On nonlinear elasticity theory for crystal defects

International Journal of Plasticity, Vol. 14, Nos. 1-3, pp. 9-24, 1998 © 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0749-6419/98 $19.00+ 0.00

Pergamon

PII: S0749-6419(97)00037-5

ON NONLINEAR ELASTICITY THEORY CRYSTAL DEFECTS

FOR

J. L. Ericksen *t 5378 Bucksin Bob Road, Florence, OR 97439, U.S.A.

(Received in final revisedform 8 June 1997) Abstract--In recent years, modifications of nonlinear elasticity theory for crystals have been used successfully to analyze phenomena associated with twinning and phase transitions involving change,; in crystal symmetry. Here, I attempt to assess how adapting the ideas to dislocation theory could bring in some mechanisms not present in linear dislocation theory, and why this might help us better understand plasticity phenomena. Conversely, twinning theory could benefit from adapting ideas used in dislocation theory. © 1998 Elsevier Science Ltd. All rights reserved

Key words: A. twinning, dislocations. I. INTRODUCTION

For some time, I have been interested in nonlinear theories of defects in materials, particularly in twinning phenomena in crystals and in defects observed in liquid crystals. It is gratifying to see the dramatic improvement in these which has occurred in recent years, although I don't claim much credit for this.* There are many unsolved problems associated with these. For some related to twinning, I believe that it will be useful to adapt ideas used in dislocation theory, for crystals subject to non-zero surface tractions, particularly. As to dislocation theory, I have been motivated to think about concepts related to this more by questions arising in discussions and correspondence with experts in this area than by personal interests. However, I have long believed that linear dislocation theory is inadequate to treat some kinds of phenomena which are observed in experiments related to plasticity. More recently, I concluded that some common ideas about basic concepts needed to be revised. This matters little for calculations made using linear theory. However, I don't believe that this is also true for nonlinear theory. From my experience with versions of nonlinear crystal elasticity theory now being used to analyze twinning and phase transitions, I see reasons to believe that nonlinear dislocation theory based on it would differ from linear dislocation theory, qualitatively, by introducing the following kinds of possibilities not contained in linear theory. (1) Predkiction of non-zero volume changes associated with dislocations. (2) Having dislocation solutions not of the Volterra or Somigliana kinds. *I dedicate this paper to the memory of my dear friend, James F. Bell. tCorresponding author. Fax: (541) 997 6399. *Work of this kind is discussed by Ball and James (1992), Bhattacharya and Kohn (1996) and Ericksen (1996) for crystals and Virga (1994), for liquid crystals, for example.

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J.L. Ericksen

(3) Occurrence of slow waves which might even come to rest. (4) Providing an alternative in elasticity theory to separate theories of Peierls-Nabarro forces. (5) Extending the theory to cover twinning. Theories of Peierls-Nabarro forces do patch some nonlinear theory to linear elasticity theory, using the idea that the periodic structure of crystals implies a periodic variation of forces. I see indications that all five of these items are likely to be of some importance in attaining a better understanding of some plasticity phenomena. My purpose is to elaborate this. In plasticity, Bell was primarily interested in what occurs at large deformations, in metals of the common f.c.c, kind. Well, they are not so large for well-annealed polycrystals crystals or single crystals, particularly when the latter are not of very high purity. If they are of high enough purity, one can get those stage I and II regimes where, in a uniaxial-stress experiment, plasticity occurs with a linear stress-strain curve. Bell was more interested in the behavior in stage III, where the stress-strain curve changes to parabolic form. In the regimes of most interest to him, applying loads causes a decrease in volume, which pretty well disappears when the loads are removed, in the kinds of experiments he did. At first, I was sceptical about out this, there being loose theoretical reasons to believe that the volume should sometimes increase, sometimes decrease, depending on the kinds of loads which are involved. So, I asked him various questions about the experiments, and did my own analysis of some of the data obtained in his laboratory for polycrystals and some obtained by Hsu (1969) for single crystals. This convinced me that Bell was right. After that, he produced much more evidence of this, both for polycrystals and single crystals, emphasizing metals of the face-centered kind.This is covered in various publications by Bell, e.g. (1973, Section 435; 1996a). It is already clear that, unlike linear theory, nonlinear theory can predict volume changes associated with dislocations. There is an old analysis by Toupin and Rivlin (1960), which produced a formula for the changes in volume occurring in an unloaded sample which has been subjected to cold working. As far as I know, Bell did not do experiments of this kind. The formula was obtained using slightly nonlinear theory, second order elasticity, with some basic assumptions which might reasonably be applied to theories which are more nonlinear. Wright (1982) collected available data on this, finding that the formula fit these surprisingly well. For Cu which had been subjected to larger loads, he noted some discrepancy, which is not very surprising, in view of the second-order approximation. Theoretically, the volume could increase or decrease, depending on material properties, but the observations found by Wright all give an increase. As is mentioned by him, this could cover some things other than dislocations, but he also mentions various things related to dislocations which might be thought to be important, but are not accounted for. It is also true that, theoretically, the volume change would occur if only dislocations were involved. Wright mentions that the calculations also should apply to twins. Since he published this, the developments in elasticity theory related to twinning theory make clear that second-order theory is inadequate to describe twins, so it is not reasonable to try to apply the Toupin-Rivlin formula to these, although the basic ideas about averaging, etc. appear to be reasonable for twins. Workers have considered crystals containing very large numbers of twins, even infinitely many, to model twin microstructures observed in shape-memory alloys, associated with Martensitic

