Some Theorems in Classical Elastodynamics LEwis T. WHEELER & ELI STERNBERG Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. N o t a t i o n a n d M a t h e m a t i c a l Preliminaries . . . . . . . . . . . . . . . . . . . . 2. E x t e n s i o n of the U n i q u e n e s s a n d the Reciprocal T h e o r e m in E l a s t o d y n a m i c s to U n bounded Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Basic Singular Solutions. L o v e ' s Integral Identity for the D i s p l a c e m e n t s a n d its C o u n t e r part for the Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. G r e e n ' s States. Integral R e p r e s e n t a t i o n s for the Solutions to the F u n d a m e n t a l B o u n d a r y initial Value P r o b l e m s of E l a s t o d y n a m i c s . . . . . . . . . . . . . . . . . . . . . 5. A U n i q u e n e s s T h e o r e m for C o n c e n t r a t e d - l o a d P r o b l e m s in E l a s t o d y n a m i c s . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 52 57 65 77 85 89

Introduction

The linearized dynamical theory of elasticity has long been a highly developed and, in large measure, complete discipline. It is therefore not surprising that most of the recent publications in this area of interest are concerned with the exploration of exact or approximative methods for the solution of relevant problem-classes and with specific wave-propagation problems. The present investigation - although ultimately motivated by physically significant applications - can make no such immediate practical claims. Our main objective is to study certain general consequences of the equations governing classical elastodynamics with limitation to mechanically homogeneous and isotropic solids. Some of the results presented in what follows aim primarily at a clarification, strengthening, and extension of theorems previously available. In contrast, the work on Green's functions, integral representations, and concentrated loads in dynamic elasticity, would appear to fill a gap in the existing literature. In Section 1 we dispose of required geometric and notational preliminaries. Here we also cite pertinent properties of Riemann convolutions and introduce the notion of an "elastodynamic state", which lends economy to subsequent developments. In Section 2 we deduce a generalized energy identity and use the latter to extend the conventional uniqueness theorem of elastodynamics to unbounded domains in the absence of artificial restrictions upon the behavior of the velocities or stresses at infinity. Further, we employ the foregoing energy identity to establish sufficient conditions for the prolonged quiescence of the far elastodynamic field belonging to a solution that corresponds to initial quiescence. This result, in turn, supplies the principal tool for a generalization of Graffi's dynamic reciprocal theorem 1 to infinite regions, which concludes Section 2. 1 Detailed references to the literature c a n be f o u n d in the b o d y of this investigation. 4*

52

L.T. WHEELER• E. STERNBERG"

Section 3 is partly expository. Here we first cite Stokes' solution for a timedependent concentrated load at a point of a medium occupying the entire space. We then examine relevant properties of this solution and of the singular solutions appropriate to force-doublets. This material is followed by a systematic derivation, based on the reciprocal theorem given in Section 2, of Love's integral identity for elastodynamic displacement fields. Finally, at the end of the section, we deduce an analogous integral identity for the associated stresses. In Section 4 we take the integral identities of the preceding section as a point of departure in deriving integral representations for the solution to the first and second boundary-initial value problem of dynamic elasticity. This task is accomplished through the introduction of suitable Green's solutions of the first and second kind. We also include here some supplementary results on properties of the Green's solutions with a view toward facilitating their actual construction. Finally, Section 5 is devoted to a uniqueness theorem for the second boundaryinitial value problem encompassing time-dependent concentrated loads with stationary points of application in the interior or on the surface of an elastic solid. This theorem is proved with the aid of the Green's solutions introduced in Section 4. 1. Notation and Mathematical Preliminaries

Throughout this investigation, lower-case light-faced Latin or Greek letters stand for scalars; lower-case Latin letters in boldface denote vectors, while lowercase Greek letters in boldface designate second-order tensors. Upper-case letters are ordinarily reserved for sets; in particular, upper-case script letters are used exclusively for sets of functions. The letter E is set aside for the entire threedimensional euclidean space. If x is the position vector of a point in E, the symbols B~(x) and S~(x) are employed, respectively, for the open spherical neighborhood (ball) of radius 6 about x and for the spherical surface of radius 6 centered at x. Thus,

n~(x)={yly~E, ly-xl<6)(6>0), S~(x)={yly~E, ly-xl--6}(6>0).

(1.1) (1.2)

Further, we agree to write B~ and S~ in place of B~ (0) and S~ (0). The symbol R, in the absence of any qualifying restrictions, will always denote an arbitrary region in E, i.e. an open connected set in E together with some, all, or none of its boundary points. The interior, the closure, and the boundary of R - in this order - will be designated by/~, R, and ~R. Further, if x~R, we agree to let R x represent the set obtained from R by deletion of the point x and w r i t e / ~ in place of (R)x. In particular, we say that R is a regular region if it is open and there is a 60 > 0 such that for every 6 > 6 o the set R c~B~ is connected and has for its boundary a finite number of "closed regular surfaces", the latter term being used in the sense of KELLOGG [1] (page 112). Note that a regular region, as defined here 1, need not be bounded and, if unbounded, need not be an exterior region since its boundary 1 Our definition of a regular region differs from, and is considerably broader than, KELLOG6'S [ll (page 113).

Theorems in Elastodynamics

53

m a y extend to infinity. In addition, the boundary of a regular region may have edges and corners. If R is a regular region, we designate by OR the subset of dR consisting of all "regular boundary points", i.e. the set of alI points of gR at which its normal is defined. We will use the symbol T for the entire real line and T for an arbitrary (open, closed, or half-open) interval of 7~. The interior and closure of Twill be designated by T and T, respectively. Finally, we adopt the notation T - = ( - oe, 0],

T+ = [ 0 , oo).

(1.3)

If a and b are vectors, a. b and a ^ b are their scalar and vector product, respectively. Standard indicial notation is used in connection with the cartesian components of tensors of any order: Latin subscripts and superscripts - unless otherwise specified - have the range (1, 2, 3), summation over repeated indices being implied. Also, if ~ and a are second-order tensors, we write ~. a for the fully contracted outer product eij trij. As usual, 6~j is the Kronecker-delta. We will frequently need to deal with scalar-valued and tensor-valued functions of position and time, having as their domain of definition the cartesian product of a set P in E and a time-interval T. If q~ is a scalar-valued function defined on 1 P x T, we denote its value at (x, t)~P x T by q~(x, t) and mean by ~0(-, t), the subsidiary mapping of P obtained upon holding t fixed. The analogous interpretation applies to ~o( x , . ) and to tensor-valued functions. As for time and space-differentiation, we write q~(m~_

(P

otra '

rp~-

(P

t ~ X i O X j .,. ~ X k

(m = 1, 2 .... ),

(1.4)

(m indi~es)

provided the partial derivatives here involved exist. Ordinarily we shall write ~b, instead of (po), 9(1). Analogous notations will be employed for differentiation of tensor-valued functions of non-zero order. We turn now to notational agreements related to the smoothness of functions. If P is a set in euclidean n-space, we denote by c~(p) the class of all tensor-valued functions of any order that are defined and continuous on P. Next, if e is a positive integer, we say that a function belongs to c~,(p) if and only if it is in c~(p) and its partial derivatives of order up to and including e exist on the interior of P and there coincide with functions that are continuous on P. If P is a set in E, T is a timeinterval, and e a non-negative integer, we let c~(,)(p x T) stand for the set of all functions in cg(p x T) having continuous partial time-derivatives of order up to and including c~ on P x T, provided each of these derivatives 2 coincides on P x 7~ with a function continuous on P x T. Finally, f#" denotes the class of all functions in ~f'(T) that vanish on T - . The order-of-magnitude symbols " O " and "0" are used consistently in their standard mathematical connotation. For example, if y s R , v is defined on R s x T, and n is a real number, we write v ( x , . ) = O ( [ x - y l " ) as x ~ y , uniformly on 1 Here and in the sequel we use the conventional notation for the cartesian product of two sets. 2 Observe that the class of functions c~(o)(p X T) is identical with ~(P x T).

54

L.T. WHEELER(~6 E. STERNBERG:

( - m , t], if and only if there exist real numbers 6 ( 0 and m(t) such that x ~ Rs c~Bb(y) implies I t (x, ~) 1< m [x -- y I" for every x ~ (-- 0% t ]. For future convenience we now recall a version of the divergence theorem that is adequate for our purposes. Theorem 1.1. Let R be a regular region and let n be the outward unit normal of OR. Suppose

f ~ ( n ) c~~(R), and assume the set { x l x E R , f(x)4:0} has a bounded closure, so that f is of bounded support. Then, I V.fdV= Iy.nda, R

(1.5) 1

$R

provided the volume integral in (1.5) exists. The truth of the preceding theorem follows trivially from the strongest form of the divergence theorem considered and proved by KELLOGG [1] i (page 119). Next, we collect here certain results from the theory of Riemann convolutions that will be needed later on. To this end we first introduce Definition 1.1. (Convolution). Let P be a set in E and suppose (pe~(P x T+),

Oaeg(P x T + ) .

We call the function 0 defined by i O(x, t)=

0

for all ( x , t ) ~ P •

q~(x,t - z ) ~(x, z) dz

for all (x, t ) ~ P x T+

(1.6)

the convolution of ~o and ~b. We also write

~=~,,6,

O(x,O=[~,qq(x,O

to denote this function and its values. I_~mma 1.1. (Properties of convolutions). Let P either be an open or closed region in E or a regular surface in E. Let

Ce(g(PxT+),

4J ~r

x T+),

co~(V x T+).

Then: (a) q~*$~(g(P x 3); (b) ~ o , r 1 6 2 (c) r 1 6 2 1 6 2

(d) r162 (e) q) * t# = 0 on P x T + implies ~p= 0 on P x T + or $ = 0 on P x T +. 1 Here Vis the usual gradient operator. 2 To avoid confusion we emphasize that KELLOGG'S"regular region" is a closed region, the boundary of which is a single "closed regular surface" (in KELLOGG'Ssense of the latter term).

Theorems in Elastodynamics

55

Property (a) is an elementary consequence of Definition 1. I. Proofs of (b), (c), (d), and (e) may be found in MI~:USlNSKFS [2] 1 book. The following two lemmas are readily inferred from Definition 1.1 and (a) of Lemma 1.1. Lemma 1.2. (Time-differentiation of convolutions). Let P be as in Lemma 1.1 and let q~~

x T+),

~ k ~ ( P x T+),

0=~o*~.

Then:

(a) 8 ~ ( 1 ) ( P x T+); (b) ,~=tp.$+q~(.,0)~b on P x T + ; (c) q~(., 0 ) = 0 on P implies Oe~(1)(P x T). Lemma 1.3. (Space-differentiation of convolutions). Let R be an open or closed region in E and let

rp ~cgl(R x T+),

~k~cgl(R x T+),

,9=tp. ~k.

Then:

(a) ,gecgl(R

x

T+);

(b) O,i=q~,i*~k+tp*~,i on [~ x T+ ; (c) ~0(.,0)=0 or ~b(.,0)=0 on R implies 8~cgl(R x T). As for convolutions of a scalar and a vector-valued function or of a scalar and a second-order tensor-valued function, we agree to write v=tp . u

a=,r

if and only if vi = tp. ui, if and only if O u = t p . ~ u .

(1.7)

Further, we adopt the notation U*V~Ui*Vi,

o,~=~,j, ~j.

(1.8)

The remainder of this section is devoted to essential preliminaries pertaining to the linearized dynamical theory of homogeneous and isotropic elastic solids. For this purpose we introduce Detinition 1.2. (States. Elastodynamic states). Let R (not necessarily open or closed) be a region in E and let T (open, dosed, or half-open) be a time-intervaL I f u and a are, respectively, a vector-valued and a second-order tensor-valued function defined on R x T, we call the ordered pair 6a = [u, a] a state on R x T. We say that 6 v = [u, a] is an elastodynamic state with the displacement f i e M u and the stress field a, corresponding to the body-force density f , the mass density p, the dilatational wave speed cl, and the shear-wave speed Cz, and write 5e = [u, ~r] e 8 ( f , p, cl, c2; R x T ) , provided:

(a) ue~2(/~ x T) c~ Cg~(R x T ) , aeCg(R x T ) , f ~Cg(R x T ) , 1 Properties (b), (c), and (d) are established in Chapter I of [2]. TITCHMARSH'Stheorem (e) is proved in Chapter II.

56

L.T. WHEELER• E. STERNBERG:

while p, el, and c 2 are constants subject to 2

p>0,

0 < - ~ - f c2 < c l ;

(1.9)

(b) u, ~, f , p, cl, and c2 on R x 7"satisfy the equations 6ij, jq-fi=PiJi,

(1.10)

ai j = p(c~ -- 2C~) ai j Uk, k + 2 p C~ U(i' j) .

