On anisotropic compressible materials that can sustain ... - Springer Link

Journal of Elasticity 35: 213-222, 1994. © 1994Kluwer Academic Publishers. Printed in the Netherlands.

213

On anisotropic compressible materials that can sustain elastodynamic anti-plane shear HUNGYU TSAI and PHOEBUS ROSAKIS Department of Theoreticaland Applied Mechanics, Cornell University, Ithaca, N Y 14853-1503, USA

Received 14 May 1993

Abstract. We characterize the class of compressible, homogeneous,possibly anisotropic elastic

materials that can undergonontrivialfiniteelastodynamicanti-planeshear motions.

O. Introduction

One of the most widely studied classes of finite deformations of nonlinear elastic materials is anti-plane shear. It involves deformations of a cylindrical region where the displacement of each particle is parallel to the axial direction and independent of its axial coordinate. Substantial analytical simplification is gained whenever the full three-dimensional field equations can be reduced to a single two-dimensional second order equation for a single scalar unknown, the out-of-plane displacement. This is indeed the case for infinitesimal deformations of isotropic materials, as well as for finite deformations of a restricted class of isotropic materials I-1,2]. In general, however, the two in-plane equilibrium equations, together with the axial equilibrium equation, constitute an overdetermined system for the out-of-plane displacement, unless certain restrictions are imposed on the constitutive law. An elastic material is said to be capable of sustaining anti-plane shear deformations provided that the in-plane equilibrium equations are satisfied by any solution of the axial one [1, 2]. Knowles established necessary and sufficient conditions in order for isotropic incompressible [1] and compressible [2] materials to admit anti-plane shear deformations. A special class of compressible stored energy functions conforming to those conditions was proposed by Jiang and Knowles [3]. These studies are limited to isotropic materials. Various deformation processes in crystalline solids, especially twinning, involve large shear deformations with displacement along certain preferred crystallographic directions. In [4, 5], twinning of cubic crystals is modelled as an anti-plane shear deformation of an anisotropic material with non-convex stored energy. The crucial role played by anisotropy in this type of problem

214

Hungyu Tsai and P. Rosakis

would motivate a characterization of anisotropic elastic materials that can sustain anti-plane shear. In general, however, such a material need not be capable of certain dynamic anti-plane shear deformations that satisfy the axial equation of motion but not its in-plane counterparts. Our objective is to find a necessary and sufficient condition in order for anisotropic elastic materials to admit finite anti-plane shear motions in the absence of body forces. These materials can automatically sustain static anti-plane shear states. Their identification would be desirable in view of possible studies of the dynamics of screw dislocations, mode III cracks or twinning in the context of finite-deformation theory, for example. Section 1 reviews the basics of finite dynamic anti-plane shear for anisotropic elastic solids. In Section 2 we derive a condition on the stored energy function, which is necessary and sufficient in order for a compressible, anisotropic material to sustain elastodynamic anti-plane shear motions. The condition is specialized to isotropic materials in Section 3. The resulting restriction is more severe than that found by Knowles I-2]. An isotropic stored energy function is exhibited that can sustain static but not dynamic anti-plane shear. We briefly consider examples involving triclinic and tetragonal symmetry.

I. Dynamic anti-plane shear for anisotropic materials Boldface letters stand for vectors or second order tensors in two- or threedimensional Euclidean space; 1 is the idem tensor with components 6 u. We let Ck(A) denote the class of k times continuously differentiable functions on the set A. Also, R and •2 stand for the real line and plane (set of ordered real number pairs) respectively. The Cartesian product of two sets A, B is A x B. The tensor (dyadic) product of the vectors a and h is a ® b. Greek indices have range {1, 2}, Latin indices range over {1, 2, 3}. A subscript following a comma indicates partial differentiation with respect to the corresponding Cartesian coordinate. Time derivatives are dotted. Summation over the range of repeated subscribed indices is implied. We follow the discussion in I-4, 5]. See [1, 2] for isotropic materials. Consider a body which occupies a cylindrical region ~ in its reference configuration. Let {el,e2,e3} be an orthonormal basis with e 3 parallel to the direction of the generators of the region ~. Consider motions (time-dependent deformations) ~(x, t ) = x + u(x, t) expressing the current position vector y at time t of a material point with reference position x E ~ ; u is the displacement field. Anti-plane shear motions are ones of the special form ~(x, t) = x + u(x, t) = x + u(xl, x2, t)e3,

x~,

t~J-,

(1.1)

where x~ = x. e~ and u : H x ~-- ~ R is the out-of-plane displacement field on the