Nonlinear defect theory

11

transformations. These have been quite successful in describing configurations observed in unloaded samples. At least for the configurations which have been analyzed, there is no change in volume, theoretically, and I don't know of any observations contradicting this. As to Bell's volume changes, the Toupin-Rivlin calculations say nothing about them, since they don't allow for non-zero surface tractions. It is only reasonable to expect from this that nonlinear theory will predict some kind of volume changes, likely to depend on which nonlinear theory is used, particularly if some involve different mechanisms than others. Here, (2) and (4) could be of some relevance. However, those volume changes seem to me to be intimately related to the constraint Bell discovered, t r U = t r V = 3, where U and V are the symmetric tensors occurring in the polar decomposition of the deformation gradient, as a replacement for the common assumption of incompressibility. Again, I have checked this, agreeing that it applies to a wide variety of the kinds of loading pr,ograms he was concerned with, in (f.c.c.) polycrystals and single crystals. However, in presenting his last thoughts on the latter, Bell (1996b) opined that a second constraint might also apply, but time ran out before he could check this out. My guess is that he would have first tried setting a linear combination of the invariants in his eqn (14) equal to zero, determining whether this gives a relation consistent with measurements. I suggest that workers interested in crystals plasticity explore this; my intuition agrees with his about this. Generally, when a theory of constrained materials applies to some material, it is because some moduli are very small compared to others. In (3), those slow waves suggest small tangent moduli, occurring as a nonlinear effect. So, my guess is that some such softening causes those odd volume changes, possibly relating these to (3). For a different kind of situation, involving softening of shear moduli at zero stress near phase transitions, one might think that this would be associated with the constraint of incompressibility, as is suggested by linear theory. However, nonlinear analyses by Ericksen (1986, 1988) gave constraints not compatible with this, one of which is much like that found by Bell. In numerous conversations we had, Bell mentioned his belief that it is important to understand why it is that macroscopic deformations which one would expect to be homogeneous are not, according to macroscopic measurements but, on a large enough scale, the differences seem to cancel out, making it possible to get reproducible experimental gross measurements. Of course, dislocation theory deals with deformation on a much finer scale, so the less coarse deformation hinted at above should still be regarded as a rather gross average. For this, I think that (2), (3) and (4) might all have some relevance. Concerning (5), plasticity effects related to deformation twinning are now receiving considerable ~.ttention, as is clear from the exposition of Christian and Mahajan (1995). One can do a bit with linear theory, by using different constitutive equations for the twins. As far as I know, the first useful results of this kind were those reported by the Woosters (1946) and Thomas and Wooster (1951), on detwinning of Quartz. However, nonlinear theory is needed and used for many twinning problems. I think that (2) is likely to be relevant to some interactions between twins and dislocations, among other things. Some theoretical background is needed for any discussion of the matters mentioned above, so I turn to this. II. ELASTICITY EQUATIONS

For analyses in elasticity theory, it is almost automatic that workers use material coordinates as independent variables, this being a more convenient way of handling most

12

J.L. Ericksen

problems. For some analyses relating to relating to defects, it really is better to use spatial co-ordinates instead, as Toupin and Rivlin (1960) did, after noting some reasons for this. For nonlinear theory, one can organize the two kinds of formulations so that the two kinds of equations look quite similar, as Ericksen (1995) did, in presenting some first thoughts on concepts relating to defects, in the context of nonlinear elasticity theory. In this section, I'll summarize forms of basic equations, as they apply to smooth solutions. Some are conventional, derivations for the others being given in the paper just cited. Elasticity theory involves comparing two kinds of configurations. There are the various configurations which a body occupies in different regions in space, here considered as equilibria, which might be attained by a body under different kinds of loadings. Things described here, like the regions or Cauchy stress tensors are of course described in terms of spatial positions. I'll use Cartesian tensor notations, Latin indices for things associated with these positions, xi denoting the spatial co-ordinates. Then there are the material coordinates X~, Greek indices being used for things associated with these. This is my way of emphasizing which things are associated with which regions. I'll interpret material coordinates in two slightly different ways. One involves a reference configuration for an homogeneous material, considered as filling all of space, X~ labelling positions here. The other is a reference configuration for a body, thought of as some particular subregion of the former. For us, the reference is to be interpreted as a perfect crystal, unstressed, a reasonably stable configuration of this kind, with reference mass density P0, some positive constant. There is the view that all real crystals contain some defects, so that such configurations are not really attained. It is not really necessary to include the reference among the aforementioned spatial configurations, but it does complicate matters if you don't accept the common compromises about this, settling for relatively pure crystals which have been well annealed, etc. to approximate the ideal. I accept such compromises so, for the reference, x = X. When we do what is conventional, use X~ as the independent variables, we think of mapping a reference configuration of a body to spatial configurations, described by functions of the form xi = xi(X~),

(1)

commonly regarded as smooth and smoothly invertible. If spatial co-ordinates are used as independent variables, we use instead the inverse functions, of the form -~o" = Xot(Xi).