(1.11) 1

lf, in particular,

T=T,

u=0

on

RxT-,

(1.12)

we say that 5 r is an elastodynamic state with a quiescent past and write

6~ [u, a ] e 81 (f, p, c,, c2; R).

(1.13)

Equations (1.10) represent the stress equations of motion - (1.11) the stressdisplacement relations of classical elastodynamics. In view of (1.10) and (1.11), the regularity assumptions under (a), though mutually consistent, are partly redundant. Note that (1.11) implies the symmetry of the stress-tensor field a on R x T since a is continuous on R x T. The wave speeds c t and c2 are expressed by c1

=1/2(1-v)F, V (l_2v)p,

c2=

(1.14)

in terms of p, the shear modulus #, and Poisson's ratio v of the elastic solid. Also, the inequalities (1.9) required under (a) of Definition 1.2 are equivalent to p>0,

-l
(1.15)

Moreover, (1.9) assure the positive definiteness of the quadratic function e that is defined by e(~o)= P

[-(c~- 2c~) (o,, ~0jj + 2 cg~0,j ~0,f]

(1.16)

for every symmetric second-order tensor 9- If 8 is the infinitesimal strain tensor associated with u, i.e. ~ i = u(i, j ) , (1.17) then e(8) represents the strain-energy density appropriate to the elastodynamic state Se. If R is a regular region, 6 ~ = [u, a] is a state on R x T, and n is the unit outward normal vector of aR, we call the vector-field s defined by si=ffijn j

on

3RxT,

(1.18) 2

the tractions of Sa acting on ~3R. 1 If g is a second-order tensor, ~v(ij)and g/rlj] are the components of the symmetric part and of the skew-symmetricpart of u respectively. 2 Recall that ~R represents the set of all regular boundary points.

Theorems in Elastodynamics

57

We now define equality and addition of states, as well as multiplication of a state by a scalar constant. To this end let R be an arbitrary region, suppose 6 ~ = [u, tr] and Se' = [u', a'] are states on R x T and let 2 be a real number. Then,

~ = S e ' r162u=u',

a=a'

6e+Sa'=[u+u',tr+a'] 25~=[-2u, 2 a ] Next, with reference to (1.4), we write . , . (~) ~ ' = 6 p(~) r u i = , i ,

on on

_,

R x T,

on

RxT,

RxT. _(~)

oij=oij

on

o

RxT,

(1.19)

and, for fixed k, adopt the notation

6P'=S(,k r

U;=Ui, k,

tr;i=trij, k

on

/~xT,

(1.20)

provided the required time and space-derivatives exist. Finally, ~o* 5 a = [-~o * u, q~ * a]

(1.21)

whenever the underlying convolutions are meaningful. 2. E x t e n s i o n of the U n i q u e n e s s and the R e c i p r o c a l T h e o r e m in E l a s t o d y n a m i c s to U n b o u n d e d R e g i o n s

The current section serves a dual purpose: here we extend NEUMANN'S [3] uniqueness theorem of classical elastodynamics to unbounded domains and subsequently generalize GRAFFI'S [4] reciprocal identity to a pair of clastodynamic states associated with an infinite region. The results thus obtained are essential prerequisites to the determination of integral representations for the two fundamental problems of dynamic elasticity carried out later on; at the same time these results are apt to be of interest in themselves. The principal tool used to establish the two theorems alluded to above is supplied by a generalized energy identity, which we state and prove presently. This lemma is an elastodynamic counterpart of a result due to ZAREMBA [5] for the scalar wave equation 1. Indeed, our method of proving the generalized uniqueness theorem is suggested by the treatment in [5] of uniqueness issues pertaining to the wave-equation. A lucid account of ZAREMBA'Spaper is given by FRITZ JOhN in [6]. Lemma 2.1. (Generalized energy identity). Suppose R is a regular region and (a) 5a=[u,a]eSo(f, p, c 1, e2;/~), (b) zrcgl(R) is a given (scalar-valued)function such that the set

(xlx~, z(x)>0} is bounded. Let q~ be the (second-order tensor-valued)function defined by q~ij(x)= zl [O-~j u,(x, z(x))+-~xiO Uj(X'Z(x))]

for a l l x ~ R .

(2.1)

1 ZAREMBA'Senergy scheme was rediscovered independently by RUBINOWtCZ [7] and by FRIEDRICHS(~; LEWY [8]. See also COURANT& HmBERT[9] (pages 659-- 661) where ZAREMBA'S result is extended and applied to the general second-order hyperbolic equation.

58

L.T. WHEELER(~;E. STERNBERG:

Then

(x)

9 (x)

~ u(x,t).s(x,t)dtdA+ S ~ u ( x , t ) . f ( x , t ) d t d V OR 0

R

0

= ~ {e(q)(x))+ 2P---~2(x,z(x))[1-c~(Vz(x)) 2]

(2.2)

P (c~- c~) [~(x, ~(x)) ^ W(x)] 2} dV, +T where s are the tractions of 6a acting on OR and the function e is given by (1.16). Proof. For convenience introduce the auxiliary vector-valued functions p and v through ~(x)

pi={~jtrij on R x T ,

vi(x)= S pi(x,t) dt

for all x ~ R .

(2.3)

0

In view of the smoothness of z stipulated in (b), and because of the regularity properties implied by (a) and Definition 1.2, v ~ c~l(g) c~ ~ ( / ] ) .

(2.4)

Further, v has bounded support by virtue of hypotheses (a), (b) and (1.12). From (2.3), (1.10), (1.11), and (1.17)follows V.v(x)= 9!(x) {~(x,t).a(x,t)+ p +p(x, z(x)). Vr(x)

~t

[u(x,t)]2-u(x,t).f(x,t)} at

(2.5)'

for all x e R ,

while (1.11), (1.16) yield o e(e(x, t)) k(x, t). a(x, t) =-~-

for all (x, t)~R x 7".

(2.6)

Now substitute from (2.6) into (2.5) and use (1.12) to infer

V. r (x)= e(~(x,9 (x)))+ p(x, z (x)). W (x) 9(,o P i~2(x,z(x))- ~ i~(x,t).f(x,t)dt. +-2 o

(2.7)

Next, note from (2.1), (1.17) that

q,.(x) = ~,~(x, ~(x)) +89E~,(~, -~(x)) ~. j(~) + ;,j(~, .~(x)) -~, ,(x)], whence (1.16), (1.11), and the first of (2.3) furnish

e(q)(x))= e(8(x, z (x)))-I-p (x, z (x)). Vz (x)

+~P (c~-c~)[a(~,

~(x)) 9 w(~)] 2 + Tp c2~ a2(x, ~(x))(w(~))

This equation, because of Lagrange's identity

~2(Vz)2= Oi. W) 2 + (fi^ W) 2, 1 Recall that ~ 9 r

aij.

2.

Theorems in Elastodynamics

59

may be written as e(8(x, ~(x)))+ p(x, r(x)) 9 V z ( x ) = e ( ~ ( x ) ) - 2 c~ ire(x, z(x))(Vz(x)) 2 (2.8)

P (c~-c~)[[,(x, z ( x ) ) ^ Vz(x)] 2

for all x e R .

Combining (2.8) with (2.7) one has + P

v-

(.)

-

(2.9)

!

for all x~R. From (2.9), the regularity assumptions contained in hypotheses (a) and (b), and the boundedness of the support of v, it is clear that V. v is properly integrable on R. Thus, integrating both members of (2.9) over R, one is entitled subsequently to apply the divergence theorem (Theorem 1.1) to the vector field v since the latter conforms to (2.4) and is of bounded support. The desired result (2.2) then follows immediately with the aid of (2.3) and (1.18). This completes the proof. Suppose now in particular R in Lemma 2.1 is bounded and restrict T to be a positive constant, say z = t. In these circumstances one recovers from (2.2) the classical energy identity of elastodynamics in the form t

t

S S[,(x, 2).s(x, 2 ) d 2 d A + ~ i t ( x , , ~ ) . f ( x , , ~ ) d 2 d V ~R o

R o

(2.10)

=~[e(,(x,t))+ p ue(x,t)]dV. As will become apparent shortly, the role played by the generalized energy identity (2.2) in connection with the extended uniqueness theorem to which we turn now is strictly analogous to that played by (2.10) in N~UMarCN'S [3] familiar uniqueness argument for bounded regions. Theorem 2.1. (Generalized uniqueness theorem). Let R be a regular region and let St', St" be two states with the following properties: (a) S t ' = [u', a ' ] e # 0 r, p, cl, c:; ~ x T+), S t " = [ u " , o " ] ~ # ( f , p, cl, c2; ~ x T + ) ; (b) u ' ( . , 0 ) = u " ( . , 0 ) , further, suppose either (c) u ' = u " on a R x T + OF

u'(.,0+)=,~"(.,0+)

on R;

,

(d) s' =s" on OR • T +, where s' and s" are the respective tractions of 6"' and St" acting on OR. Then St' = St" on R x T +. Proof. Define the state St on/~ x T by St=[u,a]=st'-st"

on R x T +,

u=~=0

on ~ x

f- .

(2.11)

60

L.T. WHEELERt~ E. STERNBERG:

From (a), (b), (2.11) and Definition 1.2 one finds without difficulty that 9" ~8o(0, p, cl, c2 ;/~).

(2.12) 1

By (2.1 I) and (1.18), since either (c) or (d) holds, u. s = 0

on ~R x T,

(2.13)

where s are the tractions of 9' acting on ~R. Now fix (x, t)~R x T+ and define the scalar-valued function z through

z(y)=t-ly-x[/2cl

for all y E R .

(2.14)

Then, evidently, ZEc~l(/~x)C~ ~(R),

[17z(y)]2= 4-~12

for all yeRx,

(2.15) 2

and because c 1 is positive by (2.12) and Definition 1.2, (yly~/~, z(y)>0} is bounded.

(2.16)

Choose rio > 0 such that B~o(x) ~ R and set

R~=R-B~(x) (0
(2.17)

In view of (2.12), (2.15), (2.16) and Lemma 2.1, one concludes that (2.2) holds for each member of the family of regular regions defined in (2.17). Thus, hearing in mind (2.13), the second of (2.15), and the fact that the body-force field of 9" vanishes identically, one has

S

S i~(y,2).s(y, 2)d2dA

s~(x) o = ~ {e(~a(y)) + ~ -

(2.18)

l12(y, "C(y))+~

(C2--C2)[U(y, "C(y)) A V'c(y)]2} dV

for every 6~(0, tSo), where the functions e and ~ are given by (1.16) and (2.1), while s now stands for the tractions of 9" acting on dry. Owing to the continuity of z on/~ and of/t, a on R x T, lim ~ ~ u(y, 2).s(y, 2)d2dA=O, 8-*0 S~ (x) 0 so that passage to the limit as fi ~ 0 in (2.18) gives {e(~p(y)) + - ~ - u2(y, z(y))+ 2

(c~-c~)~il(y, z(y))^

Vz(y)] 2}

dV=O.

(2.19)

i Here and in the sequel,we write0 in place of the body-force argument of the elastodynamic state under consideration if the body forces vanish identicallyon the appropriate space-time domain. 2 Note that the gradient of the function z given by (2.14)has an (irremovable)finitediscontinuity at y.

Theorems in Elastodynamics

61

Recall next that the inequalities (1.9), which are implied by (2.12), are sufficient for the positive definiteness of e. Moreover, (1.9) assure that all terms in the integrand of (2.19) are non-negative. Therefore, and since the integrand in (2.19) is continuous on Rx, u(y, z ( r ) ) = 0 for every y ~ R x. ~o

Finally, invoke the first of (2.15) and the regularity of u on R x T implied by (2.12), and use (2.14) to confirm that

i(x, t) =,i(x,

= 0.

Consequently, (x, t) having been chosen arbitrarily in R x 7~+, u=0

o+

on R x T

.

(2.20)

But (2.20) and (2.12) furnish u=~=0

on

/~x'F.

