the operating temperature of engines, which results in improved efficiency and lower cooling requirements for the substrate (Geiger, 1992). For example, partially-stabilized zirconia is deposited on airfoils to reduce the operating temperature of underlying nickel-based superalloy blades (Twigg and Page, 1993; Jordan and Faber, 1993). Depending on the specific film/substrate pair and on processing and service conditions, thin films may be subjected to very large compressive stresses. Although a certain level of compression in the thin film may be desirable-as in the integrated circuit industry, where small compressive stresses in the order of 0.01-0.05 GPa are deemed beneficial (Granneman, 1993)-large compressive stresses, as are likely to occur in many systems, may lead to failure by a variety of stress relief mechanisms. These include hillock formation (d’Heurle, 1989; Jou and Chung, 19931, peeling by adhesion failure at the film/substrate interface (Chopra, 19691, creep and plastic flow (Chopra, 1969), and buckling-driven delamination (Hutchinson and Suo, 1991). In this article, we specifically concern ourselves with the buckling-driven delamination mechanism, whereby a portion of the film buckles away from the substrate, thereby forming a blister (also termed buckle or wrinkle). Blisters may grow by interfacial fracture, a process which, under the appropriate conditions, may result in the catastrophic failure of the component. Blisters are often observed to adopt convoluted-even bizarreshapes and to fold into intricate patterns. A principal objective of this article is to review some recent developments based on the use of direct methods of the calculus of variations which have proven useful for understanding the mechanics of folding of thin films (Ortiz and Gioia, 1994). These developments are reviewed in Section 111, which is extracted from

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the original publication. The remaining sections are devoted to the application of these principles to the problem of predicting the shape of thin-film blisters.

11. Experimental Background

A.

SOURCES OF

RESIDUAL STRESSES

IN

THINFILMS

When the deposition temperature is greatly in excess of the service temperature, residual thermal stresses inevitably arise as a result of mismatches in the thermal expansion coefficients of film and substrate. In some applications, the temperature differential can be as high as 1000°C, and the attendant stresses and strains in the film of the order of 1 GPa and 1%, respectively (Yelon and Voegeli, 1964; Chopra, 1969; Hutchinson et af., 1992; Jordan and Faber, 1993; Jou and Chung, 1993; Twigg and Page, 1993). A second important source of residual stresses is contamination by impurities of porous films with sub-bulk density microstructures. Common impurities include water vapor, hydrogen, and oxygen (Windischmann, 1992). Oxygen can additionally lead to oxidation, specially in metals (Chopra, 1969). The resulting oxide has a larger volume per unit mass than the parent material, which results in further compression of the film (Teschke and Kleinke, 1993; Trigo et af.,1993). Processing techniques may also induce sizeable compressive stresses in the film. A case in point is furnished by oxidic coatings in optical devices, where high compressive stresses result from the application of ion-beam irradiation for the purpose of increasing the refractive index and reducing the loss of optical properties by hydrogen uptake (Wagner et al., 1993). In many systems of interest, intrinsic o r growth residual stresses arise from defects which are built into the film during processing. In films produced by evaporation, stresses are generally tensile in metallic films and can be either tensile or compressive in dielectric compounds. For example, intrinsic compressive stresses of the order of 0.1 GPa have been reported for TiO,, ZnS, and N , O , films on glass (Hoffman, 1966). The magnitude of the intrinsic stresses often depends on the sensitivity of processing and operating conditions. Thus, the compressive stresses in ZnS films are observed to decrease with increasing substrate temperature during deposition (Chopra, 1969, and references therein).

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In sputter-deposited thin films, the intrinsic stresses, which are frequently compressive, are believed to be a consequence of the lattice distortion induced in the film by energetic particle bombardment (d’Heurle, 1970). This important mechanism has been termed atomic peening. In data compiled by Windischmann (19921, compressive intrinsic stresses ranging between 0.1-3 GPa have been reported for sputtered films. Again, the intrinsic stresses depend on the sputtering conditions. For instance, stress reversal from tensile to compressive has been reported in Mo films at increasing particle energy (see, e.g., Haghiri-Gosnet el al., 1989). Similar reversals have been reported for decreasing sputtering gas pressure and decreasing angle of deposition, i.e., the angle between the substrate and the vapor source (Vink and van Zon, 1991; Windischmann, 1992, and references therein). In evaporated metallic films, a link is known to exist between film microstructure and the reduced temperature regardless of the film/ substrate pair. The reduced temperature is defined as 0, = I3/0,,,, where I3 is the deposition temperature of the substrate and 13, is the melting point of the film material. According to the structure-zone model (Thornton, 1974, 1977; Alexopoulos and O’Sullivan, 1990; Musil et al., 19931, three regimes, or ‘zones,’ may be identified. In Zone I, 0, < 0.3, a honeycomb-like microstructure is observed which is composed of columnar, tapered crystallites surrounded by voids. The latter amount to 10 to 20% of the total volume. This microstructure is controlled by deposition shadowing effects. An illustrative model of this is a grassy lawn in which the growth of any given stem is proportional to the amount of sunlight received at the tip (Bales et al., 1990). In Zone 11, 0.3 < 0, < 0.5, the same columnar structure is present, albeit in tighter contact. In this zone, the controlling mechanism is surface diffusion. Finally, Zone 111, 0, > 0.5, consists of equiaxed, roughly spherical polycrystals, which grow by bulk diffusioncontrolled recrystallization. A similar model has been proposed for oxides (see e.g., Alexopoulos and O’Sullivan, 1990). Generalized structure-zone models concerned with metallic and ceramic, polycrystalline and amorphous sputtered films, have been advanced by several authors. For example, Thornton (1974, 1977) has put forth a model which accounts for a transition Zone T, characterized by densely packed fibrous grains, between Zones I and 11; and the sputtering gas pressure as an additional variable. Interestingly, the different zones in these models appear to correlate with specific states of intrinsic stress (Thornton and Hoffman, 1977; Hoffman and Thornton, 1977; Musil et al., 1993). For instance, Zones I

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and T in Thornton's model (1974, 1977) are associated with tensile and compressive intrinsic stresses, and with low and high particle bombardment, respectively. This suggests the existence of a definite energy/film microstructure/intrinsic stress relationship (see, e.g., Windischmann, 1992). However, there may be concurrent effects which compound the prediction of residual stresses. Thus, while the intrinsic stress is expected to be tensile in Zone I, the microstructure characteristic of this zone promotes impurity contamination and oxidation, which may result in net compressive stresses (Windischmann, 1992).

B. OBSERVATIONS OF DELAMINATION IN COMPRESSED FILMS Large compressive stresses have been reported in cutting tool coatings. For example, Sakamoto et al. (1993) measured stresses of up to 5 GPa in various 0.6-1 pm sputtered Ti,,Zr,,N films applied on Si substrates. In tests performed on TIC and Tic-TiN films obtained by chemical vapor deposition on austenitic stainless steel, Asada et a f . (1993) measured residual compressive stresses of 4.25 and 2.34 GPa, respectively. Jovan and LempCrikre (1994) reported stresses of up to 12 GPa in TiN films sputtered on Si under a variety of conditions. Buckling-driven delamination has been frequently observed in these systems. Kinbara and Baba (1983) reported blisters in compressed T i c films. Gupta (1991) documented blistering in S i c films. Matuda et a f . (1981) reported stresses of about 5 GPa in 0.05-0.3 p m carbon films deposited on glass. Upon exposure to the atmosphere, these stresses led to blistering. Kinbara et al. (1981) observed blistering of C films on glass when exposed to the atmosphere. Under similar conditions, blisters have been reported by the latter authors in boron films deposited on NaCl(100). Gille and Rau (1984) reported compressive stresses of 3-6 GPa leading to blistering in carbon films deposited onto Si and quartz glass. Other studies reporting blistering in carbon films include Weissmantel et af. (1979); Nir (1984); Argon et a f . (1989a, b); Kinbara and Baba (1991); and Seth et al. (1992). Similar problems have arisen in coated integrated circuits, where high compressive stresses leading to delamination have been reported by Teschke and Kleinke (1993). High compressive stresses leading to extensive blistering are also known to occur in silicon-based protective transparent multilayer coatings deposited on optical polymers (Rostaing et al., 1993). Similar phenomena are

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observed in heavy metal films used in optical applications. In a study performed by Trigo et al. (1993), 0.3-0.6 pm-thick Zr films were sputtered under ion bombardment on Si( 100) wafers. For relatively low bombardment energies (30-70 eV/atom), adsorbed water and internal oxidation led to blistering and, eventually, to complete delamination in the course of a few days. In partially-stabilized zirconia films deposited on nickel-based superalloy substrates, Twigg and Page (1993) estimated a compressive stress of 1 GPa. Although a reduction of these stresses is possible by depositing the film at lower temperatures, adhesion is then seriously compromised, and the risk of blistering persists. In a different study, Jordan and Faber (1993) observed compressive stresses in the order of 0.1 GPa in the surface of a 250 pm Zr0,-8% Y,O, film deposited on a Hastelloy-X substrate with a 125 p m NiCoCrAlY bond coating. The system was subjected to cyclic thermal loading in the range of 400-1000°C, and was observed to relieve thermal stresses by the buckling-driven delamination mechanism.

c.

EXPERIMENTAL OBSERVATIONS OF BLISTERMORPHOLOGIES AND FOLDING PATERNS

There presently exists an extensive study of literature documenting a variety of blister morphologies and folding patterns in compressed thin films. Some of these morphologies, such as the ‘telephone cord,’ have been ubiquitously observed in many different film/substrate systems, which strongly suggests a certain universality of the underlying mechanics. The complexity of the blister shapes and their folding patterns has both fascinated and befuddled investigators, who have variously referred to them as ‘amazing,’ ‘grotesque,’ ‘intricate,’ ‘intriguing,’ and ‘spectacular.’ Some representative morphologies are reviewed in this section for subsequent reference. Blistering has often been observed to initiate in the neighborhood of the film’s edge and to propagate towards the center of the specimen (Gille and Rau, 1984; Seth et af., 1992; Eymery and Boubeker, 1994; Daniels et al., 1995). Once initiated, blisters may advance along branching bucklingfronts. Figure 1 shows an example of this type of growth reported by Seth et a f . (1992). In this case, the blister was observed to crack due to bending. Buckling fronts may eventually sweep over the entire specimen. The

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Gustavo Gioia and Michael Ortiz

FIG. 1. Buckling fronts in a 0.6 p m diamond-like carbon fdm advancing over a Si substrate. After Seth et al. (1992). Reprinted with permission of the American Institute of Physics.

blisters thus formed often fold into intricate patterns, Figure 2 (after Seth et al., 1992; see also Siddall, 1960). In other cases, propagation leads to a web of fingers which bound islands of bonded film (Yelon and Voegeli, 1964; Matuda et al., 1981). A striking example of this mode of propagation is shown in Figure 3 (after Matuda et al., 1981). As noted in that figure, the film under observation had a thickness gradient, which affords an illustration of how the blister size scales with the film thickness. Such scaling is expected from dimensional considerations alone, and has been pointed out by several authors (Matuda et al., 1981; Nir, 1984; Iyer et al., 1995). Blistering has also been observed to initiate at discrete nucleation sites, most commonly defects in the film, substrate, or interface (Kinbara et al., 1981; Nir, 1984; Ogawa et al., 1986; Argon et al., 1989b; Yu et al., 1991; Eymery and Boubeker, 1994). Initially, the blister may take the form of a circular cap or dome (Argon et al., 1989b; Yu et al., 1991; Eymery and Boubecker, 1994). After some growth, the blister loses its circular geometry, and a buckling front or fingering may ensue. Instances of both modes of growth have indeed been documented (Kinbara et al., 1981; Argon et al., 1989b; Hutchinson et al., 1992). An example of a web of blisters nucleated at a single site is shown in Figure 4 (after Kinbara et al., 1981).

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127

FIG. 2. Final folding pattern in a diamond-like carbon film as reported by Seth with permission of the American Institute of Physics.

el a/. (1992). Reprinted

Whether initiation occurs at the film’s edge or at discrete nucleation sites, the most commonly observed delamination mode entails characteristic wavy patterns which have been suggestively termed worms or telephone cords. Figure 5 (page 129) shows two fairly representative telephone-cord blisters observed by Thouless (1993) in a Si film on a SiO, glass substrate. (Albeit far less common, straight-cord blisters have also been documented; see Thouless, 1993). Telephone-cord blisters have also been observed in

FIG. 3. Web of blisters observed by Matuda el al. (1981) in a carbon film with a thickness gradient. Reprinted with permission of Elsevier Science Ltd.

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Gustauo Gioia and Michael Ortiz

FIG.4. Web of blisters observed by Kinbara et af.(1981) in an 800 A carbon film on a glass substrate. The film was exposed to the atmosphere at room temperature, and the photographs were taken after: a) 2 h; b) 4 h; c) 10 h; and d) 12 h. Reprinted with permission of Elsevier Science Ltd.

diamond-like carbon films on Si, glass and steel (Weissmantel et al., 1979; Nir, 1984; Weissmantel, 1985; Seth et af., 1992; Iyer et al., 1995); SiO films on glass and on Ni (Priest et af., 1962); Fe/Ni compositionally modulated films (Yu et al., 1991); Fe/Pt multilayered films on Si (Daniels et af., 1995); 75-25 Fe-Ni films on rock-salt (Yelon and Voegeli, 1964); bcc stainless steel films on fcc stainless steel (Eymery and Boubeker, 1994); Boron films on NaCl (Kinbara et al., 1981); ZnS films on glass (Behrndt, 1965); Mo films on glass (Ogawa et af., 1986; Mattox and Cuthrell, 1988); W films on glass and on mica (Chopra, 1969); and mica films glued on Al (Hutchinson et af., 1992). The disparity of these systems attests to the ubiquitous nature of the telephone-cord morphology. Telephone cords have been observed to arise from pre-existing straightsided blisters as a result of a boundary instability triggered by cooling of the specimen (M. D. Thouless, private communication, 1994). However, the usual mode of growth of telephone cords is by delamination at the tip, a process which has been superbly documented by Ogawa et al. (1986; see Figure 6) and pointed out by other authors (Nir, 1984; Hutchinson et af., 1992; Thouless, 1993; Daniels et al., 1995).

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FIG.5. Telephone-cord blisters in an Si/SiO, glass system reported by Thouless (1993). Reprinted with permission of the American Ceramic Society.

FIG.6. Usual mode of growth of a telephone-cord blister by delamination at the tip, as reported by Ogawa el (11. (1986) for a Mo/glass system. Reprinted with permission.

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Gustavo Gioia and Michael Ortiz

Preferred growth directions have been reported in films grown epitaxially on crystallographic planes (Yelon and Voegeli, 1964). These cords may branch forming webs of regular geometries, Figure 7 (after Yelon and Voegeli, 1964). A different example displaying a preferred direction of growth is given by arrays of cords growing parallel to each other. These arrays are often observed covering entire portions of the film, as shown in Figure 8 (A. F. Bastawros, private communication, 1994; see also Daniels et al., 1986). The experimental evidence suggests that the direction of the cords in these arrays is determined by the anisotropy of the state of stress in the film, which favors cords which are perpendicular to the axis of highest compressive stress. Based on this evidence, Priest et al. (1962) and Nir (1984) conjectured that the telephone-cord morphology itself is caused by residual stress anisotropy. However, specimens where the compressive stresses in the film are ostensibly isotropic sometimes develop telephone cords which grow in arbitrary directions, Figure 5 (see also Ogawa et af., 1986). The cords may even cross each other, seemingly without affecting their growth direction, Figure 9 (after Seth et al., 1992). Additionally, in some cases telephone cords are known to branch and form webs which

FIG. 7. Regular web of telephone cords in a 75-25 Fe-Ni film grown by epitaxial evaporation on rock salt. The (001) orientation of the cords is clearly apparent except near steps in the NaC1. After Yelon and Voegeli (1964). Reprinted with permission of Pergamon Press.

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131

FIG.8. Family of telephone-cord blisters. Unpublished photograph by A. F. Bastawros of Brown University (private communication, 1994).

FIG.9. Crossing telephone-cord blisters in a diamond-like carbon film on glass substrate as observed by Seth et al. (1992). Reprinted with permission of Elsevier Science Ltd.

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Gustavo Gioia and Michael Ortiz

FIG. 10. Web of telephone cords in a 0.92 p m thick Fe/Ni compositionally modulated film. After Yu et al. (1991). Reprinted with permission of Elsevier Science Ltd.

appear devoid of any preferred direction, Figure 10 (after Yu et al., 1991). These observations suggest that, while the direction of growth of telephone cords may be influenced by residual stress anisotropy, the telephone-cord morphology itself is more deeply rooted in the mechanics of the film/ substrate system.

Ill. Folding Patterns as Energy Minimizers The delamination of compressed thin films has been investigated by a number of researchers (see, e.g., the review of Hutchinson and Suo, 1991). The studies to date have by and large been based on conventional elastic stability theory and interfacial fracture mechanics, and have primarily been concerned with the stability of blisters of simple shapes, such as circular and straight-sided flaws (Hutchinson et al., 1992; Nilsson et al., 1993). However, conventional methods of analysis tend to become unwieldy when

Delamination of Compressed Thin Films

133

applied to arbitrary domains or to blisters in which intricate folding patterns develop. As a consequence, these problems have defied effective analytical treatment. In this section we pursue a different line of inquiry based on energy methods. We begin by noting that, in the limit of thin films, the bending energy of the film constitutes a singular perturbation superimposed on the membrane energy. This suggests a solution of the problem by matched asymptotic expansions: the outer solution minimizes the membrane energy of the film, and the inner solution is obtained by fitting boundary layers to the sharp edges which are present in the membrane solution. In pursuing this program, however, an essential difficulty is the lack of convexity of the membrane energy when regarded as a functional of the film deflections. In other fields of application, such as phase transitions and micromagnetics, this lack of convexity is tied to the emergence of fine microstructures, such as twinning or magnetic domains (James and Kinderlehrer, 1990, 1993; Kohn and Muller, 1992). In predicting these microstructures, the so-called direct methods of the calculus of variations have proven particularly effective. These methods seek to characterize solutions directly as energy minimizers, instead of as solutions of the Euler equations of the energy functional. In this context, microstructures naturally arise as a by-product of energy minimization. Similarly the intricate folding patterns exhibited by large blisters can be accorded an energetic interpretation as energy minimizers. In Section III.C, we demonstrate how the membrane energy of a straight-sided semi-infinite film can indeed be fully relaxed by appropriately folding the film. Alternative constructions have been considered by Pipkin (1986a,b) as a basis for deriving tension field theories (Reissner, 1938). Unfortunately, membrane energy minimization fails to produce a unique folding pattern unless subsidiary conditions are formulated. This difficulty is endemic in problems involving singularly perturbed nonconvex functionals, and has been addressed in the mathematical literature (Modica, 1987; Sternber, 1988). A method which originated with the pioneering work of De Giorgi (1975; De Giorgi and Franzoni, 1975) is to resort to the singular term in the energy for selecting a preferred outer solution. In the present context, the idea is to select the membrane solution containing the least possible bending energy. Because membrane solutions can exhibit sharp edges at which the bending energy density is undefined, the measure of bending energy used to select the preferred membrane solution has to be chosen with some care. An appropriate choice can be derived simply by a

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Gustauo Gioia and Michael Ortiz

local boundary-layer analysis which we adapt from the work of Modica (1987) and Sternberg (1988). By virtue of this analysis, sharp edges can be accorded a well-defined energy per unit length, or line tension. The problem is then to construct a membrane solution having the least possible edge and boundary energy. We conjecture that such preferred membrane solution is the envelope of all cones of a characteristic slope supported in the domain of the blister. The deflections are, therefore, maximal solutions of the eikonal equation. The preferred membrane solution can also be defined by Nadai’s sand-heap construction, or as the envelope of all planes tangent to the boundary having a characteristic slope. Bending can subsequently be taken into account by fitting boundary layers to the sharp edges (caustics in the geometrical optics analogy; ridges in the sand-heap analogy) of the membrane solution. An alternative (albeit approximate) means of taking bending into consideration is by subjecting the membrane solutions to Laplacian smoothing. Justification for the proposed construction is found in the excellent agreement between the analytical solutions and the deflection patterns observed by Argon et al. (1989a,b). The folding patterns that fully relax the membrane energy of the film invariably involve increasingly finer features. Bending effectively checks this process of indefinite refinement and introduces a lower bound for the spacing of the folds. Indeed, a boundary-layer analysis permits one to assign a well-defined width to the sharp edges exhibited by the membrane solution. This width sets the minimum spacing between folds, and is found to correlate closely with the wavelength of the boundary undulations observed in large blisters. This suggests that such undulations are directly induced by the internal folding of the film. In other cases, the geometry of the boundary may depend critically on the fracture properties of the interface. Such cases, most notably the telephone-cord morphology, are treated in Section IV.

A. ENERGY OF DELAMINATED FILMS For many material systems of interest, the deflections of the film following delamination are observed to be of moderate size (see e.g., Argon et al., 1989a,b), which suggests framing the analysis within the classical von KBrmh theory of moderate deflections of a plate. We begin by investigating the role played by the various contributions to the energy

Delamination of Compressed Thin Films

135

of the film, as computed from the von KBrmLn theory. We consider an infinite thin film of constant thickness h bonded to a substrate occupying the half space x j I 0, Figure 11. The film is in a state of residual biaxial compression and is debonded over a region fl, where it buckles out of its plane to form a blister. The region s1 has boundary r with outward unit normal n. We shall restrict our attention to bounded domains and presume both s1 and r to be as regular as needed. Requiring fl to be open and bounded and r to be Lipschitz-continuous suffices for most mathematical purposes. Let r: fl R 3 be the position vector of points on the midsurface of the film after deformation. The in-plane displacements and out-of-plane deflections of the film are defined in terms of r as

-

u, = r, - x u ,

w

=

r3,

(1)

respectively, where Greek indices range from 1 to 2. In von KArmAn’s plate theory, the membrane and bending strains are defined as (2)

where zqU, p ) = (uU, + up,a ) / 2 are the components of the symmetric gradient of the in-plane displacements, and e;I*pare the eigenstrains in the film, i.e., the uniform strains the film would undergo if released from the substrate. The last term of (2) is nonlinear, and couples to leading order the out-of-plane deflections to the in-plane deformations. The membrane

Fic. 11. Kinematic conventions of von Kirmin theory by Ortiz and Gioia (1994). Reprinted with permission of Elsevier Science Ltd.

