Table Of ContentExistence of Solutions ot eht Displacement
melborP for citatsisauQ yticitsaleocsiV
W. S. EDELSTEIN
Communicated by J. L. NESKCIRE
1. Introduction
In a paper published in 1909 1, OTIV ARRETLOV showed that if the relaxation
tensor
G = G (x, t)
were suitably smooth, then uniqueness for the displacement problem of linearized
quasistatic viscoelasticity followed if G were initially positive definite. Also,
using the methods of NIETSLEDE & GURTIN 2 or of ODEH & HSHKABJDAT 3,
one obtains uniqueness for the displacement problem provided G is initially
homogeneous, i.e.
G(x, )O = G(O)
and initially strongly elliptic.
In the present paper, we show that the above two sets of hypotheses are
also sufficient for existence. For the desired displacement field we obtain an
explicit representation in terms of a series of solutions of associated elastic pro-
blems. The rapid convergence of this series is demonstrated by means of Schauder
estimates derived by ,NOMGA ,SILGUOD & GREBNEmN 4 for elliptic systems of
partial differential equations.
In section 2 we define the terminology we shall use and state the main result.
Section 3 is devoted to a formal derivation of the series solutions. In sections 4
and 5 existence is established rigorously. As a by-product of the analysis in the
latter section we obtain an a priori estimate for viscoelastic displacement fields.
This work was inspired by the lectures given by ONATEAG AREHCIF at The
Johns Hopkins University 5.
Notation. R is a region in three-dimensional Euclidean space with closure R_
and boundary a R. Points in/~ are denoted by x and time by t and z. We use the
conventional indicial notation, so that unless otherwise specified, repeated
subscripts imply summation over the range ,1 2, 3. Letf be a function of x and t.
The derivatives of f are written
~,.+n f(x, t)
k(x, t)-
t~X i t~Xj ... t~Xk~t n
m indices
122 W.S. EDELSTEIN :
Let m be a non-negative integer and O<a< .1 Then for sufficiently smooth
functions of x we use the norm
g,,+~=l.u.b. Ig(x)l+l.u.b. g,i(x)l+'"+ 1.u.b. g, ij...k(X)l+
R i,R i,j ..... k,R
m indices
Ig,~j...k(X)--g, ij...k(Y) l
+ 1.u.b.
Ix-yl"
i, j, ..., k, R
m indices
x*y
Functions g whose derivatives of order up to and including the m ht are continuous
in/2 and for which all of the quotients
I g, tj...k(X)--g, ij...k(Y) (m indices i,j, ..., k)
Ix-yl ~
are bounded for x, y in ,21 (x4=y), will be said to be in C ~+m in/2.
Suppose g depends on x and t and is in C ~+m in/~ for each t in t ,1 t2. Further-
more, suppose all of the derivatives of g with respect to xi of order up to and
including the mth are continuous in/2 (cid:141) t ,1 t2. f, in addition
lim sup lg'iJ'"k(X't')--g'iJ"'k(y't') g'iJ'"k(X't)--g'~i""k(y't) =0 (1.1)
r-~t x,y x-yl ~ Ix-yl ~
for t, t' in tl, t2 and all choices of the m indices i,j, ..., k, we shall write that g
is in C~ '+~ on/2x tl, t2. Notice that condition (1.1) implies that the functions
Ig, ij...k(X, t)--g,~j...~(y, t) l
sup
Ix-yl ~
x*y
are continuous in tl, t2. If R is bounded, any function whose derivatives with
respect to ~x of order up to and including the (m + )1 ts are continuous in/2 (cid:141) t~, 2t
is in C~ '+~ on /2(cid:141) tl, 2t for 0<a<l. A vector-valued function f of x (resp.
(x, t)) will be said to be in C §m in/2 (resp. C~ '+~ in/2x tl, t2) if all of its compo-
nents fi have this property.
2. Statement of the Problem
The displacement equations of motion for linearized quasistatic visco-
elasticity, written in the notation of 6 are
t
(Gijk,(X,O) uk, t(x,t)),J+ I (Gijkl(X, r t --V) Uk,,(X,Z)),j dz= -Fi(x, t). (2.1)
0
Here, u is the displacement vector, F is the body force density per unit volume,
Gijkt
and the fourth order Cartesian tensor G with components is the tensor of
relaxation moduli. G has the symmetries
Gijkl= Gjlkl~ Gijlk.