The desired conclusion now follows from (2.11). An extension of Theorem 2.1 to mixed boundary conditions is entirely elementary. Similarly, the generalization of Lemma 2.1 and Theorem 2.1 to anisotropic and nonhomogeneous solids presents no difficulties. Next, in the first boundaryinitial value problem (surface displacements prescribed) uniqueness prevails for unbounded domains even if (1.9) is replaced by the weaker requirement that cl and c a be real, as can be shown by adapting an argument due to GURTIN & STERNBERO [10] for bounded isotropic elastic bodies 1. The relaxation of the rather stringent smoothness hypotheses involved in (a) of Definition 1.2, which render Theorem 2.1 inapplicable to certain physically important problems, is in need of further attention a. It should be pointed out that an elastodynamic uniqueness theorem valid for infinite regions may alternatively be based on the classical energy identity (2.10), following Neumann's procedure, if one introduces suitable restrictions on the orders of magnitude of the velocity and stress field at infinity. The essential advantage of Theorem 2.1 stems from the fact that it does not involve such artificial a priori assumptions. In this connection we recall that the analogous uniqueness issue in elastostatics, where the governing equations are elliptic rather than hyperbolic, is considerably more involved. For exterior unbounded domains elastostatic uniqueness theorems that avoid extraneous order prescriptions at infinity were established by FICHERA [13], as well as by GURTIN t~r STERNBERG[14]. On the other hand, the uniqueness question associated with boundary-value problems in the equilibrium theory for general domains whose boundaries extend to infinity is yet to be disposed of satisfactorily a. 1 See also GtmTIN& TOUPIN[11 ], where the result of [10] is extended to anisotropic media. 2 In this connection see a recent paper by KNoPs & PAYNE[12], which contains a uniqueness theorem for weak solutions in elastodynamics, with limitation to bounded domains. 3 For the special case of the first and second equilibrium problem appropriate to the halfspace this question was settled by TURTELTAUBt~r STERNBERG[15].

62

L.T. WHEELER&E. STERNBERG:

In preparation for a generalization of GeO,FF?S [4] dynamic reciprocal identity to unbounded regions we now proceed to Lemma 2.2. (Sufficient conditions for the prolonged quiescence of the far field). Let R be an unbounded regular region and suppose: (a) Se= [u,o]~go(f, p, cl, C2; R--); (b) for every t > 0 there is a bounded set A ( t ) = R such that

f=o

on ( R - A ( t ) ) x [0, t],

and, if OR is unbounded,

~.s=O

on ( O R - a ( t ) ) x [0, t],

where s are the tractions of 6e acting on OR. Then, for each t >0, there is a bounded set f2(t)cR, depending only on A(t), such that

u=a=0

on

(/q-f2(t)) x [0, t].

(2.21)

if 0R is bounded, if aR is unbounded,

(2.22)

Proof. Fix t > 0, let 6 > 0 be such that

ORuA(t)cB 6 A (t) c Ba and consider the set

12(t) = R c3 B~+ac, t.

(2.23)

Note that fa(t); as defined in (2.23), is a bounded subset of R. With a view toward showing that (2.21) holds, choose (y, 2) e (R - fa(t)) x (0, t]

(2.24)

and regard (y, 2) as fixed. Define the f u n c t i o n , by

"c(x)=2-lx-yl/2c,

for all xe.R.

(2.25)

Evidently,

z ~ l ( ~ y ) c3 r

[Vz(x)]Z= ~-~c'

for all x~Ry,

(2.26)

and since cl > 0, {x Ix e R, z (x) > 0} =/~ m B z a c, (Y).

(2.27)

From (2.23) and (2.24) one draws that B= a c, (Y)does not intersect B~. Thus, (2.22) and (2.27) imply

{ x l x e R , z ( x ) > O } c R - A ( t ) u OR if OR is bounded, {x [ x e g , z ( x ) > O } = K - A ( t ) if OR is unbounded.

(2.28)

Now call on (2.24), (2.25) to arrive at

,(x)
for all x e K .

(2.29)

Theorems in Elastodynamics

63

Hypothesis (a) requires u to vanish on ~ x T - . This fact, in conjunction with (2.28), (2.29) and hypothesis (b), justifies

S is(x, tl)'f(x, rl)drl=O

0 (x)

o

it(x, tl).s(x, tl)dtl=O

for all x E R , , for all xeOR.

(2.30)

Next, let ~o > 0 be such that Br ( y ) c R and put

(2.31) One concludes from (2.26), (2.27), hypothesis (a), and Lemma 2.1 that (2.2) holds for each Re in (2.31). Thus, (2.30) and the second of (2.26) yield 9r (x)

j

s~ (y) o

= S {e(q~(x))+~P-- fi2(x, T(x))+P(c~-c~)[fi(x, z(X))A Vz(x)] 2} dV, (2.32) R~

for every #e(0, ~o), where e and q~ are given by (1.16) and (2.1), while s here denotes the tractions of Sf acting on are. Since ~ is continuous on R and ti, tr are continuous on R • 7", the left-hand member of (2.32) tends to zero as ~ ~ 0 , whence S {e(q~(x)) +-~P- 1~2(X, ~'(X))'dt--~ (C2--C2)[U'(X, 27(X)) A 17T(X)] 2} R

dV:O,

(2.33)

The inequalities (1.9), which are implied by hypothesis (a), are sufficient for the positive definiteness of e and ensure that each of the three terms of the integrand in (2.33) is non-negative. Accordingly, this integrand being continuous on Rs,

u(x,z(x))=O

for all x~Ry.

Invoking once again the continuity of 9 on R and of ti on R • 7", one finds that

,i(y, ;0 =,i(y,

= o.

But (y, 2) was selected arbitrarily in ( R - I 2 (t)) x (0, t ]. Hence ti=0

on (R-f2(t)) x(0, t],

which, because of the regularity and initial quiescence of u assumed in hypothesis (a), gives u=0 on (R-f2(t)) • [0, t]. (2.34) By (2.34), and because (1.11) hold on R x T, a=0

on (g-~2(t)) • [0, t].

(2.35)

Recalling that t2(t) is closed, one shows readily 1 that the closure of R - f2(t) contains _~- f2(t). Therefore, appealing to the continuity of u and tr on ~ x ?', one

1 Cf. Exercise 1 (page 37) in [16].

64

L.T. WHEELER & E. STERNBERG"

sees that (2.34), (2.35) imply (2.21). Finally, note that (2.22) and (2.23) imply that 12(t) depends exclusively on A(t). Since t was chosen arbitrarily, the proof is now complete. It is essential to recognize that if a state with a quiescent past is characterized as the solution of a standard boundary-initial value problem in elastodynamics, the decision whether or not hypothesis (b) of Lemma 2.2 is met, is immediate from the data. Theorem 2.2. (Extension of GRAFFI'S reciprocal identity to unbounded regions).

Let R be a regular region and suppose: (a) 5e=[u, al~eo(f, p, c1, c2; R--), 5a'= [u',a']~eo(/', p, cl, c2;/~); (b) 6a satisfies hypothesis (b) of Lemma 2.2 if R is unbounded. Then, for every t > O, 5 [s*u'](x, t) dA+ 5[f*u'](x, OdV= ~ [s'*u](x, t) dA dR

R

OR

(2.36) 1

+ ~ If' 9 u] (x, t) av, R

where s and s' are the tractions of 6e and 5a' acting on OR. Proof. It is clear from the present hypotheses and Lemma 2.2 that the integrals in (2.36) are proper even if R is unbounded. Choose t > 0 and hold t fixed for the remainder of the argument. Define the vector field v by

vi(x)=[a,j*u~.](x, t)-[a~j*uj](x, t)

for all x e R .

(2.37)

In view of hypothesis (a), Definition 1.2, Lemma 1.1, and Lemma 1.3,

r ~ I ( R ) (~ ~(-~),

(2.38)

v,, ,(x)= [~,j, ,, u~] (x, 0 + [o,j * u~, ,3 (x, t ) - [~j, ,, ui] (x, t ) - [ ~ j 9 u j, ,] (x, t) for all x~R. Hence hypothesis (a), (1.10), together with the symmetry of a and a', furnish V. v (x) = p [fi 9 u'] (x, t) - [.if'. u'] (x, t) + [a 9 d ] (x, t)

-p[fi'*u](x,t)+[f'.u](x,t)-[a'

,n](x,t),

(2.39)

where On the other hand, (1.11), (1.8), and the commutativity of convolutions asserted in (b) of Lemma 1.1, imply a * 8' = a' * ~ on R • ~. (2.40) Now note from hypothesis (a) and Definition 1.2 that

u(.,0)=u'(.,0)=~(.,0)=~'(.,0)=0

on R.

Consequently, two successive applications of (b) in Lemma 1.2 give u*u'=u,~',

il,u'=u,li'

1 Recall the notations adopted in (1.8).

on Rx2~ +.

(2.41)

Theorems in Elastodynamics

65

Combine (2.39), (2.40), and (2.41) to obtain V. r ( x ) = [ f ' , u](x, t ) - [ f ,

u'](x, t)

for every x e R .

(2.42)

From hypotheses (a), (b), Lemma 2.2, and (2.37), one infers that r has bounded support. This being the case, (2.42) and the continuity o f f ' . u and f . u' on K x assured by Lemma 1.1 imply that V. v is properly integrable on R. The preceding observations enable one to apply the divergence theorem (Theorem 1.1) to v on R. In this manner and by recourse to (2.37), (2.42), and (1.18) one confirms that (2.36) holds. This completes the proof since t was chosen arbitrarily. It is worth mentioning that the foregoing argument, in contrast to GRAFFI'S [4] proof (which is confined to bounded regions), avoids the use of the Laplace transform. 3. Basic Singular Solutions. Love's Integral Identity for the Displacements and its Counterpart for the Stresses In this section, which is partly expository, we first cite the fundamental singular solution of the field equations in elastodynamics. This solution, due to STOr~.S [17], corresponds to the problem of a time-dependent concentrated load at a point of a medium occupying the entire space. We then establish certain relevant properties of STOIC~S'solution and of the associated dynamic doublet solutions. The foregoing singular states are subsequently used to establish in an economical manner LovE's [18] integral identity for displacement fields of elastodynamic states with a quiescent past, as well as an analogous identity for the associated fields of stress. The results thus obtained, which are applicable also to unbounded regions, are essential preliminaries to the construction of integral representations for the solutions of the fundamental boundary-initial value problems in dynamic elasticity, carried out in Section 4. We denote by 5ak(x, t; y Ig) = [ uk (x, t; y [g), o k(x, t; y ] g)], (3.1) oo

for every (x,t)eE, x T, the values at (x,t) of the state whose displacement and stress field is given by STOI~ES' [17] solution 1 appropriate to a concentrated load acting at y parallel to the xk-axis. Here ek g(t) is the load-vector at the instant t, if ek is a unit vector in the xk-direction. We assume the "force function" g twice continuously differentiable on ( - 0 % oo). The notation used in (3.1) is to convey that the displacements and stresses, for fixed x, t, and y, are (linear) functionals of g. Since S~k(x, t; y Ig) = Sr x - Y, t; 0 [g) for all (x, t) ~ E, x 7", (3.2) it suffices to quote STOKES' solution explicitly merely for the special choice y = 0: for every (x, t)eEo x T one has 1

41rpuk(x, t; O] g)= t

$ 2g(t-2x)d2 c-? 1

x~xk r 1 1 1 61k +--~T- [-~ g ( t - xlc,)---~2 g ( t - x/c2) j +-~c~ g ( t - xlc2), x See also LovE's [19] treatise (page 305). 5

Arch. Rational Mech. Anal., Vol. 31

(3.3)

66

L.T. WHEELER ~r E. STERNBERG:

4zc~j(x,t;O[g)_.__6c 2 [ 5X'XjXk . . . guXk+~,kXj+gjkXi .

[

x----~---

x3

1

J

l r

J 2g(t-2x)d2

Cl

+2 [ 6xixsx, _flux,+ ~IkXs+~xkXil[g(t_x/c2)- (c__Ll2g(t_x/c,)] x5

x3

J

\ cl /

2X,XjXk [~(t_xlc2)_(C2 ~a~(t_xlc,)]

--

~ik Xj'~-(~jk Xi

X3.

(3.4)

(g(t--x/c2)+~2 g(t--x/c2) 1.

Here and in the sequel x stands for [xl. The displacements (3.3) are easily seen to agree with the representative displacement field (corresponding to a force parallel to the xl-axis) appearing in [19] (page 305). The position-dependence of the integration limits in Stokes' original formulas has, for convenience, been eliminated through a change of the integration variable. The stresses (3.4) are readily found from (3.3) by use of (1.11). Stokes' solution is deduced by LOVE [18], [19] through a limit process based on a family of time-dependent body-force fields that tends to a concentrated load, in analogy to the limit treatment by KELVIN & TAIT [20] (page 279) of the corresponding elastostatic problem 1. We now adopt Definition 3.1. (The Stokes-state). Let y e E , g E ( ~ 2, and let p, cl, c2 satisfy the inequalities (1.9). We then call the state 6ak(.,. ; Ylg) defined on E~ x T by (3.1) to (3.4) the Stokes-state for a concentrated load at y parallel to the XR-axis, corresponding to the force function g and to the material constants p, cl, c2.