Gustauo Gioia and Michael Ortiz

136

and bending energy densities are then postulated to be isotropic quadratic functions of the membrane and bending strains, respectively, i.e. W"

C ?[(1 -

=

v)EapEap

+ Y ( E Y Y )21,

D W b = -[(l - v)w,apw,ap+ v<w,,,)*], 2

(4)

(5)

where

C =

Eh (1 - v 2 ) '

D =

Eh

12(1 - v 2 ) '

(6)

are the membrane and bending stiffnesses of the film, which are defined in terms of the Young's modulus E and Poisson's ratio v of the material. The membrane stresses and bending moments follow as

The membrane and bending energies of the delaminated portion of the film are obtained by integration of (4) and (9,respectively, with the result

and the total energy of the film follows as @[r] = @"[r]

+ ab[r].

(11)

The problem is, therefore, to minimize @[r]over some suitable space X of displacement fields, typically a reflexive Banach space of functions satisfying the essential boundary conditions

on

u,

=

0,

w

=

0,

(12) w,,n, = w , = ~0

(13)

r. This defines the variational problem ( P I : inf{@[r],r E XI.

(14)

Delamination of Compressed Thin Films

137

For fields of sufficient differentiability, the equilibrium equations are obtained by rendering (11) stationary. The result is

Mq3,ap

-

Nffp,p=

0’

(15)

(KpW,J,p =

0.

(16)

The equilibrium equations (15) and (16) constitute the Euler equations of the energy functional (11). Because of the lack of convexity of the energy functional under consideration, however, it will prove advantageous to resort to direct methods in the calculus of variations. These methods endeavor to minimize the energy functional directly, sidestepping any consideration of Euler’s equations. The energy functional (11) can be manipulated so as to bring it into closer correspondence with energy functionals of the type considered by James and Kinderlehrer (1990). Making use of (15), the membrane energy (9) can be expressed as

where we have denoted 1 e p

-

= ?W.,W,p

e p

EuUp = U ( 0 , p )

’

(18)

and u and w are subject to (15). Interestingly, membrane solutions relaxing

@“‘[I-]to zero result in energy equipartition in the sense that the identity C

jI1 -[(l 2

- v>€:p€:p

C

=

fn -[ 1( 5 1. Recall that the membrane solution satisfies the eikonal equation (49) which, sufficiently close to the boundary, gives w,,= k . Therefore, compatibility of slope between the inner and outer solutions demands

which leaves x( 6 ) as the only independent unknown. Inserting (109) and (111) into (1081, the line tension follows as

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Gustavo Gioia and Michael Ortiz

where primes denote differentiation with respect to 6. It bears emphasis that the line tension depends on K , x, and their derivatives with respect to the arc-length 6. The integral in (112) can be carried out explicitly with the aid of a symbolic manipulation program. The resulting expression is lengthy and will not be recorded here. Finally, the sought boundary-layer energy may now be computed as

where L is the length of r. The bending strain of (1131, which leads to the Euler equation

x follows by minimization

Again, the function 8 is lengthy and will not be recorded here. Eq. (114) expresses the equilibrium condition for the boundary-layer deflections, which in the present approach are fully determined by the function ~ ( 5 ) . Given the intrinsic geometry of r, as described by ~ ( 5 1 eq. , (114) determines the bending strains x ( 6 ) which equilibrate the boundary layer. The driving force for delamination then follows from (100). In closing this section, we assess the accuracy of the assumed kinematics (109) in the analytically tractable case of a straight boundary layer. As shown in Section III.D.2, the bending boundary layer of a semi-infinite blister can be solved exactly. The resulting deflections are w

=

k 6 log[cosh(

3 1 ,

(115)

where h is the thickness of the film, and the line tension is given by eq. (61). This exact solution furnishes a convenient benchmark for appraising the accuracy of the approximate theory developed in this section. For a straight boundary, (112) reduces to

k5 kX T=C+D-. 15x 2

(116)

Delamination of Compressed Thin Films

171

Minimization of T with respect to y, gives (117) whereupon the line energy (116) evaluates to

T

=

m k 3 .

(118)

This expression overestimates the exact line tension (61) by 8.7%. A comparison of exact and approximate deflections and slopes is shown in Figure 21.

w/h I 0.8

0.6 0.4

0.2

4

0 0

0.2 0.4

0.6 0.8

1

1.2

1.4

q k/h w’/k

1

0.8

0.6 0.4

0.2

b)

q k/h FIG.21. Comparison of the exact and quadratic a) deflections; and b) slopes for a bending boundary layer adjacent to a rectilinear boundary.

Gustavo Gioia and Michael Ortiz

172

C. EQUILIBRIUM MORPHOLOGIES: THETELEPHONE CORD Next, we turn to the problem of determining the possible shapes of blisters. This requires the introduction of additional mechanical postulates such as: a fracture criterion, when the shape of stationary blisters is sought; or a kinetic relation for delamination, if the shape of a propagating delamination front is to be determined. In this section we begin by considering the case of stationary blisters. Growing blisters are treated in a subsequent section. A commonly adopted criterion for interfacial fracture (see, e.g., Evans and Hutchinson, 1989; Hutchinson and Suo, 1991) is to suppose that the blister, now regarded as an interfacial crack, can propagate only if the driving force for delamination (100) exceeds the fracture energy of the film/substrate interface, i.e., if

9L Fc.

(119)

The observational evidence shows that .Yc depends strongly on the mode mixity at the delamination front, as measured by the phase angle I) (Trantina, 1972; Anderson et al., 1974; Liechti and Knauss, 1982; Cao and Evans, 1989; Liechti and Chai, 1992). For a thin film on an elastic substrate, Hutchinson and Suo (1991) determined the phase angle to be mMcosw tan

J+ !I

=

+ h ANsin o

-\/iZMsin w

+ h ANcos w ’

(120)

where M is the bending moment normal to the boundary of the blister, AN is the jump in the normal membrane force across the boundary, and w is a function of the elastic constants of film and substrate. Since Vw is continuous across the boundary and, in the present theory, the membrane stresses are functions solely of Vw, it follows that A N = 0 and (120) reduces to tan I)= -cot w .

(121)

Consequently, I) is constant throughout the boundary and is not a factor determinant of the shape of the blister under the conditions envisioned here. The state of mode mixity characterized by (121) coincides with that which exists at the edge of a one-dimensional blister in the limit of vanishing buckling deflections, and with the mode mixity of a blister driven by moderate values of internal pressure (Hutchinson and Suo, 1991).

173

Delamination of Compressed Thin Films

As shown in the preceding section, the requirement that the boundary layer be in equilibrium leads to eq. (114). This equation enables the determination of the bending strains x( 6 ) when the shape of the boundary, as described by the curvature K ( &),is known. An additional equation is therefore required for the calculation of the curvature itself. An assumption which yields the requisite equation is that the boundary of the blister is everywhere critical, i.e., satisfies (119) as an equality everywhere. In view of (loo), this implies that

x=

g

= - Xc,

(122)

and, consequently, X I

=

0,

XI'

=

0,

x"'= 0,

and

xil' = 0

(123)

everywhere on r. Insertion of these identities into (114) yields the reduced equilibrium equation i i ' l i 2 (

[k i ( 6 2 :

-

9k2,Cc + 2k2i:)

lo,?

+ 6Xc( Xc

i,3 [ k i ( - 1 2 i 2

l2

ic- k i ) x

- k 2 ) 2l n f l - k 2 / g c f ] +

+ 3Oki:i

-

2 2 k 2 i c i 2+ 3k3i3) -

1

12ic,(ic- k i ) 3 ln(1 - k i / i c ) 2 i 4 ( i c

- ki)'

[1 o i ; ( i c

k i ( 8 k 4 i 2 + 5%;

-

-

ki)ln(l - k 2 / i c ) + 13k5izc+

WE^)]

=

0,

(124) where i = uh,

i =x h ,

4 t/h,

(125)

4.

are normalized variables and derivatives are taken with respect to Eq. (124) is an ordinary differential equation of second order in 2 which, with appropriate boundary conditions, can be integrated to yield i ( f )and, consequently, the shape of the boundary. Eq. (124) admits the trivial solution 2 = ic= i c / k , which represents a boundary with the shape of an arc of circle of radius k / 2 c . No other

Gustavo Gioia and Michael Ortiz

174

analytical solutions of eq. (124) are known to us at present. To reveal the full range of behavior of the nontrivial solutions we proceed to solve eq. (124) numerically. Without loss of generality, we choose [ = 0 as the origin of integration and set E’(0) = 0 and E(0) > 0. The logarithmic terms in eq. (124) impose an upper bound on the admissible values of the curvature, which is restricted to the range E Ii c .In the calculations reported here we set k = 0.1 and gC= 5 X 10-6C, giving ,Cc = 0.0109545 and i c= 0.109545. Numerical solutions i (g) corresponding to values of C(0) in the admissible range are shown in Figure 22(a). In all cases the curvature decreases monotonically with [, passes through an inflection point ( 2 = O), and becomes negative over a narrow region which we term cusp. The curvature decreases without bound within the cusp and attains a limit point, which, as demonstrated subsequently, coincides with the vertex of the cusp, at

0.J 0 -0.I

-0.2

-0.3 -0.4

5

10

I5

20

I5

20

5/h

a

6o 50 40

b)

30 20 10

0

5

10

5/h FIG. 22. Distribution along the boundary of a) curvature; and b) angle between the tangent to the boundary and the xI axis, for different maximum values ~ ( 0of ) the curvature.

Delamination of Compressed Thin Films

175

which the curvature becomes negative infinity. The angle a( [) subtended by the tangent to the boundary at and the x,-axis, follows from G({) by integration of the equation

i

a'

=

2,

(126)

leading to the results shown in Figure 22(b). It is evident from this figure that the singularity of k ( at the vertex of the cusp is integrable. Indeed, the integral of G between inflection points may be identified with the angle 8 turned by the tangent through the cusp, which is always finite. It is observed in Figure 22(a) that cusps narrow as G(0) approaches 2,. As demonstrated in Figure 24(b), page 177, the width of the cusps, as measured by the arc-length between inflection points, attains a maximum for an intermediate value of 2(0), diminishes thereafter for larger values of K(O), and vanishes in the limit of K ( O) + k c . This trend is accompanied by a sharpening of the cusps. Figure 24(a) further illustrates this behavior. As 2(0) approaches 2,, the cusp angle 8 diminishes towards a limiting value 8, = 55.9". The limiting form of 2 ( { ) as 2(0) + 2, is particularly noteworthy since, as we shall see, this limit is nearly attained by many observed blisters. It is evident from Figure 22(a) that, in this limit, 2 ( f ) -+ G, everywhere except at the cusps, where 2 diverges to -a.But, as discussed earlier, the integral of k through the cusps remains finite and equal to 8, in the limit, which implies that 2 converges to Dirac-deltas of strength -8, at the cusps. Consequently, the limiting solution 2 ( g ) as i ( 0 ) + GC consists of intervals of constant positive curvature punctuated by Diracdeltas representative of sharp cusps. The parametric equations of the boundary follow from the relations

i>

-I

x,

=

cos a ,

ii

=

sin a .

(127)

Figure 23 shows the resulting boundary shapes. The solutions are reflected symmetrically with respect to = 0, and continued periodically beyond the cusp vertices as dashed lines. Remarkably, these curves closely resemble the telephone-cord morphology, Figures 5-7, which suggests that, as assumed in the calculations, the stationary part of telephone-cord boundaries, or 'wake,' is everywhere critical, i.e., the fracture criterion eq. (119) is satisfied as an identity everywhere. As may be appreciated in Figure 23, the alternating intervals of positive and negative curvature characteristic of the telephone-cord morphology are faithfully captured by the theory.

5

Gustauo Gioia and Michael Ortiz

176 I

-20

-10

I

0.074 --* I

0

10

20

x, FIG.23. Boundary morphologies for different values of

do).

Detailed comparisons between theory and observation are presented in Section 1V.D. The behavior of the boundary in the regions of high negative curvature immediately adjacent to the vertex of a cusp can be ascertained by recourse to an asymptotic analysis. To this end, we identify the cusp vertex with the origin t =0 and assume that 2 ( 8 ) -I%]" asymptotically as I l l + 0 for some unknown exponent u < 0. Substitution into eq. (124) yields dominant terms of order I 619' and I [I6'-'. The assumption that one of these terms dominates over the other leads to non-negative values of the exponent a , which is inconsistent with the required singular behavior. We therefore conclude that both dominant terms are of equal order, which necessitates a = - 2/3. Consequently, the curvature behaves asymptotically as

-

2

-)[1-2/3

(128)

Integration of eq. (126) gives

where sgn is the signum function. We note that a = 0 at the cusp vertex, in accordance with the numerical solutions shown in Figure 22(b). The parametric equations of the boundary near the vertex follows by integration of (127), with the result

The analytical structure of the cusp is displayed in Figure 25, page 178.

Delamination of Compressed Thin Films

177

a)

80 180 160 140

120

b)

100 80

60 0.02

0.04

0.06 0.08 hK(0)

0.1

0.12

hK(O) FIG.24. a) Variation of cusp; b) angles; and c) widths with the maximum curvature ~ ( 0 ) .

D. COMPARISONS WITH EXPERIMENTS

The comparisons with experimental observations presented in this section demonstrate the remarkable ability of the theory just outlined to replicate the observed telephone-cord morphology, which lends empirical support to the theory. Figure 26 shows a telephone-cord blister in a Si film applied on a SiO, glass substrate (M. D. Thouless, 1993). This blister, and

178

Gustavo Gioia and Michael Ortiz

-6 -8

-io

-**L

-14

-I

-0.5

0.5

I

0 -0.5 '

-I:-

xfi FIG.25. The geometry of a telephone-cord cusp. a) Curvature vs arc length; b) angle vs arc length; c) cusp shape.

Delamination of Compressed Thin Films

179

FIG.26. Comparison of telephone cord in a Si/SiO, glass system reported by Thouless (1993) and boundary shape predicted by theory. Reprinted by permission of the American Ceramic Society.

many others, developed in a solar cell under service conditions (Thouless, private communication, 1994). The film is approximately 18 p m thick, and the estimated eigenstrains correspond to a characteristic slope k = 0.15. Given these values, a solution of eq. (124) can be readily fitted to the observed cord shape. Two adjustable parameters are available to effect the fit: K c , or, equivalently, ic= kZc; and K(0). The solution shown in Figure 26 appears to optimize the fit, and it is obtained for K, = 0.147874 and 130) + K c . The agreement between theory and observation is quite remarkable. It may be of some practical import to note that the fracture energy gcof interfaces can be estimated by means of the procedure above. Indeed, from the identities Kc = h d m / k and D = Eh3/12(1 - v'), F, follows as Fc =

hk '2; E 24(1 - v ' )

(131)

Taking E = 105 GPa, v = 0.19, and the value of Kc obtained by fitting the boundary of the blister, the estimate gc = 40 N/m is obtained for the Si/SiO, interface and the mode mixity characterized by (121). Figure 27 shows a telephone-cord blister in a diamond-like carbon film on a glass substrate (after Nir, 1984). The blister originated at a slit visible on the left side of the figure, and appears to have reached a steady state of

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Gustauo Gioia and Michael Ortiz

FIG.27. Comparison of telephone cord in a diamond-likecarbon/glass system reported by Nir (1984) and boundary shape predicted by theory. Reprinted with permission of Elsevier Science Ltd.

propagation towards the upper right corner. The dimensions of the blister were not reported. For an estimated characteristic slope k = 0.2, it is possible to find a solution of eq. (124) which fits the observed geometry of the blister. The resulting solution is shown in Figure 27 and corresponds to taking K, = 0.11095, 2(0) + k, and an estimated cord width to film thickness ratio of 10. The unknown thickness h can be estimated from eq. (131) by setting gC= 5.5 N/m, E = 150 GPa and Y = 0.2 (Gille and Rau, 19841, with the result h = 1.7 p m . Finally, the telephone cord width follows as 17 p m . The resulting fit between theory and observation is again quite remarkable. Our final example concerns a telephone cord in a compositionally modulated Fe/Ni thin film, Figure 28 (after Yu et al. 1991). The film is 0.92 p m in thickness, and the dimensions of the blister are shown in the

FIG. 28. Comparison of telephone cord in a compositionally modulated Fe/Ni film reported by Yu et al. (1991) and boundary shape predicted by theory. Reprinted with permission of Elsevier Science Ltd.

Delamination of Compressed Thin Films

181

figure. An excellent fit of the observed geometry is obtained for an estimated k of 0.12 and by taking kc = 0.0182574 and k ( 0 ) 4 k c , Figure 28. As in the first example, this value of 2, can be taken as a basis for estimating the fracture energy Fc through eq. 131. With E = 200 GPa and v = 0.2, the result is Sc = 0.038 N/m. Ideally, if the maximum curvature ~ ( 0 in ) the preceding examples could be measured directly from the micrographs, the geometry of the boundary would follow in its entirety. Regrettably, the micrographs are not of sufficient resolution to enable such direct measurements, and one is thus ) an adjustable parameter to be determined by reduced to treating ~ ( 0 as curve fitting. Interestingly, in all three examples presented in this section, values of ~ ( 0 very ) close to the maximum admissible curvature K, = { m / k are arrived at by this means. In this type of cord, the positive curvature of the boundary scales with the square root of the ratio between the fracture energy and the bending stiffness of the film. Additionally, the cusps have been found to narrow to a point as ~ ( 0 approaches ) K ~ Figures 22(a) and 24(c). Near this limit, the cusps increasingly appear as points of discontinuity of the normal, Figure 23. It bears emphasis that, despite appearances, the normal to the boundary is always continuously turning provided that ~ ( 0 < ) K ~ and , the cusps have the well-defined analytical structure shown in Figure 25.

E. GROWTHOF TELEPHONE-CORD BLISTERS As pointed out in Section II.C, telephone-cord blisters are commonly observed to grow at the tip. In this region, the driving force 9 must therefore exceed the fracture energy Sc. The resulting net driving force imparts the boundary a normal velocity V , which causes the tip to propagate further. The work conjugacy relation (94) suggests that V is driven by F through a kinetic relation of the type

subject to the restrictions imposed by the second law (Abeyaratne and Knowles, 1991). Kinetic relations such as (132) have been extensively treated within the theory of phase transitions (Abeyaratne and Knowles, 1988, 1990, 1991, 1992). In the context of dynamically growing cracks, relations such as (132) are often interpreted as crack-tip ‘equations of

,

182

Gustavo Gioia and Michael Ortk

motion’ (Freund, 1990), and have been investigated experimentally for a variety of bimaterial systems (Tippur and Rosakis, 1991; Liu et al., 1993; Lambros and Rosakis, 1995). For the quasistatic type of growth often observed in thin-film blisters, by contrast, there is a paucity of experimental data to guide the formulation of appropriate kinetic relations (see Ogawa et al., 1986; Kinbara and Baba, 1991, for notable exceptions). In the absence of a generally acceptable kinetic relation for quasistatic interfacial crack growth, we adopt the simple linear ‘overstress’ model

(g- g c ) / B if 5 > gC otherwise,

K=i0

(133)

where B is a kinetic coefficient. It should be noted that, in this model, arrest and the subsequent attainment of a stationary boundary requires g to approach the fracture energy gCfrom above. Consequently, as the tip grows it may be expected to leave behind a wake which is everywhere critical, Le., one on which LY = gcidentically everywhere, Figure 29. This is precisely the case treated in the preceding section. Consequently, the telephone-cord geometries described therein resurface here in the wake of the tip. Conversely, the boundary of the tip may be expected to tend asymptotically to the stationary telephone-cord geometries governed by eq. (1241, which act as ‘guides’ for the tip as it wriggles forth, Figure 29. The general transient problem is governed by eq. (1141, which establishes the equilibrium of the film, and the kinetic relation (132). The numerical solution of these equations proceeds as follows. Suppose that the parametric equations x(5) of the boundary are known at time t. Then, the bending strains ,y( 6 ) can be computed from eq. (1141, e.g., by recourse to a finite difference scheme. This determines the distribution of driving force .Y( 5 1 over the boundary and, through the kinetic relation (1321, the

FIG.29. Different stages of growth of a telephone-cord blister.

Delamination of Compressed Thin Films

183

corresponding normal velocities V,( ). The parametric equations of the boundary at time t + A t can then be approximated as x( 6) + At%( 5 )n( 5 1, where n( 6 is the unit normal to the boundary. Evidently, by a recursive application of these steps the growth of the boundary can be followed to any desired extent. This general treatment of the transient problem will not be attempted here. Instead, we shall focus on the analytically tractable case of a telephone-cord blister for which K K, everywhere except at sharp cusps. In this case, we surmise that the shape of the tip remains nearly unchanged as it pivots around a cusp. Let w be the angular velocity of rotation of the tip about the cusp; x( 6) the parametric equation of the tip; t(s) the unit tangent vector; n ( 6 ) the unit outward normal; and V , the normal velocity of advance of the tip. Then, the following identity holds:

-

In addition, we expect K > K , at the tip, and hence, the boundary layer thickness k/X is bounded above by 1 / ~ i.e., , k

1

-
0. A combination of (2.3) and (2.5) leads to U, =

-L(yK

+ 8).

This partial differential equation governs the interface motion.

(2.6)

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Equation (2.5) contains the special case 9 = - yK, known as the Laplace-Young relation for liquid films (e.g., Isenberg, 19781, where 9 is the pressure difference between the two neighboring bubbles. This relation results from the equilibrium of a liquid film under the pressure difference and the surface tension. Such interpretations are misleading for a phase boundary in solid state.