(2)

Generally, the material description is more convenient, because the region in the reference configuration associated with a body is known and fixed. However, one can be interested in designing a body to have a certain shape, when it is subject to certain loads, clamping conditions, etc. Then the situation is reversed, so it is better to use the spatial description. Occasionally, problems of this kind have come up in my conversations with engineers. I happen to remember the first time this occurred, a long time ago, in a conversation with Ray Mindlin, concerning the design of better piston rings. For somewhat similar reasons, the spatial description can be better for some analyses of defects, as I will mention later. For the material description, we introduce the strain energy per unit reference volume W, as a function of the deformation gradient xi.,~. The requirement that it not be changed

Nonlinear defect theory

13

by superposing rigid body motions on any configuration reduces it to a function of the usual Cauchy-Green tensor Cotlg = xi, axi, fl = Cfla,

(3)

so, we have

(4)

w = W(xg,.)=

Essentially, this is an idea used by Cauchy in his development of linear elasticity from macroscopic considerations; he also treated molecular theory. As is customary, I assume that W = 0 in the reference configuration. The Piola-Kirchhoff tensor, measuring force applied to the actual configuration per unit reference area, is given by aW Tai - - aXi, a '

(5)

and the conventional equilibrium equations are T,~i,~, + p ~ = 0,

(6)

where J} is the body force per unit mass. Then there is that (referential) configurational stress tensor Pot s = W~ot.o -

TotiXi,~,

(7)

generally not symmetric. The first index goes with the reference vector element of area, the second with the force. It is straightforward to show that, when eqn (6) is satisfied, this satisfies Pa~,a + Pog~ = O,

gg = --fixi,#.

(8)

Conversely, if eqn (8) is satisfied, so is eqn (6), granted the usual condition related to invertibility, i.e. det [] xi,a II> O.

(9)

So eqn (8) can also be used as the equations of equilibrium. If you superpose rigid rotations, the Pa~ and g~ don't change, but Tag and f- do, reflecting the fact that rotating the actual configuration rotates the forces acting on it. As is traditional, T~g and J~ are regarded as mechanical forces and they are conjugate to xg. The configurational forces are instead conjugate to X~, which means that they do work in quite different ways. For almost all, if not all analyses of defects, body forces are assumed to vanish, so T and P both have zero divergence. Eshelby and many other writers then introduce the displacement u = x - X and add T to P to get a slightly different version, e,~t~ = W S , ~ - ra×ur. ~ .

(10)

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J.L. Ericksen

As a young student, I was taught not to add tensors of different kinds, as is done here, and it does have some unpleasant consequences for nonlinear theory. For one thing, rotating the body amounts to applying a rotation to x but not X, so u does not rotate as a vector would. While P ~ is not affected by such rotations, P ~ is. Also satisfying div P = 0 does not always imply that div T = 0. For such reasons, those interested in nonlinear theory tend to use P rather than P, as is the case the recent discussions of configurational forces by Gurtin (1995) as well as Maugin and Trimarco (1995). Both cover ideas about configurational forces not discussed here, and omit some covered here. Let a denote the stress tensor used in linear theory. Calculating the linear approximation to these tensors, one gets T~a,

P~-a,

P~O.

(11)

Here, I don't use indices, since a approximates different kinds of tensors. What is done in linear theory, following Eshelby,* is to use the linear approximation for T, giving the usual linear equilibrium equations as an approximation of eqn (6). Then use eqn (10), with T replaced by tr, W being taken as the quadratic form used in linear theory. So this becomes a quadratic function of displacement gradients, familiar to those who know a bit about defect theory. This satisfies eqn (8), with the approximation g ~ - f . For this, what is changed by using P is indicated in the linear contribution indicated in eqn (11), which does of course have a vanishing divergence, when body forces vanish. For singular solutions, with P or P used to calculate the configurational force exerted on defects, it could make a difference which one uses, depending on what side conditions are imposed on tr, near singularities. Eshelby believed very strongly that forces acting on defects are configurational or, as he often put it, not real. By this reasoning, a force calculated using ~r would be real, so this should give zero, as a contribution to the force (or moment of force) acting on a defect. I think it accurate to say that this has become the conventional practice, for calculating forces on defects. So, for linear theory, we have the quadratic (12) to be used for calculating force on defects. Granted this, it doesn't really matter whether one uses P or P, for such linear theory. Conceptually, there is no analog of the balance of moments of mechanical forces, for the configurational forces, in general. Actually, for isotropic materials, P turns out to be symmetric for nonlinear but not linear theory, giving the obvious analog for nonlinear theory. Conceptually, crystals are not isotropic, so this quirk of isotropic materials is not relevant. One thing about the linear theory is worth noting. If one did a systematic approximation of P or P, retaining all terms quadratic in displacement gradients, one would get what is described above, plus additional terms. This might better describe P, but it won't have a vanishing divergence, in general. So, at least tacitly, the judgment is that it is more important to have the vanishing divergence. Toupin and Rivlin (1960) do use systematic approximations, but make no use of any ideas about configurational forces. Effectively, they make a different judgment. Incidentally, there is a routine which leads to eqn (8), rather automatically. From the old work of Noether (1918), we know that, ifa Lagrangian is invariant under an n-parameter *See e.g. Eshelby(1951, 1956). There is somenonlineartheory in the latter.

