Journal of Elasticity, vol. 2, no 3, September 1972 Wolters-Noordhoff Publishing Groningen Printed in the Netherlands

On a theorem of Muskhel ishvi l i

MICHAEL HAYES University o f Eas t Angl ia , Norwich

(Received February 23, 1972)

1. Introduction

In the static plane theory of homogeneous isotropic elastic bodies subjected to surface trac- tions alone, Muskhelishvili [1, page 132] stated the following result:

" I f some part of the body (i.e. a subregion which may even be arbitrarily small) is in a 'natural state' i.e. if no stresses occur there, then the whole body is in a natural state or in other words no stresses occur anywhere".

The purpose of this note is to extend this result to the full three dimensional theory of in- homogeneous anisotropic elastic bodies.

2. Basic Equations

In the classical linear theory of homogeneous elastic bodies not subject to internal constraint the constitutive equation is

tlj = ciikzUk,~. (1)

Here t u are the components of stress in a rectangular Cartesian coordinate system x , CijkZ are the elastic constants - usually assumed to have the symmetries

Cijkl = Cjlkl ~" Cijlk = Ckllj , (2)

the displacement components are denoted by u~ and the comma denotes partial differentiation with respect to x~ i.e. uk, z = OUk/~X~. The displacements U~(Xk, t) are always assumed to exist and to be real twice continuously differentiable functions of Xk and t, where t(>~ 0) denotes time. The tractions t(~) across a surface element with unit outward normal n_ are given by

t(~)i = tij n j = cijkt Uk, t n j . (3)

The equations of motion in the absence of body forces are

tij ,y = pO2ui/~t 2, (4)

where p is the density. Using (1), equation (4)becomes

(Cijkl Uk, l), j = P~2Ui/~t2, (5)

this form of the equations being adopted since later the Cijkl will be assumed to vary.

3. Elastodynamics

T~EOR~M 1. Suppose the motions of a continuous homogeneous elastic body ~ satisfy equations (5). I f all the stresses are zero in a finite three dimensional subregion of the body for

Journal o f Elasticity, 2 (1972) 201-204

202 Michael Hayes

a finite time then all the stresses are identically zero everywhere in the body for all time, if and only if

det (Cijkl,~j};l--p~2t~ik) ~ 0 V real _2, tp s.t. 2i,~i--}-(~ 2 ~ O. (6)

Proof. The equations of motion (5) form an elliptic system [2, page 136] if (6) is satisfied. Now any solution of class C 2 of the second order linear elliptic system (5) which has real ana- lyric coefficients consists of real analytic functions in ~ for 0 ~< t < ~ . Thus the solutions ui(xj, t) of (5) are real analytic functions. So too are the stresses tij. Hence if t~j = 0 in a finite subregion of ~ for a finite time,

ti~ = 0 (7)

everywhere in ~ '¢t • 0 ~< t < oo. To prove that (6) is also necessary suppose it is not satisfied or equivalently, that for some = ¢ = ¢ , ,

det (Ciju2*A*--p~*2~ik) : 0, (8)

where ~b* is assumed > 0 without loss of generality. Then 3R # 0_ s.t.

-pq~ 6,k)Rk = 0. (9)

Now consider the displacement field

u, = Ri(2*x j - qS*t) ~, (10)

where n is any real number : n > 3. The equations of motion (5) are satisfied by (10) in view of (9), for all n > 3.

Consider the displacement

{Ri(2*XiodP*t)3 [ ).*xj-q~*t <= O, t >= 0; (11) ui = 2*xj -~*t > 0, t > 0.

Here _u e C 2 and satisfies the equations of motion (5). For ~' take any finite continuous body bounded by 2sx j = T~b*, where T > 0, and lying in the region 2*xj-Tda* > 0. The stresses are zero everywhere in ~' for t • 0 ~< t < T, but for t > T, the stresses are not zero everywhere in ~ .

REMARK 1. The condition (7) means that there are no real characteristic surfaces. Hence discontinuities may not propagate. It is hardly surprising therefore that if part of the body is quiescient for a finite time it will always remain quiescient.

REMARK 2. Nowhere in the proof of the Theorem 1 have the symmetry conditions (2) been used. It follows that the result applies in more general contexts e.g. in the theory of small de- formations superposed on large of Caucy elastic materials.

REMARK 3. Theorem 1 may be extended to inhomogeneous elastic bodies whose elastic properties change with time provided the Cijkt are assumed to be analytic functions of_x and t throughout ~ Vt/> 0. The solutions of (5) are then analytic functions of x and t provided (6) holds at every point of ~ Vt ~> 0. Sufficiency follows as before. To show necessity consider the particular case of a homogeneous body with time independent CljRZ and use (l 1).

REMARK 4. In Theorem 1 if the word "stresses" ~s replaced by "displacements" the resulting theorem is equally valid. The proof goes through as in Theorem 1 with minor modifications.

On a theorem of Muskhelishvili 203

4. Elastostatics

Muskhelishvili proved the result quoted in the Introduction in the plane theory of elastostatics. The corresponding result for three dimensional anisotropic media is the

THEOREM 2. Suppose a continuous homogeneous elastic body ~ is in static equilibrium in the absence of body forces. If all the stresses are zero in a finite three dimensional subregion of the body then they are identically zero everywhere in the body if and only if

det (Cijkl~j21) # 0 ~f real ~ # O. (12)

Proof. (12) is the condition that the equilibrium equations

(c,jk, uk,,), j = 0 (13)

be elliptic. The Cljkl a r e constants and hence the solutions ui of (13) are analytic functions. Thus t u are analytic and hence if t u are zero in a finite subregion of ~ they are zero everywhere in

To prove necessity note that if (12) is not satisfied i.e. if for some _~ = _2",

det (Cijkl,~j •I ) = O, (14)

then 3R # _0 s.t.

ciju 2j 2t RR = 0. (15)

Hence, as before, we have the C z solution:

* 3 * Ri()~ j x j ) > 0 ; , 2j xj _ (16)

U i ~- t 0 , "* < 0), Aj Xj =

to the equilibrium equations (13). The corresponding stress field is

t * * 2 tu = 3CijuRk2 t (2~xp) , 2*x~ > O; (17)

t 0 , 2pXp -<_ 0.

The tractions are continuous across 2*Xp = 0. For M take the unit sphere

xpxp < 1. (18)

The hemisphere

XpXp <<_ 1, 2*xp < 0 (19)

is unstrained and unstressed, whilst in the hemisphere

xpx~ < 1, 2*xp > O, (20)

t u ~ O. Note that t u = 0 on 2px~ = 0. Thus a finite subregion of ~ is unstrained and un- stressed - but the stress is not zero everywhere in M.

The remarks 2, 3 and 4 again apply.

Acknowledgement

I am indebted to R. J. Knops for his comments on an earlier draft.

204 Michael Hayes

R E F E R E N C E S

[1 ] Muskhelishvili, N. L, Some basic problems of the mathematical theory of elasticity. (Translated by J. R. M. Radok) Noordhoff, Groningen (1963)

[2] John, F,, Plane waves and spherical means applied to partial differential equations. Interscience, New York (1955)