C. WEAK STATEMENT AND GALERKIN METHOD The partial differential equation (2.6) is not a good way to look at the general problem for several reasons. First, (2.6) is incorrect when surface tension is anisotropic. Second, because the problem in general has to be analyzed approximately, a partial differential equation need not be a good starting point. The following weak statement circumvents the difficulties of anisotropy, and leads to the Galerkin method in numerical analysis. Other merits of the weak statement will become evident as the subject develops. Completely ignore Section B, and start from Section A again. Replace the driving pressure 9 in (2.2) with the interface velocity u,, by using the kinetic law (2.3), giving

Make the following statement: the actual velocity, u,, must satisfy (2.7) for virtual migration Sr, of arbitrary distribution on the interface. Following the terminology of variational calculus, we refer to this as the weak statement of the problem. One may find an approximate interface velocity that satisfies (2.7) for a family of virtual motions (instead of arbitrary virtual motions). Obviously, the larger the family, the more accurate the approximation. This consideration leads to the Galerkin method, a formal presentation of which follows. Model the surface with n degrees of freedom, writing q l , . . . q, for the generalized coordinates, and q , ,... q, for the generalized velocities. For example, a sphere has one degree of freedom, its radius; a rod has two degrees of freedom, its radius and height; a general surface may be modeled by an assembly of triangles, with the positions of the vertexes being the generalized coordinates. Describe a surface by expressing the position vector on the surface, x, as a function of two surface coordinates,

Motions of Microscopic Surjaces in Materials

203

s1 and s2, and the time t . Using the generalized coordinates, we express the position vector as x(sI, s2 ;q l , . . .,q,), with the time implicitly con-

tained in the generalized coordinates. The free energy is a function of the generalized coordinates, G(ql ,q 2 ,q 3 ,.. .), The generalized forces, f i ,.. . ,f,,,are the differential coefficients of the free energy, namely SG

= -fi

Sq,

--f2

Sq2 -

-f, Sq,

.

(2.8)

Once the free energy function is known, the generalized forces are calculated from fi = - d G / d q , . The virtual motion of the surface, Sr, , is linear in the variations of the generalized coordinates:

The shape functions N, depend on the generalized coordinates. The interface velocity is linear in the generalized velocities: 0, =

c44;.

(2.10)

Substituting the above into the weak statement, (2.71, we obtain

C Hij4j 6qi = C 6Sqi i,j

7

(2.11)

i

where

H,,

=

Ni q j ---a. L

(2.12)

Equation (2.11) holds for arbitrary virtual changes Sq, , so that the coefficient for each Sq, must equal. Thus, (2.13) Equation (2.13) is a set of linear algebraic equations for the generalized velocities. Once solved, they update the generalized coordinates for a small time step. The process is repeated for many steps to evolve into the surface. Because the matrix H and the force column f depend on the generalized coordinate column q, (2.13) is a nonlinear dynamical system.

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z. suo

The physical interpretation of the matrix H is evident from (2.13): the element of the matrix, H i j , is the resistant force in the q,-direction when the state moves at unit velocity in qj-direction. We will call H the viscosity matrix. From (2.12), H depends on the generalized coordinates but is independent of the generalized velocities or the positions on the interface. The viscosity matrix is symmetric and positive-definite.

D. GEOMETRIC VIEW One can visualize the above formulation in geometric terms (Sun et al., 1996, Yang and Suo, 1996). Imagine a hyperspace with the free energy as the vertical axis, and the generalized coordinates as the horizontal axes. The free-energy function, G ( q , ,q 2 ,q 3 , .. .), is a surface in this space, to be called the energy landscape. A point on the landscape represents in general a nonequilibrium state of the system, described with a set of values of the generalized coordinates and a value of the free energy. The bottoms of valleys on the landscape represent equilibrium states of the system. A curve on the landscape represents an evolution path of the system. Thermodynamics requires that the system evolve to reduce the free energy, and therefore the evolution path be a descending curve on the landscape. Starting from any point other than a valley bottom on the landscape, infinite descending curves exist. Consequently, thermodynamics by itself does not set the evolution path. Nor does thermodynamics select one valley as a final equilibrium state among several valleys. The evolution path is set by thermodynamics and kinetics acting together. At a point on the landscape, the slopes of the landscape represent the generalized forces. The Galerkin procedure assigns a viscosity matrix H at every point on the landscape. The generalized velocities are determined by q = H - ' f , which gives the direction and magnitude of the incremental motion on the landscape. The evolution path is thus determined incrementally. This global, geometric view does not add any new information to the problem, but does give an intuitive feel for a complex system. If the energy landscape contains several valleys, the one that will be reached by the system as the final equilibrium state will also depend on kinetics. A change in the kinetic parameters, without changing the energy landscape, may shift the system from moving to one valley to another. A n example is given in Section V1.B. In the language of nonlinear dynamics, we say that the

Motions of Microscopic Sugaces in Materials

205

change of kinetic parameters changes the basins of attraction, Clearly, this is a universal theme of material processing. E. VARIATIONAL PRINCIPLE

The following variational principle is equivalent to the weak statement in Section C. In numerical analysis, these two forms lead to an identical set of ordinary differential equations. In the remainder of the article, we will use the weak statement exclusively. The variational principle is included here for completeness. Let W be a virtual interface velocity distribution, which need not satisfy any kinetic law. Associated with the virtual velocity, the free energy changes at rate 6. Introduce a functional -

n =G +/ Z d A , W2

(2.14)

which is a combination of the virtual free-energy rate and a term associated with the virtual rate. The functional is purely a mathematical construct, and has no clear physical meaning. Given an arbitrary virtual velocity distribution 0 , one can compute a value of I1. The variational principle is now stated: Of all the virtual velocity distribution W,the actual distribution u, minimizes n. The proof of the principle follows. According to (2.2), the virtual free energy rate is G(0)

=

-pod,

which is linear in the virtual velocity. Consequently, the difference in

n is

The actual velocity satisfies the kinetic relation u,/L =9. According to (2.21, the sum of the first two terms vanishes. Thus,

This is nonnegative for any virtual velocity distribution, hence the proof.

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JUNCTION; EQUILIBRIUM OR NONEQUILIBRIUM F. TRIPLE

1. Force on Triple Junction If the solid particle is polycrystalline, the free energy becomes G

=

YsAs

+ ybAb

gV.

(2.15)

Here ys is the surface tension of the solid-vapor interface, yb the surface tension of the grain boundaries, A , the area of the interface, and A , the area of the grain boundaries. For simplicity, both surface tensions are taken to be isotropic. As an example, Figure 2 shows that a grain boundary and two surfaces form a triple junction, i.e., a line in the third dimension. The length of the he., the dihedral junction is 1, and the two surfaces meet at angle angle). Move the junction by S y and the surfaces by 6 r n , resulting in a virtual change in the free energy, SG. Define the driving force on the triple junction, f , and the driving force on the surface, 9, simultaneously bY j

~

+~j S Sy r , , d~

=

-SG.

(2.16)

This is an extension of (2.2). We may postulate separate kinetic laws for the junction and surface motion:

y

=

L,f,u,

=

L9,

(2.17)

where y is the velocity of the triple junction, L , the junction mobility, and L the surface mobility. Equations (2.16) and (2.17) complete the modification. One can find an explicit expression for the force on the triple junction. When the junction moves by distance 6 y , the area of the grain boundary

Motions of Microscopic Surfaces in Materials

207

changes by lay, and the area of the two surfaces changes by - 2 cos(q/2)1Sy. Consequently, associated with the virtual motion of the junction and the surfaces, the free energy changes by

A comparison of (2.16) and (2.18) gives the expressions for the two driving forces: (2.19) The driving force on the junction, f, has a clear interpretation: it is the sum of the surface tensions projected along the y-axis. In this example, because of symmetry, we only need to consider the motion in the y direction. If the junction can move in both x and y directions, there are driving forces in both directions. One can also include junction motion into the weak statement. Replace the forces in (2.16) with the velocities by using the kinetic laws (2,171, giving

3

-isy

Lt

+

un

- srn d~ = -

L

s ~ .

(2.20)

The actual velocities, y and u, , satisfy (2.20) for arbitrary virtual motions, Sy and 6rn .In this weak statement, the surface tension for both the grain boundary and the surfaces can be anisotropic, provided the free energy G is evaluated by a surface integral of the surface tension. 2. Equilibrium Triple Junction The triple junction is commonly assumed to be in equilibrium at all times, even when the surfaces still move. That is, the driving forces on the triple junction vanish at all times. For the present example, setting f = 0 in (2.19) results in the well-known expression for the equilibrium dihedral angle U,: *e

Yb

cos - = -. 2 2Ys This relation fixes the slope of the surfaces at the junction.

(2.21)

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2. suo

The assumption of equilibrium junction is justified by the relative rate of the junction motion and surface motion. It only takes a small number of atomic adjustments to reach the equilibrium angle, so that the time needed for the overall grain shape change is limited by the surface motion. The idea can be made definite as follows. Let d be a length scale that one likes to resolve from the model, e.g., the depth of the surface groove caused by the grain boundary underneath. The effect of the junction mobility is negligible if L , d / L >> 1. Assume equilibrium junction is equivalent to prescribing an infinitely large junction mobility. Consequently, the first term in the weak statement (2.20) drops, which then becomes identical to (2.71. The weak statement (2.7) simultaneously determines the surface velocity and enforces the triple junction equilibrium. In applying the Galerkin procedure to the weak statement (2.71, one need not fix the dihedral angle to the equilibrium value. Rather, the equilibrium dihedral angle comes out as a part of the solution, approximately in a short time, consistent with the level of approximation of the entire surfaces. In the terminology of variational calculus, the equilibrium dihedral angle is a natural boundary condition, which is enforced by the weak statement itself. The position of the end of the surface is an essential boundary condition, which must be enforced in addition to the weak statement. 3. Nonequilibrium Triple Junction

Situations exist where the junction mobility plays a role. For example, impurities segregated to the junctions may reduce the junction mobility, retarding the overall surface motion. The effect should be pronounced if the grain size is very small. From (2.211, the junction may reach equilibrium only when Y b < 2ys-that is, when the grain boundary is energetically more favored than two surfaces. If Y b > 2ys, equilibrium will not be reached until the grain boundary is completely replaced by two parallel surfaces. In this case, a finite junction mobility prevents the junction from running at an infinite velocity. Another example involves atomic decohesion along a grain boundary when the body is subject to a tensile stress normal to the grain boundary. The triple junction may be out of equilibrium, and the dihedral angle between the two free surfaces approaches O", instead of the equilibrium

Motions of Microscopic Surfaces in Materials

209

value of (2.21). The unbalanced force at the triple junction may drive a reaction that leads to the environmentally-assisted cleavage. Section VII1.C discusses a similar situation.

111. Interface Migration Driven by Surface Tension and Phase Difference

This section gives examples of interface migration under surface tension and free energy density differences between the two phases. Finite element schemes have been formulated on the basis of the weak statement (Cocks and Gill, 1995; Du et al., 1996; Sun et al., 1997). It is too early to judge them critically. Instead, this section gives an elementary demonstration of the Galerkin procedure, and describes several analytical solutions. A.

SPHERICAL PARTICLE IN A

LARGEMASS

OF VAPOR

When a small solid particle is introduced into a large mass of a vapor, the particle may change both shape and volume, as atoms evaporate to, or condense from, the vapor. We start with the simplest situation where the surface tension is isotropic and the solid particle is spherical. The system has only one degree of freedom, the radius of the sphere. 1. Free E n e w The introduction of a spherical particle of radius R into a large mass of vapor changes the free energy by G

=

4rR2y

4

+ -3 r R 3 g .

(3.1)

Here y is the surface tension, and g the free-energy density difference between the two phases; y is always positive, but g can be either positive or negative. If g > 0, the volume term reduces the free energy when the particle shrinks. If g < 0, the volume term reduces the free energy when the particle grows. We will concentrate on the case g < 0. Figure 3(a) sketches the free energy as a function of the particle radius. As R increases, G first increases when the surface term in (3.1) dominates, and then decreases when the volume term dominates.

2. suo

210

A

4%

I

FIG.3. A small solid particle in contact with a large mass of its vapor (g < 0). a) Free energy as a function of the particle radius. b) Particle of different initial radii evolve with the time.

The free energy maximizes at a finite particle radius, R,. The significance of this maximum is readily understood. Imagine a particle of radius R # R,. Thermodynamics requires that the particle change size to reduce G . If R < R , , the particle shrinks to reduce G . If R > R,, the particle grows to reduce G. The critical particle radius is determined by setting dG/dR = 0, giving RC

=

-2- Y

(3 2 )

g The two energetic parameters, y and g, have different dimensions; their

ratio defines this length. 2. Kinetics Section 1I.B applies because the surface tension is isotropic. The driving pressure on the surface of the spherical particle of radius R is

(3.3)

Motions of Microscopic Sur$aces in Materials

21 1

The kinetic law (2.3) relates the surface velocity to the driving pressure: dR =

-L(;

+g).

(3.4)

dt

This ordinary differential equation governs the particle radius as a function of the time, R ( t ) . The energetic competition shows up again: the particle growth rate is positive if R > - 2 y / g , and negative if R < - 2y/g. Let R , be the particle radius at time t = 0. The solution to (3.4) is (R

-

R,)

+ R , In

I

R - R, R, - R,

1

=

-Lgt

(3.5)

Figure 3(b) sketches the radius as a function of the time. The behavior depends on the initial radius. A supercritical particle ( R , > R , ) grows with the time without limit. A subcritical particle ( R , < R , ) shrinks and disappears. B. ANISOTROPIC SURFACE TENSION:ROD- OR PLATE-SHAPED PARTICLES

In the example above, the free energy alone decides whether the particle grows or shrinks, and the kinetics sets the time. This division in roles between energies and kinetics comes about because the system has only one degree of freedom. As discussed in Section II.D, when the system has more than one degree of freedom, the free energy alone does not determine the evolution path or the final equilibrium state. The following example has two degrees of freedom, and is used to illustrate the Galerkin procedure. Imagine a crystal having anisotropic surface tension such that it grows to a prism with a square cross section. When a small particle of such a crystal is introduced into its vapor, it has two degrees of freedom: the base side B and the height C . The surface tensions on the prism bases and sides are y , and y 2 , and the mobilities are L , and L,. When the crystal grows by unit volume at the expense of the vapor, the phase change alone increases the free energy by g. The total free energy of the system, relative to the vapor without the particle, is G ( B , C ) = 2y,B2

+ 4y2BC + gB2C.

(3.6)

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2. suo

Associated with the virtual changes SB and SC, the free energy varies by

SG

=

(4y,B

+ 4y2C + 2gBC) SB + (4y2B + g B 2 ) SC.

(3.7)

The kinetic term on the left-hand side of (2.7) is

1$ Sr,&

BBC

= -SB

CB2

+ -SC.

(3.8) L2 2 4 The weak statement requires that the sum of the two equations above vanish. Collect the coefficients of SB and SC, giving B=-L2

(‘:I

-+-

~ Y +Z 291,

B

c

=

-L1(

%+

2g).

(3.9)

These are coupled ordinary differential equations, to be integrated numerically once the initial particle dimensions are given. No numerical results will be presented here. Brada et al. (1996) have used this approach to study coarsening of grains of a crystal with high surface-tension anisotropy. See Carter et al. (1995) for a demonstration of the effects of surface-tension anisotropy. C . SELF-SIMILAR PROFILE:THERMAL GROOVING

In a polycrystalline particle, a grain boundary intersects with the particle surface, forming a triple junction, Figure 2, When heated, the surface grooves at the triple junction. The problem was solved by Mullins (1957). The surface motion reduces the grain-boundary area but increases the surface area, so that the total free energy decreases. Mass relocates by either evaporation or surface diffusion. This section summarizes Mullins’ analysis for evaporation; Section V1.C will summarize his analysis for surface diffusion. When the groove depth d is so small that gd/ y + 0, the effect of g on grooving is negligible. Mullins assumed that g = 0, namely, the vapor is in equilibrium with the flat solid surface. The groove depth is taken to be much smaller than the grain size, so that the two grains in Figure 2 are infinitely large. This is a two-dimensional problem in the plane normal to the triple junction. In Figure 2, the x-axis coincides with the surface remote from the groove, and the y-axis with the grain boundary. Describe the surface shape at time t by function y ( x , t ) . The curvature of the surface is

Motions of Microscopic Sufaces in Materials

213

The velocity normal to the surface is

The equation of motion (2.6) becomes (3.10) The initial and boundary conditions are as follows. The surface is initially flat, i.e. y(x,O)

=

(3.11)

0.

The surface remote from the groove is immobile at all times: y(fco,t)

=

(3.12)

0.

The triple junction is taken to be in equilibrium during grooving, so that the two surfaces meet at the equilibrium dihedral angle, 'Pe, given by (2.21). This dihedral angle fixes the slope of the surface at the triple junction:

The partial differential equation (3.101, the initial condition (3.10, and the boundary conditions (3.12) and (3.13) determine the evolving surface profile, y ( x , r ). The initial geometry has no length scale, but the time and the mobility set a length scale, Consequently, the groove grows with a selfsimilar profile. Define the dimensionless coordinates:

\lLrst.

x=-

X

JLrst

,Y=-.

Y

(3.14)

)JGF

Describe the groove profile by a function Y ( X ) . The partial differential equation (3.10) becomes an ordinary differential equation 2 dx2

+ [ 1 + (fj2](xf

-Y)

=o.

(3.15)

The initial condition (3.11) and the boundary condition (3.12) both become Y(W) = 0.

(3.16)

z. suo

214

The boundary condition (3.13) becomes

d - Y(0)= rn. dx

(3.17)

The boundary-value problem (3.1343.17) is integrated numerically (Sun et al., 1997). Figure 4 shows the groove profile for various dihedral angles. For a system with a larger ratio y b / y s , the dihedral angle ‘Pe is smaller, and the groove is deeper. When the ratio y b / y s is small, the slope of the surface, dY/dX,is small. Dropping the high-order term (dY/dxI2in (3.13, the ordinary differential equation is linear, so that the groove depth must be linear in rn. Mullins’ (1957) calculation showed that, under the small-slope approximation, the groove depth is

d

=.

1.13rnm.

(3.18)

Figure 5 plots the numerical solutions of the groove depth determined by both the exact and the linearized equations, indicating that Mullins’ linear approximation is good for most purposes. The exact nonlinear solution has been used as a benchmark to check the accuracy of a finite element code (Sun et al., 1997).

-1

-1.6

I

X

FIG.4. The profile of the groove over a grain-boundary caused by evaporation.

215

Motions of Microscopic Surfaces in Materials

0.75

-

0.5

-

0

0.25

0.5

0.75

1

m FIG.5. The groove depth as a function of parameter m , which relates to the ratio y b / ys by (3.13).

D. STEADY-STATE PROFILE 1. General Solution Mullins (1956) studied the steady-state surface motion, i.e., the entire surface moves in the same direction at the same velocity. The motion is motivated by surface tension, and the mass transport mechanism is evaporation-condensation. Both the surface tension and the mobility are taken to be isotropic. The governing equation is (2.6), setting g = 0. Figure 6 shows a surface moves in the y-direction at velocity u. The coordinates x and y move with the surface. A plane problem is considered where the surface shape is invariant along the axis normal to the x-y plane.

I

-

x

FIG.6. Geometry of a steadily moving surface.

z. suo

216

Let ds be the curve element, 8 the angle of the element from the x-axis. Surface tension drives the surface to move toward the center of the curvature. Consequently, the surface must concave in the direction of the velocity, and the slope is restricted between - ?r/2 I O Ir/2. According to our sign convention, the curvature, K = d 8 / d s , is positive on the entire curve, and the normal surface speed is v, = - v cos8. Equation (2.6) becomes LydO v c o s 8 = -. (3.19) ds Observing that cos 8 ds = dx, one readily integrates the above equation, giving LY X = --ex,,, (3.20) U

where x o is a constant to be determined by the boundary conditions. Similarly, with sin 8 ds = dy, one integrates (3.19) and obtains

y

=

-

LY

- ln(cos 8 ) + y o , U

(3.21)

where y o is another constant to be determined by the boundary conditions. Equations (3.20) and (3.21) together describe the shape of the steadily moving surface, with O as a parameter. The following paragraphs illustrate simple applications. 2. Steady-State Grooving

When the groove size becomes appreciable relative to the grain size, the grooves of two adjacent grain boundaries interact, and the self-similar solution in Section 1II.C is no longer valid. The particle surface may recede with a profile and velocity independent of the time. Consider an idealized geometry with periodic grain boundaries of spacing D , Figure 7. The dihedral angle relates to the ratio yJyS by (2.21). The surfaces move down to decrease the area of the grain boundaries, with no further

FIG.7. Steady-state grooving over periodic grain boundaries caused by evaporation.

Motions of Microscopic Sur$aces in Materials

217

change in the surface area. The slopes are 0 = +(T - *)/2 at the two adjacent triple junctions x = +_D/2. With these as the boundary conditions, (3.20) determines the velocity: (3.22) which is inversely proportional to the grain size. Equation (3.21) determines the groove depth:

(3.23)

3.

