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AN EXISTENCE AND UNIQUENESS THEOREM IN LINEAR ELASTODYNAMICS Paolo Podio Guidugli z

S O M M A R I O : Si studiano le questioni di esistenza e unicith per i/flroblema al contorno di tipo ellittico che risu#a dal/a trasformazione secondo Laplace rispetto al tempo de/problema nei valori iniziali e al contorno della elastodinamica lineare. S U M M A R Y : Laplace transform with respect to time is applied to the boundao,-initial-value problem of linear elas/od3,ttamics in order to produce a simpler elliptic botmdary-value problem. E x istence arm tlniquettess of the weak solution of/he latter problem are proved.

1.

I n t r o d u c t i o n .

In this paper we deal with the boundary-value problem which results from the general evolution problem of linear elastodynamics via Laplace trasform with respect to time. It is well known that the boundary-initial-value problem of elastodynamics can be formulated in the framework of the variational theory of hyperbolic operators, and solved on finite time intervals under physically reasonable hypotheses (cf., e.g., [1]). Unfortunately, these hypotheses are by no means as general as it would be desirable. Moreover, the variational approach to hyperbolic operators is somewhat more delicate than in the standard elliptic case, and partly shares the elegant simplicity of the latter only when the initial displacement vanishes. On the other hand, L-transform takes "automatically" care of dependence on time and initial data whatsoever, and reduces the elastodynamic problem to a one-parameter family of elliptic boundary-value problems. Of these problems we give a variational formulation for two reasons. First, the requirements of unambiguous conceptual setting, as well as the demands of numerical applications, are most satisfactorily met in this way. Second, existence and uniqueness of weak solutions are usually proved by applying arguments as Lax & Milgram's that depend on checking continuity and coerciveness of integrodifferential forms defined on suitable function spaces, and this reveals by no means trivial in most cases. Now, a peculiar feature of our approach to elastodynamics is that continuity and coerciveness of our family of forms easily ensue when the same properties hold for the single integrodifferential form of linear elastostatics, i.e., whenever the classical inequalities of Poincar6 and Korn can be shown to be valid.

1 lstituto di Scienza delle Costruzioni, Universitb di Pisa. 194

In the variational theory of partial differential operators, an essential tool in showing that a problem is properly posed consists in a suitable a priori bound to be used both to prove continuous dependence of the solution on the data, and to take the first steps towards regularization. Although establishing such a bound is undoubtedlypossible in the present context, we here confine ourselves to treating the existence issue, as this is not only preliminary to others but also appropriate to serve as an illustration of our methods. Stability and regularity will he discussed elsewhere, as well as other properties of the solution which are invariant under inverse L-transform.

2.

L-transform of the elastodynamic problem.

In this section we show that the usual setting of the elastodynamic problem is equivalent to the one obtained by application of the Laplace transform. We aim only to motivate the study of the weak formulation of the Ltransformed problem (see next section). Accordingly, we content ourselves with proving the equivalence abovementioned in a quite formal way, and purposely skip listing the underlying smoothness hypotheses. Let .¢~ denote a regular, bounded region of R", with boundary 0 8 , and let n designate the outward unit normal to 0 8 . Let further R+(R0 +) denote the positive half line (R + deprived of the null element). We stipulate that the following data are assigned on ~i~: the elasticity field, L [ . ] (x). Here L [ . ] is a mapping from Lin (the space of second-order tensors) into Sym (the subspace of symmetric elements of Lin); - -

- - the density field, Q(x)> 0; the initial displacement and velocity fields, u0(x) and

- -

v0(x). We further stipulate that a body force field b(x, t) is assigned on ~/~ X Rg, as well as a surfacial displacement field, u"(x, t), on 01~ × R+;

- -

a traction field, s^(x, t), on 0z~ × R~'. Here 01~, 02~ are subsurfaces of 0 8 , such that 0°~ = 01~'0 0~.~, and

- -

0 ~ ' n 0 ; 8 = S~.

In the classical formulation (el., e.g., [3]) the mixed problem consists in finding a vector field u(x, t) on ;~ X R + that solves the displacement equation of motion div L [grad u] -t- b = oil

on

B~J × Rff

(2.1)

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and satisfies the

initial conditions

u ( . , 0) = u0 ;

fi( -, 0) = v0

on ,~,

(2.2)

i) (minor symmetries) Nul L = Skw (the subspace of skewsymmetric elements of Lin). As L : Lin--> Sym, it follows that A . L[B] ----sym A . L [sym B], V A, B ~ Lin3

the

displacement condition (,,,ajar symmetry)

ii) u = ~

on

0 1 ~ × R~-,

(2.3)

A . L[B] = B . L[A], and the

traction condition iii)

L [grad u] n = s"

on

029;3 × R~"

(2.4)

When 01~' -----0 ~ (0zSY = ~ ) , (2.1) . . . . . (2.4) describe the

displacementproblem, when c92~= 0 ~ (0x,.~= ~ ) the traction problem. The following lemma establishes the equivalence between the above formulation of the problem of elastodynamics and the formulation in which the initial conditions are incorporated in the field equation via L-transform. Let W(x, .) be any tensorial field defined on R +, and let

~ ( x , p ) = Ioe-vt t]F(x, t) dt

L e m m a . u corresponds to a solution of the mixed problem if and only if ~ solves the equation divL [grad~]Wf=@2~

(positiveness) A.L[A]>0,

A ~0.

lm,l,,r such

lmA.A< A.L[A]._- 0.