Nonlinear defect theory

15

continuous group, there are n vector-like quantities such that their divergence vanishes whenever the Euler-Lagrange equations are satisfied, and there is a routine for calculating these. Here, eqn (6) gives the Euler-Lagrange equations, when f = 0. For homogeneous materials, the Lagrangian W is invariant under the 3-parameter group of translations of X~. Turn the', crank and you get eqn (7) as a formula for the corresponding divergenceless quantities. Use the invariance under rotations and you get what amounts to the usual balance of moments, for nonlinear theory. For the Sl:,atial version, it is pretty much a matter of using the same kinds of arguments, interchanging the roles played by xi and x~. So we use eqn (2) in place of eqn (1), replace W by w, the: strain energy per unit present volume, to be regarded as a function of X~,i. What does not get interchanged is the invariance under finite rotations, which means that w reduces tc, a function of C ~ , given by eqn (3) or, better, its inverse D~a, given by D,~: = X~,iX:,i,

(13)

w = w(X~,i) = ff(O~).

(14)

so we have

Here, in calculating forces, we use the present elements of area. If dS~ denotes the vector element of area of this kind, dS,~ the correspondent in the reference configuration, the usual kinematical relation is dSi =jXa,idSa,

j = det II x~.~ II •

(15)

So we introduce spatial measures of configurational and mechanical stresses as indicated by pi~dSi = P&,dS~,

tjidSj

=

T~idS,~,

(16)

tji being the familiar Cauchy stress tensor. The prescriptions for these work out to be

Pi~ = OX,~,i '

(17)

tji~--W~ji--pj~Xa, i

(18)

and

If you put in the function ff for w, you can check that this gives tji as a symmetric tensor, as it should. The new equilibrium equations are 9i, j + p f . = O ,

(19)

and Pia,i + Pg~ = 0,

(20)

where p is the present mass density. As before, eqn (19) implies eqn (20) and vice versa. It might seem curious, but this gives a formula for p~ which looks more like eqn (5), for Ti~,

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J.L. Ericksen

while eqn (18) looks more like eqn (7). In this sense, the roles played by these stresses get reversed. So, if we use the same kind of reasoning we did before, for linear theory, we should linearize the configurational stress and use the argument used before for P to get an analogous quadratic formula for t. Suppose that I didn't think much about what I was doing, and confused the material co-ordinates X~ for spatial co-ordinates and vice versa. I would then introduce as the displacement A = X - x , really - u . Suppose that I am involved in a calculation based on linear theory, so I use the linear equations, and do not worry about deducing them from nonlinear theory. However, one can do the derivations. With the understandings, the linear elastic strains are taken to be ,

1

gik :

,

,

-~(Ui, k -'1- Uk,i) ,

(21)

and plugging this into the linear constitutive equations gives the corresponding linear estimate of stress a'. Working out the linear approximations gives p _-__~r',

t ~- -tr'.

(22)

Comparing this to eqn (11), I get trapped into thinking that p is the Piola-Kirchhoff stress, t the configurational stress. Actually, since u' = - u , I should use a = - ~ to compare the two formulae. However, I carry on, to calculate forces on dislocations. Will my blunder cause me to get wrong answers? Try this for yourself. Take any such calculation you find in the literature. Most likely, the author(s) won't tell you which co-ordinates they are using, so assume they are spatial, and interpret what they call displacement as u', probably not their intent. My experience is that you will get the same result, formally. So simply reinterpret the conventional (10), and you get the analogous tj i = W ~ j i _ _ ~rjkUk, t , i = W~ji -- t3rjkUk, i ,

(23)

really the same equation, differently interpreted. This is a bit strange, since it is not symmetric, as the Cauchy stress should be. As before, eqn (23) does not include all the terms which you would get from a systematic expansion such as Toupin and Rivlin (1960) used, but its divergence will vanish. As far as I know, I am the only one who has realized that there are these two interpretations of linear theory which are conceptually very different. Had Eshelby known this, it would be like him to acknowledge it and give some argument for preferring one to the other. Later, I'll discuss this a bit more. IlL KINEMATICS OF DISLOCATIONS

Eshelby contributed much to working out procedures for setting up and solving defect problems using linear theory, this being treated in some detail by Eshelby (1951), for example. Concerning dislocations, he wrote there that "The fundamental property is

~

ui,jdxj

*His equation number.

= bi

(22)*

Nonlinear defect theory

17

for any closed circuit embracing the dislocation line." While he doesn't use the words here, there is no doubt that the circuits are what are commonly called Burgers circuits,/~ being the Burgers vector. For these, the atomistic view is that you look at the atoms in a crystal containing defects, constructing Burgers circuits which are closed curves, consisting of straight lines connecting atoms, avoiding the defects. In the continuum view, we no longer see the atoms. For the analog, we look at some configuration actually taken on and, in it, construct a closed oriented curve, not passing through any singularities in the corresponding elastic solution. For this to make sense, we need to use the spatial representation of such a curve, say X i = Xi('t'), 0 < "t" < TO,

xi(O) = Xi(~O).