Velocity of an Abnormally Growing Grain

Under certain conditions, in a polycrystal one grain grows much larger than the others, at the expense of the neighboring grains (Hillert, 1965). In Figure 7 replace the vapor phase with the large grain, and keep the small grains of size D. For one reason or another, the small grains do not grow, but the boundary between the large grain and the small grains moves with the mobility L b . All the grain boundaries have the same surface tension y b , so that the dihedral angle is = 2 ~ / 3 .Equation (3.22) becomes

*

(3.24) The large grain grows at a velocity inversely proportional to the size of the small grains. IN A THINFILM;EFFECTOF MIGRATION E. GRAIN-BOUNDARY SURFACE EVAPORATION

Consider a polycrystalline film on a single crystal substrate. The grains have a columnar structure. Due to crystalline anisotropy, some grains have lower film-surface tension and film-substrate interface tension than other grains. When the film is heated, the grains with low combined surface and interface tensions grow at the expense of other grains. The survival grains may have (in-plane) diameters much larger than the film thickness. For example, Thompson et al. (1990) studied a thin Au film on a (100) surface of NaCl substrate. When the film is deposited at room tempera-

218

2. suo

ture, the Au grains are very small and are of several orientations. After anneal at 325°C for three hours, the grains grow and the survival grains are predominantly of (111)Au II (001)NaCl, with two in-plane orientations, [liOIAu I1 [llOJNaCI and [liO]Au )I [liOINaCl. The two types of grains are crystallographically equivalent, and therefore have the same free energy. When the equivalent grains impinge, they stop growing. The NaCl substrate has a fourfold symmetry, and the (111) Au grains have a threefold symmetry. Minimization of the free energy in this case does not require lattice matching. Yet another phenomenon may intervene: grooving at the intersections of the grain boundaries and the film surface may break the film. Assuming that the grain boundaries are immobile, Srolovitz and Safran (1986) and Miller et al. (1990) showed that the film breaks into islands if the ratio of the grain size to the film thickness exceeds a critical value. Miller et al. (1990) demonstrated that the prediction is consistent with the observation of a ZrO, film on a single crystal Al,O, substrate. Clearly the two processes-grain-boundary migration and surface grooving-compete to determine the fate of a polycrystalline film. Grainboundary migration may lead to a large-grained, continuous film. Surface grooving may break the film into islands. Mullins (1958) analyzed the effect of grooving on grain-boundary motion, where the surface grooves via surface diffusion. He obtained a steady-state solution for a moving triple junction, but left the velocity of the motion undetermined because the grain-boundary motion was not analyzed. In simulating grain growth in thin films, Frost et al. (1992) modeled the effect of grooving by setting a threshold curvature in the kinetic law, below which grain boundaries remain stationary. Brokman et al. (1995) analyzed a grain boundary moving in a thin sheet, including both surface diffusion and grain-boundary migration, which allow them to determine the steady-state velocity. In an independent study, Sun et al. (1997) analyzed a similar problem with either surface diffusion or surface evaporation. The following discussion draws on these studies, assuming surface evaporation. Surface diffusion will be discussed in Section V1.D. 1. Grain-Boundaty Motion When the Su$ace Remains Flat First imagine that the surface of the film is immobile and remains flat as the grain boundary migrates, Figure 8(a). The in-plane grain size is much larger than the film thickness, so that we focus on one grain boundary and

219

Motions of Microscopic Surfaces in Materials

h

(4

FIG.8. A grain boundary migrates in a thin film. a) The free surface is immobile and remains flat. b) The free surface grooves due to evaporation.

ignore all the others. The two grains, labeled as + and -, have different surface tensions y', and ys- , and interface tensions y: and yl: . Denote the grain-boundary tension by yh , and the grain-boundary mobility by L , . The grain boundary is taken to migrate to the right. Because the film surface and the film-substrate interface are immobile, at the triple junctions the surface tensions do not balance in the vertical direction. Junction equilibrium in the horizontal direction determines the two angles in Figure 8(a): sin 4

Y: - Y , =

, sin 0" =

Y, - Y,'

(3.25)

Yb

Yb

In the steady-state motion, the grain boundary is concave to the right, so that the two angles must satisfj 4 > 8,). Using (3.23, this condition becomes Y,+

Y i < Y:+

r:.

Thus, grain - must have smaller free energy than grain boundary to migrate to the right.

(3.26)

+ for the

grain

220

2. suo

The general solution (3.20) determines the grain-boundary velocity: (3.27) In the limiting case when both

4 and

8, are small, (3.25) and (3.27) give

This limiting result reproduces that of Thompson et al. (1990). 2. Simultaneous Grain-BoundaryMigration and Surface Evaporation Now two kinetic processes occur simultaneously: the grain boundary migrates at mobility L , , and the surface evaporates at mobility L s , Figure 8(b). Assume that the vapor is in equilibrium with a flat-film surface, but atoms at the triple junction can evaporate. The entire configuration moves at a uniform speed u to the right. All the moving surfaces must concave in the direction of the velocity. The surface of the new grain must be straight, because a curved surface would concave to the wrong direction. Evaporation causes the new grain to be thinner than the parent grain by d . Equilibrium at the top triple junctions in both horizontal and vertical directions requires that

The film-substrate interface is immobile, so that the angle 4 is the same as in (3.25). The steadily moving grain boundary must concave in the direction of the velocity. Consequently, the two angles must satisfy 4 > 8, namely (3.29) An application of (3.20) to the grain boundary and to the surface of

grain

+ gives

u(h - d )

=

L b y b ( + - 81, ud

=

L: yTa.

Motions of Microscopic Surfaces in Materials

22 1

Solving the equations, we have the velocity (3.30) and the groove depth (3.31)

The thickness of the new grain depends, among other things, on the mobility ratio. The effect of surface evaporation on the grain-boundary motion may be appreciated as follows. In the limiting case L , -=cLb, the groove depth is negligible compared to the film thickness (3.31). Even in this case, a tiny amount of evaporation significantly affects the grain-boundary motion by rotating the surfaces at the triple junction. Take, for example, y', = ys- = y b . Without evaporation (Figure 8(a)), O,, = 0"; the grain boundary can move steadily to the right if 4 > o", i.e., if the two grains have infinitesimal differences in the film-substrate interface tensions. With evaporation (Figure 8(b)), 8 = 30"; the grain boundary can move steadily to the right if 4 > 30", i.e., if the two grains have a finite difference in the film-substrate interface tensions. On the basis of the weak statement (2.71, Sun et al. (1997) used finite elements to simulate the non-steady motion. When > 8, an initial configuration quite different from the steady-state quickly settles down to the steady-state. When 4 < 8, the grain boundary drags the triple junction toward the substrate, and finally breaks the film.

F. STEADY-MOVING INTERFACE DRIVEN BY SURFACETENSION AND PHASE

DIFFERENCE

Under the small slope assumption, Brokman et al. (1995) gave an approximate steady-state solution for an interface driven by both surface tension and phase difference. The exact steady-state solution to the full nonlinear equation (2.6) follows. The problem has a length scale, 1=

r/g.

(3.32)

z. suo

222

With the reference to Figure 6 , u, becomes

=

- u cos 6 and K = de/ds, (2.6)

(3.33) with the dimensionless constant being c=-

U

Lg. Noting that dx

X

- -

1

-c8 + -ce +

=

cos ds and dy

=

sin 8 ds, one can integrate (3.33) to give

1 C

J

c - cos

S

1 C

G

tanh-' ( J F T s i n e ) c - cos 8

_Y -1

e

1

- - In C

7

+-,

Yo

IC cos 8 - 11 + -

1 '

(3.34) c2>1

(3.35)

Here xo and y , are integration constants to be determined by boundary conditions.

IV. Interface Migration in the Presence of Stress and Electric Fields In many material processes, elastic and electrostatic fields allow additional means of free-energy variation. For example, during a phase transition, the difference in the crystalline structures of the two phases induces a stress field (e.g., Eshelby, 1970; Abeyartne and Knowles, 1990; Lusk, 1994; Rosakis and Tsai, 1994). In a polycrystalline film, grains of different orientations have different elastic energy densities due to elastic or plastic anisotropy (e.g., Sanchez and Arzt, 1992; Floro et al., 1994). In a ferroelectric crystal, domains of different polar directions have different elastic and electrostatic fields (e.g., Pompe et al., 1993; Roytburd 1993; Jiang, 1994). The main concepts in Section 1I.A still apply, with the modification that the free energy G includes the elastic energy and electrostatic energy. This, in turn, requires that the stress and electric fields be solved as boundary-value problems. After the fields are solved and the free energy

Motions of Microscopic Sugaces in Materials

223

C computed, (2.2) defines the driving pressure on an interface, 9. The kinetic law (2.3) then updates the position of the interface. The weak statement (2.7) still applies. A two-step finite element method would proceed as follows. At a given time, the first step solves the boundary-value problem of the stress and electric fields by using a conventional finite element code. The second step updates the interface position according to (2.131, where the driving forces on interface nodes can be calculated with a procedure described by Socrate and Parks (1993), and the viscosity matrix calculated according to (2.12). The whole procedure repeats for the next time increment. The approach would allow a relatively crude mesh to determine the elastic and electric field. Often, the mismatch strain is too large to be accommodated elastically, and dislocations appear to partially relieve the stress. Similarly, electric charge carriers diffuse to partially accommodate the polarization mismatch. Finite element approach could also treat relaxation due to combined plastic deformation, electric conduction, and interface migration. These important effects are beyond the scope of this article and will be ignored. This section collects basic equations and gives elementary demonstrations. A. FREE ENERGY

1. Field Equations Subject a solid insulator to a field of displacement u and electric potential 4. The strain tensor e and the electric field vector E are the gradients:

The conventional index notation is adopted. The insulator is separated into domains by interfaces (or domain walls). Consider an interface between two domains labeled as + and - ,with the unit vector normal to the interface, n, pointing to domain f . Force t and charge w are externally supplied on per unit area of the interface. The body force and the space charge inside the domains are taken to be negligible. In a domain both stress tensor CT and electric displacement vector D are divergence free: cr.. ' 1 . 1.

=

0 , Dl,i = 0.

(4.2)

z. suo

224

Across the interface, they jump by

n.[a:' 'I a:'] 'I = t j , ni[D;- D;]

=

(4.3)

/ DiEi dV.

(4.4)

Applying the divergence theorem, one obtains

/ tiuia 2 / aijeijdV, / =

-w+

dA

=

The integrals extend over the interface area A and the volume V . The equations above hold for any constitutive law. 2 . Free-Energy Density Function We will make the standard local equilibrium assumption: a free-energy function exists for every phase in the crystal, even though the crystal as a whole is not in equilibrium. At a fixed temperature, the Helmholtz free energy per unit volume of a phase, W , is a function of the strain and the electric displacement, W(e,D). When the state of the crystal varies, the energy density varies by dW

=

+

aijdeji EidD,.

(4.5)

Once the energy density function is prescribed, the field equations and the boundary conditions define the boundary-value problem. The crystal also stores energy in interfaces. Denote the surface energy per unit area of an interface by y . An interface is assumed to be a sharp transition within a few atomic layers, so that electro-mechanical field is unaffected by the interface tension, and the interface tension unaffected by the electro-mechanical field. 3. Free Energy of a Polydomain Crystal

+

Prescribe a distribution of traction t and electric potential on the external surface of the crystal. On the part of the surface where the electric potential is not prescribed, e.g., the interface between the crystal and the air, we assume that negligible electric field lines escape from the crystal. This is a good approximation for a crystal having a large permittivity, where the prescribed electric potential does work on the crystal, not on the air.

Motions of Microscopic Su#aces in Materials

225

The combination of the energy function and the field equations defines an electro-mechanical boundary-value problem. Once the field is solved, the Gibbs free energy of the entire crystal is calculated from G

=

1

ydA

+ 1W d V -

1

@wdA - /tiUidA.

(4.6)

The first integral extends over the interfaces, the second over the volume of the body, the third over the potential-prescribed surface, and the fourth over the traction-prescribed surface. 4. Deep- Well Approximation

So far, the free-energy density W can be an arbitrary function of the electric displacements and strains. A useful approximation has often been adopted. Figure 9(a) illustrates the free-energy density for a onedimensional model of a ferroelectric crystal at a fixed temperature. When the temperature is far below the Curie point, the free-energy density has two deep wells at 0, and - D s , corresponding to the spontaneous polar states. Due to crystal symmetry, the spontaneous states have the same free energy, g. Figure 9(b) shows the D-E curve derived from the free energy function, E = dW/dD. The peak, E , , is the field needed to switch polarization uniformly over the entire crystal, which is often much larger than the field needed to cause domain wall motion. Consequently, the state in each domain is near one of the spontaneous states, with approximately a linear D-E relation. Consider in general a spontaneous state with the strain e”), the electric displacement D”), and the Helmholtz free energy per unit volume g(’). Expand the free energy density function by the Taylor series around this spontaneous state, retaining up to the quadratic terms:

(4.7)

The first-order terms vanish because the stress and the electric field vanish at the spontaneous state. The coefficients C, p, and h characterize the elastic, dielectric, and piezoelectric responses near the spontaneous state.

2. Suo

226

I FIG. 9. a> The free-energy density as a function of the electric displacement. The free energy has deep wells when the crystal is in the ferroelastic state, far below the Curie temperature. b) The elastic field vs electric displacement curve. Only the linear parts of the curve near the spontaneous states are realized in a crystal.

The stress and the electric field are differential coefficients, (4.5), so that

These linear relations are valid inside each domain. Together with the field equations, they define a linear, coupled, electro-mechanical boundary-value problem.

Motions of Microscopic Surfaces in Materials

227

B. ELLIPSOIDAL TRANSFORMATION PARTICLE IN INFINITE MATRIX UNDER REMOTELOADING

Numerical analysis is usually required to solve the boundary-value problem above. Fortunately, many analytic solutions exist for an ellipsoid inclusion in an infinite matrix subject to remote loads; see Osborn (1945) for dielectric, Eshelby (1957) for elastic, and Dunn and Wienecke (1996) for piezoelectric inclusions. The shape of ellipsoids is versatile enough to model many phenomena. A nice feature common to this class of problems is that all fields inside the ellipsoid are uniform. Here we will not list these solutions, but will discuss the calculation of the free energy. The discussion parallels that of Eshelby (1957) for an elastic inclusion. Consider a transition from one solid phase to another. Without the constraint of the parent phase, the new phase would have a spontaneous strain es, a spontaneous electric displacement Ds,and a free energy change per volume g. All the three quantities are relative to the parent phase. When a small particle of the new phase grows inside the parent phase, both phases have stress and electric field. No dislocations, free charges, or other defects are present to relieve the field. Model the new phase particle by an ellipsoid, and denote its surface area by A and volume by V. Model the matrix as an infinite medium, and load it such that a uniform stress crif and a uniform electric field EP prevail far away from the particle. The free energy of the matrix in the absence of the particle, remotely loaded as described above, is the reference state. The free energy G defined by (4.6) is the sum of five contributions. a) Surface energy The phase boundary increases the free energy by G,

=

j ydA.

(4.10)

The integral extends over the ellipsoid surface. When the surface tension is isotropic, G, = y A . The surface energy resists particle growth. b) Energy due to phase difference When both phases are free from the stress and electric fields, the free-energy change due to the phase difference is G,

= gV.

(4.11)

228

2. suo

In our sign convention, the phase difference resists particle growth if g > 0. The following three terms arise from various fields. Owing to the linearity of the problem, the free energy must be a bilinear form of the spontaneous quantities efj and Df and the applied loads mi; and E A . They may be grouped according to their physical significance. c) Work done by the applied load through spontaneous strain and electric displacement The work done by the applied load on the spontaneous strain and electric displacement changes the free energy by

A positive work reduces the free energy, and thereby motivates particle growlh.

d) Energy due to strain and polarization misfit In the absence of the applied loads, the spontaneous strain and electric displacement cause fields in the matrix and the inclusion. Let aijand E/ be the fields in the inclusion; they are linear in efj and Of, and various coefficients may be found in the above cited papers. The free-energy change due to the misfit is

G,,,

1 = --

2

( aijefj + E/D:)V.

(4.13)

This contribution is a positive-definite quadratic form of e:j and O f , and resists particle growth. e) Energy due to heterogeneity (i.e., modulus difference) Imagine two infinite bodies, each subject to a,? and EA at the infinity. One body is an infinite matrix without inclusion, and the other body is an infinite matrix containing an ellipsoidal inclusion. The constitutive laws are given by (4.8) and (4.9) with the spontaneous strain and electric displacement removed; C , @, and h are moduli for the matrix, and C * , @*, and h* for the inclusion. The first body has uniform strain e and electric displacement D everywhere, and the second body has strain e* and electric

Motions of Microscopic Surfaces in Materials

229

displacement D* in the inclusion. The free-energy difference between the two bodies is 1 GH = - [(C,,,, - C$q)e,,e,*, + ( Pf, - P,: )W;”

5

+ ( h f k ,- hTk,)(e,,D:‘+ e i , ~ , ) ] ~ .

(4.14)

This contribution is quadratic in (T~; and EP, and either motivates or resists particle growth, depending on the relative moduli of the two phases.

c.

GROWTHOF

A SPHERICAL PARTICLE OF

DILATION

As an illustration, consider two phases of an identical chemical composition but with different crystalline structures. Without the constraint of the parent phase, the new phase would have a pure volume expansion with linear strain e,, and a free-energy change per volume g, both being relative to the parent phase. The parent phase is loaded remotely by a hydrostatic stress (T.The electric effect is absent. We will assume that the new phase grows like a spherical particle in an infinite matrix. The system has one degree of freedom, the radius of the particle, R . Such an assumption excludes shape change, which may be important in some phenomena; for example, Johnson and Cahn (1984) showed that the spherical particle is unstable against shape change under certain conditions. The elastic stress field for a spherical inclusion in an infinite matrix can be readily solved. For simplicity, we first assume that the two phases have similar elastic constants, with Y being Young’s modulus, and Y Poisson’s ratio. The free energy change due to the introduction of the particle into the matrix is

The physical significance of each item is evident from the previous discussion. The free energy has the same dependence on the radius as in the problem studied in Section III.A, so that the previous discussion applies. The driving pressure is given by

g =-

dG 2Y - --g 4 r R 2 dR R

+ 3ae,

Yef

-

-.

1 - v

(4.16)

z. suo

230 The radius changes at rate

(4.17)

1-v

The solution to this ordinary differential equation is similar to (3.5). Consider the case that the elastic constants for the two phases are different. Denote the bulk modulus and shear modulus of the parent phase by B = Y/3(1 - 2v) and p = Y/2(1 + v), and B* and p* for the corresponding quantities for the particle. The fourth term in (4.15) should be modified to (4.18) A fifth term should be added to (4.151, -1

(4.19) This term motivates particle growth if the new phase has a lower bulk modulus than the parent phase. D. GROWTH OF

A

180" DOMAININ BARIUMTITANATE

Barium titanate (BaTiO,) undergoes a phase transition at 130°C. Figure 10 shows the unit cells of the two phases. Above 130"C, the crystal is cubic, and the ions lie symmetrically in the unit cell. Between 0 and 130°C the crystal is tetragonal, and the ions lie asymmetrically in the unit cell. We next concentrate on the changes at a fixed temperature between 0 and 130°C. Depending on the position of the titanium ion relative to the center of the unit cell, the crystal may have polar direction of any one of the six uariunts. A load may shift the position of the titanium ion, and thereby rotate the polar axis from one direction to another. An electric field may rotate the polar direction by either 180" or 90",but a stress may only rotate it by 90". A 180" polar rotation does not result in any strain; a 90" polar rotation results in a strain.

Motions of Microscopic Surfaces in Materials

231

C

a paraelectric, T > 13OOC

0Ti4+

ferroelectric, O°C < T < 13OoC Ba2+

002-

FIG. 10. The crystal structures of barium titanate (BaTiO,). The high-temperature phase is nonpolar. The low-temperature phase is polar and the Ti ion is off the cell center.

The crystal changes its state by domain-wall migration. The loads needed to move the domain walls are much lower than the loads theoretically predicted to uniformly switch the crystal. In fact, the latter has never been observed. Miller and Savage (1959) observed that the domain walls in BaTiO, move at a wide range of velocities (10-9-10-' m/s). The new domains tend to start as spikes. In the following we review a model study of the growth of a small 180" domain, assuming that the growing domain is elliptic (Landauer, 1957; Loge and Suo, 1996). Rosakis and Jiang (1995) showed that sharp tips can emerge from the growing domain; their analysis will not be reviewed here. Figure 11 illustrates the cross section of a cylindrical domain in a large parent domain having the opposite polarization. Because both domains have the identical spontaneous strain, the elastic and the piezoelectric effects may be ignored compared to the dielectric effects. The problem is further simplified by assuming isotropic domain wall energy, permittivity, and mobility. To avoid solving an electrostatic problem for complex-shaped inclusions, we approximate the cross section of the domain by a sequence of ellipses, evolving the domain with two generalized coordinates, the semi-axes cyl and a*.

z. suo

232

t

D5

fl-

tDs

+++ + +++

FIG.11. A 180” domain grows in a parent phase driven by an electric field.

The free energy due to the introduction of the nucleus into the parent crystal is

Here, s is the perimeter of the ellipse, and E is the permittivity. The first term is the domain wall energy, which resists the growth and tends to make the domain circular. The second term is the work term associated with polarization reversal, which drives the nucleus to grow and tends to make the nucleus circular. The third term is the depolarization energy induced by the discontinuity of the spontaneous polarization, which strongly resists the growth in the a1 direction, but weakly resists the growth in the a2 direction. The problem has a characteristic length, I , = y&/D;, which we will use to normalize the semi-axes of the ellipse. Figure 12 shows the contours of constant levels of free energy, normalized as G/(27rI,y). The loading level for the simulation is EEJD, = 0.05. The free-energy surface has a saddle

Motions of Microscopic Sufaces in Materials

233

point at a 1 = 131, and a2 = 5001,, indicated by SP in Figure 12. The physical origin of this saddle point is evident. Along either the axis a1 = 0 or a2 = 0, when the needle-shaped domain elongates, both the work term and the depolarization energy vanish, and the domain-wall energy increases the total free energy. Along a path with a large aspect ratio a 2 / a , , the total free energy is low for both a very small and a very large nucleus, and reaches a peak for an intermediate one. The fate of a nucleus depends on its initial position on the thermodynamic surface. To decrease the free energy, a very small nucleus shrinks, and a very large nucleus grows. For a nucleus near the saddle point, its fate is determined by both the energetics and the kinetics. In all cases, the free-energy landscape alone does not determine the evolution path. We next calculate the evolution path and rate. The differential equations (2.13) become (4.21)

FIG.12. Free energy contours for a 180" domain nucleus.

z. suo

234

DimensionlessTime, t/(YE2/LDs4) FIG.13. The semi-axes of a 180"domain nucleus as functions of the time.

The expressions for the generalized forces and viscosities are given in Loge and Suo (1996). Given initial semi-axes of a nucleus, we trace its evolution by numerically integrating (4.31). The problem has a characteristic time, t o = l i / ( L y ) , which is used to normalize the time. Figure 13 shows the evolution of a nucleus of initial axes a , = 5001, and a 1= 2001,. The &,-axis increases almost linearly with the time after some initial adjustment. The a,-axis decreases first, and then increases slowly relative to the &,-axes. The domain grows to a long needle in the direction of spontaneous polarization, because of the large effect of the depolarization energy.

E. EXPLICIT FORMULA FOR

THE

DRIVING PRESSURE

Eshelby (1956, 1970) called the following quantity the energy momentum tensor

+ and domain -, with the unit normal vector n pointing toward domain +. Denote the sum of the principal curvatures of the interface by K. Assume that no external force or charge

An interface separates domain

Motions of Microscopic Sugaces in Materials

235

lie on the domain wall. Eshelby showed that, when the interface moves in the direction n by distance 6 r , , the free energy of the crystal changes by

The interface tension y is taken to be isotropic. A comparison of (4.23) with (2.2) gives the driving pressure on the interface

9=

-

yK

+ n j ( P j T - Pj;)nj.