Quasistatic yticitsaleocsiV 321
We say that G is initially positive definite (IPD) on /~ if and only if, given
any symmetric second order tensor ?,
Gi jkl(X, )O iT j Vkl> A ~ij i~ j
for all x in/~. Here, A is a positive constant which does not depend on ? or x.
The tensor G is initially strongly elliptic (ISE) in R if and only if, there is a positive
constant A such that given any pair of vectors ,~ g, and any x in R,
~G j k, (x, 0) i~ k~ 1q qt > A ~ t~ qj qj.
Notice that a tensor G which has the symmetries mentioned above and is IPD,
is ISE.
Finally, we call G initially independent of x (IIX) if
Gi j k t(X, 0) = Gi j ~ 1(0).
In order to apply the Schauder type estimates of 4, we define a special
class of regions R. Suppose R has the property that a positive number d exists
such that given any point P within the distance d of a R there exists a neighborhood
pU with the properties: (a) it contains the sphere of radius d/2 and center P;
(b) the set Up ~c R is the homeomorphic image of the closure of a three-dimensional
hemisphere of radius 1 under a mapping Tp which, together with its inverse,
is of class C 2+~. Suppose also that over their respective domains both T, and
its inverse have finite 2 + (cid:12)9 norms which are bounded by a constant K independent
of P and that
A =infAe>0,
P
where Ap is the minor constant 1 for pU n 0 R. Such a region R we shall call ADN-
regular.
We are now in a position to state the main theorem.
Theorem. Suppose R is ADN-regular and bounded, F is in .~C in K x 0, oo)
for some ~(0<~< 1), and that G is in C 3 in Rx 0, oo) and is either IPD or both
ISE and IIX in R. Then there is a unique vector u in C~ ~+ on Rx 0, ~) which
satiesfies (2.1) in R x 0, oo) together with the condition
u=0 on 0Rx0, oo). (2.2)
The boundary value problem (2.1), (2.2) constitutes what was referred to
in the introduction as the displacement problem. It is easy to see how to transform
to the present case when non-zero boundary data is prescribed.
3. Formal Solution
We associate with the tensor G an "elasticity" operator E defined by
(E u)i = ( iG j k, (X, )O .kU t), j. (3.1)
x The minor constant depends on the initial value system (3.2), (3.3) given below. Its defini-
tion (p. 43 of 4) is too involved for statement in the present brief note.
421 W.S. NIETSLEDE :
Let F be the inverse operator for E, i, e., the operator which assigns to any func-
tion f which is C a in R the vector u which satisfies
E u =f in R (3.2)
and
u=0 on dR. (3.3)
We now consider the system
t
,x(io~ t)+ S k)'~}G (1 x, t--~)(Fq,)k,,(x, z),s dz = -Fi(x, t). (3.4)
0
By comparing (2.1) and (3.4) we see that a formal solution u of the displacement
problem is obtained by taking
u(x,t)=Fq~(x, t) in Rx 0, oo). (3.5)
Define the transform A by
t
(a,p),(x, t)=
Gi ") j k t ,X( t -- )Z (F ,k)P( t ,X( Z), j d z. (3.6)
0
Then (3.4) becomes
q~+A tp= -F. (3.7)
This equation has the formal solution
~= -(I+A)-'F= - ~ (-A)"F (3.8)
n=0
where I is the identity operator and
A~
By combining (3.5) and (3.8) we obtain the representation
,x(u -F~ F')A-(
t)= (3.9)
n=0
= ~ (-1)~+IFA'F. (3.10)
n=O
4. Some Preliminary Results
For boundary value problems involving the operator E we have the following
Theorem A 2. Suppose that the components G~jkz(X, )O are in C ~ on R for all
values of their indices and that G is either IPD or both ISE and IIX on R. Let R
be bounded and ADN-regular. Then, given any function f which is in C ~ in
(0<~<1), there exists a unique vector u=u(x) in C ~+2 in R such that u and f
satisfy (3.2) and (3.3).
