On the determinacy of motions for the displacement problem in the dynamics of elastic bodies with...

Journal of Applied Mathematics and Physics (ZAM P) Vol. 24, 1973 Birkh~iuser Verlag Basel

On the Determinacy of Motions for the Displacement Problem in the Dynamics of Elastic Bodies with Viscosity By Lewis Wheeler and R. Ray Nachlinger, University of Houston, Houston, Texas, USA

1. Introduction

In this paper, we establish uniqueness of the solution to the displacement initial-boundary value problem appropriate to the dynamics of elastic bodies with viscosity. Of special significance is the fact that our results are free of linearizing assumptions. Thus, neither is the material response taken to be linear, nor is the motion restricted so as to justify the linearizing of kinematical quantities.

For the case of purely longitudinal motions, the matters of uniqueness, existence, and stability were treated in [-1] by Greenberg, et al. and in [-2] by Dafermos. Although [1] preceded [-2] into the literature, the latter allows for nonlinear dependence of the stress upon the strain rate, whereas it is assumed to be linear in [1]. In the present paper, we deal with possibly three-dimensional motions, but we do not go into the questions of existence and stability. The approach we use in this investigation might be termed 'the method of energy integrals'. In this respect it is related to the one used by Dafermos in [2] to prove uniqueness.

2. Notation

Vectors will be denoted by lower case latin letters in boldface. Points in euclidean three space will be identified with their position vectors, which we ordinarily denote by x, y, or z. Upper case boldface latin letters are used for tensors of order two; for tensors of order greater than two, we use upper case sans serif letters in boldface. Standard indicial notation, with the usual summation and differentiation con- ventions, is employed in connection with the cartesian components of tensors.

The letter D is to stand for a bounded domain in euclidean three space. Thus D always represents an open connected set. For the closure and boundary of D we write, respectively,/) and OD. Because the divergence theorem is applied in the sequel, we require the notion of a regular domain. For our purposes, it suffices to let 'regular' mean the divergence theorem is valid for vector fields that are continuously differentiable on/) .

ZAM P 24/38

602 Lewis Wheeler and R. Ray Nachlinger ZAMP

Finally, we say that a function is of class c#, if it is continuous, or that it is of class ~ where m > 0 is an integer, if it is m-times continuously differentiable. Occasionally, it is important to specify precisely the set where smoothness is assumed. In such cases, we write c#(p) or c#,,(p) where P is a subset of the appropriate finite- dimensional euclidean space. Thus, cg(p) stands for the class of functions defined and continuous on P, while p ~ c#,, (p) if ~o ~cg(p) and every partial derivative of order up to and including m is the restriction of a function in c#(p) to the interior of P.

3. Regular Motions. Balance Laws in Reference Form

For our purposes, it is convenient to employ the referential description of motion. Hence the body is assumed to occupy at time t = 0 a domain D, the points of which serve to identify the particles that comprise the body. The location of those points at times t > 0 may then be described by a vector-valued function Z defined o n / ) x [0, ~) , i.e.

z =X(x, t) (3.1)

is the position vector at time t of the particle initially at x. By a regular motion we mean a function ~ with the properties

Xe~2( / )x [0, oo)), xe3=~X(x,O)=x, det F > 0 on D x [0, oo), (3.2)

where F = gl . The displacement

u(x, t)= z(x, t )-x

and its gradient H = Vu obey

u~Cg2(bx [0, oo)), x~D~u(x,O)=O, d e t ( H + l ) > 0 on / ) x [0, oo), (3.3)

if the motion is regular. In terms of the first Piola-Kirchhoff stress tensor S, the laws of balance of

linear and angular momentum assume the respective forms 1)

V.S+pb--pii, SFr=FS r, (3.4)

where b denotes the body force and p stands for the density in the reference domain D. We say that a pair {S, u} is a dynamical process corresponding to p, b if u is the displacement due to a regular motion, SEcgl(/5 x [-0, oo)), p~c#(b), p > 0 on D, and (3.4) hold on D x (0, oo).

1) A discussion of the Piola-Kirchhoff stress tensors and the equations of motion they give rise to may be found in Section 43A of [3].