(4.24)

If medium - is taken to be a traction-free but strained solid, and medium + the vacuum, (4.24) becomes W.

9 =- y K -

(4.25)

Asaro and Tiller (1972) obtained this formula in analyzing surface motion. The equation of motion (2.3) becomes U, =

-L(yK

+ W).

(4.26)

V. Diffusion on Interface: Formulation

This section formulates mass diffusion on an interface. The interface may be either a free surface, or a grain boundary. The diffusion species are taken to be electrically neutral, so that only mass conservation need be enforced. The free energy has the same contributions as before, e.g., surface tension, external work, and elastic energy. A. GENERAL CONSIDERATIONS 1. Virtual Motion, Mass Conservation, and Inter&ace Motion

Figure 14 illustrates in three dimensions a surface that represents either a free surface or a grain boundary. Denote the unit vector normal to a surface element by n. An arbitrary contour lies on the surface, with the curve element dl, and the unit vector in the surface and normal to the curve element m.At a point on the contour, m and n are perpendicular to each other, and both are perpendicular to the tangent vector of the curve at the point.

2. suo

236

FIG.14. An interface in the third dimension. Also drawn is an arbitrary contour lying on the interface.

-

Let 6 1 be a vector field on the interface, such that 6 1 m is the number of atoms crossing unit length of the curve. As before, 6 indicates a virtual motion, namely, the number of atoms is small and need not obey any kinetic law, Following Biot (1970), we refer to 6 1 as the virtual mass displacement, to distinguish it from the atomic flux used below. Let 66 be the number of atoms added to the interface per unit area. Consider the interface area enclosed by the contour in Figure 14. Atoms move only on the interface, so that the number of atoms added to the area equals the number of atoms flowing in across the contour. Thus,

jsgdA +$61*mdl=0.

(5.1)

The first integral extends over the area of the interface enclosed by the contour, and the second over the contour. Equation (5.1) holds for any contour on the interface. Recall the surface divergence theroem, $ 6 I * mdl = 1 V * (61) dA,where the operator V is carried on the surface. (Some writers signify the surface operator with V,.) Mass conservation requirements can also be expressed in terms of the surface divergence: 66

+v

*

(61)

=

0.

(5.2)

The atomic flux, J, is a vector field on the interface, such that J * m is the number of atoms across per length per time. Let be the number of atoms added to unit area of the interface in unit time. Mass conservation requires an expression similar to (5.11, / i d A +$J

- mdl= 0,

(5.3)

Motions of Microscopic Su$uces in Materials

237

and an expression similar to (5.2),

& +V

J

=

0.

(5.4)

We next connect & to the motion of the free surface and the grain boundary. The expressions are similar between S( and the virtual motion of the interfaces. On a free surface, atoms of the solid diffuse from one part of the surface to another. Atoms added to a surface element cause the element to move in the direction toward the vacuum at the velocity on = a&.Here is the volume per atom. A grain bounduly is taken to be in local equilibrium. The atoms inserted to the grain boundary instantaneously crystallize, rendering the atomic structure of the grain boundary invariant. Yet the inserted atoms may add to either one of the two grains. Evidently, 8 only determines the relative motion of one grain with respect to the other, but not the migration of the grain boundary itself. Denote the velocity of one grain relative to another by Au,,, being positive when the two grains recede from each other. The atoms added to a grain-boundary element cause the two grains to drift apart at velocity Av,, = a&. The migration of the grain boundary is a degree of freedom independent of the relative motion of the two grains, and should be treated by the interface migration kinetics in the previous sections. Relative sliding of the two grains are often taken to be fast; see Section VI1.B. Cocks (1992) considered a locally nonequilibrium grain boundary, which will not be reviewed here. 2. Defining Difision Driving Force Associated with the virtual motion, S I and S t , the free energy of the system varies by SG. Define the driving force for diffusion, F, as the reduction of the free energy associated with one atom moving unit distance on the interface. That is.

The integral extends over the interface. Equation (5.5) holds for arbitrary virtual motion. The force F is a vector on the interface, and has a unit of force/atom.

238

z. suo 3. Kinetic Law

Following Herring (1950, we adopt a linear kinetic law:

J=MF.

(5.6)

This equation defines the atomic mobility on the interface, M , which is a second-order tensor at any one point on the interface, and may also vary from point to point. In this article we will assume that the mobility is independent of the crystalline direction. The mobility is determined in practice either by observing a phenomenon such as surface grooving, or by an atomistic simulation. The mobility relates to the self-diffusivity by the Einstein relation, M = D S / l l k T , where D is the self-diffusivity on the interface, S the effective thickness of atoms that participate in mass transport, fz the volume per atom, k Boltzmann’s constant, and T the absolute temperature. The self-diffusivity is approximately D = ub2 exp( - q / k T ) , where u is the frequency of atomic vibration, b the atomic spacing, and q the activation energy. Atomic mobility on an interface is sensitive to impurities. When the impurity atoms segregate to the interface, the interface has a much higher impurity concentration than the bulk crystal. For example, adding a few percents of copper to aluminum substantially slows down aluminum diffusion on grain boundaries (Ames et al., 1970). This empirical fact has been used to make electromigration-resistant interconnects in integrated circuits. B. DIFFERENTIAL EQUATIONS

The considerations above specify the surface diffusion problem. At a given time, the free-energy variation determines the driving force, the kinetic law relates the driving force to the flux, and the flux then updates the surface shape according to mass conservation. The procedure repeats for the next time increment. These general considerations lead to two approaches for computation. One approach, due to Herring (19511, defines the chemical potential on the surface, leading to partial differential equations. This subsection lists these equations. The following subsection formulates an alternative approach on the basis of a weak statement. The two subsections can be read independently, in any order.

Motions of Microscopic Surfaces in Materials

239

1. Chemical Potential First consider an interface which is a closed surface in the third dimension. Herring (1951) defined the chemical potential of an interface element, p, as the increase of the free energy associated with the addition of one atom to the element. Thus,

The integral extends over the surface. The chemical potential has a unit of energy/atom. Using (5.6) and the divergence theorem, one obtains that

A closed interface does not have a boundary curve, so that the line integral vanishes. A comparison of (5.5) and (5.8) equates the two area integrals for arbitrary distribution of 61, so that the two integrands must be identical:

F = -vp.

(5.9)

The driving force is the negative gradient of the chemical potential. As expected, atoms diffuse from an interfacial element with high chemical potential to an interfacial element with low chemical potential. Next consider the continuity conditions at a triple junction. As discussed in Section II.F, the local equilibrium assumption requires that the freeenergy variation associated with the translation of the junction vanish. Consequently, the three interfaces meet at angles determined by the surface tensions. These considerations apply here. In addition, the local equilibrium assumption requires that the chemical potentials on the three interfaces be equal at the triple junction. To see the last statement, consider three interfaces that meet at a straight line of length 1. On the three interfaces I , , Z2, and Z, are the components of the I vector pointing to the junction. The junction is

240

2. suo

neither a mass sink nor a mass source, so that the net mass coming to the junction vanishes, S I , + SZ, + 61, = 0. Recall that the chemical potential is the free-energy change associated with adding one atom. The free-energy change due to the atoms moving to the triple junction is SG = -I( p l 61, + ,u281, p 3 81,). A combination of the above two equations give SG = -1( p l - p 3 )SZ, - f( p2 - p 3 )81,. The local equilibrium assumption requires that 6G = 0 for any virtual mass displacements SZl and 61,. Consequently, the chemical potential is continuous across the triple point, p , = p 2 = p3.

+

2. Free Surl'ace

Mass conservation relates the velocity normal to the free surface to the flux divergence: (5.10) U , + RV * J = 0. As stated in Section IV.E, associated with adding atoms on the interface, the free energy varies by SG

=

/(yK

+ W)R @dA.

(5.11)

The surface tension y is isotropic, the sum of the two-principle curvature K is positive when the surface is convex, and W includes energy density due both to stress and electric field. A comparison between (5.7) and (5.11) gives the chemical potential on the surface, p =

R( y K

+ W).

(5.12)

The diffusion driving force is F

=

- V ( R y K + RW).

(5.13)

A combination of (5.6), (5.10), and (5.13) gives u, = MR2V2(yK

+ W).

(5.14)

This partial differential equation governs the motion of a free surface when the surface tension is isotropic. 3. Grain Boundary

Mass conservation relates the relative velocity of the two grains to the flux divergence: Av, + R V * J= 0. (5.15)

Motions of Microscopic Surfaces in Materials

24 1

Let a, be the normal stress component on the grain boundary. To insert one atom to the grain boundary, the normal stress does work, varying the free energy by 6 G = -lunfl66dA.

(5.16)

Consequently, the chemical potential is p =

-Ran.

(5.17)

The driving force for diffusion on the grain boundary is F

=

avo-,.

(5.18)

Atoms diffuse on the grain boundary from an element of low-normal stress to an element of high-normal stress. A combination of (5.6), (5.151, and (5.18) gives All,

=

-MR2V2an.

(5.19)

This partial differential equation governs the normal-stress distribution in the grain boundary.

C. WEAKSTATEMENTS 1. Weak Statement When Interface DifSusion Is the Sole Rate Process Ignore Section V.B and start from Section V.A again. Consider a polycrystal particle with grain boundaries and free surfaces. We first assume that the grain boundaries do not migrate and grains are rigid, so that diffusion on interfaces is the only kinetic process. Replace the force in (5.5) with the flux using the kinetic law (5.61, giving

jy

dA

=

-6G.

(5.20)

The integral extends over all the interfaces in the system. Different interfaces, of course, may have different mobilities. The actual flux J satisfies (5.20) for all virtual motions that conserve mass, dictated by (.5.1)-(5.4) on each interface and by flux continuity at every triple junction. This formulation circumvents the differential equations in Section V.B, and the local quantities such as the chemical potential, the curvature of a free surface, and the normal stress in a grain boundary. The statement also

2. suo

242

enforces local equilibrium at the triple junctions, namely, (a) the interfaces meet at a junction with angles determined by the surface tensions, and (b) the chemical potentials of all the interfaces are equal at the junction. Should for any reason the two types of junction mobilities be finite, one could add them to the weak statement in the manner described in Section 1I.F. 2.

Variational Principle

Needleman and Rice (1980) formulated a variational principle that includes grain-boundary diffusion, and devised a finite-element method on the basis of the variational principle. Extensions have been made to analyze several phenomena involving interface diffusion (e.g., Bower and Freund, 1993, 1995; Cocks, 1994; McMeeking and Kuhn, 1992; Sofronis and McMeeking, 1994; Suo and Wang, 1994; Svoboda and Riedel, 1995). Following the steps in Section II.E, one can prove the following variational principle. Of all virtual flux j that conserves mass, the actual flux minimizes the functional (5.21) The weak statement and the variational principle lead to identical ordinary differential equations that evolve the generalized coordinates.

3. Galerkin Procedure Interface diffusion differs from interface migration in one significant way. For interface diffusion, mass conservation is expressed by partial differential equations, (5.2) and (5.4). When the shape of the surface is axisymmetric or invariant in one direction, the surface divergence involves one-dimensional differentiation, which can be integrated readily. The Galerkin method proceeds as follows. Model the surface with n degrees of freedom, writing q , , . . ., qn for the generalized coordinates, and q l , .. .,q, for the generalized velocities. Following the same procedure as in Section 1I.C to compute the generalized forces f j , the virtual displacement of the interfaces 6 r n , and the velocity of the interfaces u, . Integrate (5.2) and (5.41, and one obtains

Motions of Microscopic Surjfaces in Materials

243

where Qi plays the similar role as the shape functions. The weak statement (5.20) then leads to the same equation as (2.13), with the viscosity matrix being

The shape of the surface is updated as before. We will demonstrate this method in later sections. 4. Include Mass Conservation in Weak Statement

The procedure above, however, fails for a general surface in three dimensions, because the surface divergence in (5.2) now consists of differentiation of two surface coordinates. Consequently, one cannot integrate (5.2) to relate 6 I to Sq, , .. . ,S q , . What happens physically is clear. When the virtual motion of the surface is prescribed, mass conservation does not fully determine the virtual mass displacement. That is, a general surface requires degrees of freedom for 81, in addition to the degrees of freedom for the surface shape. The following notes may be useful in this connection. Mass conservation is a constraint, much like incompressibility in deformation analysis. One may use one of several methods in finite element methodology to include mass conservation in the weak statement. Here we use a penalty method as an illustration. Consider a closed surface for simplicity. Allow S t and SI to vary independently, and associate a driving force A with the new degree of freedom S t + V . (811, writing I F . SIdA

+ /A[St+

V.(SI)]d4

=

-SG.

(5.22)

The integrals extend over the closed surface. Prescribe an independent kinetic law for this new degree of freedom: ~+v.J=M,A.

(5.23)

The mobility M , is an adjustable parameter in the finite element analysis; when M , is very small, mass conservation (5.4) is recovered approximately. The weak statement becomes

z. suo

244

Incidentally, one can confirm that the parameter h has a simple physical interpretation, h = - p .

D. Multiple Kinetic Processes Consider a grain boundary which both migrates and acts as a diffusion path. Let Sr, be the virtual migration of the grain boundary, 6 1 be the virtual mass displacement on the grain boundary, and SG be the freeenergy variation associated with the combined virtual motion. Define the migration driving pressure 9 and the diffusion driving force F simultaneously by

The integrals extend over the grain-boundary area. Equation (5.25) holds for any mass-conserving virtual motion. Replacing the driving forces by the kinetic laws of the two processes, (2.3) and (5.61, we have

J SI I-&+/'rn a% -SG. *

un

L

M

=

(5.26)

The actual migration velocity u, and flu J satisfy this weak statement for arbitrary mass-conserving virtual motion. Other kinetic processes can be similarly added to the weak statement. Take, for example, a system of interfaces that move by diffusion on the interfaces, and creep in the grains. The free energy consists of the external work and various interface tensions. The problem was first treated by Needleman and Rice (1980). Denote the virtual displacement field in the grains by 6 u i , and the actual velocity field in the grains by u i . We will assume that the solid is incompressible, i.e., ui.i = 0.

(5.27)

Define the stress tensor, uij,and the diffusion driving force F, on the same basis, namely, as the energy-conjugates of their respective kinematic quantities. Thus,

j

+ 1Fi SIi dA = - 6 G .

uijS U ~dA , ~

(5.28)

Motions of Microscopic Surfaces in Materials

245

Interface diffusion obeys the kinetic law (5.6). For this demonstration, the grains deform according to a linear creep law: a,, =

am

4)+ d U , , j

+ U,,J

(5.29)

Here, amis the mean stress; 77 is the viscosity of the material; and a,, = 1 when i = j, SIJ = 0 when i # j. Replacing the diffusion driving force with the flux by (5.6), and the stress with the velocity gradient by (5.29), we obtain

(5.30) The actual velocity and flux satisfy this weak statement for arbitrary virtual motion.

VI. Shape Change due to Surface Diffusion under Surface Tension This section gives examples of shape changes motivated by surface tension. Most examples invoke surface diffusion as the only mass-transport mechanism. One example involves simultaneous grain-boundary migration and surface diffusion. A. RAYLEIGHINSTABILITY

Over a century ago, John William Strutt Rayleigh noted that a jet of water is unstable and breaks to droplets under the action of surface tension. Similar phenomena occur in a solid state; see Rodel and Glaeser (1990) for an experimental demonstration and literature survey. For example, at a high temperature, a crack-like pore in a solid undergoes a sequence of morphological changes until the crack breaks into many small cavities. The crack first blunts its edge, from which finger-like channels emerge, and the channels then break into small cavities. The morphological changes shorten the distance over which mass transports, and therefore accelerate the crack healing. Assume that the surface tension is isotropic, and the free energy of the system is the surface area times the surface tension. Of all figures of the same volume, the sphere has the lowest surface area, and therefore the lowest free energy. Why, then, does a cylinder evolve into many small spheres, rather than one, single large sphere?

246

z. suo

Consider a long cylinder of radius R. Perturb the surface along the longitudinal direction of the cylinder. It can be shown that the perturbation reduces the surface area if its wavelength exceeds 2rrR. Further, Srolovitz and Safran (1986) compared a row of identical spheres with the long cylinder having the same total volume, and noted that the spheres have a smaller total surface area than the cylinder if the sphere radius exceeds 3 R / 2 . This sphere radius corresponds to an initial perturbation wavelength of 9R/2. From these geometric (energetic) considerations, one expects that a sequence of configurations exists, from a cylinder to a row of spheres of large enough radii, with decreasing surface areas. But these energetic considerations do not answer the question raised above. The answer has to do with kinetics. It takes a short time for a cylinder to evolve into a row of spheres. The spheres break the masstransport path, preventing the system from reaching the minimal energy configuration, a single, large sphere. Here we have assumed a certain kind of mass-transport mechanism, such as fluid flow or solid diffusion. If, instead, the cylinder is sealed in a small bag, it will evolve to a single, large sphere via vapor transport. Nichols and Mullins (1965a, b) carried out a linear stability analysis of a cylinder using several mass-transport mechanisms. For surface diffusion, they showed that a perturbation of wavelength, A, = 2fi.rrR, amplifies most rapidly. If the initial imperfections of all wavelengths have a similar amplitude, it is reasonable to expect that the finite sphere size corresponds to wavelength A,. In what follows, the Rayleigh instability is used to illustrate the application of the weak statement. Surface-tension anisotropy is also included in the end of the analysis. 1. Free Energy Figure 15 illustrates a long cylinder of initial radius R with isotropic surface tension y . Perturb the cylinder to a wavy surface of revolution

where r is the radius of the perturbed surface, z the axis of revolution, t the time, pR the average radius, E R the amplitude, and A the wavelength. If the family of the assumed virtual motion contains the exact solution, the Galerkin procedure leads to the exact solution; otherwise the

Motions of Microscopic Surjfaces in Materials

247

FIG. 15. Perturb a cylindrical surface to a surface of revolution with an undulation along the axial direction.

Galerkin procedure leads to an approximate solution. In this case, the family (6.1) happens to contain the exact solution of Nichols and Mullins (1965a, b). Mass conservation requires that the volume be constant. Thus, r r 2 dz

[A

=

rR2A,

(6.2)

JO

which, to the leading term in profile is

E,

gives p

=

1 - s 2 / 4 . Thus, the surface

(6.3) The wavelength A is fixed in the linear stability analysis, so that this model has only one generalized coordinate: the perturbation amplitude E . The free energy of the column is the surface tension integrated over the column surface. In one wavelength the free energy is

and to the leading term in G

=

E,

2rRAy

+ T2 R * y [

(y)’ -

I]&’.

(6.5)

If the quantity in the bracket is negative, the free energy decreases as E increases. Consequently, the amplitude of a perturbation grows if its wavelength exceeds a critical value A,

=

27rR.

This reproduces the condition established by Rayleigh.

(6.6)

z. suo

248

2. Kinetics

Because of symmetry, J = 0 at z = 0. Mass conservation relates the flux J ( z ) to the rate of the change of the volume between 0 and z . Thus, 27rrRJ

/ dt

d = --

z

rrr2 dz.

(6.7)

0

To the first order in the perturbation, the above is R h sin[

--

J =

T) i.

(6.8)

255-0

The weak statement (5.20) becomes E

& =

-

(6.9)

7’

with the characteristic time 7 =

7

being

--[IR4

-

(-4*]-’( 27rR

27rR

-2

*

hf12M

For the initial condition

E =

~ ( 0 at ) t

E(t) =

=

(6.10)

0, the solution to (6.9) is

E(O)exp(

I;).

(6.11)

Figure 16 shows the trend of the characteristic time (6.10). When A < 277R, the perturbation increases the free energy, T < 0, and the perturbation

FIG.16. The characteristic time as a function of the wavelength.

Motions of Microscopic Surfaces in Materials

249

diminishes with the time. When A > 2 r R , the perturbation decreases the free energy, T > 0, and the perturbation amplifies with the time; T minimizes at A,, = 2\/2.rrR, agreeing with the analysis of Nichols and Mullins (1965a, b). The conclusion above is made on the basis of the linear stability analysis, where high-order terms of E have been ignored. A complete simulation of the surface evolution is necessary to take into account the actual initial imperfection and large shape change (Nichols, 1976). 3. Surface-Tension Anisotropy

Imagine a crystal having transversely isotropic surface tension. The long cylinder of circular cross section has constant surface tension y o . When the cylinder becomes a surface of revolution, the surface tension is nonuniform along the longitudinal direction. Denote 8 as the angle of the normal vector of an arbitary crystal plane, measured from the normal vector of the perfect cylinder. For small 8, the surface energy y can be expanded in the powers of 8, assuming y is a smooth function of 8. The crystal is assumed to have such a symmetry that the crystal plane at 8 and the crystal plane at - 8 have the identical surface energy. Consequently, the expansion only contains the even powers of 8. Take only the first two terms:

Here the dimensionless number a indicates the anisotropy. When a > 0, the crystal plane of the perfect cylinder has the largest surface tension of all the neighboring crystal planes. Perturb the cylinder to a surface of revolution with profile (6.3). To the first order of E , 8=

dr

__

d2

2rR A

= -

sin(

The free energy (6.41, to the leading order in

G

=

2rz h).

E,

2rRAyo + r RAyo[(l - 2 a ) ( 2

(6.13)

is

F)2 -

(6.14)

250

2. suo

The cylinder is unstable when the perturbation wavelength exceeds a critical value A,, given by

(A) 2

=1-2ff.

(6.15)

If the surface tension is very anisotropic, i.e., a > 1/2, the cylinder is unstable for perturbations of any wavelength.