Anisotropic compressible materials

215

cross section H of ~ in the (xl, xz) plane, during the time interval 9- = [to, tl] c •. We assume uE C2(1-I × Y-). The deformation gradient tensor F is defined by F(x, t) = V~(x, t) = 1 + u.,(x 1'

X2'

t)e3 ® e,,

x ~ ~ , t ~ J-.

(1.2)

We define the (two-dimensional) shear strain vector y as follows: = ~e~ = u,~e~ on FI x J-.

(1.3)

Then the deformation gradient tensor in (1.2) can be expressed as F = 1 + y~e 3 (~) % = 1 + e 3 t~) y

on ~ x J .

(1.4)

This implies that anti-plane shear deformations are volume-preserving since det F = 1 regardless of ~ . The right Cauchy-Green tensor C = FTF is then given by C ( x , t) : C('~ 1, ~2) : 1 + y ® e 3 + e 3 (~) y + y ~) y

on ~ × J .

(1.5)

In terms of the nominal or Piola-Kirchhoff stress tensor tL balance of linear m o m e n t u m without body forces takes the form V'~=pii

on~xJ,

(1.6)

where p > 0 denotes the constant mass density in the reference configuration. We assume that the body is composed of hyperelastic, compressible material and is homogeneous but possibly anisotropic in the reference configuration. The nominal stress is then determined by the constitutive relation O t~ = ~-~ W(F);

(1.7)

the scalar valued function W is the stored energy or elastic potential function, defined for tensors with positive determinant. The objectivity requirement on W demands its dependence on F only through the right Cauchy-Green tensor C, w(v) = if(c),

¢ =

vTv,

which in turn ensures balance of angular m o m e n t u m . Here if" is

(1.8) a C2

function


216

of symmetric, positive-definite tensors. By the chain rule and the symmetry of C, the constitutive relation (1.7) can be written as o = 2F O ff'(C)

or

c~C

aij = 2Fik -c~ff'(C) OCkj

(1.9)

With the aid of (1.4) and (1.5) the constitutive law (1.9) yields a~ = 2 ~

(C),

E

a33 = 2 ~

(C) + ~p ~

(C) ,

(1.10)

and regarding the shear response, t~W 0"3a = ~ (~1' ~2);

(1.11)

tJ/~

here we define the restricted stored energy function for anti-plane shear deformations as A

A

w(r,, ~,~) = w(c(~,~, rg),

(1.12)

with C given by (1.5). Clearly, u, F, C and o are independent of the axial coordinate x3. Hence, in view of (1.10) and (1.11) the equations of motion (1.6) reduce to the in-plane equations

a ~ p , a = 2 I d ~ ( C ( u , , u . 2 ) ) 1,~=0

onH xY,

(1.13)

involving only tr~a and the axial equation of motion ~2 W

a3~,~ =

(u,l, u,2)u,~a = pii

on II x Y-.