(24)

The next step is to construct the image of this curve in a perfect crystal. I won't belabor how this is done in the atomistic scene, but in the continuum analog, we plug eqn (24) into eqn (2), to get the corresponding curve in the reference configuration. This may or may not be a closed curve, depending on what dislocation lines are embraced by the Burgers circuit. The Burgers vector for the circuit is ro

ro

Bc~ = X~(ro) - Xa(O) = I dXct = J xc~,idxi , 0

(25)

0

the latter in~legral being taken on the curve given by eqn (24). If you like, you can replace X by my u' in the last integral. Try to interchange the role played by the perfect and defective crystal here, to put the closed curves in the perfect crystal, and it doesn't make sense. So, for the aforementioned quotation to make sense, we should interpret Eshelby's co-ordinates as spatial co-ordinates, his u as what I call u'. Often, he didn't say whether his co-ordinates should be interpreted as material or spatial, in his writings but, from various clues, I don't doubt that he was interpreting them as material, generally. From this, I assume Proposition 1: Burgers circuits and dislocation lines should be described, using spatial coordinates. The Burgers vector is a vector in the reference configuration. Certainly, Eshelby knew about those atomic pictures associated with Burgers vectors and circuits. It used to be common for those lecturing on dislocations to explain this and I'm sure he heard at least as many of these as did I. What is strange is that he overlooked the logical implications for elasticity theory, summarized in my Proposition 1. So have his followers, as far as I know. I have used familiar ideas wrongly, without thinking much about it, kicking myself when the light dawned. Apparently, he did something like this. Conceptually, it does have important implications. One that would have shocked him is that a force capable of doing work in changing the dislocation line must conjugate to x and the P he recommended is not. He was knowledgeable about such matters and was generally careful about them. Mechanical forces could so do work, and commonly do. The latter cannot do work in changing the Burgers vector, but configurational forces might. I'll come back to such questions later. There is the problem of deciding what kinds of singularities should be considered to represent defects, dislocations in particular. For the latter, the usual picture of an isolated

18

J.L. Ericksen

defect involves a surface often thought of as a slip plane, with a jump in displacement occurring there, a more violent singularity occurring on all or part of the boundary of this surface, what is interpreted as the dislocation line. Actually, a slip plane is more likely to be a curved surface, but I accept this common abuse of language, Assuming no other singularities occur, the Burgers vector should have the same, non-zero value for all similarly oriented circuits enclosing this line, and vanish for all circuits not enclosing it. With these conditions, it is feasible to allow or exclude jump discontinuities in displacement gradients. From Proposition 1, we should use spatial co-ordinates as independent variables. Kinematically, one can deduce conditions on jumps, using the conditions on Burgers vectors, where the discontinuity surface is smooth. Label with a plus sign limits of functions taken from one side, a minus sign those taken from the other, let vi denote the unit normal, outward directed relative to the region labelled +, and let [f] = f + - f _

(26)

denote the jump in the function occurring in the square bracket. Then, one gets [X~] = const.

(27)

and that, for finite jumps in derivatives, there are functions A~ such that = A

v,.

(28)

Briefly,with the zero Burgers vector condition, you get some integrals independent of path, which can be used to define a function Y~ near the surface which is continuous across the surface and would be X~, except that it doesn't match the non-zero Burgers vector condition. Apply the customary conditions of compatibility to it and you get eqn (28). Here, I touch upon reasoning which needs to be used to construct the region associated with the body in the reference configurations, to take care of little chunks of matter which might have been taken from one place in a body and put in another. It won't be the same body you had before those defects occurred, a remark that also applies to mine. So, for a body containing many defects, it is not easy to keep book on all that is needed to construct the corresponding reference region. For various problems, it is not really necessary to do this "damage assessment". Linear theory almost always uses the Volterra model of dislocations, which means taking A~ = 0. Eshelby gave some thought to approximating these by Somigliana dislocations, which also have A~ ----0. One can show that, with the usual assumption that the strain energy function is strictly positive, A~ must vanish. Briefly, discontinuities like these with A~ ~ 0 are associated with stress waves, which must move with calculable, non-zero speeds. So, having A,~ ~ 0 would be a genuine nonlinear phenomenon. This is important for twinning, commonly considered to fit eqn (28) and a special case of eqn (27), [X~] = 0, or the equivalent of such conditions. Concerning the dislocations, some things have entered my mind. One is that, with the Volterra model, one is really dealing with multi-valued functions, which means that the discontinuity surface can be chosen rather arbitrarily; one can put the cut where you like, within reason, which seems to me a bit unphysical, given that slip planes really are particular crystallographic planes. Concerning this, Nabarro (1987, p. 21) writes that "In the

Nonlinear defecttheory

19

theory of dislocations in an elastic continuum there is no such special plane, and it is in fact an essential part of this theory that the state of the body depends only on the line and its strength b, and not on the cut bounded by this line which was made in order to introduce the dislocation." My view is that nonlinear theory should be capable of distinguishing those slip planes, perhaps by allowing jumps with A~ # 0 near the line. From experience with twinning theory, we have some understanding of what properties constitutive equations should have, to make it possible to have such jumps. One needs a band of deformations too unstable to be observed, separating more stable regimes. In the latter, near the unstable regime, some acoustic speed gets small, vanishing as one enters the unstable region or, in static terms, some normally positive tangent modulus goes to zero, becoming negative in the unstable regime. One can state the conditions more precisely, but I don't want to get into technical details. Also, with such dislocations, it could then be easier to understand conversions of dislocations to twins, which are considered in deformation twirming. Referring to the introduction, this bears on (2) and (3). For plasticity, there are indications of some such softening and of slow waves in the observations of what Bell (1973, Section 4.31) called the Savart-Masson effect, along with the McReynold slow waves. This is better evidence, but still weak. For the theory considered, we are considering deformation on a finer scale, somewhere between atomic spacing and the distance between dislocations. The measurements are on a much larger scale. So, it is a bit tricky to interpret what the measurements imply about events on the finer scale. Also, most of the observations mentioned above are on polycrystals, not single crystals. However, as is discussed in some detail by Bell (1981), there is evidence that quite similar phenomena do occur in Stage III deformation of single crystals, including X-ray observations, made using high-speed scanning measurements, taken during loading. For the latter, one is getting into regimes where the crystal is obviously damaged by large numbers of defects, making X-ray pictures fuzzy. Perhaps an expert could decide whether the phenomenon at hand is visible in the haze, but I can't. Theoretically, the phenomenon is one which might be produced by loading and either remain or disappear when loads are removed; I am pretty sure that one could construct theories of either kind. So, we have a genuinely nonlinear effect which might or might not be helpful in understanding plasticity. I'd like to see workers find other possibilities and improve our theoretical understanding of them. IV. FORCES