B. A Row

OF

GRAINS-A MODELWITH Two DEGREES OF FREEDOM

An important distinction exists between a system of one degree of freedom and a system of multiple degrees of freedom. For a system of one degree of freedom (e.g., the spherical particle studied in Section III.A), the free energy is a function of the generalized coordinate (i.e., the particle radius), represented by a curve in a plane with the free energy as the vertical axis and the generalized coordinate as the horizontal axis. A point on the curve represents a nonequilibrium state; a minimum point on the curve represents an equilibrium state. Energetics requires that the state descend on the curve. Consequently, energetics alone determines the final state. Kinetics is restricted to the role of determining the time needed to approach the equilibrium state. For a system of two degrees of freedom, the free energy is a function of two generalized coordinates. This function is a surface in a threedimensional space, with the free energy as the vertical axis, and the two generalized coordinates as horizontal axes. A point on the surface represents a nonequilibrium state in general; the bottom of a valley represents an equilibrium state. Energetics requires that an evolution path be a descending curve on the surface. There are, however, countless descending curves on a surface from any point other than a bottom of a valley. Consequently, when a system has two or more degrees of freedom, energetics by itself does not determine the evolution path. Kinetics plays a more significant role than just timekeeping. In the analysis above of the Rayleigh instability, the system is modeled with only one degree of freedom, the amplitude of the perturbation, E . It gives the sensible predictions when the perturbation amplitude is small, but cannot predict the spacing of the final spheres. In fact, the system has

Motions of Microscopic Surjaces in Materials

25 1

infinitely many final equilibrium configurations, and simply cannot be modeled with one degree of freedom. We next illustrate these general points in some detail with a row of grains (Sun et al., 1996). Similar problems arise in an electrical interconnect (Srolovitz and Thompson, 1986), powder sintering (Cannon and Carter, 1989), and a fiber constrained on a substrate (Miller and Lange, 1989). Figure 17(a) illustrates a fiber of bamboo-like grain structure. The fiber consists of a row of identical grains, initially cylindrical in shape and connected at their ends, each grain being of length Lo and diameter D o . The grains change shape by mass diffusion on the surfaces and grain boundaries, under the action of surface and grain-boundary tensions, ys and yb . The fiber is unconstrained in the longitudinal direction. The grains are assumed to remain identical to one another (Figure 17(b)). They will evolve to either one of two equilibrium configurations: the isolated spheres (Figure 17(c)), or connected disks of truncated spheres (Figure 17(d)).

000 m c) Isolated Spheres

d) Truncated Spheres

FIG. 17. (a) The initial cylinder-shaped grains. (b) Barrel-shaped grains approximate an intermediate, nonequilibrium state. (c) Grains pinch off and spheroidize, approaching an equilibrium state, a row of isolated spheres. (d) The array contracts as atoms diffuse out from the grain-boundaries and plate onto the free surfaces, approaching another equilibrium state, a touching array of truncated spheres.

252

2. suo

The final equilibrium state is selected by an interplay between the free energy and the kinetic process. For most materials, Yb < 2 y , , and the isolated spheres in Figure 17(c) have higher free energy than the truncated spheres in Figure 17(d). For the fiber to groove along the triple junction, pinch off, and spheroidize, atoms need only diffuse on the surfaces of the grains. For the fiber to become truncated spheres, atoms must diffuse out of the grain boundaries to allow the grain length to shrink. If the atomic mobility on the grain boundary is much lower than that on the surface, Mb 2 ] (3n+1)/2n

- (fi//u>

(3n + l ) / n

(2.29)

2.

The non-dimensional torque ratio is plotted as a function of / / u in Figure 4, for p = 1 and 2, and n = 3 and 5. For all cases shown there is a significant enhancement in torque by a factor of 3 to 5 when / / a is increased from zero to unity. We further note that the elevation in torque

-

-n=3

3-

0' 0

'

I

0.2

I

I

0.L

t

I

0.6

I

I

0.8

I

I 1

(/a FIG.4. Elevation in torsional strength due to strain gradients. The wire is of radius a and the torque Q at a given twist is normalized by the torque Q, required to achieve the same twist for a conventional solid.

Strain Gradient Plasticity

313

with increasing / / a is greater for p = 1 than for p = 2, and is greater for n = 3 than for n = 5. At small values of / / a , a greater size effect is shown for p = 1 than for p = 2. This is made explicit by the asymptotic expressions for Q/Qo at small / / a . For p = 1, Q/Qo increases linearly with / / a according to

Q _ -1+

( 3 n + l)(n

Qo

whereas for p

=

n(2n

+ 1)

+ 1)

(&//a)

+ o(e2>

(2.30)

2. Q / Q o increases quadratically with / / a according to

-Q= I + Qo

3 (3n + 1) (E'/d2 2 n

+

(2.31)

The measured torsional response shown in Figure 2(a) may be used to deduce a value for the material length scale E'. Follow the same procedure as outlined by Fleck et al. (1994), we find that the magnitude of / is somewhat sensitive to the value adopted for p. For the case p = 2, / = 4 pm fits the data, while for p = 1, / - 2 pm.The value p = 2 will be used in most of the examples in the sequel as this value allows for a more robust numerical implementation of solution procedures in more complicated problems than is the case when p = 1. Nevertheless, the option of using (2.23) with a value such as p = 1 should not be foreclosed in constitutive modelling in the future, as the linear combination of strains and strain gradient terms may reproduce behavior better than the harmonic sum of these terms. Gradient contributions become numerically larger at smaller values of the gradient invariants relative to the strain invariants for the linear combination than for the harmonic sum, and this difference may be physically significant as suggested by (1.1). Indeed, the distinct differences in the elevation of the torque between p = 1 and 2 in the range of small //a which is evident in Figure 4 and in (2.30) and (2.31) may provide an opportunity to establish the choice by direct comparison with experimental data. 2.

m e Grain Size Effect on Polyclystalline Yield Strength

Ashby (1970) has argued that the grain size dependence of flow strength can be explained in terms of the anisotropy of slip from grain to grain. Consider a pure polycrystalline metal under uniaxial tension. If each grain were unconstrained to deform freely under the applied stress then any chosen grain would deform to a different shape from that of its neighbors due to the difference in its crystallographic orientation. To ensure compat-

314

N. A. Fleck and J. W. Hutchinson

ibility of deformation from grain to grain, the strain state within each grain is non-uniform, and geometrically necessary dislocations are generated within each grain in order for them to fit together. The simplest physical arguments reveal that the density pG of the geometrically necessary dislocations scales with the average strain in each grain divided by the grain size d. (Measurements of dislocation densities in deformed polycrystals do seem to follow this law: the density increases linearly with tensile strain, and, at a given strain, the density scales with the reciprocal of the grain size. See for example Essmann et al., 1968; McLean, 1967). For small grains the density pG will exceed that of the statistically stored dislocations p s , and relation (1.1) suggests that the elevation in flow strength scales with d-’/’. This is consistent with the observed Hall-Petch grain-size relationship, whereby the elevation in yield strength due to grain-boundary strengthening varies as d-’/’. We note in passing that Nix has shown that the grain-size effect on strength is enhanced when the material exists in the form of a thin layer on an elastic substrate (Nix,1988). A crystal plasticity version of the strain gradient theory is introduced in Section V below and can be used to explore the grain size effect in quantitative detail. Smyshlyaev and Fleck (1996) have used the linear limit of this crystal theory to predict the effect of grain size d on the macroscopic shear modulus of incompressible face-centered cubic (fcc) polycrystals. The linear result provides a useful qualitative guide to the effect of grain-size strengthening of non-linear polycrystals. Also, the linear result is the first step in the estimation of the grain-size effect for non-linear polycrystals: in order to use the non-linear variational principle of Ponte Castenada (1991, 1992) the effective properties of the linear solid are required. In the linear analysis of Smyshlyaev and Fleck (1996) the crystals are oriented in a uniform manner, so that the macroscopic response of the polycrystal is isotropic and can be described by a single shear modulus p*. It is assumed that each fcc crystal contains 12 independent slip systems, and that each slip system deforms in shear with an associated shear modulus p . The strain energy density W for each slip system is expressed in terms of both the elastic shear strain y and the spatial gradient of shear strain Vy according to the assumed relation (2.32) where the material length scale / ensures dimensional consistency. Hashin-Shtrikman bounds on the macroscopic shear modulus are shown in

Strain Gradient Plasticity

315

Figure 5 as a function of the grain size d . It is found that the macroscopic stiffening varies approximately as d P ' l 2 ,in support of the Hall-Petch relationship. However, the effect is not a strong one: as Q d is increased from zero to unity, the macroscopic shear modulus increases by about 10%. It is well known that a strong Hall-Petch effect is not observed in fcc polycrystals due to the availability of a large number of slip systems for each crystal. Further calculations are required to determine the predicted grain-size effect for non-linear fcc polycrystals and for other crystal structures. 3. Strengthening of Metal Matrices by Rigid Particles The macroscopic strength of particle-reinforced metal-matrix composites is found to depend upon particle diameter in addition to volume fraction, for particle diameters in the range 0.1 p m to 10 pm. For example, Lloyd (1994) has tested a composite of silicon carbide particles in

FIG.5. Hashin-Shtrikman upper and lower bounds on the macroscopic shear modulus p* for an isotropic face-centered cubic polycrystal. Each grain of dimension d is assumed to deform by slip on 12 independent slip systems. The slip response is taken as linear elastic with a strain energy density function given by expression (2.321, involving the material length scale .! Taken from Smyshlyaev and Fleck (1996).

316

N. A. Fleck and J. W. Hutchinson

an aluminium-silicon matrix. He observed a 10% increase in the strength when the particle diameter was reduced from 16 p m to 7.5 p m with the particle volume fraction fixed at 15%. These particles are sufficiently large that it is thought that plastic deformation is by the interaction of dense clouds of dislocations with each particle rather than by individual dislocation interactions. Nevertheless, conventional continuum plasticity (Bao et al., 1991) predicts that the size of the particles (with the volume fraction fixed) should have no effect on the composite yield strength. As in the previous examples, this follows from the absence of a material-length scale in the conventional theory using simple dimensional considerations. Steep plastic gradients adjacent to each particle necessitate the existence of geometrically necessary dislocations and associated local hardening, as argued by Ashby (1970) and Brown and Stobbs (1976). These gradients provide the rationale for turning to an approach based on strain gradient plasticity when the particles are micron-sized or smaller. Fleck and Hutchinson (1993) predicted a strong size effect for a dilute volume fraction p of isotropically distributed rigid spherical particles of radius a in a matrix of power law strain gradient solid characterized by (2.2) and (2,l).In uniaxial tension, the average macroscopic stress of the composite is related to its average macroscopic strain 2 by (2.33)

where the factor p(n + l ) f p / n is the relative strengthening due to the particles at a given strain E. Fleck and Hutchinson (1993) calculated the factor f p for the CS solid in (2.20), and the results are repeated here in Figure 6: f p is plotted as a function of / / a for selected values of n as solid line curves. It is noted that f p increases dramatically with increasing / / a . The role played by the strain hardening index n on this factor is more modest, with a small decrease in f p with increasing n. Additional calculations have been carried out to investigate the role played by stretch gradients acting in concert with rotation gradients on the strengthening due to a dilute concentration of rigid particles. Eq. (2.33) remains valid, and results for f p are included in Figure 6 as dashed line curves for the SG solid in (2.20). The contribution of /~q$$) to the strain energy density w leads to enhanced strengthening at finite / / a . For example, at / / a = 1 the full-strain gradient theory predicts 50% more strengthening than that predicted for the couple-stress solid, for all values of n considered.

317

Strain Gradient Plasticity 10

I

I

I

I

-CS - solid ---

S G - solid

/ /

/ / //

n=l

/

'

///

/ // /

-

The influence of the parameter p in (2.23) on fp is displayed in Figure 7 for the two solids. Numerical difficulties in the solution process are encountered for values of p close to 1, and thus the smaller value of p used in Figure 7 is 1.25. The trend of the prediction for the smaller value of p can be approximately reproduced using p = 2 if a large value of / / a is chosen. However, as was the case for wire torsion, one concludes that the form of the interaction between strain hardening and strain gradient hardening as reflected by the parameter p may be deserving of further attention, particularly because of the differences in behavior at small values of / / a . There is experimental and theoretical evidence to suggest that the size of each phase in a two-phase alloy has an effect on the macroscopic strength (Funkenbusch and Courtney, 1985; Funkenbusch et af., 1987; Smyshlyaev and Fleck, 1994, 1995). Specifically, it is observed that the

N. A. Fleck and J. W . Hutchinson

318 20 )

'

I

/

I

fP

I

I

I

I

I

0.2

0.4

0.6

0.8

1

Va FIG.7. The influence of the parameter p in (2.23) on fp, for particle strengthening, n

=

3.

smaller the size of each phase, the greater is the strength. The two-phase alloy bridges the two extremes of a metal matrix composite containing rigid particles and a single-phase alloy. Since a size effect is noted at both limits, it is not surprising that a size effect is also observed for the two-phase alloy. 4.

Void Growth and Softening: The Role of Void Size

A common fracture mechanism of ductile metals is nucleation, growth, and coalescence of voids. There exists a well-defined mechanics of void growth based on conventional continuum plasticity theory (e.g., Rice and Tracey, 1969; Gurson, 1977; Needleman et al., 1992; Tvergaard, 1990). None of the widely used results from the literature on void growth involve any dependence on void size, even though they are sometimes applied to voids of micron or even sub-micron size. There is some indirect evidence that voids in the micron to sub-micron size range are less susceptible to growth at a given stress state than larger voids (private communication from A. G. Evans, 19961, but a careful experimental examination of this issue remains to be carried out. It is of interest to study theoretically the

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effect of void size upon void growth and the associated macroscopic softening within the context of the present class of strain gradient theories. We shall see that void growth strongly distinguishes between gradient effects tied solely to rotation gradients versus those arising as well from stretch gradients. Moreover, void growth phenomena may provide a robust means for confronting strain gradient plasticity predictions with experiment . Consider the problem of an isolated spherical void of radius a in an infinite, incompressible power law matrix which has been addressed in most prior work, but with a matrix characterized by w in (2.21) with the effective strain (2.15). The solid is subjected to uniform remote axisymmetric loading specified by u; = S and u: = uz = T with S > T , as depicted in the insert in Figure 8(a). As in previous work, it is convenient to + $T and the remote deviatoric employ the remote mean stress uz = stress urn= S - T , and to introduce the non-dimensional measure of stress triaxiality as X = u , / u m The . remote strain follows from uij = d w / d c i j as

v

Calculations have been performed on the volume expansion increment of an isolated void of volume V in the infinite matrix subjected to an increment of axisymmetric loading specified by hy3 . Calculations have been performed for each of the two strain-gradient solids given by (2.20). The problem for the CS solid, for which w depends solely on the rotation gradients, was previously analyzed by Fleck and Hutchinson (1993). Results for the normalized dilation rate, P/hY3V, have the functional form

v

(2.35)

It is through the dependence on !/a that void size produces behavior different from that for the conventional power law solid. Representative results are given in Figure 8(a) for the normalized dilation rate as a function of /,/a for the stress triaxiality in uniaxial tension, X = 1/3, and the triaxiality representative of that directly ahead of a mode I crack tip in plane strain, X = 2.5. This figure illustrates the sharp differences in the void growth behavior implied by the two constitutive models, CS and SG, which were alluded to above. The limit for / / a = 0 is the result for the conventional solid. There is very little effect of

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320

I

I

10

1

0.1 conventional solid,{ = 0 CS - solid, (/a 1 SG - solid. (/a = 1

X (b)

FIG.8. Growth rate of an isolated void under remote axisymmetric loading: axial stress S and transverse stress T . The stress triaxiality X is defined by X = ( S + 2 T ) / 3 ( S - TI. a) Effect of void radius a upon normalized void growth rate, for n = 3. b) Effect of stress triaxiality X upon normalized void growth rate.

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void size upon the normalized dilation rate for the CS solid at both triaxialities. This is readily understood when it is recognized that the dominant deformation field in the vicinity of void producing its expansion is the spherically symmetric radial displacement field. The radial displacement field is irrotational inducing no rotation gradients and, therefore, no strain gradient contribution to the strain energy density w for the CS solid. By contrast, the spherically symmetric radial field does induce significant gradients of stretch, leading to the exceptionally strong reduction of growth rates for smaller voids in the SG solid relative to larger voids seen in Figure 8(a). In effect, the expanding void is surrounded by a shell of hardened material due to the presence of both strain and strain gradient hardening. The authors are unaware of any existing experiments against which to test these predictions. Typically, metals contain a distribution in size of pre-existing void-nucleating defects such as soft inclusions and particles with weak interfaces. When stretch gradients influence hardening, strain gradient theory suggests that only voids larger than some cut-off size will grow and coalesce; smaller voids should experience little growth. Several other phenomena will be discussed later in this survey which display a similarly strong selectivity to stretch gradients over rotation gradients due to the character of their deformation fields. One is the closely related phenomenon of cavitation instability, and another is the indentation hardness test. It seems reasonable to hope that careful experimental observations related to these phenomena, in conjunction with tests where rotation gradients are dominant, should provide a means to identify which of the competing constitutive models best reproduce small scale behavior. For completeness, void growth is plotted in Figure 8(b) as a function of triaxiality ratio for three solids: the conventional solid (i.e., / / a = 01, the CS solid with / / a = 1, and the SG solid with / / a = 1. Results are presented for both n = 3 and n = 5. It is clear from the figure that the couple-stress solid displays almost the same rate of void growth as the conventional solid at all stress triaxialities. In contrast, for all triaxilities, the rate of void growth is reduced by about an order of magnitude for the SG solid compared with the conventional solid. There is a close connection between the reduction in void growth due to strain gradient effects and the reduction in macroscopic softening due to the presence of the voids, as the counterpart to particle strengthening. The same power law metal matrix discussed above is considered which contains an isotropically distributed dilute population of spherical voids of radius a .

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N. A. Fleck and J. W: Hutchinson

The porous solid is subject to the overall, or macroscopic, axisymmetric stress state characterized by S and T with triaxiality X as defined above. With Eij and Eij as the macroscopic stresses and strains and @ ( O i j )as the strain potential such that E j j = d@/aa,,, Fleck and Hutchinson (1993) have shown that a dilute concentration p of voids alters @ to

The function f, can be computed from the solution for the isolated spherical void in the infinite matrix ( F in (2.35) is related to f,, by F = d f , / d X ) . The larger is f,,, the larger is the strain at a given stress, or, equivalently, the greater is the softening at a given strain. Plots of f, versus the triaxiality factor are given in Figure 9 for the same three solids considered in the previous figure: the conventional solid (//a = 0) and the CS and SG solids, each with / / a = 1. Again it is seen that there is little difference between the softening predicted for the conventional solid and the CS solid, with both predicting a large increase in softening as triaxiality increases. When stretch gradients are assumed to contribute to

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hardening, as in the case of the SG solid, softening is significantly moderated, especially at large triaxiality. In other work, Gologanu et al. (1995) have taken the first steps to address a strain gradient theory for porous metals by generalizing the Gurson (1977) model to include the effect of macroscopic strain gradients. They do not address the issue of local strain gradients on the growth of the voids, as done here. Instead, these authors retain the assumption that the material surrounding the voids is a classical elastic-plastic solid. They derive a modified yield function for applications where the gradient of strains is such that the deformation scale begins to become comparable to the void spacing. Their results are somewhat surprising in that the dominant effect of the strain gradient is not influenced by the void volume fraction or void spacing, and, moreover, the gradient effect persists as the volume fraction approaches zero. 5.

Cavitation Instabilities

A void in an elastic-plastic solid will grow unstably at sufficiently high mean stresses in what is usually described as a cavitation instability (Bishop et al., 1945). Strain gradients will delay cavitation to larger mean stresses when the void size is comparable to &, assuming the solid experiences hardening due to stretch gradients. In this section, results from M. Begley (work in progress) will be presented for the effect of strain gradients on the cavitation instability of a spherical void of initial radius a in an infinite, incompressible solid subject to a spherically symmetric radial loading at infinity. In the notation used above, the remote stress is specified by S = T so that the remote mean stress is :u = S . An isotropic solid containing a spherical void undergoes spherically symmetric deformations such that there are no rotations and, therefore, no rotation gradients. Thus, of the two solids in (2.201, only the SG solid will produce a strain gradient effect through its dependence on stretch gradients. The only strain gradient term to survive in (2.19) is the term with coefficient &,: and B reduces to 2

(2.37) Cavitation is where r denotes the radial coordinate and, as before, & driven by the elastic energy stored in the remote field, and therefore it is essential to include linear elastic strains in the constitutive model. To this

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end, the power law SG solid, (2.21) with (2.371, is coupled to an incompressible linear elastic range according to

w

=

1 -Eg2 2

for 8 < go

where E is Young's modulus and the connection 2, = E g o is required. In uniaxial tension, this gives E = a / E for a < Xo and E = 270(a/20)n for a > So.Here, 2, is to be regarded as the initial tensile yield stress and go as the associated tensile yield strain. The inclusion of a strain gradient dependence in the linear elastic range through B is only for mathematical convenience. It has essentially no influence on the cavitation stress since the strain gradients in the elastic region are very small. The cavitation analysis for the strain gradient solid parallels that given for the conventional elastic-plastic solid given by Huang et al. (1991). The results presented below are obtained using an exact finite strain analysis in which ze is the logarithmic strain and r is the radial coordinate in the deformed state. However, Begley established that finite strain effects do not have a major influence on the solution. The cavitation stress, a,, is the remote mean stress at which the void grows without bounds, i.e., dV/da," + m. It has the following dimensionless form (2.39) ac/2, = F ( N , 2,/E, / / a > where N = l/n. Values of ac/X0 for the conventional solid ( / / a + 0) are presented by Huang et al. (1991); [acl,,/XO lies between 4 and 8 for values of N and Z 0 / E typical of metals. Here we display plots of the normalized cavitation stress as the ratio a,/[ac]clp=O, thereby emphasizing the effect of the strain gradients. The normalized cavitation stress is shown as a function of 8 / a for various N with 8, = 0.003 in Figure 10(a), and for various 8,, with N = 0 in Figure 10(b). The effect is a strong one according to the assumed material model. Voids with radii less than about 2 8 will have a significantly enhanced resistance to cavitation relative to larger voids. 6. Indentation Hardness Testing

Size effects have been recognized in indentation hardness testing for some time. Various factors can give rise to a hardness measurement which depends on the size of the indentation, including surface effects and the

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1

1

0

0

0.2

0.5

0.6

0.8

1

1.2

I

I

I

1.L

Va

(a) 3,

I

I

I

2.5

a, -

2

[q.O 1.5

1

0.5 I

0

I

0.2

I

0.L

I

0.6

I

0.8

I

L

1

1.2

1

/a (b)

FIG.10. a) The normalized cavitation stress uc/[a,]/=, as a function of //a. a) effect of strain hardening exponent N , with go= 0.003, b) effect of yield strain go,with N = 0.

absence of nearby dislocation sources for nano-scale indents. However, as mentioned in the Introduction in connection with the hardness data in Figure 1, there appears to be clear evidence of a strong indentation size dependence in the range of micron to sub-micron indents which is due to the increasing dominance of geometrically necessary dislocations as indents become smaller (Poole et al., 1996; Stelmashenko et al., 1993; Ma

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N. A. Fleck and J. W.Hutchinson

and Clarke, 1995; Nix, 1988). The scale is sufficiently large for very large numbers of dislocations to be involved: consequently, this is another example where a continuum plasticity approach would appear to be required for quantitative analysis. The simplicity of the indentation test and the availability of equipment for conducting micro-indentations suggests that this test may be a good candidate for measuring the material length scale, k, in the strain-gradient constitutive model. Work is underway to analyze the test for several indenter head shapes, applied to both polycrystalline and single-crystal materials. At this writing, the only results available are those by Shu and Fleck (1996) for axisymmetric frictionless indenters applied to the CS material. Three head shapes were considered: flat-ended, conical, and spherical and only a minor effect of head shape was found for indentation of the non-linear solid. In addition, results were obtained for a flat-ended circular punch with sticking friction. Shu and Fleck (1996) obtained their indentation results by the finite element method, with elements designed to capture the straingradient dependence. The calculations have been performed without accounting for finite strain effects, but earlier studies on indentation of conventional elastic-plastic solids indicated that finite strain effects should not alter the hardness predictions by more than a few percent (Bower et al., 1993). The results of Shu and Fleck (1996) are repeated here for the flat-headed indenter of radius a. The indenter is pushed into a semi-infinite half-space of the elastic-plastic CS material specified by (2.38) and (2.20). Two conditions at the interface between the indenter and the half-space have been considered: frictionless contact and fully sticking contact with no sliding. The load P applied to the indenter approaches an asymptote P,,, as the indenter is forced down into the half-space. The hardness is defined to be H = P,,,,,/(.rra2).For the problem posed it has the form H

=

Z o F ( n , / / a , Z0)

(2.40)

where the dimensionless function F depends on which of the two contact conditions are operative. The dependence on the initial yield strain gois weak; the results presented in Figure 11 have been computed with Zo = 0.01, The results in this figure bring out the role of strain gradients by normalizing H at a given / / a by the corresponding hardness prediction for the conventional solid (with / = 0). There is an appreciable dependence on the assumed contact condition. The size dependence for the frictionless indenter is rather small, while that for the indenter with no sliding is somewhat more substantial. Indentation is not unlike void growth

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Strain Gradient Plasticity

or cavitation in that the indenter forces a ‘radial’ outward expansion of the material. The contribution to the deformation field from this outward expansion produces relatively small rotations. Thus, as in the case of the other two phenomena, indentation is not expected to give rise to a very large strain gradient effect for the CS solid. The sticking indenter induces more shearing and rotation than the frictionless indenter, and this is the qualitative explanation for the difference between the two cases. It would appear that the indentation hardness test is another instance for which a hardness dependence on stretch gradients will greatly influence the predicted size effect. Work is currently underway to calculate these effects for indenters, conical and flat-headed, forced into the SG solid. The expectation is that the size effects for this solid will be considerably larger than those evident in Figure 11. 7.