Vol. 24, 1973 On the Determinacy of Motions for the Displacement Problem 603

4. Material Response

The purpose of the present section is to specify the class of materials with which our results are concerned, as well as an auxiliary restriction on the material response that is needed in Section 5.

In [4], B.D. Coleman and V.J. Mizel investigated constitutive equations that allow for dependence of the stress upon the deformation gradient and its rate, as well as the temperature and its gradient. It is with such materials that we propose to deal, but in the absence of thermal effects. We are thus led to a constitutive equation of the form

SFT = T(H)+ T(H, L), (4.1) 2)

where

L = F F -~. If the dissipative part, T, is in class cga, we may define

V(H, L)= T L (H, L). (4.2)

In connection with V, we will assume the existence of a constant 3) 2 > 0 such that

Mij Mkl Vijkl ~ t~ M(ij) M(ij) (4.3)

where M is arbitrary and the parentheses refer to the symmetric part.

5. Uniqueness of Solutions

The argument employed by Neumann to establish uniqueness of solutions in linear elastodynamics rests on an energy identity. It is therefore perhaps not sur- prising that an analogue of Neumann's identity plays an essential role in the present investigation.

Theorem5.1. Let {S, fi}, {S, ~} be dynamical processes corresponding to the A - -

mass density p and to the respective body-force fields b, b. Then,

~ {tr [ ( S - S) ( / ~ r - fiT)] + 2 ~ - (5.1)

= ~ p ( ~ - b ) . ( ~ - ~ ) d V + ~ ( g - ~ ) . ( ~ - ~ ) d A , D O D

provided D is a regular domain and

g=~Sn, -g=Sn on ~3D• (5.2)

where n denotes the unit outward normal vector.

a) Note that (3.4)2 implies symmetry of T, T. 3) We have not rendered explicit the possibility that the material be inhomogeneous, in which case ~r

and Tdepend upon position as well as H and L. However, it is our intention here that 2 be independent of these variables.


Proof By hypothesis,

V. ( S - S ) + p ( b - b ) = p ( ~ - u ) .

Thus,

_. p (~-~). fv. (~- $)] +p(f~-b). (u- u)=T V ~- I,~-r

which in turn furnishes

-" P ~ ~ - '2 V. [(~ T - ~ r ) (~ _ u)] - t r [ ( ~ - S) ( ~ T _ ~ r ) ] + p (~ _ b) . (u -- u) = ~ - - ~ - [u -- u l �9

The desired conclusion now follows from the divergence theorem and (5.2).

TheoremS.2. Let {S, fi}, {S, ~} be dynamical processes corresponding to the ^ -

mass density p and the respective body-force fields b, b. Assume that (4.1) holds, and let ~ecgl (~--), Te~2 (~-- x ~'-), where .Y- denotes the set of all tensors of order two. Suppose that D is a regular domain, let t o >_ O, and assume

u(', to)= u(', to) (A) a (', to) = ~( ' , to),

(B) b = b o n D x [ t o , OO);

(C) f i=~ on~?Dx[to,OO);

(D) T conforms to (4.2), (4.3).

Then,

~=~, cd=~ on/SXEto,OO).

on D;

(5.3)

Proof For convenience, put u = f i - ~ , S = g - S , etc. Because of (A) and (4.1), it suffices to show that h defined on [to, oo) by

h(t)= ~ IHI 2 dV, [HI=V~rHH r (5.4) D

vanishes on [to, oo), in order to conclude (5.3). Assume to the contrary that there

exists t*e(to, oo) such that

h( t*)>0, (5.5)

and let

T=sup{slh=O on [t o, s]}. (5.6) 4)

Then,

t o < T < t*. (5.7)

4) The existence of Tis ensured by the fact that h(to)=0.


Hypotheses (A), (B), (C), and Theorem 5.1 furnish

~o[Pli~(x,t)12+/tr(Sl:Ir)dQdV=O, (5.8)

for every teET, t*]. By (4.1),

S = S - S = T(/~) (F- ~)r _ ~(~) (if- ~)r + T(fl, L) (F- ~)r- T(H, L) (F- ~)r

= [~(/~)_ ~(~)] (/~- a)r + ~(~) (~- 1)T H T (~- 1)r (5.9)

+ I- T(/-), L ) - T(/~, L)] (F- 1)T + IT(/1, L)- T(H, L)] (F- 1)T

+ T(H, L)(~-~)~ H ~ (F-1)~.