With the misinterpretation pointed out in the last section, one should look carefully at interpretations of many calculations. It is a tricky matter to correctly interpret calculations of forces acting when a dislocation moves through a material. For one thing, this changes the slip plane, adding to or subtracting from it. This involves changing the Burgers vector for some circuits from zero to a non-zero value or vice versa. As Ericksen (1995) notes, configurational forces can do work in this process. For another, for edge or mixed dislocations, one needs to bring in or remove vacancies or atoms, so one should consider forces between these and the dislocation, or maybe the slip plane. I have not thought through for myself how best to view these. Then, there is the point that changes in the slip plane are firmly linked to changes in the line, so changes in position of the latter are not really independent of those changes in Burgers vectors. In addition, there is the atomistic view that Burgers vectors should be integral linear combinations of (reference) lattice vectors, which means that they change only if the line shifts by an integral number

20

J.L. Ericksen

of lattice spacings, roughly speaking. This is important in the considerations of PeierlsNabarro forces, in particular. For linear theory, the Peach-Koehler forces are among those considered, and it is hard to know how to define these, for nonlinear theory. However, 1'11 attempt to interpret the symbolism. The idea is to try to estimate how surface tractions influence motion of dislocations. In Eshelby's (1956, Section 9) treatment of this, he considers dislocation loops, along with a smooth linear elastic solution with this stress tensor denoted by tr r, describing the loading. He considers an infinitesimal variation of a small part of the dislocation line, adding an infinitesimal area to the slip plane. If d S represents the vector element of area for this, his calculation of the change in energy gE gives for it ~E = +bioi~dSj ,

(29)

with bi the Burgers vector. One can fix the sign by adopting conventions for orienting the vectors, as he does. To interpret this, consider the slip plane to be located in the spatial configuration, so d S is the spatial version and, from Proposition 1, the Burgers vector is in the reference configuration. The interpretation of tr r fitting these is the linear approximation to the spatial configurational stress. So, I would rewrite this as 8E = +B~pi~dSi,

(30)

and interpret this as the work done in changing the Burgers vector from zero to B~ in this process. Eshelby doesn't, skipping to the next step, using d S = ls A ~,

(31)

where I is the length of the line segment varied, s is its direction, and ~ is the infinitesimal displacement vector, indicating how it is displaced. With this, one can put eqn (30) in the form 8E = IF. ~,

(32)

as a justification for regarding F as a force per unit length acting on the dislocation; work this out and you get the Peach-Koehler formula, as I interpret it. Effectively, this converts a configurational force to one which is conjugate to x. With the singularity at the dislocation, one should not use the total stress in this formula and, with nonlinear theory, one can't simply add smooth solutions to singular solutions, so this seems to make some sense for linear theory only. In writing about his memories of Eshelby, Nabarro (1985) writes that "Eshelby maintained this distinction* rigorously. When he calculated the force between parallel disclinations in a nematic liquid crystal and found that 'the supposedly configurational force in a nematic is in fact a real force exerted on the core of the dislocation by the surrounding medium', he was very disturbed, and he circulated the draft of his paper t to many colleagues before publishing it." As was acknowledged by Eshelby (1980), I was one of the *He refers to the distinctionbetweenmechanicaland conflgurationalforces. tMy referenceto Eshelby(1980).