The Stress Field in the Vicinity of a Sharp Crack Tip

Attempts to link macroscopic fracture behavior to atomistic fracture processes in ductile metals are frustrated by the inability of conventional plasticity theories to adequately model stress-strain behavior at the small scales required in crack-tip models. This does not appear to be an issue for metals whose fracture process is void growth and coalescence since the process zone is usually measured in tens or even hundreds of microns. It is 2.2 I

2.0 1.8

I

I

I

I

/.,/:

/

-

=0

1.6 -

. -.---

{/a FIG.11. The normalized indentation hardness H / [ H ] , , for a flat-ended circular punch. Results are presented for both the frictionless and sticking cases, for a range of values of strain hardening exponent n.

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N. A. Fleck and J. W.Hutchinson

a major issue when the fracture process is atomic separation. Conventional plasticity theory is unable to explain how stresses at a sharp crack tip can reach levels necessary to bring about atomic decohesion at the tip of a sharp crack where the relevant scale is far below the micron level. The high-strain gradients invariably present near the crack tip in an elasticplastic solid suggest that there should be an annular zone surrounding the tip within which geometrically necessary dislocations play a role in elevating the local hardening and, therefore, the stress levels near the tip. A discussion of a number of the open issues surrounding the goal of bridging from the macroscopic level where loads are applied to the crack tip where the fracture process occurs is given in the article by Bagchi and Evans (1996). Some progress has been made in applying strain-gradient plasticity theory to the estimation of crack tip fields by Huang et al. (19951, Xia and Hutchinson (19961, and Schiermeier and Hutchinson (1996). Thus far, only the CS solid has been considered in these studies. We begin the discussion in this section by first considering the more general elastic-plastic deformation theory SG solid depending on both rotation and stretch gradients, such as the power-law solid in (2.21) where the generalized strain quantity 8 is defined by (2.19). For plane strain and mode I11 crack problems, a J-integral exists for this material which equals the energy release rate of the crack when evaluated on a contour circling the crack tip in the usual manner. Application of the J-integral to the crack-tip problem implies that the energy density w must have a r-l singularity, where r is the distance from the tip. This same conclusion is reached for the conventional solid, leading directly to the general form of the HRR crack tip fields for a power-law material. The form of the singular fields is different for the general strain gradient solid. For the general solid, the strain gradients q will dominate the strains E as the crack tip is approached. Since 8 must have a r - n / ( n singularity for w + r - ' , it follows that q must also have a r - n / ( n + l )singularity. The dominant quantities in the crack tip singularity field can be written in a form analogous to that for the HRR fields for the conventional power-law solid: +

(2.41)

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329

where C , is a normalizing constant and (r, 0) are planar polar coordinates centered at the tip. The @variations, +jijk and ? j j k , depend on the choice of normalization and on which of the modes, I, I1 or 111, pertains. They also depend on the particular combination of the material length parameters, , invoked for the material model (e.g., as in (2.20)). In the general case, the stress and strains are less singular than the quantities in (2.41). More importantly, however, is the fact that the higher order stresses T~~~ contribute to the tractions in proportion to their gradient-see ahead to eq. (3.8) in Section 111. Thus, tractions t acting on material surfaces will be strongly singular according to t + r-(n'2 ) / ( n + I). It follows that the tractions near the tip of a sharp crack in the general strain gradient solid will be significantly higher than the tractions near the crack tip in the conventional solid.

/,

The mode ZIZ crack The analytical features outlined above have been detailed for the mode I11 crack (Schiermeier and Hutchinson, 1996). For the antiplane shearing deformations of mode I11 no stretch gradients occur. Thus, the general solid reduces to a solid which depends only on rotation gradients in anti-plane shear, and, as a consequence, the problem fits within the framework of couple stress theory. The mode I11 crack study investigated the influence of the two rotation-gradient invariants in (2.19), x,',,~;,and x,!~x$,with the finding that the x;jx$ invariant plays a secondary role. Thus, with little loss in generality, the mode I11 crack in the general solid can be replaced by the problem for the crack in the CS solid, similar to the exact reduction which applies for wire torsion. The full details of the singular crack-tip fields in (2.41) have been worked out for the mode I11 problem. Numerical results have been obtained using a finite element scheme for the full field, merging the singular fields with the outer HRR field whose amplitude is specified as J . (For the pure power material, the HRR field is a solution to the field equations at large r.) The transition from the outer HRR solution to the inner singular field (2.41) occurs smoothly over the annular region centered at the tip roughly equal to 1 T / < r < 5f.This transition is illustrated by the behavior of the shearing displacement of one face of the crack relative to the other, 6, displayed in Figure 12. The normalization of 6 on the ordinate is a consequence of the choice r// as the abscissa. The three curves shown are the HRR solution, the asymptotic solution from the singular field (2.40, and the full finiteelement solution to the problem. The r-dependence of 6 associated with (2.41) is 6 -+ r(n+2)/(nt'),and it can be seen that the full numerical-

N. A. Fleck and J. W; Hutchinson

330 10 8

2

0

0

0.5

1

2

1.5

2.5

r/!

FIG.12. The shear displacement profile near the tip of a mode I11 crack in the CS-solid.

solution asymptotes to the near-tip solution. The full solution also asymptotes to the HRR solution once r / / exceeds about 5 .

Mode Z and ZZ crack solutions Although no results have yet been obtained for mode I or I1 cracks in the general strain gradient (SG) solid, it is expected that many of the qualitative features of the solution described for the mode I11 crack will apply to the plane strain-crack problems as well. We note in passing that for the general case of plane-strain deformation, such as the mode I or I1 crack-tip fields, the combined strain quantity, 8, and hence the strain density, w , can be written solely in terms of the invariants &ij&ij, T$$)T$;) and x:j x;j since the invariant ,y:j xii vanishes identically. Thus, the most general material in this class is represented by (2.19) with only two length parameters, k‘, and a second parameter proportional to (4/,2/3 8 f , 2 / 5 ) ’ / 2 . Quite another result from that described above has been obtained for plane-strain cracks in the CS solid, whose strain-gradient contribution to hardening, one should recall, depends only on the rotation gradients. Unexpectedly, it turns out that the singular field for the plane-strain problems is irrotational (Huang et al., 1991; Xia and Hutchinson, 1996). Consequently, rotation gradients vanish in the dominant singular field, as does the invariant xi;.xi;.. The argument leading to (2.41) no longer applies for the plane-strain crack in the CS solid, since the strain-gradient contribution to 8 does not dominate the contribution from E, as r + 0. (This possibility is excluded in the mode I11 problem because no irrotational singular field exists.) The outcome of the full analysis of the plane-strain crack problems for the CS solid is that E, is dominant in the singular field and the requirement that w -+ r - l leads

+

Strain Gradient Plasticity

331

to a singular field with E + r - n ') and u + r-l/(n+l).This is the same r-dependence for the stresses and strains in the HRR field. Nevertheless, the two fields are not the same. The HRR field is not irrotational; moreover, the gradients of 7 in the CS crack problem are of the same order as u,and therefore make their presence felt in the singular field. As in the case of the mode I11 problem, the plane-strain HRR solution satisfies the field equations for the power law CS solid as r + m. Given the observations above, the general form of the solution to the full problem for a semi-infinite plane-strain crack in the CS solid with the HRR field imposed as r + 03 can be written as follows (Xia and Hutchinson, 1996):

Here, m is the couple stress tensor whose components represent a subset of the components of T as specified in Section 111. The quantities Eij and Gij approach the corresponding HRR quantities as r / / 4 03 (if the normalizing factor I,, is the same as that for the HRR fields) and approach those associated with the singular field as r//0. The quantities gij and k i j vanish for both r//+ 0 and 03, but are non zero in the intermediate region connecting the remote field to the crack-tip singular field. Plots of Gv and Goo are given in Figure 13 for the mode I crack with n = 5. The transition from the HRR solution to the dominate singularity again occurs smoothly over the range from r//= 5 to about 1/5. Surprisingly, the normal stress acting in the plane ahead of the crack tip, aoo(8= 0) is hardly affected by the strain gradient effects in the CS solid. It is anticipated that this is one aspect that will change for a solid whose hardening depends on stretch gradients. A better understanding of stress elevation required to produce decohesion will require the investigation of the stress fields for the mode I crack for materials whose hardening depends on both the stretch gradients and rotation gradients. As discussed above, the nature of the crack-tip singularity will change to the form given in (2.41) when stretch gradients are included. It seems likely that the more general theory will result in a significant elevation of tractions ahead of a sharp crack, but that remains an open question.

N. A. Fleck and J. W.Hutchinson

332

2,

I

I

1

mode I , n = 5

1.s

1

L 0

I

I

I

L5

90

135

180

0("1 (a)

FIG.13. Non-dimensional stress field around the tip of a mode I crack for the CS solid, at selected values of radius r from the tip. The non-dimensional stresses are defined in (2.41).

Strain Gradient Plasticity

333

8. Implicationsfor Further Development of the Theory

Among the examples discussed above are several for which stretch gradients are absent, such as wire torsion and the Mode I11 crack, and several for which rotation gradients are either absent or relatively unimportant, such as cavitation, void growth and, possibly, indentation. In the other examples, both types of gradients are present and have the potential for contributing to size effects due to strain gradient hardening. The two sets of examples which are dominated by one type of gradient over the other may provide the best means for the experimental determination of the material-length parameters governing the two types of hardening in the theory. As has been emphasized in this survey, the objective for developing a strain gradient theory of plasticity is the extension of continuum plasticity to the micron and, possibly, sub-micron range for application to phenomena involving large numbers of dislocations. There is ample evidence that conventional plasticity theories fail to capture the strong size effects which become important at these small scales. Plasticity effects are important in many applications of thin films and multilayers. The thickness of films and layers in some of the applications fall within the scale for which the present theories are intended. Shear lag in thin metal films and layers at edges and corners where they attach to substrates gives rise to significant shear and rotation gradients. Recent experiments on thin film/substrate systems which are cycled thermally to produce plastic deformation display a large film-thickness dependence on yielding and continued plastic flow (private communication from A. G. Evans, 1996). Such effects are well known for very thin epitaxial films where dislocation nucleation is the controlling factor. However, in these experiments the films are polycrystalline and contain large numbers of dislocations, making them candidates for the present class of theories.

111. The Framework for Strain Gradient Theory

A. TOUPIN-MINDLIN THEORY Toupin (1962) and Mindlin (1964, 1965) have developed a theory of linear elasticity whereby the strain energy density w per unit volume depends upon both the symmetric strain tensor cii = i ( u i , + u,, i ) and

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-

the second gradient of displacement 7)ijk = did,u, = uk , i j , where di is the forward gradient operator. Their theory furnishes stress quantities which are work conjugate to the generalized strain variables E and q, and also provides a principle of virtual work. The work increment per unit volume of solid due to an arbitrary variation of displacement u is given by (2.4) where the symmetric Cauchy stress crij is the work conjugate to the strain variation Scij and the higher order stress measure Ti,k ( = T j i k ) is the work conjugate to the strain gradient variation 87)ijk. For the special case of a deformation theory solid we may write w = w ( c i j ,?l;,k), giving uij= d w / d & i , and ri,k = d w / d q , , k . Following the strategy of Toupin and Mindlin, the work increment for a volume, V , written

or, via the divergence theorem, as

+

/,[

nirijk( ' j " k ) ]

dS

(3.2)

aj

where ni is the unit normal to the surface S of the body. The term auk appears in the last integral on the right-hand side of (3.2). We note that dj auk is not independent of auk on the surface s because, if 6uk is known on S, so is the surface-gradient of Su,. In order to correctly identify the independent-boundary conditions in a variational principle we resolve the gradient dj auk into a surface-gradient, oj auk, and a normal gradient, n j D auk,

where the surface-gradient operator Dj is

(3.4)

Strain Gradient Plasticity

335

and the normal gradient operator D is D

-

= nkdk

(3.5)

To proceed, we substitute (3.3-3.5) into (3.2) and make use of Stokes's surface divergence theory (see for example, Mindlin, 1965) to get the final form for the principle of virtual work

(3.7) The surface traction tk on the surface S of the body is

and the double-stress traction rk on S is rk = ninjrijk

(3.9)

We conclude from (3.6) that the displacement field u must satisfy three equilibrium equations given by relation (3.7) and six boundary conditions given by (3.8) and (3.9). For the special case where the surface S has edges, an additional term must be added to the right-hand side of (3.6). Suppose S has an edge C, formed by the intersection of two smooth surface segments S") and S(') of S. The unit normal to segment S(') is designated n(l) and the unit normal to the segment S(') is designated d2). The unit tangent c(') along the edge C is defined with segment S ( ' ) to the left, and the unit tangent c(') is defined with segment S"' to the left. We define the unit outward normal to C and lying within the surface S(l) by k(') = cC1) X dl);similarly, the unit outward normal to C and lying within S(2)is written as k") = c(') X d2).Then, the additional term to be added to the right-hand side of (3.6) upon application of the surface divergence theorem to (3.4) is

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N. A. Fleck and J. W. Hutchinson

where dl)is the arc length along the edge C in the direction dl).The line load Pk is given by

In the more general case, the piece-wise smooth surface S can be divided into a finite number of smooth parts, S,, each bounded by an edge, C, . The line-integral contribution (3.10) becomes (3.12)

and the line load p k along any sharp edges C,, is given by the jump in value A of ( n i k j T i j kon ) each side of C,,: Pk

=

A( nikjri

(3.13)

The complete virtual work expression becomes

(3.14)

Global equilibrium of forces dictates that the net force on the surface of the body vanishes. In order to obtain an explicit expression for the net force on the body we impose a uniform virtual displacement Suq . The associated internal work in (3.2) vanishes and the external work is given by (3.15)

Since this expression vanishes for all Suq we conclude that the net force on the body, given by j s [ n i ( a j kj)]dS, vanishes. A similar argument is used to obtain an expression for the net moment on the body. For a body in equilibrium, the net moment on it vanishes, implying that the work done through an incremental displacement field 6u = 68" x x vanishes for an arbitrary uniform rotation SO". The internal virtual work vanishes

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Strain Gradient Plasticity

since a uniform rotation induces vanishing strains and vanishing strain gradient. The external work follows from (3.2) as S w E X T= 68". /,[x

x n -(cr

-

2.7)+

3

where e is the alternating tensor. Thus, the vanishing net moment on the body is the vector quantity working through 68" in the equation above. B. CONNECTION WITH COUPLESTRESSTHEORY

It is instructive to rearrange the principle of virtual work into a form which separates out the work terms associated with couple stresses and those associated with double forces per unit area. This decomposition leads to a transparent reduction of the general framework to couple stress theory when the constitutive behavior depends only on the rotation gradients. We shall consider first the general case of a compressive solid, and then specialize the results later for the incompressible limit. Following the development of Smyshlyaev and Fleck (1996) the strain gradient tensor q is partitioned into a symmetric tensor qs representing stretch gradients, and a curvature measure q A : q

=

qs

+

(3.17)

q A

In parallel with the definitions (2.10) and (2.11) for the deviatoric tensor q', we adopt the definitions (3.18) and qlyk

%Jk - q;!i

2

+

Telkp x p ~

2 Fe]kpXpr

(3.19)

and note that q s and q A are orthogonal to one another. The q A - x relation may be inverted to give

x '.I . = -21e .

qP

14'

(3.20)

and so q A is a useful third-order tensorial representation of the secondorder curvature tensor. Each has eight independent components, since Xii = 0.

N. A. Fleck and J. u! Hutchinson

338

A similar decomposition applies to the higher-order stress T into a symmetric tensor r s defined by 1 ‘Tik -(7ijk + 7jki + r k j j ) 3 and a remainder such that

r . We

can split

(3.21)

(3.22)

+ 7.4 rjk

7i.l k. = 7. i l5 k:

On noting that r s is orthogonal to an arbitrary q A and that T i $ is orthogonal to an arbitrary 77’ the work increment 6 w can be written simply as

6w

= (+.. rj

a&.. + 7i.l k. ij

677.. ilk

= (+.. rj

a&..+ 7si l k i]

677s ilk

+ 7i.7. k 6 7i7l4k

(3‘23)

&&

The work term T;$ represents the work done by the couple stresses m acting through the curvature increment 6 x, (3.24) 674 rjk = m.. j r ax. ij On substituting (3.19) into (3.24) we obtain an explicit expression for m in terms of 4 4 mjP = -3e .1kP 7.4 = (3.25) Ilk 3e .1kP 7I .l k. 7 i. lAk

and similarly, substitution of (3.20) into (3.24) gives the inversion formula 1 1 (3.26) 7i qPr = -4e .w m11. . + -4e . 11‘. m 41 The higher-order stress terms T appearing in the reduced form of the principle of virtual work (3.14) can now be re-expressed in terms of couple stresses m and the symmetric tensor 7’. The derivation is by successive application of Stoke’s surface divergence theorem and follows that laid down by Toupin (1962) and Mindlin (1964). Here, we quote only the result: uij 8 E j j

+ mji axjj + r;k [qk

aqi$k]

dV=/[fk

Suk]dV+

V

$

6 8 k ] d s + / [ ? 6 & f q ] d S+ S

n

/ r i k S

[Pk

8Uk]ds

6 u k ] d s (3.27)

C,

where the body forces fk are in equilibrium with stresses within the volume V according to the relation (3.71, which may be rephrased as 1 (3.28) ( + j k , i - -2e .I l k m 1.1., i.I - 7.5: r l k , i.l. + f k = 0

Strain Gradient Plasticity

The reduced surface traction satisfy

tk

339

has three independent components which

+ nknpTpSkl(Dlnl)+ nknpnsnlrpSks(DJnI)

(3.29)

while the two reduced torque tractions ijk are tangential to the surface of the body and are given by

qk = nlmrk+ 2 n , n , n p e k p q ~-, ~nqk n p ( n r m l -p+

2n,nJnqe,qr~l~ (3.30) r)

The single double force 7 is the work conjugate to the normal strain increment _= n,n, 8 q I and is related to 7' via 7

The line force satisfies

fjk

fjk =

=

n,n,n,.rJk

(3.31)

along the edges C , of the piece-wise smooth surfaces S,

[

nikj'tk

+ ninjnkkq'Jq

+

1 yninpnqkreqrkmrp

]

(3.32)

The version (3.2743.32) of the principle of virtual work makes explicit the contribution to internal and external work from an increment in strain, an increment in rotation gradient (curvature) and an increment in stretch gradient. Following the argument of Koiter (19641, the spherical part of the couple stress tensor rn enters neither the virtual work expression for the body nor the constitutive law (since x is a deviatoric tensor), and without loss of generality we may take mrr= 0. Now consider the limit when the strain energy density w depends only on strain and rotation gradients and has no dependence on qs. The double forces per unit area 7' vanish, the term / s [ 7 S ~ N l d in S relation (3.27) disappears and the virtual-work expression reduces to that of couple-stress theory as given by (B15-Bl8) in Fleck et al. (1994) C . THE INCOMPRESSIBLE LIMIT

In the incompressible limit, the mean stress viiand the mean higherorder stress tensor T~ do not appear in the constitutive law. We shall show that a consequence of the kinematic constraint imposed by incom-