(1.14)

In general, Eqs. (1.13) and (1.14) form an overdetermined system for the single unknown function u. One expects that there will be solutions of (1.14) which do not satisfy (1.13), unless additional restrictions are placed on the stored energy function W. The analogue of this issue appropriate for statics of isotropic materials was explored by Knowles I-1,2]. Its importance is clear, since the reduction of the system of equations of motion (1.6) to the single axial equation (1.14) for u is perhaps the main analytical motivation for the study of anti-plane shear. We adapt the definition of Knowles I-2] to allow for dynamics and anisotropy: We say that a stored energy function W sustains (nontrivial)


217

anti-plane shear motions along the direction e 3 /f, for any plane domain I-I and time interval ~--, every solution u of the axial equation (1.14) also satisfies (1.13). Note that some anisotropic materials may sustain anti-plane shear motions only along special directions. In the following section we will characterize the class of materials that sustain anti-plane shear motions. It will be assumed that the restricted stored energy function w in (1.12) satisfies the condition of uniform strong ellipticity, i.e., for some constant c > 0 and all values of (71,72),

(~2w(y 1, 72) ~Y~7# d~d# >i cdzd x V(dl, d2) E[~2.

(1.15)

It is often assumed that the reference configuration is stress-free. By (1.9) this occurs provided

t3C (1) = O.

(1.16)

2. Materials that sustain anti-plane shear motions Whenever (1.15) is valid, (1.14) is a quasi-linear second-order hyperbolic equation for u. The following lemma is a special case of a result by Hughes, Kato and Marsden [6], concerning the well-posedness of the initial-value problem for (1.14). In what follows, C~ is the set of all Ck(E 2) functions with compact support, while H is any plane domain. LEMMA. Assume that the restricted stored energy function w E C6(H 2) satisfies the uniform strong ellipticity condition (1.15). Given any initial conditions U(X 1' X2' O) = ~O(Xl,

X2)'

il(X 1' X2'

O) = ~/(X 1' X2)

on II,

(2.1)

with ~o and ~k in C'~ and C 3 respectively, there exists a t, > 0 such that (1.14) has a unique solution u~C2(l "1 × I - - t , , t,-I) satisfying (2.1) at t = O. The assumed smoothness of w is the weakest allowed by Theorem 3 in [6], from which the lemma follows. In case W itself (and not merely w) is of class C 6 and uniformly strongly elliptic, short-time existence and uniqueness for the three-dimensional equations of motion (1.6) follows from Theorem 3 in [6]. In that case, suppose that the body is subject to initial displacement and velocity which are along the e 3 direction and independent of x 3. If the material sustains anti-plane shear motions, then by the above result, the subsequent motion will also be of anti-plane shear type (1.1), at least for short times. For other materials

218


one can find an initial anti-plane shear state giving rise to a motion which is not an anti-plane shear. Next, we state our main result. PROPOSITION. Assume that the restriction of ITVto anti-plane shear deformations, w(7 l, 72), defined in (1.8), is in C6(R 1) and satisfies the condition of uniform strong ellipticity (1.15). Then ~V sustains anti-plane shear motions along the direction e, if and only if, relative to some orthonormal basis {el, e2, e3} with e3 =

e,

o¢v

a,a = ~

(c'(71, 72)) = const.

V(71, 72) E •2.

(2.2)

Moreover, if the reference configuration is stress-free ((1.16) holds), then (2.2) becomes

a~t~ = ~

(C(71, 72)) = 0

V(~,1, 72)~R 2.

(2.3)

We now prove this claim. The sufficiency of (2.2) for W to sustain anti-plane shear motions is clear; since now the in-plane stress components a,a are constant regardless of the shear strains, (1.13) holds trivially for all such motions. Conversely, let W sustain anti-plane shear motions. By the lemma, a solution u of (1.14) may be found on H x I - t , , t,] satisfying (2.1) for any smooth ~o and ~b. For such materials, (1.13) must be satisfied at time t = 0, whence by (2.1), I OO-~,p((~(~o,1, ~0,2))10 , =~

Vq~eC~.

(2.4)

It is convenient to introduce the functions

~,a (7 i, 72) = ~

(C(71, 72)).