Nonlinear defecttheory

21

many, so I know something about his worries. Certainly, he felt very strongly that forces on defects should be configurational, although I have never understood his reasons for this. So, he started believing that the liquid crystal workers had made some error, which he would find and correct. However, after a careful examination, he concluded that it was he who was wrong and, being an honest and scholarly person, he made this public. I doubt that it even occurred to him that he might be somewhat wrong about this, for solids. At the time, I had not thought much about these calculations. However, questions relating to this came up in later correspondence and discussions with others, inducing me to collect my thoughts, which are presented by Ericksen (1995). In taking a hard look at the calculations, I found the conceptual error described in (Section III). I then argued that, in particular, the force on a dislocation line should sometimes be regarded as mechanical, to be calculated using the Cauchy stress tensor. Involved here are forces not calculated using elasticity, but from the microscopic view of atoms, perturbed a bit from the periodic arrangement, what are commonly called Peierls or Peierls-Nabarro forces. Actually, a kind of continuum theory is often used for these, motivated by the atomic picture. Remember the elementary arguments used to make plausible the notion that crystals must have a finite shear strength. One version I remember being exposed to in more than one lecture on dislocation theory, long ago, considers a one-dimensional shear stressstrain curve associated with a picture of atoms arranged periodically. Pick the crystallographic direction of shear in a way making it easy to see what should happen and it's intuitively obvious that this should be periodic curve. So, start from a place where the shear stress vanishes, and the slope of the curve is positive. For a small positive shear stress, associLated with small shear strains, one gets a point closer to a maximum on the curve, thought of as a limit of metastability, it being conceded that one should get pretty close to this, physically, in a perfect crystal. Note that this implies that the tangent modulus vanishes at the limit and should at least get small for deformations viewed as attainable, at least in the perfect crystal. With the Peierls-Nabarro forces, the aim is to do more realistic calculations using this idea, perhaps with atomistic theory. Of course, one does not use linearized theory, for this part of the calculation. The question is whether the dislocation line will move through the material, when forces are exerted on it by other defects, walls, etc. The idea is to assume that it does not, so one then uses elasticity theory to calculate forces acting on it and determine whether these can be balanced by the Peierls-Nabarro force. As was already mentioned in (Section III), the force acting on the dislocation line should be conjugate to x. To me, having the dislocation line there is not really different from having some foreign object embedded in the material, when it is considered rLot to move through the material. Given this, it seems to me reasonable to use the Cauchy stress tensor to calculate that force. Also, we should match this force to the Peierls-Nabarro force. Conceptually, I have no idea how to calculate such a force, in atomistic theory, if it is not to be interpreted as mechanical. There is another point in that, with the forces regarded as mechanical, one should also balance couples. I do share the common view that the equilibrium theory is not what should be used, to analyze what happens when a dislocation moves through the material. Physically, I don't see a good way to fit the Peach-Koehler forces into such calculations since they involve moving the dislocation through the material. So, as an amateur in this business, I'll leave this to experts. In using nonlinear elasticity theory to do analyses of problems relating to twinning and phase transitions, workers have found it important to use a theory of material symmetry

22

J.L. Ericksen

which is quite different from that used for linear theory and, by most workers not concerned with these phenomena, for nonlinear theory as well. Essentially, it better accounts for the periodic nature of crystals, building in those periodic shear stress-strain curves as a contribution to the stress, for example. So, from this view, the Peierls-Nabarro forces can be cancelled out by requiring resultant stresses to vanish, as was assumed by Toupin and Rivlin (1960). Workers have found some tricks for simplifying theory in some cases, compromising with the periodicity, but saving the part of this which is important for the phenomena considered. One is not really interested in all those maxima and minima in that shear-stress strain curve, but it is good to see at least one. So, there is some experience in using continuum theory analogous to that involved in the consideration of PeierlsNabarro forces. From this view, some of the latter kinds of atomistic calculations could be viewed as substitutes for experiments we don't know how to execute, providing some information about constitutive equations. So, this elaborates item (4) in the introduction. My experience is that research moves ahead faster when workers with different kinds of expertise get involved, and there are real experts in continuum theory of this kind. Admittedly, this is still quite speculative,and it will take hard research to better understand what such theory can do, concerning the dislocations. My reinterpretation of linear theory and thoughts about different kinds of forces both suggest one side condition associated with configurational stress, which I'll state in words as Proposition 2: Resultant configurational forces exerted on a dislocation line vanish. Interpret this in the same way as workers have done with the analogous statement for mechanical forces, as the vanishing of the integral over tubular surfaces surrounding the line, of configurational surface tractions. For linear theory, it is clear that this does not force Peach-Koehler forces to vanish, for example. Concerning those surfaces of discontinuity discussed in (Section III), extremal conditions in the calculus of variations suggest, for one thing [pi]vi = 0,

(33)

in the notation used before. This is satisfied automatically for Volterra dislocations and for the kinds of twins most commonly analyzed, those occurring in unstressed bodies. With my proposal for linear theory, this becomes the commonly used jump condition for the stress tensor. In brief, the assumption is not obviously wrong, as far as I can see, but neither has it been put to a very good test. Actually, in considering the less than perfect mobility of transformation twins subjected to loading, I found a reason to reject eqn (33) for these. However, I don't see any similar reason to reject it for dislocations. Toupin and Rivlin (1960) did not use it, but did use the usual condition on the Cauchy stress tensor associated with jump discontinuities, [tji]vj = 0,

(34)

in my notation. Reasonably, this applies to likely generalizations of the Volterra and Somigliana dislocations. Combining this with eqns (17), (27) and (28), one gets [W] ----Aapic~Vi,

(35)

Nonlinear defect theory

23

where p~ can be evaluated on either side. Again, this is satisfied in a trivial way by Volterra dislocations, etc. So, eqns (33) and (34) are consistent with common practices, for whatever that is worth. Given that eqn (33) is unfamiliar, it is natural to view it with more suspicion, although eqn (34) might also fail to apply because surface energies are important. For edge dislocations, one might introduce a surface energy, in describing why atoms again draw closer when a half plane is removed, but is hard to see why this should have any effect after the gap closes. What is harder is to decide what to say about singularities associated with dislocation lines. Fixing the Burgers vector for circuits embracing the dislocation line obviously does force X~.i to behave rather badly, near the dislocation line. The considerations of periodic shear stress-strain curves, etc. suggest that the stresses, etc. don't really behave as badly as is depicted by linear elastic solutions. Tame these a little, and one could make sense of resultant fc,rces on surfaces intersected by dislocation lines, which would be helpful. It would take more taming to get energy integrals to converge, but this could happen, I think, with reasonable nonlinear constitutive equations. Given all the doubts about such matters, I t]hink it premature to try to propose definite assumptions for such singularities. However, I will endorse one used by Eshelby (1951, Section 7), viz. Proposition 3: An elastic body reacts to applied forces in the same way whether it is selfstressed or not.