Theorem 3.1. (Properties of the Stokes-state). The Stokes-state 6ak(.,. ; Y l g) of Definition 3.1 has the properties: (a) 6ak( ", "; yl g)r 80(0, p, Cl, C2; E~,);

(b) uffx,. ; Yl g)=O(Ix-yl-~),

,~(x,. ; Yl g)=O(Ix-y1-2)

as x ~ y, uniformly on ( - Go, t]for every t e ( - oo, oo);

(c) lim

S

s~(x , ' ; y l g ) d a x = g e~

on (-oo, oo),

~'-',0 S,~(y)

lira

~ (x-y)Ask(x,.;ylg)dA~=O

on ( - o o , oo),

(3"5)2

~0 Sn 0') where sk(., 9; y] g) stands for the traction vector of 5ek(., . ; y[ g) acting on the side of S,~(y) that faces y, ek denotes the unit base-vector in the x,-direction, and the preceding limits are attained uniformly on ( - oo, t ]for every t ~ ( - oo, oo); 1 See STERNaERG & EUBANKS [21] for an explicit version of this limit process. Equations (3.3), (3.4) reduce to the solution of Kelvin's problem if g ( t ) = 1 (-- oo < t < o0). 2 A subscript attached to an "element of area" or an "element of volume" in a surface or volume integral indicates the appropriate space variable of integration.

Theorems in Elastodynamics

67

( d ) / f h ~ @ 2, then

h * ~ k ( ", " ; Y i g ) = g * ~ k ( ", " ; y l h )

on E , x ~ .

Proof. In view of the translation identity (3.2) it suffices to take y = 0. To verify (a), note first that (3.3), (3.4), together with the assumed regularity of g, imply that uk(', " ;0l g) and ak(.,. ;0[ g) satisfy the smoothness requirements in part (a) of Definition 1.2. Moreover, since g vanishes on T - , one draws from (3.3) that

Uk( ., . ; 0 [ g ) = 0

on E o x T - .

To complete the proof of (a) substitute from (3.3), (3.4) into (1.10), (1.11). Property (b) follows at once from (3.3), (3.4) and the hypothesis that gE ~2. Consider now part (c). After a brief computation based on (3.4) and (1.18) one finds that

I sk( x, z; Ol g) dAx='~ s.

g('c-~I/Cl)+2g(z-~l/C2) (3.6)

-~l g(T,--~lcl)-[--~2 g(T--~]lc2)] ek for every ~ ( - - o o , oo) and every ~/>0, so that lira Ssk(x,z;OIg) dAx=g(z)e k

forevery ~ ( - - o o , oo).

~-~0 S~

The uniformity of this limit follows from the inequality

~ sk(x, z; 0l g) dAx-- g(z)

ek =<

20

max

C2 ( - Qo,t]

Igl,

which holds for every t ~ ( - oo, ~ ) and every t/>0, provided z e ( - oo, t], by virtue of (3.6) and since g vanishes on ( - o o , 0 ] and is continuously differentiable on ( - o o , oo). The second of (3.5), for y = 0 , is immediate from (3.4) and (1.18). Finally, property (d) is readily inferred from (3.2), (3.3), (3.4), Definition 1.1, (1.21), and the assumption that g and h are both in ~2. This completes the proof in its entirety. Definition 3.2. (Dynamic doublet-states). Let y ~ E, g ~ (9a, and let 5ak(.,. ; y I g) be the Stokes-state of Definition 3.1. We call the state defined on Ey x T by

~ k , ( . , . ; y i g)= [,k,(.,. ; ylg), o~,(.,. ; y i g)] =~,,~ (., "; ylg)

(3.7) 1

the dynamic doublet-state for the pole y, corresponding to the xi-axis and the xt-axis , the force function g, as well as to the material constants p, cl, c2. From (3.7), (3.2) follows 6ekt(x,t;ylg)=Sak~(x--y,t;Olg)

for every ( x , t ) e E ~ x T .

(3.8)

1 Recall the differentiation convention (1.20). For functions of more than one position vector, the space differentiation so indicated is always understood to be performed with respect to the coordinates of the first position vector. 5*

68

L . T . WHEELER & E . STERNBERG:

We list next the cartesian components of displacement and stress belonging to

5Pkl('," ; O[ g), which may be computed from (3.3), (3.4) by means of (3.7). For Qo every (x, t)~E o x T one thus obtains 1

4.pu:i(x,t;OIg)=-3

[%~x,

<~,,x,+<~,,x,+<~,,x, x3 J1 i 22g(t-2x)d2 r

_[.o~,~, ~,~x,+~,,~,+~~.,x,~ [~.~,-~ ~,~,~-~<~,-~,c2~] ~.9~ X i X k XI

, t;OIg)=6C 2 [ 35XiXJ xkx! 4~aij(X, X7

5 (~i j Xk XI "~-(~i I X j X k ~l- ~k I Xi X j "Jff~j k Xi XI -~ (~j I Xi Xk "~-~ik Xj Xl) X5 1

+

xa

! 2g(t-2x) d2-2c2

45XiXjXkXlx7

r 6 (6 i j X k X l "~-6 i I Xj X k -~ (~k I Xi Xj "~ 6j k Xi Xl -~ 6j | X i X k "~ 6 i k Xj Xl) X5 Jr.,~i/6kl'~-Oik6jl't-6jk61!

2

+m C2

[

1

6ijXkXl'~-OilXjXk+6klXigj+6jkXiXl+OjlXiXk+OikXjX X4

l

(3.10)

1o~,,,,~1~o~ [~<, x,,:> (~) '~`,x,~,>] L---~

,,<,~, j

~-~-j

+ [3(6~xjxt+,~j~x~x3 ,~tk,~j~+6j~,6n] x'

+

6,~xsx,+~j~x,x,

--

..

X3

J [g(t--x/C2)+~2 g'(t--x/c2) ]

9

_ ~,~,,,,x,x, [~,_x,~_ (~)'~,,_x,~,~] C2 X 5

(~I 2]

Theorems in Elastodynamics

69

We observe that if g(t) = 1 (to < t < oo), (3.9), (3.10) reduce to the corresponding elastostatic doublet-states 1 for t > t o +x/c 2 . In analogy to Theorem 3.1 one has Theorem 3.2. (Properties of the dynamic doublet-states). The dynamic doublet-state 6ak l(.,. ; y ] g) of Definition 3.2 has the properties: (a) 6ak'(., 9 ; y] g)e~0(0, p, Cl, C2; E,); (b) uk'(x, . ; Yl g ) = O ( l x - y l - 2 ) , " ; Yl g ) = O ( I x -

y 1 - 3 ) as x --, y,

uniformly on ( - oo,t]for every t e ( - ~ , ~ ) ; (c) lira

S ski( x ' ' ; y l g ) d A x = O

on ( - - ~ , o o ) ,

~--,0 S~ (y)

lira

~ (x--y)Askl(x, . ; y l g ) d A x = g e j k l e j

on (--oo, oo),

~I'-'0 S~ (y)

where sk l(., . ; y I g) are the tractions of 6Pk t ( . , . ; y I g) acting on the side of S~ (y) that faces y, while e~ is the unit base-vector in the xj-direction, ejk t denotes the usual alternating symbol, and the preceding limits are attained uniformly on ( - oD, t ] for every t e ( - oo, oo); (d) i f h ~ a, then h*,gakt( ", ";y]g)=g*,gakl( ", " ; y l h )

on E y x T .

Proof. Property (a) is a direct consequence of Definition 3.2, Theorem 3.1, and Definition 1.2. Properties (b) and (c) may be established by the same procedures used to verify their counterparts in Theorem 3.1. Finally, (d) may be confirmed directly with the aid of Definition 3.2, Lemma 1.3 and part (d) of Theorem 3.1. A physical interpretation of the dynamic doublet-states is easily arrived at on the basis of (3.7) and (3.2). In this connection we refer also to LOVE'S [18] discussion of the singular solutions under consideration. In preparation for a proof of LovE's integral identity, we introduce next Lemma 3.1. Let yeE, ct>0,

= [u,

(f, p, cx, c2; B,(y)),

and suppose SPk ( . , . ; y [ g) is the Stokes-state of Definition 3. I. Then, for each r e ( - oo, oo),

(a) lim

~ Is 9 uk ( . , . ; y I g)] (x, t) dAx = O,

~r-,O Sn (y)

(b) lira

j" [sk(.,. ; y I g)* U] (X, t) dA== Fg* Uk] (y, t),

,1--,o s~ (y)

where s and sk(., . ; Y l g) are the tractions of 5" and 6"k(., . ; Y l g) acting on the side of S,(y) that faces y. Proof. The truth of (a) and (b) for t e ( - ~ , 0 ] is at once apparent from Definition 1.1 and (1.8). Thus choose t >0, hold t fixed for the remainder of the argument, and let fle(0,~). With a view toward proving (a) for the present choice of t, set ikQ/)= ~ [ s . u k ( . , . ; y l g ) ] ( x , t ) d A = forevery r/e(0, fl] s. (y)

Cf. [21] (page 150).

70

L.T. WHEELER & E. STERNBERG:

and appeal again to Definition 1.1 and (1.8) to see that t

I~0/) =

~s(x,t--z)'uk(x,T;Ylg)dxdA,,.

j

s~ (,r) o

Therefore, bearing in mind the present hypotheses, one has the estimate

II~(rl)l~4rc~12tM~(~l)M~(tl)

forevery ~/e(0, fl],

(3.11)

where M~ (r/)=max Is(x, T)I,

(x,Q~S,(y)x[O,t],

M~ (t/) = max I uk (X, 9 y I g) l,

(X, Z) e S, (y) x [0, t].

(3.12)

The function M1 is bounded on [0,fl] by virtue of (1.18) and the continuity of a on Ba (y) x [0, t ], whereas M~(t/)=O(t/-1) as ~ 0 because of (b) in Theorem 3.1. Hence (3.11), (3.12) imply conclusion (a). Next, set I~(t/)=

S [sk( ", . ; y l g ) * u ] ( x , t ) d A x

forevery t/e(0, fl]

s~ (y)

and define an auxiliary function v through v(x, ~) = u (x, z ) - u (y, x)

for all (x, x) ~ Ba (y) x [0, oo).

(3.13)

Accordingly, I

t

I [ I (3.14) k] dz .

]1~ (rl)- [g * uk'l (y, t) l < s~$0,) o sk(x, t - z ; Y I g)" v(x, ~) d~ dAx

I + iu(y, t-z). 0

[ ~ sk(x,x;ylg)dA,--g(T)e Sn (Y)

The second term in the right-hand member of (3.14) tends to zero with ~ since this limit may be taken under the time-integral x and because of (c) in Theorem 3.1. Consequently,

[Ik(~l)--[g*uk](y,t)]<=4ZCrl2tMz(tl)Mk(tl)+O(1)

as t / ~ 0 ,

(3.15)

where M2 (t/) = max Iv (x, z) I,

(x, z) e S,(y) x [0, t],

M~ (t/) = max Is k(x, x; y I g) l,

(x, z) e S~(y) x [0, t]

(3.16)

for every t/e(0,fl]. From (3.13) and the continuity of u on Bp(y) x [0,t] follows M2(t/)= o(1 )

as t/--, 0.

On the other hand, (1.18) and (b) of Theorem 3.1 imply M~(r/)=O(r/-2)

as r / ~ 0 .

Thus (b) follows from (3.15), (3.16). The proof is now complete. 1 See MIKUSINSKI [2] (page 143).

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71

Theorem 3.3. (LOVE'Sintegral identity for the displacement field). Let R be a regular region. Suppose: (a) Se= [u,o]~8o(f, p, cl, c2; R); (b) u~CC(z)(OR x T),

a~cC(2)(OR x T),

f ~ cr

x 3).

Further, let 6ek(.,. ; Yl g) be the Stokes-state of Definition 3.1 for a concentrated load at y parallel to the xk-axis, corresponding to the force function g and to the material constants p, cl, e2. Then, for every (y,t)~R x (-o% 00), 3

Uk(y, t)= ~ i=l

S [U~(X, t; ylsi(x, .))--S~(X, t; y[ui(x , -))]dA x dR

3 + E ~ u~(x, t; ylf,(x, .))dVx, i=1

(3.17)

R

where s and sk(., . ; y] g) are the tractions of SP and Sek(., . ; y] g) acting on dR. ProoL Note that the integrands in (3.17) involve Stokes-states with the respective force functions si(x, .), ui(x, .),f~(x, .) and that these integrands may be written in fully explicit form by making the appropriate substitutions for g in (3.3), (3.4) and by recourse to (3.2), (1.18). The validity of (3.17) for (y, t)ER • (--o%0] is evident from the fact that both Sc and ~ k ( . , . ;Y[ g) have quiescent pasts. Choose (y,t)~R • oo), hold (y,t) fixed until further notice, take 0~>0 such that B~(y)=R, and set

R,=R-B~(y)

for every t/~(0,~).