(2.5)

From (2.4) and the chain rule we find (tP,I,tP,2) ~P,~a= 0

(2.6)

for all q~ in C~. Now specify the function q~ on a neighborhood X of the origin to satisfy q~(x 1, x2) = a~x~ + ½g~ax~xa,

(x 1, x2)~ ~V',

(2.7)


219

where a~ and 0~a = #p~ are arbitrary constants. Evaluating (2.6) at the origin, one finds that they must hold for ~p,~= a~ and ~p.a, = #p~. Since 0p, = 0,a are arbitrary, the only possibility is that ~X~p/dT~ in (2.6) must be skew in ~, ft. This, together with the symmetry X~a = Ea~ from (2.5), is easily seen to dictate that dE"a (a 1, a2) = 0

(2.8)

for all (al, a2) in R z, since a~ in (2.7) are arbitrary. It follows that the functions E~p are constant a n d - - v i a (2.5)--we arrive at the necessity of (2.2). If we further assume that the reference configuration is stress-free, then by (1.5), (1.16) and (2.5), E~a(0, 0) = 0. This reduces (2.2) to (2.3), completing the proof. The sufficiency of (2.2) in order for W to sustain dynamic anti-plane shear persists even in the absence of the smoothness and strong ellipticity hypotheses above. Without the existence result 1-6] cited in the above lemma, proving the necessity of (2.2) would have been quite difficult. An analogous result for static anti-plane shear seems substantially more challenging. For isotropic materials sustaining static anti-plane shear, the issue was settled by Knowles I-1, 2-1.

3. Examples (i) lsotropic materials We specialize our results to isotropy and compare them with those of Knowles [-2]. The stored energy for isotropic materials takes the form W(F) = ec(l 1, Iz, I3),

(3.1)

where the deformation invariants lk(C) are given by I1(C) = tr C = C~R,

I2(C) = l[(tr C) 2-tr(Cz)],

I3(C) = det C.

(3.2)

For anti-plane shear deformations we have [1] 11 = 12 = 3 + ~'~7~,

13 = 1.

(3.3)

By means of (1.10), (3.1) and (3.3), it can be shown [2, 3] that a~a=2

- ~ 1 ( I , I, 1 ) + ( I - 1 )

dec

x 6~a-2~a ~

d,,. 2 ( l , l , 1)+ ~

(I, I, 1),

(1, 1, 1) (3.4)

220


where I = 3 + ?=?=. Condition (2.2) of the proposition then specializes to ~I~ 814' (I, I, 1) + t3W (I, I, 1) = const. (no sum) ~I"-'-~(I, I, 1)+(2+?=?~) -~2 t3I3

(3.5)

and

dI----~-(I, I, 1)7172 = const.

(3.6)

for all (71, ?2) in •2. Since the value of 7172 can be chosen independently of I, the coefficient of 7172 in (3.6) must vanish. This, together with (3.5) and the Proposition, shows that the followin 0 two conditions are necessary and sufficient in order for an isotropic material to sustain anti-plane shear motions: 8I~' (I, I, 1) +

(,

I, 1)

=

const,

and

-~2 (I, I, 1)

=

0

VI/> 3;

(3.7)

here we have assumed that W conforms to the smoothness and strong ellipticity hypotheses made above. The constant in (3.7) is zero in case the reference configuration is stress-free. The conditions of Knowles [2], appropriate for isotropic materials that sustain static anti-plane shear, are less restrictive than (3.7). Any material that sustains anti-plane shear motions also sustains static anti-plane shear; however, the converse is false; see (ii) below. Jiang and Knowles I-3] introduce a class of materials with stored energy functions given by ff'(l l, 12, 13) = Wo(l,) + Wg(l l) f (13) + g(13),

(3.8)

where primes indicate derivatives, while f and g are required to satisfy f(1) = 0 ,

f'(1) = - 1 ,

0(1) = 0 .