N. A. Fleck and J. W.Hutchinson

340

pressibility is that the number of independent boundary conditions is reduced by one. Further, we may take T~ to vanish identically without affecting any of the terms in the virtual work principle or constitutive law. The arguments are as follows. Consider again the reduced form of the principle of virtual work for the compressible solid, as given by relation (3.13). Incompressibility places a kinematic restriction on the normal component of D Sui at the surface S of the body: n j D S u j = -D,Suj (3.33) since

-

dj

Suj = Dj Suj + n j D Suj = 0

(3.34)

In other words, given a distribution of displacement increment Su, on the surface of the body, the normal components D S u j are constrained by (3.331, and only the two tangential components of D Suj remain arbitrary. Consequently, only the two tangential components of the double-stress traction rk can be specified as independent external tractions on the body. Somewhat lengthy manipulation of (3.14) and making use of (3.34) gives [Ui>S&,!j+

r ~ j k S 7 ) : l k ] d V = j [ f k S U k ] d T / +/ [ ( i k + H n k ) S u k ] d S V

S

(3.35) where H is a combined measure of the hydrostatic stress according to 1 H G - ukk

1

(3.36)

- 2Tjkk, j

The body force f k is in equilibrium with the deviatoric stresses ( a ‘ , ~ ’ ) and the distribution of H according to + uik,~ -

-k

H,k

The three independent surface tractions incompressible body are

t;,

fk

ik

=

nj(u:k - T1,k.j ’ )

T;jk,ij

=

(3.37)

on the surface S of the

+ Dk(n,njn,T;j,) - Dj(niTijk)

+ [ ninjr;jk - n , ( n i n i n p ~ i ; . , ) ] ( D q n q )

(3.38)

Strain Gradient Plasticity

341

and the two independent double-stress tractions ?k tangential to S are ?k

=

njnjrjjk- n 1. nI . nP n k 7.’. 11P

(3.39)

We emphasize that the constraint of material incompressibility has reduced the number of independent boundary conditions associated with the displacement gradient from three to two. The line load h k in (3.35) is given by (3.40) Note that the hydrostatic stress + a k k and the hydrostatic higher stress r H enters the virtual work statement (3.35) only via the term in H appearing in the surface traction term of (3.35) and in the equilibrium statement (3.37). The relative magnitude of f a k k and T~ is arbitrary: only the combination H as defined in (3.36) is known. Therefore, we may simplify the virtual work statement by taking T~ to vanish identically and put 7 = 7’. The number of independent boundary conditions in the alternative statement of virtual work (3.27) may similarly be reduced by one upon enforcing incompressibility. The incompressibility condition (3.33) implies that normal strain increment is not an independent kinematic quantity: it can be re-expressed in terms of direct strains tangential to the surface S. Thus, the term involving can be eliminated from (3.27) by making use of the identity

(3.41)

IV. Flow Theory A flow theory version of strain gradient plasticity is now outlined, based on the physical argument that the current strength of the solid is dependent upon the accumulated strain and strain gradient. The resulting formulation results in a higher-order set of differential equations than conventional plasticity theory, with an associated increase in the number of boundary conditions. Consequently, the theory predicts the existence of

342

N. A. Fleck and J. W. Hutchinson

boundary layers of deformation near to stiff interfaces. An alternative strategy adopted by Bassani and co-workers (Acharya and Bassani, 1995) is to assume that the current tangent-hardening modulus is increased by the presence of accumulated strain gradients. The resulting formulation in rate form is then only a slight modification of conventional theory, with no additional boundary conditions, and no boundary layers near to interfaces. Additional experiments are required in order to determine which strategy is the more accurate. We believe the flow theory outlined below has a firm physical basis, and is consistent with the observed physical phenomena described in Section 11. In this section we first review conventional J, flow theory for an elastic-plastic solid. A strain gradient version of J , flow theory is then proposed, which is the complement to the deformation theory defined above. Stability and minimum principles follows in a straightforward fashion. Briefly, the strain gradient version of J2 flow theory is generated by the following prescription. In the absence of higher-order stresses T , the deviatoric, symmetric Cauchy stress u’ may be represented by a fivedimensional vector. When higher-order stresses are present the role of u r is replaced by that of the 23-dimensional vector I: = (u’, 7 ’ ) ; I: is made up of the five symmetric components of u ’, the eight independent components of couple stress T ’ and ~ the ten independent components of double force per unit area T”. In like manner, the deviatoric strain tensor E ’ is replaced by the 23-dimensional vector B = (E’, qr).

Review of conventional J, flow theory In conventional J , flow theory, higher-order stresses are absent and the strain tensor E is decomposed additively into an elastic part eel and a plastic part &PI. The elastic strain is related to the Cauchy stress u via the linear relation &el r ) =&.. rjkl

kl

(4.la)

where

Here, E is Young’s modulus and v is Poisson’s ratio. The plasticity relations of conventional J2 flow theory provide a connection between the plastic strain rate E P ’ and the stress rate 6.:the plastic strain &PI is determined by integration of E P ’ with respect to time. In

Strain Gradient Plasticity

J2 theory, written as

CP1

343

is taken to be incompressible and the yield surface CP is CP.(a,Y)= a, - Y = 0

(4.2)

,/%,

and Y is the where ue is the von Mises effective stress, a, = current flow stress. For a hardening solid, the material response is plastic when CP = 0 and 4 > 0; and the response is elastic when CP < 0, or CP = 0 and be I0. The plastic strain rate E P 1 is assumed to be linear in the stress rate u , and to lie normal to the current yield surface, giving (4.3)

where the hardening rate h is chosen such that the uniaxial tensile response is reproduced. This dictates that h equals the tangent modulus of the stress versus plastic strain curve in simple tension. The work rate U per unit volume of the elastic-plastic body is U=

(T11

&11 = 0-11 &el 11

+ s 11 &PI 'I

(4.4)

and so U may be partitioned into an elastic part Uel = u l 1 i ~ and ' a plastic part U p ' = s,,&,~l. Substitution of (4.3) into U P ' = sl,.4t' gives UP1 = a,ci,/h which may be re-written as U P 1 = a,&$ where the effective plastic strain by direct evaluation, rate '!i = & / h . Observe that 2:' = making use of (4.3).

d m

Flow theory for strain gradient solid Now assume the existence of higher order stresses in the elastic-plastic body. We define the elastic strain energy density we' for a purely elastic response of the isotropic, compressible solid by 2

U

we'= E ( 2(1 + v ) ( l

-

2v)

(€;:)

+

1

2(1

+ v)

E elg 'l

e!

I (4.5)

where the five elastic length scales L, have no physical significance but are introduced in order to partition the strain gradient tensor q into an elastic part and a plastic part 77

= $1

+

$1.

(4.6)

344

N. A. Fleck and J. W Hutchinson

It should be noted that the elastic part of the strain gradients cannot be expressed as gradients of the elastic strains alone when plastic strains occur. A sensible strategy is to take L ,

k'p'

(4.14)

by (4.15)

When the solid is subjected to a uniaxial tensile stress u , C = u and yield occurs when u = Y by (4.10); we interpret Y as the uniaxial flow stress. Equation (4.10) is a natural generalization of (4.2) once it is assumed that u is replaced by C = (a 7 ' ) and u, by C in the higher-order version of the theory. Plastic straining is assumed to be normal to the yield surface and the plastic-strain rate is taken to be linear in the stress rate: (4.3) then generalizes to (4.16) where &?Pi = ( & P ' , + l P ' ) has already been defined. In the case of uniaxial tension, where the axial stress is u and the plastic strain is &PI, we find 'c = (+ and (4.16) reduces to (4.17) Thus, the interpretation of h remains the same as for conventional J2 flow theory.

346

A? A. Fleck and .I. W; Hutchinson

The plastic work rate U P ' is, via (2.41, (4.18) into (4.18) we get L j p l = On substitution of the expression (4.16) for &I' Z % / h which may be rewritten as O P l = XkP' where the overall effective plastic strain rate kp' = %/h.

The main constitutive relations in the strain gradient formulation are now summarized in index notation. Plastic flow is normal to the yield surface such that

and (4.19b) by (4.11) and (4.16). The rate of overall effective stress rate form of (4.10,

% is given by the (4.194

The elastic strain rates follow from (4.1) and (4.8) as (4.20) and (4.21) where K = f - ' . Note that the current formulation predicts that higherorder stresses remain present in the case of a purely elastic response with vanishing plastic strains. This is for purely mathematical convenience, and is given no physical significance. Indeed, the magnitude of the elastic higher-order stresses may be made arbitrarily small by choosing the ratio to be sufficiently small. In some cases it may even be convenient to

347

Strain Gradient Plasticity

take L , / f , to be on the order of unity (e.g. the cavitation problem in Section II.C.5). If the strain gradients are small in the elastic range, the choice of the L, will hardly matter. In the strain gradient versions above of J , deformation theory and J2 flow theory, proportional loading occurs at a material point when all stress components of ( u , ~increase ) in fixed proportion. That is, with ( a ' , ~ ' ) fixed and A as a monotonically increasing scalar quantity, the components (a,T ) = Moo,T O ) increase proportionally. When proportional loading is experienced at a material point, it is readily shown that the predictions of the strain gradient versions above of J , deformation theory and J, flow theory coincide.

B. MINIMUMPRINCIPLES The yield surface (4.10) is convex in the stress space ( u , T ) and the plastic strain rate is normal to the yield surface. Hence the strain gradient version of J , flow theory (with h > 0) satisfies the generalized Drucker's stability postulates (Drucker, 1951) &..&.PI I] I]

+ +..r ] k ?.tPI] k - > 0

(4.22a)

for a stress rate ( u ,i) corresponding to a plastic strain rate ((+" ll -

+ (Tijk

~ ' I* ) & 11~ p l

-

T$k)+$L

20

(€PI, GP'),

and

(4.22b)

for a stress state (u, T ) associated with a plastic strain rate (&PI, q p l ) , and any other stress state (u*, T * ) on or within the yield surface. Minimum principles are now given for the displacement rate and for the stress rate, for the strain gradient version of J , flow theory. These minimum principles follow directly from those outlined by Koiter (1960) for phenomenological plasticity theories with multiple yield functions, and from the minimum principles given in more general form by Hill (1966) for a metal crystal deforming in multislip. The presence of higher-order stresses can be included simply by replacing u' by Z and kP' by kP1, as outlined above. Consider a body of volume V and smooth surface S comprised of an elastic-plastic solid which obeys the strain gradient version of 5, flow theory (4.19-4.21). The body is loaded by the instantaneous stress traction rate ip and double stress traction rate 3; on a portion S, of the surface

348

N. A. Fleck and J. W . Hutchinson

(see relations (3.6-3.9) for the definition of stress tractions and double stress tractions). The velocity is prescribed as Ihp and the normal velocity gradient is prescribed as Dzip on the remaining portion S, of the surface. Then the following minimum principles may be stated. 1. Minimum Principle for the Displacement Rate Consider all admissible velocity fields u j which satisfy ui = up and Dlij = Drip on S,. Let k j j = + u j , i )and q i j k = u k , j j be the state of strain rate derived from u j , and define (u,i>to be the stress rate field associated with (E, q) via the constitutive law for the strain-gradient version of J , flow theory (4.19-4.21) with a hardening modulus h > 0. Then, the functional F(u), defined by

is minimized by the exact solution (u,E, q, u, i) The .exact solution is unique since the minimum is absolute. 2. Minimum Principle for the Stress Rate Consider instead all admissible equilibrium stress rate fields (u,i) which satisfy the traction boundary conditions ii = iP and ij = i: on S,. Let rip and Dlip be prescribed on the remaining portion S,, and define (E,$ to be the state of strain rate associated with the stress rate ( u , i) via the constitutive law (4.19-4.21) with h > 0. Then, the functional H(u,i), defined by,

+,

is minimized by the exact solution (h, E, u, 7 ) . Uniqueness follows directly from the statement that the minimum is absolute. The proofs of the minimum principles for the displacement rate and stress rate require three fundamental inequalities, which are the direct extensions of those given by Koiter (1960) and Hill (19661, and are stated here without proof. Assume that at each material point a stress state (a,7 ) is known; the material may, or may not, be at yield. Let (E, be associated with any assumed ( u ,i) via the constitutive law (4.19-4.21).

Strain Gradient Plasticity

349

and

(&*a,* + ,$..&. - 2L...k.* + +* +* + 77,Jkiijk - 2+jk+;k) 'J 11 11 ' J ) ( Ilk I l k 11

1J

2 0 (4.25~)

The equality sign holds in the three expressions above if and only if u* - b and i* = i.

V.

Single-Crystal Plasticity Theory

We shall use the notions of statistically stored dislocations and geometrically necessary dislocations to provide the physical basis for continuum theory of single-crystal plasticity. Slip is assumed to occur on specific slip systems in a continuous manner. The increment in flow strength of any given slip system depends upon the rates of both the strain and the first spatial gradient of strain. The crystal theory fits within the framework of Toupin-Mindlin strain gradient theory described in Sections 3 and 4.It will be seen that the theory leads quite naturally to a dependence on gradients of rotation and stretch. Attention is restricted to the class of theories usually referred to as small strain theory for which it is tacitly assumed that stresses are small compared to the incremental moduli as well as small strains.

The relationship between plastic strain gradient and dislocation density in single crystals has been explored by Nye (1953) and Kroner (1958, 1961, 1962). We summarize the theory for the case of small deformations. In order to calculate the density of geometrically necessary dislocations, a crystal lattice is embedded within the solid. We assume that the material shears through the crystal lattice by dislocation motion, and that the lattice (and attached material) undergoes rotation and elastic stretching as shown

350

N. A. Fleck and J. W. Hutchinson

duf --L

lattice planes

elastic stretching and rotation of lattice

plastic slip

FIG. 14. The elastic-plastic deformation of a single crystal. Here, dus is illustrated for single slip which in the text duf is defined for the general case of multi-slip.

in Figure 14. Consider the relative displacement du, of two material points separated by dxj, in a Cartesian reference frame. The relative displacement dui is decomposed into a displacement duf due to slip, a displacement dur due to rotation, and a displacement duf due to elastic stretching, duj = duf

+ duf + duf

(5.la)

akj

(5.ld)

where

and

duf

= c;'

Here, duf is linearly related to dxj via the slip tensor yij, dur is related to dxj via the lattice rotation tensor and duf is related to dxi via the elastic strain tensor E: . A particular slip system, a,is specified by the vectors (d"),m("))where s(") is the slip direction and m(")is the slip plane normal. The slip tensor yij is associated with an amount of slip y ( " ) on each of the slip systems, hence

+;,

where the summation is taken over all slip systems. The total strain cij = i ( u i , j+ u ~ ,at~ a)material point equals the sum of the elastic strain

351

Strain Gradient Plasticity E;' and the plastic strain that

E$

, where

E{'

is the symmetric part of yij such

where

The density of geometrically necessary dislocations is related to the net Burger's vector Bi associated with crystallographic slip. Make an imaginary cut in the crystal in order to produce a surface S of outward normal n. Define Bi as the resulting displacement discontinuity due to slip on completion of a Burger's circuit around the periphery r of the surface S. In other words, Bi completes the circuit when r is traversed in the sense of a right-handed screw motion along n. Thus, Bi is

Bj =

#r dus

=

(5.4)

y i j dxj

which may be rewritten using Stokes's theorem as

Bi=

1crinnndS

(5.5)

'nkjyij, k

(5.6)

S

where =

In (5.6) enkjdenotes the alternative tensor. The tensor a is Nye's dislocution density tensor or torsion-flexure tensor. It gives a direct measure of the number of geometrically necessary dislocations. Kroner's OL tensor, here labelled a K ,is related to Nye's tensor a by a K = -aT,where the superscript T denotes the transpose. Nye (1953) has related 01 to the distribution of individual dislocations within a crystal as follows. Suppose there exist dislocations parallel to the unit vector r with Burger's vector b. Let there be N of these dislocations crossing unit area normal to r. The number crossing a unit area normal to the unit vector n is Nn r, and the associated Burger's vector is N(n * r)b. Hence in suffix notation Bi = Nnjrjbi and from (5.5),

-

~ y . .=

'I

Nb.r. ' I

(5.7)

352

N. A. Fleck and J. W. Hutchinson

If there are other sets of dislocations present with different values of N, b, and r, then the total crjj is obtained by summing the values of Nbjrj from each set. It must be emphasized that, if the distribution of dislocations is continuous, the net displacement discontinuity,

vanishes along any closed path within the material, because the incompatibility in displacement

due to slip is exactly matched by an equal and opposite displacement mismatch

as demanded by (5.la). It is clear from (5.la) and (5.5-5.7) that the density of geometrically necessary dislocations is defined unambiguously only when a crystal structure is embedded in the material. We treat aij as the fundamental measure of the total density of geometrically necessary dislocations. An alternative version of the expression (5.6) for aij may be derived through introduction of the unit vector t = s X m.Note that (s, m,t) forms a right-handed triad with t in the slip plane and orthogonal to s. Substitution of the relation m = t X s into (5.2) gives via (5.6) aYin =

c s~a)r(,.)(Sp)ty

- t k( a ’sn ) (a))

(5.8)

a

The dependence of a upon the slip gradient in the slip direction s, and in the transverse direction t is made explicit by (5.8). (A slip gradient in the slip direction s is associated with edge dislocations lying along the t direction with Burger’s vector parallel to the s direction. Similarly, a slip gradient in the t direction is associated with screw dislocations lying along the s direction and Burger’s vector parallel to the s direction.) Note that a slip gradient in the m direction does not contribute to a, as shown explicitly by relation (5.8).

Strain Gradient Plasticity

353

, ~ ~to the slip tensor y i j , lattice The strain gradient 7 , k ; = u ~ is related rotation tensor Cpij and elastic strain tensor c'.; by differentiating (5.1) with respect to x k to give q1k1 . . = u1 ,.i k. = y .11.k .

+ 4 11.k ..

+&;!k

(5.9)

This expression for q can be simplified by using compatibility to eliminate the term c$jj,k as follows. First, the lattice rotation tensor 4ij is rewritten in term of the lattice rotation vector 8, as Cpjj = e k j i O k . By continuity of displacement e P j k q j k ; = e p j k u jik, vanishes, and thus (5.9) reduces to epjk3/ik,j

+ epjkeilkOl,j + e p j k E 2 , j

=

(5.10)

This relation may be inverted to give (5.11) and (5.9) can thereby be rewritten as

The strain gradient tensor q may now be decomposed additively into an elastic part qe'and a plastic part q p ' as q$

= &;,k - Ejekl,;

+ &el lk,I

(5.13a)

and

We note in passing that the lattice curvature Op,i is closely connected to Nye's measure of the total density of geometrically necessary dislocations, q j :upon making use of (5.6), the relation (5.11) can be re-expressed as (5.14)

The plastic strain gradient qSican be expressed in terms of the sum of the gradient slip over N slip planes as (5.15)

N. A. Fleck and J. W. Hutchinson

354

where the resolution tensor JI is given by

B.

STRESS

MEASURES FOR ACTIVATING SLIP

We employ a work statement in order to define the appropriate stress measures to activate yield on any slip system a. With the work conjugate of the slip rate +(") as and the work conjugate of the gradient of slip rate );+: as the plastic work rate per unit volume k p ' over the active slip systems is

QF),

(5.17) In terms of macroscopic quantities, the internal plastic work rate follows from (2.4) as

(5.18) where 2;' and i$ denote the plastic part of the strain rate iijand strain rate gradient respectively. On equating the two expressions above for the plastic work rate for arbitrary +(*) and +,(;), we obtain explicit expressions for d a )and via (5.3 and 5.19,

eijk,

(5.20)

We note from (5.19) that the work conjugate of the slip rate +("I is the resolved shear stress on the slip plane and in the slip direction. In similar fashion, the work conjugate of the gradient of slip rate +,(;Iis the resolved double force per unit area QF)on the slip plane. We adopt the notation that the slip gradient along the slip direction d a ) is +,(;), the slip gradient along the normal m(OL)to the slip plane (a)is +:,$, and the slip gradient along the transverse direction t ( a )= d o ) X mca)to the slip plane

Strain Gradient Plasticity

355

and

QP)= ( ti")m:.'J)sp)+

s (1U ) t ( " 1) m ( k U )

- s j a ) m $ , a ) t i a )) T j j k

(5.21~)

respectively, with no sum on LY in the expressions above. The notation of statistically stored and geometrically necessary dislocations suggests that the amount of strain hardening of a slip system is governed by the accumulated slip and the accumulated slip gradient on that slip system (and on all slip systems if latent hardening is taken into account). By analogy with the classical theory, we assume that yield of a given slip system (a) occurs when some combination of the resolved QP), Qg),and QP) attains a critical value. The stress-like quantities da), precise details of the formulation are not yet clear but are being pursued.

Appendix. Jt Deformation Theory and Associated Minimum Principles

We begin by writing the effective stress Z as the work conjugate of 8, with

With the particular choice for w ( 8 ) given by (2.21) note that I; is a power law function of 8:

Assume the solid is incompressible and can support Cauchy stress, couple stresses and double stresses. Then, the work done per unit volume equals the increment in strain energy,

356

N. A. Fleck and J. K Hutchinson

where the primes denote deviatoric measures. The work relation above and the definition (2.15) enables us to write the deviatoric stress (u', 7 ' ) in terms of the strain state of the solid as

and

where '(1) = 2 '(1) ' i j k - F'IVijk

(no sum on I)

(A5b)

Note that the three stress measures T,$' are the unique orthogonal decomposition of rijk,and they are the work conjugates to the three-strain gradient measures ~i$),giving

An explicit formula for the overall effective stress measure Z is derived by

substituting (A4-A5) into (2.13, viz

Minimum Principles Principles of minimum potential energy and minimum complementary energy can be written in a straightforward manner. The proofs are omitted as they follow immediately the development of Fleck and Hutchinson (1993) for the non-linear couple stress solid. Consider a body of volume V and surface S comprised of the non-linear strain gradient solid. In general the solid may be taken as compressible. A stress traction t z and a double-stress traction r{ act on a portion S, of the surface of the body. (Recall the relation between surface tractions and stresses given in (3.8) and (3.91.) Body forces are neglected in the current development. On the remaining portion S, of the surface the displacement is prescribed as u$ and the gradient of displacement normal to the surface is prescribed as Dug. Then, the following minimum principles may be stated.

Strain Gradient Plasticity

357

Principle of minimum potential energy Consider all admissible displacement fields u, which satisfy u , = up and Duk = Dui on a part of the boundary s,. Let E,] = $