Let h e ~ 2 and assume h does not vanish identically on [0, oo). From (3.2), (3.3), (3.4) one then infers

uk(.,.;y[h)=ok(.,.;ylh)=O

on

(E-Bc~,(y))x[O,z ]

for every

z>0,

while (a) of Theorem 3.1 ensures that the body-force field of 6ek('," ; y Ih) vanishes on E s x T. In view of the preceding observations and hypothesis (a), one is entitled to apply the reciprocal theorem (Theorem 2.2) to the pair of states 5~ and sek(.,. ; y l h ) on R~. Thus

[s * uk(.,. ; y lh)] (x, t) dA~ + ~ [f * uk(.,. ; y lh)] (x, t) dV~ ~Rn

Rn

= ~ [sk(',';ylh)*u](x,t)dA~

forevery

q~(0,0t),

(3.18)

ORn

where s and sk('," ; y]h) are the respective tractions acting on OR~. Next, pass to the limit as ~/~ 0 in (3.18) and use Lemma 3.1 to conclude that [h * Uk] (y, t) = ~ [ f * uk(., 9; Y l h)] (x, t) dVx, R

+ ~ {[s.u,(.,.;ylh)](x,t)_[sk(.,.;ylh).u](x,O}dA~. OR

(3.19)

72

L . T . W H E E L E R & E . STERNBERG:

From (3.19), hypotheses (a), (b), conclusion (d) in Theorem 3.1, as well as (1.21), (1.8) and (b), (d) in Lemma 1.1, one now draws 3

[h * u k] (y, t)= ~ S [h * up(.,. ; Y lfi(x, "))] (x, t) dVx 3 + •

i= 1 a S [h 9 {up(.,. ; y l s,(x, . ) ) - s~(.,. ; y lu,(x, .))}] (x, t) dA~.

(3.20)

i = 1 OR

If R is unbounded, it follows from (3.2), (3.3), and the fact that 5~ is a state with a quiescent past, that 3

up(x, ~; y If,(x, . ) ) = 0

(3.21)

i=1

for every (x, -c)~(K- Be, t(Y)) x [0, t]. Similarly, if in addition OR is unbounded, 3

3

Y~ up(x, z; y ls,(x, .)) = Z s~(x, ~; y lu,(x, .)) = 0 i=I

(3.22)

i=l

for every (x,T)e(OR-Bc,t(y))x [0,t]. Because of (3.21), (3.22), the integrands in (3.20) are of bounded support. Interchanging the orders of the space-integrations and convolutions in (3.20), as is permissible in the present circumstances ~, and using again the distributivity of the convolution ((d) in Lemma 1.1), one arrives at 3

[h * {uk(y, ")- ~

~ [up(x,. ; y lsi(x, "))- s~(x,. ; y lui(x, "))] dax

i = 1 OR

3

(3.23)

- E ~ up(x,. ; ylf,(x, .))dVx}] (t)=O. i=1 R

Since (y, t) was chosen arbitrarily in R x (0, ~), (3.23) holds for all (y, t) e R x (0, or). The term within braces in (3.23) is readily shown to be continuous on R x [0, ~), whereas h, by assumption, is continuous on [0, ~ ) and does not vanish identically. Thus, the desired conclusion now follows from (e) in Lemma 1.1. This completes the proof. The integral identity (3.17) represents an extension to elastodynamics of the corresponding formula due to KIRCHHO~ [22] (1882) for the scalar wave equation. At the same time (3.17) is a dynamic counterpart of SOMIGLIANA'S [23] (1889) integral identity in the equilibrium theory z. A result similar to (3.17), but confined to two-dimensional elastodynamics, was deduced by VOLTER~ [24] (1894). LOV~ [18] (1904) sketched a proof of (3.17), applicable to bounded regions, with the aid of BETTFS elastostatic reciprocal theorem, treating the inertia forces as body forces. A somewhat more detailed derivation of (3.17) along these lines may be foundin a recent dissertation by DEHooP [25] (1958). SOMIGLIANA[26] (1906) arrived at a closely related integral identity by different means, taking K m c r m o ~ ' s formula as his point of departure. a This reversal is trivially justified for the surface-integrals in (3.20) because of the regularity of the integrands; in the case of the improper volume integrals, whose integrands are singular at y, the reversal is easily legitimized by an elementary limit process. 2 See also Low [19] (page 245).

Theorems in Elastodynamics

73

A precise statement of Theorem 3.3, which also covers unbounded domains, is not available in the previous literature, so far as we are aware. Further, the present proof, which rests on the dynamic reciprocal theorem, would appear to be more direct and more explicit than the proofs referred to above. Our next objective consists in establishing an identity analogous to (3.17), for the stresses of an elastodynamic state with a quiescent past. To this end we require Lemma 3.2. Let y ~ E, ~ > O, 5 r = [u, ~r] e 80 (f, p, ca, c 2 ; B~(y)),

and suppose 6akt(., .; Yl g) is the dynamic doublet-state of Definition 3.2for the pole y, corresponding to the xk-axis and the xraxis, the force function g, as well as to the material constants p, cl, c2. Then, for each t e ( - 0% oo), S I s . u k t(.,. ; y I g)] (x, t) dAx

(a) lim

~ 0 S n (y)

= ~ [g * {(3 -- 8 c 2) u,,i 6~, + 2 (3 + 2 c z) U(k,,)}] (y, t), (b) lim n-~o

S [:'(.,.;ylg),u](x, sn (y)

Oda,

= ~s [g * {(3 - 8 c 2) ui, i 6kt -- (9 -- 4 C2) U(k,,)}] (y, t) -- [g * Utk' ,]] (y, t),

where c=c2/cl, while s and skt(.,. ; Yl g) are the tractions of Se and Sekt(.,. ; y[ g) acting on the side of S~(y) that faces y. Proof. If t e ( - o o , 0 ] , conclusions (a) and (b) follow at once from Definition 1.1 and (1.8). Thus fix te(0, ~ ) for the remainder of the argument. Bearing in mind that g e (#a, one infers from (3.8), (3.9), (3.10), (1.18), after a tedious computation, that lim

~ Uikl (X,.,. Y l g) nj(x) dA x

~-~o s~ (y)

1 -15pc-~2 [(C2--1)(t~ij6kl'l"6il6jk)"l'-(C2"l"4)6ik6jl]g lim

(3.24) on

[0, t],

~ (xj--y~) sk'(x,. ; Y l g) dAx

rr-*O Sq ($)

=-~[(3--8C2) 6ijfk~--Z(6--C2)6ik6j,+(3+Zc2)6i,6jk]g

on [0, t], (3.25)

where nj are the components of the inner unit normal of S~ (y), and the limits in (3.24), (3.25) are attained uniformly on [0, t]. Next, let fle(0,~), set ik,(q)=

~ [s.uk,(.,.;ylg)](X,t)dA~ Sq 0')

forevery

q~(0, fl],

and define q~ through ~(x,z)=a(x,z)-a(y,z)

forall

(x,z)eB#(y)x[O,t].

(3.26)

74

L . T . W H E E L E R & E . STERNBERG:

Then Definition 1.1, (1.8), (1.18), and (1.11) enable one to conclude that I] s(t/) - t~s [g 9 {(3 - 8 c 2) us, s 6ks + 2 (3 + 2 c 2) U(k, 0}] (Y' t) t

=

us ( x , z ; y ] g ) t P s j ( x , t - z ) n j ( x ) d z d A ~ S~ (y) 0 t

+ Iosj(y,t-z) 0

{ ~ u~t(x,z;y[g)nj(x)dA~ S n (y)

1

1 5 ~ 2 I-(c2 - 1) (~5sj 6 k s+ fist ~Sjk) + ( c2 + 4) c5,k 3j S] g (X)}d z. Equation (3.24) consequently furnishes the estimate Ilk '(r/) - ~ [g,. {(3 - 8 c 2) us, s 6kS + 2 (3 + 2 C2) U(k, 0}] (Y' t) l <41r r/2 t Ml(rl) M~SO1)+o(1)

as

(3.27)

r/~O,

where

(x. z) e s,(y) • [0. t].

M101) = max l/~o (x, z). ~o(x, z),

(x, z) E S.(y) • [o, t]

M~S(r/) = m a x [ukS( x, ~; Y[ g)l,

(3.28)

for every ~/e(0,fl]. F r o m (3.26), (3.28), and the continuity of a on B#(y)x [0,t] follows Ml(t/)=o(1) as r / - , 0 , whereas (3.28) and (b) of Theorem 3.2 imply M]l(r/) = O (r/- 2) as

r/-, 0.

Thus, combining (3.27) and (3.28), one confirms (a). To verify (b), let I~'(r/)=

S [skt(',';Ylg)*u-](x,t)dax

forevery

t/~(0, fl].

(3.29)

Sn (y)

From the assumed regularity of u on

B~(y)•

oo) one draws, for every

(x. ~) ~ B. (y) • ( - oo. oo). u (x, z) = u (y, z) + u,, (y, z) ( x s - Y3 + v (x, z),

(3.30)

where

v~:C2(B~(y)x~), v(x,.)=O(Ix-yl 2)

as

x-*y,

(3.31)

uniformly on [0, t ]. On the basis of (3.29), (3.30), the first of (3.31), Definition 1.1, (1.8), and (b) in L e m m a 1.1, one arrives at 11 s(r/) - t~ [g * {(3 - 8 c 2) us, s ~Sks- (9 - 4 c 2) u~, o}] (Y, t) + [g 9 Utk' q] (y, t) t

t

=Su(y,t-z). j" s~'(x.z;ylg)dAxd~+ ~ ~v(x.t-O o s. ~) t "sk~(x, x ; Y l g ) d ~ d A ~ + $ us, j ( Y , t - z ) { 0

s~ ~y) o ~ (xj--Yj)S~l(x,z;Ylg)dA~ Sn (~)

---~S[(3--8C2)6S~f~S--2(6--C2)fSkfj,+(3+

2C2)f, Sfjk]

g(O}dz.

(3.32)

Theorems in Elastodynamics

75

The first and third terms in the right-hand member of (3.32) tend to zero with r/ because of (c) in Theorem 3.2 and (3.25), respectively. Hence

[I~t(rl)-~[g. {(3--8C2) U~.,6kt--(9--4C2)U~k.t)}](y, t)+[g*Utk, J ( y , t)] <=4rrrl2tm2(q)m~t(q)+o(1) as q-~O,

(3.33)

where

M2 (q) = max I v (x. ~) l.

(x. z) ~ S, (y) x [0. t].

M~Z(r/) = m a x [skt(x, Z; Yl g)l,

(X, z)~S,(y) x [13, t],

(3.34)

for every qe(0,fl]. Now invoke (3.31) to see that m2(q)=O(q2)

as q ~ 0 ,

and call on (1.18), as well as (b) in Theorem 3.2 to justify that

Mk2Z(q)=OO1-a)

as

q~0.

Conclusion (b) thus follows from (3.33), (3.34). This completes the proof. Theorem 3.4. (Integral identity for the stress field). Let R be a regular region. Suppose:

(a) 5e=[u,a]ago(f, p, Cl, c2; R); (b) u e~(a)(0R x 2~), o~cg(a)(OR x 2~), f ~'~3)(.1~ x T).

Further, let sPkl(.,. ; y[ g) be the dynamic doublet-state of Definition 3.2for the pole y, corresponding to the xk-axis and the xraxis, the force function g, as well as to the material constants p, ci, c2. Define the state

~ ' ( ' , - ; yl g)- [u-k,(.,. ; yl g), ~k,(.,. ;yl g)] through f f k , ( . , . ;y[ g)=p(c2_2c2)6:~,(.,. ;y[ g)~kt

(3.35)

+2pc2Sptkl)(.,.;ylg ) on E, xT". Then, for every (y, t)eR x ( - 0% oo), 3

trkt(y, t)= E

S [~k'( x, t; yIU,(X, "))--ffkt(x, t; yIs,(x,

i = 1 OR

3 _ y' Sffk'(x,t;ylf,(x,.))dVx,

"))]

dAx (3.36)

i=lR

where s and ~kl(., . ; y ] g) are the tractions of Se and ~k l(., . ; y ] g) acting on OR. Proof. Since the following argument is quite similar to the one used in proving Theorem 3.3, it may be summarized in condensed form. If (y,t)~R x ( - o o , 0 ] , (3.36) is a consequence of the fact that Se and ~k ~( . , . ; y I g) have quiescent pasts. Hence choose (y, t)e R x (0, oo) and hold (y, t) fixed until further notice. Take ~ > 0 such that B~( y ) c R and set

R~=R-B~(y)

for every r/~(O,~).