(3.9)

It is clear that the materials characterized by (3.8) and (3.9) satisfy conditions (3.7) and can sustain anti-plane shear motions. (ii) A counterexample Consider the stored energy function I7¢'(I~, 12, 13) = all 1 + a2(3I 1 -- 212)13 + g(la),

(3.10)

with non-zero constants a I > 0, a 2 > - a ~ , and g an arbitrary function. It is


221

easily shown that this energy function satisfies the conditions of Knowles [2] and thus sustains equilibrated anti-plane shear deformations. However, it violates (3.7) and cannot sustain some anti-plane shear motions. For such a material, although a nontrivial anti-plane shear deformation is possible, a suitable perturbation may change the subsequent elastodynamic state into one which is not of anti-plane shear type. (iii) Arbitrary shear response Suppose the restricted stored energy w is of class C 6 and satisfies (1.15), but is otherwise arbitrary. We ask the question: Does a three-dimensional I?V exist that sustains anti-plane shear motions and whose restriction (1.12) is the given w? Consider the following stored energy function:

I~(C) = ~ [(I t - 3) + g(13) "[- h(C33)'] "1- (I)(Cl3, C23),

(3.11)

where/z = const. > 0, while g and h are as smooth as desired and satisfy g'(1) = - 1 ,

9(1) = h(1) = h'(1) = 0.

(3.12)

Note that (3.11) satisfies (2.3)--thus sustains anti-plane shear m o t i o n s - - for any choice of O, 9 and h, in view of (3.12) and (3.2). Next observe that by choosing

o(~,, ~,~) = w(~,,,

~,~) -

/z

g (~,] +

~,~)

(3.13)

and appealing to (3.3), we ensure satisfaction of (1.12). Thus the answer to the above question is affirmative. (iv) Tetragonal materials In cases of physical interest one would insist in addition to the above that W conform to a special type of anisotropy. For example, the stored energy is said to have tetragonal symmetry provided I~(C) = ff'(RCR T) whenever R is a rotation of 1t/2 about e 3 or of ~z about e 1. In that case it follows from (1.12) that w must be symmetric in its arguments and even in each, namely,

W(~D ~2) = W(~2' ~1) ~- W("["~ 1, -'~"~2) (~1' ~2) e ~ 2 .

(3.14)

Given a w that abides by (3.14), we now seek a three-dimensional I?¢ that, in

222

Hun#yu Tsai and P. Rosakis

addition to sustaining anti-plane shear motions along e 3 and conforming to (1.12), has tetragonal symmetry. Let if" be given by (3.11), in which (I) is defined by (3.13) and (~2W

/~ = ~

(0, 0).

(3.15)

Here and in (3.11) p is a shear modulus. Once again, (2.3) holds. The tetragonal symmetry of if" in (3.11) can now be shown from (3.14) and (3.13), which imply that w and (I) have four-fold symmetry. This delivers the requisite stored energy. Rosakis (Eq. (3.11) in [4]) proposed a special function W in order to obtain a type of restricted stored energy w satisfying (3.14). A claim made in I-4] that this stored energy sustains anti-plane shear is erroneous as pointed out by Fosdick [7]. This problem can be alleviated if instead a stored energy of the form (3.11) is employed. The assumed cubic symmetry in I-4] is not essential and may be substituted by tetragonal.

Acknowledgement The authors would like to thank R. L. Fosdick for pointing out an error in [4]. His comments prompted this investigation. This work was supported by the National Science Foundation through Grant No. MSS-9009730.

References 1. J.K. Knowles, On anti-plane shear for incompressible elastic materials. Journal of the Australian Mathematical Society Series B 19 (1976) 400-415. 2. J.K. Knowles, A note on anti-plane shear for compressible materials in finite elastostatics. Journal of the Australian Mathematical Society Series B 20 (1977) 1-7. 3. Q. Jiang and J.K. Knowles, A class of compressible elastic materials capable of sustaining finite anti-plane shear. Journal of Elasticity 25 (1991) 193-201. 4. P. Rosakis, Compact zones of shear transformation in an anisotropic solid. Journal of the Mechanics and Physics of Solids 40(6) (1992) 1163-1195. 5. P. Rosakis and H. Tsai, On the role of shear instability in the modelling of crystal twinning. Mechanics of Materials 17 (1994) 245-259. 6. T.J.R. Hughes, T. Kato and J.E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Archive for Rational Mechanics and Analysis 63 (1977) 273-294. 7. R.L. Fosdick, Private communication.