2 Theorem A can be shown to follow from Theorem B by means of the methods o f .REDUAHCS
A description of these methods can be found, for example, in .7
citatsisauQ yticitsaleocsiV 521
Thus F is well defined in its prescribed domain. We also need the following
Schauder estimate:
Theorem B. Suppose that G and R satisfy the hypotheses of Theorem A.
Then there exists a constant c>0 depending only on G~jktla(0), A, A, ~, K, and d
such that for any function f which is Holder continuous ni R with exponent ~ (0 < ~ < ,)1
3
I(Ff)j2+~<c~ If~l~. (4.1)
i=1
Theorem B is a specialization of Theorem 9.3 of 4 combined with Remark 2
on the same page. If we had assumed only that G is ISE, then (4.1) would take
the form
,(Ff)j2+~<c(,=~f,l~+ =~ ,(Ff),,o). (4.2)
In order to eliminate the term
3
I(Ff)i Io
i=1
from the right-hand side of (4.1), one must assume (cf. Remark 2 on page 74 of
4) that a uniqueness theorem holds for the boundary value problem (3.2),
(3.3). It is shown in 8 and 9 that such a uniqueness theorem follows from either
of the hypotheses on G stated in Theorem B.
We shall make frequent use of the following.
Lemma. Let G and R be as ni Theorem B and suppose that f is ni C~ (0<~< )1
on R x 0, oo). Then Ff is ni r2C ~+ on R x ,0 oo).
Proof. By Theorem A, Ff is in C §2 on R for each t in 0, oo). The rest of
the Lemma is immediate from (4.1).
5. Proof of the Main Theorem
In order to simplify the calculations, we prove the main theorem only for
the case of a homogeneous material, i.e., the case where G does not depend on x.
The proof in the inhomogeneous case is conceptually the same.
We shall establish the following assertions:
(i) for n > 0, A n F is in C~ in R x 0, oo);
(ii) for any fixed to>0, the series (3.8) converges uniformly in the compact
set/~ x ,0 to;
(iii) the function ~q defined by (3.8) is in C~ in Rx ,0 oo);
(iv) for t~O, oo)
3
Y. I(rA n F)j h § <= c e a (or l.u.b. ~ IF i xI (Q (5.1)
n=0 0, t i=l
where
a(t)= 261 c l.u.b. I i~~t1) j k l(z) I. (5.2)
"e0, t
i, j, k, 1
126 W.S. EDELSTEIN :
By (ii), the function pt defined by (3.8) exists and is continuous. By (iii) and
the Lemma, the function u defined by (3.9) exists, satisfies (2.2), and has the
desired smoothness properties. Assertion (iv) implies that
A (-Are= (-1)'A '+n e,
n=O n=O
so that r rigorously satisfies (3.4), and hence, u satisfies (2.1). Inequality (5.1)
also establishes the representation (3.10) of u and hence the following visco-
elastic a priori estimate:
3
lujl2+~(t)<ce ~ 1.u.b. ~ IFil~(z). (5.3)
o,t t=1
Thus, pending verification of (i), (ii), (iii), and (iv), the proof is complete.
We first show that for n > 0,
3 t 3
I(An+t F)~l~(t)<a(t) ~ ~ I(AnF)jl~(z)d~. (5.4)
i=l 0 j=l
In fact,
t
(A n F) i (x, t) = ~ G}~)k,(t-- z) (FA n F)i,, j (x, z) d z
+1
0
so that
t
I(An+lF)i(x, t) l <33 K(t) ~ l.u.b. I(FAnF)k.tj(x, r)l dz.
0 j,k,l,R
Here,
K(t)- 1.u.b. I C~.~,(T) I. (5.5)
* e 0, t
i,j,k,l
Applying inequality (4.1), we get
3 t 3
2 1.u.b. I(A"+'F),(x, t)l<=34cK(t) S ~ I(A"F),I~(T)dz. (5.6)
i=1 ~ oi=1
Furthermore, for x 1= = y,
I (A" + 1 F), (x, t)- (A" + ~ F), (y, t) I
Ix-y ~
t n n
)l< <l)z--I t(tkI jiG I(FA F)k, lj(x,z)--(FA F)k, tj(y,z)l dr (5.7)
o Ix-yl ~
t 3
<33 cK(t) ~ ~ I(AnF)il,(z)dz.