76

L.T. WHEELER&E. STERNBERG:

Let h e (~a and assume h does not vanish identically on [0, ~). Observe from (3.8), (3.9), (3.10), and (a) of Theorem 3.2 that, for every t/e (0, ~), 5:k z(.,. ; y Ih) qualifies as a candidate for the state ~ of Theorem 2.2 on R~. Thus, in view of the present hypothesis (a), the reciprocal theorem (Theorem 2.2) is applicable to the present pair of states 6* and 5Pk ~( . , . ; y Ih) on R,. On passing to the limit as ~/--}0 in the resulting identity, and using Lemma 3.2, one arrives at [h * uk,~] (y, t) = - S I f * u~ '('," ; Y I h)] (x, t) dVx R

+ S ([sk~(',';Y[ h) * u](x, t ) - Is * ukt(.,.;y[h)](x, t)} dA x . (3.37) OR

From (3.37), hypotheses (a) and (b), conclusion (d) in Theorem 3.2, and (b), (d) in Lemma 1.1, one now draws 3

[h * Uk, t] (Y, t)= -- ~. ~ [h * u~t(., . ; y lfi(x, "))] (x, t) dVx 3 ~=1 ~

(3.38)

+ y, ~ [h. {s~'(.,. ;ylu,(x, .))-u~'(.,. ;yls,(x, .))}] (x, t) d,4~. i=1

OR

After permissible reversals of the space-integrations and convolutions involved in (3.38), one finds that 3

[h 9 {u~,,(y,.)- ~

S [~/~'(~, 9 y Ius(x,.))- u,~'(~, 9 y IstC~,-))J dax

i=1

0R

3 + ~'. ~f u ,kt(x , , . ; y l f , ( x , .))dV~}](t)=O. i=I

(3.39)

R

Since (y,t) was chosen arbitrarily in R x(0,c~), equation (3.39)holdsforall (y, t ) e R x (0, ~). But the term within braces in (3.39) is continuous on R x [0, ~), while h is continuous on [0, oo) and does not vanish identically, so that (3.39) and (e) of Lemma 1.1 furnish 3

uk,,(y, t)= E J [s~'(x, t; y lu,(x, .))-uf'(x, t; y ls,(x, "))-Idh,, i=1

OR

3

(3.40)

j[ u ki~t f x , t ; y l f ~ ( x , . ) ) d V ~

-~ i=1R

for every (y, t ) ~ R x (0, oo). The desired conclusion now follows from (3.35), (3.40) and (1.11). This completes the proof. It is clear that (3.40) may be obtained formally from LovE's identity (3.17) by differentiating the latter under the integral signs and by making use of the relations

d--d--- u~ (x, t; y lg)= -u~Z(x, t; y ]g) , ayt

oy, a~(x, t; y lg)= -~J(x, t; rig), which hold for every (x,t)eEy x T because of (3.2) and (3.7). A rigorous proof of Theorem 3.4 based on this alternative procedure is, however, quite cumbersome.

Theorems in Elastodynamics

77

Finally, we remark that (3.40) enables one to write down immediately formulas analogous to (3.17) and (3.36) for the dilatation and rotation fields of an elastodynamic state with a quiescent past. The linear combinations of doublet-states entering the formulas just alluded to are those characteristic of a dynamic center of dilatation and a dynamic center of rotation. Closely related integral identities for the dilatation and rotation were obtained by TEDONE [27]. 4. Green's States. Integral Representations for the Solutions to the Fundamental Bonndary-initial Value Problems of Elastodynamics

In the present section we aim at integral representations for the displacements and stresses of the solutions to the first and second fundamental boundary-initial value problems in classical elastodynamics. The appropriate boundary data consist of the surface displacements in the first problem and of the surface tractions in the second problem. Further, we confine our attention at present to elastodynamic states with a quiescent past. 1 The integral identities (3.17) and (3.36) involve both the surface displacements and the surface tractions on the boundary of the region at hand. In order toarrive at the desired representations, we need to eliminate from the integrands in (3.17), (3.36) the surface tractions inconnection with the first problem and the surface displacements in connection with the second problem. This purpose may be accomplished by means of suitable elastodynamic Green's states. With a view toward the first boundary-initial value problem we introduce Definition 4.1. (Green's states of the first kind). Let R be a regular region, y~R, and let gE(# 3. We call

~k(.,.; rig)= [ik(., 9 y lg), ~k(.,-; y Ig)] the displacement Green's states of the first kind and s;k'(', " ; r i g ) = [ilk,(.,. ; Y I g), ~k,(.,. ; r l g ) ]

the stress Green's states of the first kind for the region R and the pole y, corresponding to theforce function g and to the material constants p, ci, c2, provided: (a) 6 ~ k ( . , . ; y l g ) = S e k ( . , . ; y l g ) + f f ~ k ( . , . ; r l g )

~k,(.,.;ylg)=~Tkz(. ,.;ylg)+s~,(., . ; y l g )

on R, x T , on ~,xT",

where s;k(., . ; Yl g) and ~k~(., . ; Y[ g) respectively denote the Stokes-state of Definition 3. I and the linear combination of doublet-states (3.35);

(b) P~(.,.;ylg)=[i~(.,.;ylg), ~k(.,.;ylg)]~o(O,p,c, c2;~), sek'(', " ; Y l g ) = [ik'(', " ; Y I g), ~k,(.,. ; Y lg)] e~f0(a, p, C~, C2; R), bk(., .; y ] g ) ~ 2 ) ( a R (c)

x ~) '

ik( ", ";Yig)=--uk( ", ";Yig) ikt( " , ' ; y i g ) = - ~ k ' ( " , ' ; y l g )

~RZ(., . ; y l g ) ~ l ) ( ~ R x ~ ) ; on a R x T , on ~ R x ~ .

1 See the end of Section 4 for a relaxation of this restriction upon the initial conditions.

78

L.T. WHEELER& E. STERN'BERG:

The regular parts S#k(.,. ;Yl g) and s~kl(.,. ;Yl g) of the displacement and stress Green's states of the first kind are each evidently defined through requirements (b), (e) as the solution to a first boundary-initial value problem for R. Moreover, they are uniquely determined by these conditions because of Theorem 2.1. In contrast, the existence of these regular states, and hence of the corresponding Green's states, is contingent upon the existence of a solution to the first problem for the region under consideration in the presence of sufficientlysmooth boundary data. Before proceeding with our immediate task it is convenient to have available Lemma 4.1. Let R be a regular region, y ~ R, g~ (~3, h ~ ~3, and let ffok(.,. ; y lg), ~kt(., . ; y] g) be the regular parts of the Green's states of the first kind introduced in Definition 4.1. Then: (a) h * • k ( ., " ; y l g ) = g * p k ( ", " ; y l h )

on ~ •

(b) h . S ~ k ~ ( . , . ; y l g ) = g . S T ~ t ( . , . ; y l h )

on R x ~ ' .

Proof. Consider first (a). From Definition 1.2 and (b) in Definition 4.1 one obtains after two successive applications of Lemma 1.2 and Lemma 1.3, h * S~k(., 9; y I g) e 8 0 (0, p, Cl, C2; R),

g,~ak (.,

(4.1)

. ; y Ih)~go(0, p, cl, c2; R).

Next, call on (c) in Definition 4.1 and (d) of Theorem 3. I to see that h*~k( ", " ; Y l g ) = - - h * u k ( ", ' ; Y l g )

(4.2)

=--g*uk( ",';ylh)=g*~k(",';ylh)

on aRxT".

Conclusion (a) now follows from (4.1), (4.2) and the uniqueness theorem (Theorem 2.1). The proof of (b) is strictly analogous. We are now in a position to turn to Theorem 4.1. (Integral representation for the solution of the first boundary-initial value problem). Let R be a regular region. Suppose:

(a) Se--[u, ~]E~O(f, p, Cl, C2; R); (b) ueC~4)(aRxT),

~rEc~tg)(ORxT),

/ec~t4)(RxT).

Further, let S~k(., 9; y Ig) and .~k t(., . ; y lg) be the Green's states of the first kind of Definition 4.1 for the region R and the pole y, corresponding to theforce function g and to the material constants p, cl, c2. If these Green's states exist for all y e R and all ge(~ 3, then for every (y, t ) e R x 7", 3

uk(y,t)= E [Sfik(x,t;ylf~(x, .))dVx- S~sk(x,t;ylu,(x, .))dAx], i= 1

R

(4.3)

OR

3

ak,(y, 0 = - E I-Sfi~'(x, t; ylf~(x, .))dV~- ~ ~sf'(x, t; ylu,(x, .)) dA,], 1= l

R

OR

(4.4)

Theorems in Elastodynamics

79

where ~k(., .; Ylg) and ~u(., .; Ylg) are the tractions of 6~k(., .; Ylg) and s)kZ( ", " ; Y lg) acting on OR. Proof. If (y, t)eR x T-, (4.3) and (4.4) follow trivially from (1.18), Definition 4.1, and hypotheses (a) and (b). Define a function h by setting /04 h(t)=

[4!

foreveryt~(-oo,0] for every t~(0, 00)

(4.5)

and observe that hef~ 3. Choose y~R and note from (4.5), (c) in Definition 4.1, (3.2), (3.3) that if dR is unbounded, ~(., .;ylh)=0

on (dR-Bc, t(y))x[O,t ]

forevery t > 0 .

(4.6)

From (4.6) and (b) in Definition 4.1 one concludes that 6~k(., . ; y lh) satisfies the conditions imposed on the state S" of Theorem 2.2 in hypotheses (a) and (b) of that theorem. Thus, and because of the present hypothesis (a), one may apply the reciprocal theorem (Theorem 2.2) to the pair of states 5 a, sek(., .; ylh) on R. Accordingly, and by virtue of (c) in Definition 4.1, j" [s 9 uk(., 9 ; y ] h)] (x, t) dA~ = j"I f , ~ k ( . , . ; y lh)] (x, t) dV, ~R R _ j" [~k(., ";ylh)*u](x, t) dA~ for every t~(0, 00).

(4.7)

~R

From hypotheses (a) and (b), (3.2), (3.3), (4.5), (b) of Lemma 1.1, and Lemma 1.2 there follows

a , j , u~(., 9 ; y Ih) E (r

x ~).

(4.8)

Furthermore, hypotheses (a) and (b), Lemma 1.2, (b) of Lemma 1.1, and (b) in Definition 4.1 imply

f, ~(.,.

; y rh) ~e(5~(R x

:~),

(4.9)

8~j(.,. ; y Jh), UaaCC(s)(dR x ~ ) . Let to>0. If R is unbounded, then (4.6), (b) of Definition 4.1, and Lemma 2.2 ensure that there is a bounded set t2 (to)c ~ such that ~k(., .;y]h)=Sk(., " ; y ] h ) = 0

on (R-t2(to))X]-0, to].

(4.10)

On the other hand, (3.2), (3.3), and (4.5) furnish

uk( ", . ; y l h ) = 0

on (E-Bclto(Y))x[O, to].

(4.11)

Assertions (4.8) to (4.11), together with (1.18), justify five successive time-differentiations of (4.7) under the integral signs on the interval (0, to). Since t o was chosen arbitrarily in (0, oo), one thus has

[s . uk(., . ; y lh)](5)(x, t) da~= ~ [ f . ilk(.,. ; y lh)]tS)(x, t) dVx OR

R

_ j" [~k(., .;ylh).u]r OR

for every re(0, oo).

(4.12)

80

L . T . W H E E L E R & E . STERNBERG:

Next, appeal to hypotheses (a) and (b), (1.18), (d) in Theorem 3.1, and (a) in Lemma 4.1 to see that 3

[s 9 uk(.,. ; y lh)] (x, t)= 2 [h * uk(.,. ; y I S,(X, "))] (X, t) i=1

for every (x, t) ~ OR x ~F, 3

[ f . ffk(.,. ; y lh)] (x, t) = 2 [h * h~(., .; y [f~(x, "))] (x, t) i=1

(4.13) for every (x, t) ~_R x T,

3

[ ~k ( . , . ; y l h )

,

u](x,t)=2[h*~k(., ",ylu,(x,'))](x,t) i=1 r

for every (x, t)~ OR x T. Now note that for ~,e~(T +) equation (4.5) and Lemma 1.2 imply [h 9 ~b](s) = ~b

on (0, ~ ) .