and suggest that you examine what he wrote concerning this. Of course, one needs to use some judgment, in interpreting statements of this kind. For linear theory, it excludes the possibility of using concentrated force solutions to equilibrate unbalanced surface tractions in representing defects, for example. With the suggested side conditions on configurational forces, we have almost enough information to do a copy of the Toupin-Rivlin procedures for averaging the Cauchy stress, which should yield additional information. One also needs some kind of boundary conditions on configurational surface tractions. I think that these might somehow be involved in helping or resisting motion of defects from the boundary to the interior or vice versa, but I haven't digested literature relevant to this. Thus, I won't try to suggest anything definite for these. So, I have discussed some of my reasons for thinking that those nonlinear mechanisms listed in the introductions are built into versions of nonlinear crystal elasticity theory now being used, and are likely to be of some importance for understanding some plasticity phenomena. The next step should be to formulate and tackle some more definite problems of this kind. My thoughts on this are not yet clear enough to present, so I will not expend more ink.

REFERENCES Ball, J. M. and James, R. D. (1992) Proposed experimental tests of fine microstructures and the two-well problem. Phil. Trans. R. Soc. London A338, 389. Bell, J. F. (1973) The experimental foundations of solid mechanics. In Handbuch der Physik, vol. Vial1, ed. C. Truesdell, pp. 1-813. Springer-Verlag, Berlin-Heidelburg-New York. Bell, J. F. (1981) A physical basis for continuum theories of finite strain plasticity, Part II. Arch. Rational Mech. Anal. 75, 103. Bell, J. F. (1996a) The decrease in volume during finite plastic strain. Meccanica, (in press). Bell, J. F. (1996b) The kinematics of large plastic strain in cubic single crystals: a new look in the laboratory at G. I. Taylor's analysis of finite shear on face diagonals. In Contemporary Research in the Mechanics and Mathematics of Materials, eds R. C. Batra and M. F. Beatty, pp. 11-39. CIMNE, Barcelona.

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Bhattacharya, K. and Kohn, R. V. (1996) Symmetry, texture and the recoverable strain of shape-memory polycrystals. Acta Mater. 44, 529 542. Christian, J. W. and Mahajan, S. (1995) Deformation twinning. Progress in Mat. Sei. 39, 1. Ericksen, J. L. (1986) Constitutive theory for some constrained elastic crystals. Int. J. Solids Structures 22, 951. Ericksen, J. L. (1988) Some constrained elastic crystals. In Material Instabilities in Continuum Mechanics and Related Mathematical Problems, ed J. M. Ball, pp. 119 136. Clarendon Press, Oxford. Ericksen, J. L. (1995) Remarks Concerning Forces on Line Defects. Zamp 46(special issue), $247. Ericksen, J. L. (1996) Thermal expansion involving phase transitions in certain thermoelastic crystals. Meccanica, (in press). Eshelby, J. D. (1951) The force on an elastic singularity. Phil. Trans. R. Soc. London A244, 87. Eshelby, J. D. (1956) The continuum theory of lattice defects. Solid State Physics 3, 79. Eshelby, J. D. (1980) The force on a disclination in a liquid crystal. Phil. Mag. A42, 359. Gurtin, M. E. (1995) The nature of configurational forces. Arch. Rational Mech. Anal. 131, 67. Hsu, N. N.-H. (1969) Experimental studies of latent work hardening of aluminum single crystals. Ph.D. dissertation, The Johns Hopkins University, Baltimore. Maugin, G. A. and Trimarco, C. (1995) The dynamics of configurational forces at phase-transition fronts. Meeeanica 30, 439. Nabarro, F. R. N. (1985) Material forces and configurational forces in interaction of elastic singularities. In Proe. Int. Syrup. on Mechanics o f Dislocations, 1983, eds E.C. Aifantis and J. P. Hirth, pp. 1-3. Michigan Technical University, American Society of Metals, Metals Park. Nabarro, F. R. N. (1987) Theory of crystal dislocations. Dover Publications Inc., New York. Noether, E. (1918) Invariante Variationsprobleme. Naehr. Akad. Wiss. Goettingen, Math.-Phys KI 2, 235. Thomas, L. A. and Wooster, W. A. (1951) Piezocrescence-the growth of Dauphin+ Twinning in quartz under stress. Proc. R. Soe. London A208, 43. Toupin, R. A. and Rivilin, R. S. (1960) Dimensional changes in crystals caused by dislocations. J. Math. Phys. 1, 8. Virga, E. G. (1994) Variational theories for liquid crystals. Chapman and Hall, London-Glasgow-Weinbein-New York-Tokyo-Melbourne-Madras. Wooster, W. A. and Wooster, N. (1946) Control of electrical twinning in quartz. Nature 159, 405. Wright, T. W. (1982) Stored energy and plastic volume change. Mechanics o f Materials 1, 185.