The assumptions ~-effl ( j ) , T ~ff2 ( j - • 3-) ensure the existence of positive constants M1, M2 such that o n / ) x [T, t*] the Lipschitz conditions

[ T ( / t ) - T(H)I < M11HI, IT(fI, L)-T(H,L)I<MzlHI (5.10)

are obeyed. The assumed differentiability of T further implies the existence of a constant M 3 > 0 such that U defined by (Taylor's expansion)

U~j= T~j(ffI, L)- T~j(fl, L)-Lk, ~jkt(fl, L) (5.11)

satisfies

IUI <Ma 1/tl 2 on /9 • I-T, t*]. (5.12)

Next, note that

L = L - L = h ~ - ' - ~ P -1 = a ~ - I - ~ - I HF - ' (5.13)

By (5.8)-(5.13),

r (5.14) -< ~ i [(Mx + M 2 + 3 M]) M4 IHI I/-ll + M3 IHI 3] dr dg,

D T

provided

M4 = _max [[ lr(H)l + [ T(H, L)[ + t~[ + {/'-1[ + IF- 1[ +IV(H, L)[ + 1]. O x [T, t*l

If there are numbers ml >0 and m z > 0 such that on [T, t*]

~ I:1,,,~7,a l:lk,~; 1 V~jk~(I~l,L)dV>ma ~ l:l~t:I~dV-m2 ~ li~l II:tl dV, (5.15) D D D

then (5.14) gives t

t

< ~ ~ [(M~ + M 2 + 3 M 3) M4 IHI [/~l + Ma [HI 3 + rn2l ti[ I/tl] dr d V D T

606 Lewis Wheeler and K Ray Nachlinger ZAMP

for every te[T, t ,] . Application of the inequality

2ab<a2 +b 2 (5.16)

to the right-hand member yields

p . t dz~ r (5.17)

"3 3m~ fi2 <--~ ST [2~1 (MI+M2+3M3)2M~IHIz+--~-`I21Iz+M3]H[ +~ml I ' ]d"cdV.

Next, note that (5.5), (5.6) imply H(. , T ) = 0 on D. Thus, s ince / t is continuous on b x [T, t*-], there exists ~ (T , t . ] such that

M 3 I/5/1 < _ ~ - on D x I-T, t--].

Consequently, and in view of (5.17), (5.4), t

(5.18) t

3M~ ( i l + i 2 + 3 i 3 ) 2 ~ h ( r ) d z 3m~ t + f (z ) -< 2ml r Am1 T

for every te IT, t-J, provided

�9 p m3=rn~n~- and f(t)=~lfi(x,t)12dV.

D

It follows from (5.4), (5.6) that

H(-, T ) = 0 o n D .

Accordingly,

h(t)= j]Hl2 dV = IHl2 dzdV=2 j Str(Hl:tr) dzdV, D D T D T

so that (5.16) furnishes t

h(t)<4~nl ih(z) dz+mt 3 o : I/-/12 dzdV.

Therefore, and by (5.18), 3 I - " t 2 t

m3 f (t) + h (t) <2-~ml [ 2 + (Ml + M2 + 3 M3)2 M~ ]j r~ h(z) dz + 3~m~ ~ f (z) r

for all te[T, i]. It is therefore clear that there exists a positive constant M such

that if tE [T, t-J, then t

f(t) + h (t) <_ M ~ If(z) + h (z)] dr.


It now follows from Gronwall's lemma 5) that

f = h = 0 on IT, t---I,

which contradicts (5.6). In order to complete the proof, (5.15) needs to be established. Let G = / t ( ~ - ~ ) r,

and appeal to (4.3) to get

I iLIim ~ ml I~Ikn Flint Vij kl(~-~, K') dg>_,~ I G(ij) G(ij)dV. (5.19) D D

It is readily verified that

2 G,ij, G,j)= Gij G 0 + (/2/i k ~ - ,)z + t/I,/2/jk Fjj~n I F//k 1), m -- (~/i Him FjJm 1 Elk 1),k

Since ri =0 on ~?D x [to, oc), the divergence theorem thus furnishes

2 ~ G(ij) Gtij) dV D

= I [G,j G,j+(IZI, R ~;1)2--it, I-:Ijk(~7, ,' ~'~X),m+iZ, IZljm(~m ~ ~k- ~), k] dV (5.20) D I [Gij GiJ - ~li ls m (~ ml F/ik 1),m -~- ~li I~Ijm (Fjm I F//k 1), k ]d V.