Therefore, and by (4.12), (4.13), 3

E

3

S uk( x, t;yls,(x, .))dA.= E S~k( x, t ; y l f , ( x , .))dV.

i = 1 OR

i=1 R

3 - E ~ "sk(x,t;ylu,( x, "))dAx

for every t~(O, oo).

(4.14)

i = 1 ~R

Finally, combine (4.14) with (3.17) and use (a) of Definition 4.1 to conclude that (4.3) holds for every (y, t)ER x T. The verification of (4.4) is easily carried out in a strictly analogous manner with the aid of the reciprocal theorem (Theorem 2.2) and the integral identity (3.36). Turning to the second boundary-initial value problem, we adopt Definition 4.2. (Green's states of the second kind). Let R be a regular region, y~R, and let ge(r We call ffk('," ; Y I g)= [uk('," ; Y I g), ~k( ", 9; Yl g)] the displacement Green's states of the second kind and

s~kl(.,. ; y[ g) = c~k l(.,. ; y] g), &kt(.,. ; Y[ g)] the stress Green's states of the second kind for the region R and the pole y, corresponding to the force function g and to the material constants p, c 1, c2 , provided: (a)

~k(o,';y]g)=SPk(~176176176 6~k~(.,~176

on R y x T , on ~ y x T ,

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81

where 6ek(., .; Ylg) and 67kt(., .; Ylg) respectively denote the Stokes-state of Definition 3.1 and the linear combination of doublet-states (3.35);

(b) ~ ( . , . ;ylg)--[~(.,. ;ylg),?r~(.,. ;ylg)]~o(O,p, cx, ce;~), s~k '('," ; Yl g)= I-uk'(', "; Yl g), ~kt(.,. ; Ylg)] ~ 80 (0, p, CX, CZ; R); (c)

ik(.,.;ylg)=-s~(.,.;ylg)

on dRx~', $

ikt(.,.;ylg)=--s-kt(.,.;ylg

)

on d R x T ,

where ~k(., . ; Ylg) etc. denote the tractions of S~k(., 9; Ylg) etc. acting on OR. The regular parts of the displacement and stress Green's states of the second kind are uniquely characterized, in view of (b), (c), and the uniqueness theorem (Theorem 2.1), as solutions to second boundary-initial value problems for R. The existence of the Green's states of the second kind evidently depends on the solvability of the second dynamic problem on R for sufficiently regular surface tractions. The following lemma is a counterpart of, and may be proved in the same way as, Lemma 4.1. Lemma 4.2. Let R be a regular region, yeR, g~(g3, hefts, and let ~ k ( . , 9; Ylg), S~kz( ", " ; Y lg) be the regular parts of the Green's states of the secondkind introduced in Definition 4.2. Then: (a) h * p k ( ' , ' ; y l g ) = g * p k ( ' , ' ; y l

h)

(b) h . ~ ' ( . , . ; y l g ) = g * ~ k ' ( . , . ; y l h )

on ~ x T ; on ~ x ~ .

Theorem 4.2. (Integral representation for the solution of the second boundaryinitial value problem). Let R be a regular region. Suppose: (a) 6a= [u, ale ,8o(f, p, cl, c2; -~);

(b) uec6t4~(~R • T), aec~t4)(aR • T), /ec~r

• 3).

Further, let ~ (., 9 ; y Ig) and ~k l ( . , . ; y ]g) be the Green's states of the second kind of Definition 4.2 for the region R and the pole y, corresponding to the force functiong and to the material constants p, cl, c2. If these Green's states exist for all y e R and all ge(9 3, then for every (y, t ) e R x T, 3

u~(y,t)= E [~ ak(x,t;ylf~(x,.))dV~+ ~ ~k(x,t;yls,(x,.))dA~], i=1

R

(4.15)

dR

3

a~,(y, t ) = -

Y [~ a,~'(x, t; ylf,(x, .))ave+ $ a~'(x,t;yls,(x,.))dA~], (4.16) t=1

R

OR

where s are the tractions of 6a acting on dR. The truth of this theorem may be confirmed with the aid of Lemma 4.2 by an argument parallel to that employed in the proof of Theorem 4.1. The smoothness restrictions imposed under (b) of Theorem 4.1 and Theorem 4.2 may be relaxed somewhat at the expense of more elaborate regularity hypotheses. As will become 6a

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L.T. WHEELERr E. STERNBERG:

clear at the end of this section, the foregoing two theorems may be used to generate representations of the solution to the first and second elastodynamic problem in the absence of a quiescent past. Finally, integral representations for the solution of mixed boundary-initial value problems in elastodynamics, similar to those contained in Theorem 4.1 and Theorem 4.2, are easily established by means of suitable generalizations of the Green's states of the first and second kind. Equations (4.3), (4.4) in Theorem 4.1 and (4.15), (4.16) in Theorem 4.2, for a fixed choice of the pole y, involve elements of the relevant Green's states corresponding to an infinite family of force functions (depending on the position parameter x). Accordingly, the representation at a single point of the given region of the solution to either fundamental problem of elastodynamics would seem to require that one solve an infinity of boundary-initial value problems in order to determine the requisite families of displacement and stress Green's states. We show next that this apparent difficulty is easily overcome, and in this connection consider first the representation of states whose body forces and surface displacements or surface tractions are separable functions of position and time. Thus, suppose the state 5~ in Theorem 4.1 is such that

u(x, t)=~(x)p(t)

for every

(x, t)~OR x 7",

for every

(x,t)eRxT.

o

_

f(x,t)=f(x)q(t)

(4.17)

ao

Then, as is clear from (3.3), (3.4), Definition 4.1, and Theorem 2.1, Equations (4.3), (4.4) give way to

llk(y , t)= S f~ 9~k(X, t; y] q)dV~-- S ~(x) . sk(x, t; y] p)dAx, R 0R O'ks(y, t)= -- S f ( X ) . ~kt(x, t; y]q)dV~+ ~ ~(x). g~t(x, t; ylp)dA,,. R

(4.18)

OR

Similarly, if the state 5p in Theorem 4.2 has the separable data co

s(x,t)=~(x)p(t)

for every

(x, t) ~ c~R x T,

for every

(x,

o

f(x,t)=f(x)q(t)

(4.19)

x

then (4.15), (4.16) may be replaced by

Uk (y, t) = j"f~

9 ~k(X, t; Y lq) dV=-1- j' ~ (X)" ~k(x, t; Y lP) dA=,

R

OR

t) = - I f~

9

R

t; y I q) d V x - I

t; y Ip) d & .

(4.20)

OR

In order to facilitate the construction of integral representations for states whose data are not necessarily separable we insert here Theorem 4.3. (Standardization of the force function in the construction of Green's states). Let R be a regular region, and let y e R . Further, let 5~k(., .; Ylg) and DR ! (', " "~Y Ig) be the Green's states of the first kind of Definition 4.1 or the Green's states of the second kind of Definition 4.2, and let h be the function defined by h(t)={

~4 for every t 4 ! for every

te(-~,O] te(O, co).

(4.21)

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83

Then: on

~,x(0, ~);

(b) 5 e k l ( . , . ; y l g ) = [ g * S f k Z ( ' , ' ; y l h ) ] (5) on

.~,x(0, ~ ) .

(a) s e k ( - , . ; y l g ) = [ g * S e k ( . , . ; y l h ) ] ~ 5 )

Proof. If 6~k(., 9; ylg) and S~kt( ., 9; Ylg) are Green's states of the first kind, then (1.21), (d) in Theorem 3.1, (d) in Lemma 1.1, Lemma 4.1, and Definition 4.1 yield

h,6~k(.,.;ylg)=g,g~(.,.;ylh) h*S~kt(.,.;ylg)=g*S~kt(.,.;y[h )

on ~,x~, on

(4.22)

~yxT.

On the other hand, (4.22) hold true also for Green's states of the second kind by virtue of (1.21), (d) in Theorem 3.2, (3.35), (d) in Lemma 1.1, Lemma 4.2, and Definition 4.2. Further, note from Lemma 1.2 that for the present choice of h, every function ~keqf(T +) obeys the identity

~b=[h,~]~5)

on (0, oo).

Thus, conclusions (a) and (b) follow from (1.21), (4.22), (1.19), and the regularity properties of the Green's states of the first and second kind implied by Definition 4.1 and Definition 4.2. This completes the proof. Theorem 4.3 enables one to generate directly the Green's states of the first and second kind for a given region and a fixed pole, corresponding to an arbitrary (sufficiently smooth) force function from those corresponding to the standard force function h given by (4.21). For example, (4.3) may now be written as u k (y, t) = S I f * fik ( "," ; y I h)] (') (x, t) d V, - j" [u * ~ k( . , . ; y I h)] (') (x, t) dAx. R

OR

Additional properties of the Green's states are supplied by Theorem 4.4. (Symmetry of the Green's states). Let R be a regular region, let x and y be distinct points in R, and let g e ~ a. Further, let ~ k (. , . ; y [g), S~k t ( . , . ; y Ig) be the Green's states of the first kind of Definition 4.1 or the Green's states of the second kind of Definition 4.2. oo Then, for every tm T, (a) a k ( x , t ; y l g ) = a ~ ( y , t ; x l g ) , (b) b ~ ] ( x , t ; y l g ) = O^'ik t ( y , t', xlg). Proof. It will be sufficient to illustrate the proof of this theorem by demonstrating merely (a) for the case in which s~k( ., 9 ; y[g) is a Green's state of the first kind for the region R and the pole y. If t e T - , (a) is immediate from Definition 4.1. Also, (a) holds trivially for every tE~'if g = 0 on [0, ~). Hence, assume that g fails to vanish identically on [0, oo). Now choose ~ > 0 such that B~(x)=R, B~(y)=R, while B~(x)c~ B~(y) is empty. Then, for each r/e(0, ~), the region

R. = R - ~ . ( x ) - B.(y) is regular and, by hypothesis and Definition 4.1, sak(', 9; y I g)e 8o(0, p, c~, c2; R,), f f i ( . , . ; xl g) e ~'o(0, p, ca, c2; R~). 6b

Arch. Rational Mech. Anal., Vol. 31

(4.23)

84

L.T. WHEELER& E. STERNBERG"

Further, (a) and (c) in Definition 4.1 imply

~k(.,.;ylg)=O

on

aRxT,

so that the state 5~k(., .; Ylg) conforms to condition (b) imposed on 6" in Lemma 2.2. Accordingly, Theorem 2.2 may be applied to the pair of states in (4.23), whence [-~k(.,. ; y l g ) . ~,(.,. ; x lg)] (Z, t) dA~ Sn (y)

+ S Esk("';Ylg)*ui("';xlg)](z't)dAz S n (x)

= ~ [~'(.,.;xlg)*~(.,.;ylg)](z,t)d.4~ Sn

(y)

+ ~ [~'(',';xlg)*~k(',';Ylg)](z,t)dh~ S n (x)

for every t > 0. Next, pass to the limit as r/--+ 0 in this equation, bearing in mind Lemma 3.1 and Definition 4.1, to arrive at

[g* uk('," ;xlg)](y,t)=[g.a~(.,.;ylg)](x,t) ^' for every t > 0. Conclusion (a), for the displacement Green's states of the first kind, now follows from (e) in Lemma 1.1. Conclusion (b), for the stress Green's states of the first kind, as well as both conclusions for the Green's states of the second kind, may be reached in a strictly analogous manner. Theorem 4.1 and Theorem 4.2 presuppose that the state to be represented has a quiescent past and possesses regularity properties beyond those introduced in the definition of an elastodynamic state with a quiescent past (see Definition 1.2). We conclude this section with a theorem permitting one to obtain from the results established already representations of states that are free of the restrictions just mentioned. Theorem 4.5. (Regularization of elastodynamic states). Let R be a regular region and let ~ = [ u , a] ~ 8 ( f , p, c 1, c2; R x T+).

Let n > 2 be an integer and let q~ be the function defined by tp(t)= ~" 0 forevery t"/n! for every

t~(-oo,0] te(O, oo).

Suppose further

y'=[~',o"]=m,~

on gx:~.

Then: (a) 5e' ~8o(f', p, cl, c2; R), where, for every (x, t)~R • T,

f'(x, t)= [~p,f] (x, t)+p,e(t) a(x, o+)+p ~(t) u(x, o); (b) u'eC~t"- 1)(t3R x T), a'ecg(~-l)(OR x T), f'Eqf("-2)(R x T);

(C) u'Ec~(n+I)(/~xT+), a'ecg(~+x)(/~xT+), 6*--6" 'on+l) on

~x(O, oo).