0 i=1
Combining (5.6) and (5.7), we obtain (5.4). By hypothesis, A ~ F =F is in .~C in
Rx 0, oo). If we assume that nA F is also, then (5.4) implies that A ~+n F is in C ~
on t~ for any ts 0, oo). Let
B = FA n F.
Quasistatic Viscoelasticity 721
By the induction hypothesis and the Lemma, Bi, jk is in C~ on R x 0, oo) for all
choices of the indices. Therefore, since
(y, t') I
(A" + 1 F)i (x, t) - (A n + x F)~
<33 t K ( t) l.u.b. I Bk, t j(x, )r- -- Bk, t j(Y, -r) +
j,k,l
~eO,t
+33K(t) 1.u.b. ng, tj(y,-r) It-t'l+
~xO,t
j,k,l
+33 t' 1.u.b. IBk, t j(~, )r- 1.u.b. Gi (1) jkl ( t ---r)-- G i (~) jk t(t'--r)l
~xO,t ~eO,t
j,k,l i,j,k,I
for any x, yeR and t, t'eO, oo), A"+~F must be continuous in t~xO, oo).
Also, for t, t'>0,
) ' t ' Y ( i ) F I + " A ( - ) ' t ' x ( i ) F X + " A ( Ilx_yl~
(A" + 1 F)~ (x, t)- (A" + ~ F)i (y, t)
sup
y=l=x Ix-y (cid:12)9
<I t
ijkt( -- X
(+)
0
t-r t-r
Bk, lj X, --Bk, lj ,
Bk, t j (X, )r- - Bk, z j (Y, )r-
x sup d'r+
x~:y Ix-yl" Ix-yl"
lj t / k, lj d-r.
+i (~)" , t',~(a) ( ~-) sup
0 Gijkl('t--'C)----t-Uijkl t'-- x*.v Ix-yl"
By the Lebesgue dominated convergence theorem, the right-hand side of this
inequality tends to zero as t' ~ t. Thus, (i) is established.
From (5.4), it follows by a routine induction argument that
3 (a(t) t) n
I(AnF)il~(t)< nt 1.u.b. ~ I )r-(~l~F (cid:12)9 (5.8)
i=1 0, t i=1
This inequality implies (ii). By (i) and (ii), pt is continuous in R x 0, ~). By (5.8),
,x(~o~I t)-qh(y, t)l < ~ I(Z"F)~(x, t) - (a" F), (r, t)l
x-yl ~ ,=o ~ Ix-yl"
(5.9)
3
<e a(')' 1.u.b. ~ IFil=(-r)< oo.
0, t i=1
Thus pt is in C ~ in R for te 0, ~). By (5.9), the series
1.u.b. 1 (A"F)i(x't')-(A"F)'(y't') (A"F)i(x't)-(A"F)i(Y't) I
,=o ,:,y Ix-yl" Ix-yl"
converges uniformly for t, t'e 0, to. Therefore, by (i), (iii) is established.
9 Arch. Rational Mech. Anal., Vol. 22
128 W.S. :qrIETZLEDE Quasistatic Viscoelasticity
By (4.1) and (5.8),
(FA"F)jI2+~(t) <c ~ ~ )t(~I,)F~A(I
n=O n=O i=1
<c t)" 1.u.b. 2 I F, l~(z)
n=o n! t0,t i=1
-
Assertion (iv) follows.
etoN dedda ni .foorp A special case of the above problem has been studied by :~E~AVALH &
UNAELEDERP .01 Roughly speaking, they establish existence and uniqueness of a weak solution
of the displacement problem for that class of homogeneous isotropic materials in which the
hereditary bulk modulus is proportional to the hereditary shear modulus.
.tnemegdelwonkcA The author is indebted to Professors G. AREHCIF and J. L. NESKCIRE
for their encouragement and their helpful criticisms of the manuscript. This research was support-
ed by a grant from the National Science Foundation to Johns Hopkins University.
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Illinois Institute of Technology
Chicago, Illinois
devieceR( yraunaJ ,7 )6691