D

Hence, it is clear that there exists a constant m 2 >0 such that

2 ~ G,,,) Gt,j)dV>_~ (G,j 2m2 II:l]) dV. o G ; J - - 7 - I~il (5.21)

Next, note that if 2m~/2 is the minimum over/9 x IT, t*] of the smallest eigenvalue of P-~(~-I ) r, then m~ >0 and

2m1 �9 �9 Gii Gij ~ - - - Hij Hi i on D x IT, t*]. (5.22)

The desired inequality (5.15) follows from (5.19)-(5.22), and the proof is now complete. If, instead of (4.3), we assume that

Mij Mkl Vijkl >-- ). Mij Mij, (5.23)

then a minor modification of the proof shows that uniqueness holds with Hypoth- esis (C) supplanted by

(C') ( g - ~ ) . ( f i - u ) = 0 on c~Dx [to, oo).

The advantage of (C') over (C) is that it significantly extends the class of boundary conditions to which the theorem pertains. On the other hand, (5.23) unfortunately

~) [5, p. 286].


does not in general hold, because (4.1), (4.2), and (3.4)2 imply

Vijkl = V(ij)kl. (5.24)

Concern ing (4.3), we should men t ion that a l though there appears to be no cr i ter ion

by which we can judge its physical correctness, it does not a p p e a r to have any

obvious shor tcomings in this respect. F o r the special case in which T is independen t

of H and l inear in L, (4.3) and the mechanical dissipation inequality [4, p. 1121]

t r TD>_O

each imply

Viikl Mij Mkl >_0

for every second-o rde r tensor M. Thus, in this instance, it is na tu ra l to view (4.3)

as a requ i rement that the d i ss ipa t ion inequal i ty be strict for a veloci ty grad ien t

wi th non-van i sh ing symmet r ic par t .

Acknowledgment

We grateful ly acknowledge the suppor t p rov ided

F o u n d a t i o n th rough G r a n t GK-31070.

by the N a t i o n a l Science

References [1] J.M. GREENBI3RG, R.C. MACCAMY and V.J. MIZEL, On the Existence, Uniqueness, and Stability of

Solutions of the Equation tr'(ux) uxx+ AUxtx=Po uu, J. Math. Mech. 17, 707-728 (1968). 1-2] C.M. DAFERMOS, The Mixed Initial-Boundary Value Problem for the Equations of Nonlinear One-

Dimensional Viscoelasticity, J. Diff. Eq. 6, 71-86 (1969). [3] C. TRUESDELL and W. NOLL, The Nonlinear Field Theories of Mechanics, Encyclopedia of Physics,

S. FL~GGE, Ed. Springer, New York, Vol. 3, Part 3, 1965. [4] B.D. COLEMAN and V.J. MIZEL, Existence of Caloric Equations of State in Thermodynamics, J. Chem.

Phys. 40, 1116-1125 (1964). [5] J. DIEUDONN~, Foundations of Modern Analysis, Enlarged and corrected printing (Academic Press,

New York 1969).

Summary For a class of materials in which the stress depends upon the deformation and its rate, we

show - without the aid of linearizing assumptions - that the solution to the displacement initial boundary- value problem is unique.

Zusammenfassung Fiir eine Klasse von Materialien, deren Spannung von ihrer Verformung und der Geschwindigkeit

derselben abh~ingt, Zeigen wir - ohne die Hilfe yon Linearisierungsannahmen -, dass die L6sung des Anfangs- und Randwertproblems fiir die Verschiebungen eindeutig ist.

(Received: January 30, 1973)

On the determinacy of motions for the displacement problem in the dynamics of elastic bodies with...

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