(4.24)

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85

Proof. Observe that the function tp has the properties q~e~n-1 c~cr

r

'

q~(n+l)=0

on

(0, oo). (4.25) I

The first of (4.25), in conjunction with the conditions imposed on Sain Definition 1.2 and the properties of convolutions given in Lemmas 1.1, 1.2, 1.3, enable one to reach conclusion (a) without difficulty. Next, appeal to the first of (4.25) and the continuous differentiability of u on x T + to see that

~ ( . , 0 + ) + 6 u(., 0)~ ~e~n-2)(~ •

~).

Thus (b) follows from the first of (4.25), the above definition of f', the continuity of u, a, a n d f on R x T +, and Lemma 1.2. Finally, (c) is a consequence of (4.25), the continuity of u and o" on R • T +, as well as Lemma 1.2. This completes the proof. The preceding theorem owes its usefulness to the fact that, while the state 6ais not assumed to conform to hypotheses (a) and (b) in Theorems 4.1, 4.2, it is conveniently recoverable in the manner of (4.24) from a state that does meet these hypotheses, provided n > 6.

5. A UniquenessTheorem for Concentrated-load Problems in Elastodynamics As a further application of the Green's states introduced in Section 4 we treat in this section a uniqueness issue associated with the second boundary-initial value problem of elastodynamics in the presence of concentrated loads acting at fixed material (interior or boundary) points of the body. The uniqueness theorem arrived at here asserts the completeness of a direct formulation of concentratedload problems that rests on prescribing-in addition to the body forces, regular surface tractions, and initial c o n d i t i o n s - t h e orders of the displacement and stress singularities at the load points, as well as the stress resultants of the latter singularities. This formulation of the singular class of problems with which we are concerned clearly lies beyond the scope of ordinary uniqueness theorems in dynamic elasticity, such as Neumann's theorem or Theorem 2.1 in the current investigation. The uniqueness theorem constituting our present objective is a dynamic analogue of a recent elastostatic result due to TURTELTAUB& STERNBERG[28] (see Theorem 5.2 of [28]) and will be proved by parallel means. With a view toward clarifying the relevance of the theorem presented in what follows, we emphasize that the idealization of a "concentrated load" in elasticity theory derives its physical significance from a limit definition of the solution to problems involving such loads. Accordingly, the solution to the singular problem under consideration would have to be defined as the limit of a sequence of regular solutions, corresponding to distributed body forces and surface tractions that tend to the given concentrated loads. A program aimed at confirming the equivalence of the direct and the limitformulation of concentrated-load problems may be pursued in three stages. First, one would seek to demonstrate the existence of the limit solution by proving the 1 We write ~uE'c~~176 (T) if ~u~m(T) for every positive integer m.

86

L.T. WHEELER & E. STERNBERG:

appropriate convergence of the sequence of approximating regular solutions. Next one would examine the limit solution and attempt to verify that it possesses the properties underlying the direct formulation of the problem; in particular, one would have to determine the orders and stress resultants of the singularities inherent in the limit solution at points of application of concentrated loads. Finally, one would aim at showing that these properties suffice to characterize the limit solution uniquely. The direct formulation of the singular problem at hand has the advantage of obviating the need for a limit process that is apt to be highly cumbersome in actual applications. The program outlined above was proposed in [21] for the equilibrium theory and was carried out rigorously in [28] with limitation of the first two stages to concentrated surface loads acting on finite bodies with sufficiently smooth boundaries. The limit treatment of internal concentrated loads in elastostatics is in essence disposed of by the derivation due to KELVIN & TAIT [20] (page 279) of the solution to KELVIN'Sproblem 1. Further, the requisite properties of Kelvin's solution, which is in elementary form, are trivially inferred. Similarly, LovE's [19] (page 304) derivation through a limit process of the Stokes-state verifies its physical significance. Moreover, conclusions (a), (b), (c) in our Theorem 3.1 furnish the pertinent properties of Stokes' solution. In contrast, a limit treatment of concentrated surface loads in dynamic elasticity - even under very stringent restrictions upon the body geometry - represents an extremely difficult task with which we do not propose to cope at present. Thus, we rely solely on Stokes' solution as a motivation for the a priori assumptions regarding the order of the singularities at the points of application of concentrated loads introduced in Theorem 5.1. (A uniqueness theorem for elastodynamic problems involving concentrated internal and surface loads). Let R be a regular region and assume that for each y e R there is at least one g e ~ 3 , not identically zero on ( - ~ , oo), such that the displacement Green's states of the second kind exist for the region R and the pole y, corresponding to the force function g and to given material constants p, ct, c 2. Let P = {a 1.... , a~}

be a set consisting of n distinct points in R. Further, let S~', 6~'' be two states with the following properties: (a) S:'=[u', o']ES(f, p, cl, c2; ( R - P ) x T + ) , r e " = [u '', a " ] ~ 8 ( f , p, cl, c2; ( R - P ) • (b) as x --, a k ( k = 1. . . . . n),

a'(x,.)=O(Ix-a~l-2), a"(x, . ) = O ( I x - a ~ l - 2 ) ,

u'(x, . ) = O ( I x - - a k l - 1 ) , u"(x, " ) = O ( I x - - a k l - ' ) , uniformly on [0, t ] for every t > 0; (c) lim

~ s'(x,.)dA=lk,

T/-*O Ak(a)

lim

S s " ( x , ' ) d A = l k on I - 0 , o o ) ( k = l , . . . , n ) ,

~/-'0 Ak(~)

1 For an explicit version of the underlyinglimit process see [21].

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87

where lk(k = 1, ..., n) are given vector-valued functions of the time, Ak (tl) = n n S, (a k)

(k = 1.... , n) ,

while s', s" are the tractions of S~', Sf" acting on the side of A k 01) that faces the point a~, and the preceding limits are attained uniformly on [0, t ]for every t > 0 ; r

o

(d) u (., 0)= u,

t!

u"'( . , 0 + ) = ~ ,~ s'=p,

o

9

, (.,0)=,,

s"=p

on

,"(.,0+)=~

on

R-P,

( d R - P ) x[O, oo),

provided s', s" here denote the surface tractions of SP', Sa'' whereas ~, ~, and p are functions prescribed on their respective domains of definition. Then, 6a'=6 a'' on ( R - P ) x l - 0 , oo). Proof. Choose y e R and hold y fixed. Let

~ ' ( ' , - ; rig) = [a'(-, -; rig), ~'(.,-; rig)] be the displacement Green's states of the second kind for R and y, corresponding to g, p, cl, and c2 , where ge ~a and fails to vanish identically. It is clear from Definition 4.2 and conclusion (a) in Theorem 3.1 that 6~'(', 9 ;Yl g)E 8o (0, p, c 1, c2; Ry), *

9

s

on

(5.1)

o0

dRxT,

if ~i(., 9 ; Ylg) are the tractions of g i ( . , 9 ; Ylg) acting on dR. Next, define the state 6a = [a, a] on ( R - P ) x T through S e = S a ' - 6 a'' on ( R - P ) x ( 0 , c~),

u=a=0

on ( R - P ) x ( - o o , 0 ] .

(5.2)

Then, by hypotheses (a), (b), (c), (d), and Definition 1.2, oq' E 80 (0, p, cl, c2; R - P ) ,

U(X,.)=O(Ix--akI-~),

(5.3)

~(X,')=O(Ix--ak1-2) as x--'ak

( k = l . . . . . n),

(5.4)

uniformly on [0, t ] for every t > 0, lim

~ s(x,.)dA=O

on

[-0, oo)

(k--1,...,n),

(5.5)

t/--* 0 A k (~/)

this limit being attained uniformly on [0, t ] for every t > 0, and 00

s=0

on

(dR-P) xT,

(5,6)

where s are the appropriate surface tractions of Sa. Take r/o > 0 such that any two spheres (balls) of radius r/o centered at distinct points of P are disjoint and do not intersect B,o(y ), while, for every r/e(0, r/o), B~(y) ~ R and the region

R ~ = R - U B~(aD-B,(y) k=l

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L.T. WHEELER& E. STERNBERG:

is a regular region. Evidently, (5.1) and (5.3) now permit an application of the reciprocal theorem (Theorem 2.2) to the pair of states ~ t ( . , . ; Ylg), 60 on R,. Because of (5.6), the second of (5.1), and the vanishing of the body forces of g ~ ( ' , "; Ylg) and 6a, one finds in this manner that S [s* ~ i ( . , . ; y [ g)] (x, t) dA, + k= 1 Ak (~)

=~

S Is * fi'(., 9; y I g)] (x, t) dA, Sn (y)

~ [~'(.,.;ylg),u](x,t)aA~+ $(y) [~'(.,.;rlg).uq(x,t)dA~

k = 1 A k (~)

(5.7)

Sn

for every r/e(0, r/o) and for all t~(0, oo). At this stage hold t > 0 fixed and invoke (5.5), bearing in mind the uniformity on [0, t] of the limit in (5.5), to see that for k = 1. . . . . n,

[s* ~'(., . ; ylg)](x, t)dA, ak (~) t

= ~ ~s(x,t-z).[~'(x,z;ylg)-~'(ak,z;y[g)]dzdA,+o(1)

as

r/+0.

Ak 0t) 0

Hence (5.4) and the continuity of ~ ( . , 9; y [g) on R, x ~ yield lim

~

[s,~'(.,.;ylg)](x, OdA,=O

( k = l .... ,n).

(5.8)

~--,0 Ak (~)

On the other hand, (5.4) and the continuity of ~i(., .; Ylg) on Ryx T furnish lim

~

[~'(.,.;ylg),u](x, OdA,=O

(k=l,...,n).

(5.9)

t/~0 Ak (~)

Now pass to the limit as ~/+0 in (5.7) and appeal to (5.8), (5.9), together with Lemma 3.1 and Definition 4.2, to arrive at

[ g , u] (y, t) = 0. But t > 0 was chosen arbitrarily, so that [g * u] (y, . ) = 0

on

(0, oo).

(5.10)

Since, by hypothesis, g does not vanish identically on [0, oo), one infers from (5.10) and (e) in Lemma 1.1 that u ( y , . ) = 0 on [0, oo). Recalling that y was chosen arbitrarily in R - P , one draws u=0

on

(R-P) x[O, oo).

(5.11)

Moreover, (5.11), (1.11) imply that a vanishes on ( R - P ) x [0, oo). The desired conclusion now follows from the continuity of u, a on ( R - P ) • [0, oo) assured by (5.3) and from (5.2). The preceding theorem is at once broader and more restrictive than Theorem 2.1. While Theorem 5.1 encompasses a class of singular elastodynamic states, not covered by Theorem 2.1, it presupposes the existence of the displacement Green's states of the second k i n d - a n d hence the solvability of a class of

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89

regular second

boundary-initial value p r o b l e m s - f o r the region at hand. N o such existence hypothesis is involved in T h e o r e m 2.1. It follows f r o m T h e o r e m 5.1, in particular, that the Stokes-state is uniquely characterized by (a), (b), and the first of (c) in T h e o r e m 3.1. On the other hand, (a) together with b o t h of (c) in T h e o r e m 3.1 fail to characterize the Stokes-state uniquely. To see this, consider the state

5ek(-,- ; y I g)+ 5P"('," ;yl h),

(5.12)

where sek(., . ; Y lg) is the Stokes-state of Definition 3.1, h e ffs and is not identically zero, while 50"( ., .; y [g) is the linear combination of doublet-states (appropriate to a dynamic center of dilatation) accounted for t h r o u g h Definition 3.2. The state defined by (5.12), in view of T h e o r e m 3.2, evidently conforms to (a) and (c) in T h e o r e m 3.1 but is distinct f r o m the Stokes-state; it possesses, however, displacement and stress singularities at y of a higher order than those inherent in 6ak (., . ; y Ig). This example makes clear that hypothesis (b) in T h e o r e m 5.1 c a n n o t be omitted; n o r can it be relinquished in favor of the weaker requirement that, uniformly on [0, t] for every t > 0 , lim S (x-ak)AS'(X,')dA=O r/--.o Ak(r/)

on

[0,~)

( k = l .... , n ) ,

lim ~ (x-ak)AS"(x,.)dA=O ~/--.o Ak(r/)

on

[0,~)

( k = l .... , n ) ,

without invalidating the conclusion. A n analogous counter-example related to a concentrated surface load on the b o u n d a r y of an elastic half-space is easily constructed.

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Acknowledgment. This investigation was carried out under Contract Nonr-220(58) of the California Institute of Technology with the Office of Naval Research in Washington, D.C. One of the authors (L.T.W.) is also indebted to the National Science Foundation for a Graduate Traineeship. We are both most grateful to M.J. TIYRTELTAtm,who read the manuscript and offered valuable suggestions and criticisms. California Institute of Technology

(Received July 8, 1968)