V A, B ~ Lin.

(u.u ÷ v u .

(2.5) and the affiliated 'norm where f = 1~+ O(v0 + p u 0 ) ,

(2.6)

and satisfies the boundary conditions

u = ~

on

0 1 ~ × (po, co),

(2.7)

L [grad u] n = s"

on

az~3~ × (p0, co).

(2.8)

R e m a r k 1. We have given in the Lemma the strong formulation of the L-transformed mixed problem. The L-transformed displacement and traction problems can be read off (2.5) . . . . . (2.8) in the obvious way. We close this section by recalling that the choice of the elasticity tensor L is usually subject to some requirements of physical plausibility and mathematical convenience. For completeness we list those restriction on L that we assume to be satisfied hereinafter: SEPTEMBER 1975

Ilull¢ = f ~ ( u

• u ÷ v u . v u ) = Ilullo ÷ lul¢;

here Itullo = (ulu) is the natural norm of L 2, and lull is the seminorm of the first partial derivatives. Let further V be a closed subspace of /-/1, which is defined by a set of boundary differential operators, and possibly by some adscititious integrability conditions, in a manner that will be made precise in the next section. We denote by V ' the dual of V, and by ( . , • ) the pairing between elements of V ' and V. We consider the family of forms on V

av(u, v) = p 2 ( u [ v)0 + b(u, v),

(3.1)

where p is a real parameter, and b(u, v) is the energy integral relative to the problem of linear elastostatics, i.e.,

2 Here we designate by a dot the inner product of Lin, and by sym the mapping which associates to any element of Lin its symmetric part. 195

b(u, v) = .fa~e grad v • L [grad u].

(3.2)

It is clear by the above definition that b( •, • ) is a firstorder, real, bilinear integrodifferential form. Under the hypotheses on the elasticity tensor that we have previously listed, it is not difficult to show that b ( . , • ) is also symmetric, continuous and coercive on V, i.e., respectively,

b(u, v) = b(v, u), (3.3)

Ib(u, v)l < ~ ltult, I1%, b(u, u) > m Ilull~,

VU, V ~ V;

(here, A,/, m are two positive constants). We recall that some restrictions on the choice of V are essential to applying the inequalities of Poincar~ and Korn in the process of establishing (3.3)2.s (cf. [2], [4]). It follows from (3.1)and(3.3) that the bilinear form a v ( . , .) is symmetric, continuous and coercive on V for p < + oo fixed. Therefore, if f3, is a one-parameter family of elements of V ', the generalized Lax & Milgram argument we referred to in the Introduction applies, and we conclude that, for any p < + oo fixed, there exists one and only one solution up ~ V of the equation

ap(u, v) = (fj,, v),

V v ~ It.

(3.4)

If we now identify u with the L-transformed displacement field,s equation (3.4) admits of an interpretation as the weak formulation of the differential problem (2.5) . . . . , (2.8). I n fact, the form av(. , • ) univocally corresponds to the second-order, variational, elliptic operator

Indeed, the problem of finding the minimum of a quadratic functional of type (3.8) is known to be equivalent to the existence problem for an equation of type (3.7) involving a second-order, variational, elliptic operator.

Remark. It is a simple matter to check that q)p{ • } is the functional called "transformed energy" by Benthien & Gurtin. In their paper [5] (cf. also [3], sections 65 and 66), q~v{" } is defined in a rather complicated way by successive applications of convolution and L-transform, and is associated with a certain variational principle for elastodynamics. Benthien & Gurtin note an analogy between Op{ • } and the functional of the principle of minimum potential energy for elastostatics. This analogy consists in showing, as they do, that ~p{ • } assumes its minimum value at a solution of the mixed problem. In the light of our present developments, the same result appears in a more natural and complete form.

4.

Specification of the space of solutions. N o n h o m o g e n e o u s problems.

In this section we substanzialize our treatment by making precise the definitions of the space of solutions V, and of the datum fv, for the three types of problems which are summarized in (3.4). Moreover, we indicate how to reduce the cases with nonhomogeneous boundary conditions to suitable homogeneous problems.

i) Displacement prob/em. I f the boundary data vanish we chose:

- - Epu -----dip L [grad u] - - Open

(3.5) v = Hi,

in the sense of the distributions on V, i.e., av(u,t8) = (Evu, q~), V u ~ H 1 and Vq~ ~ C~. (3.6)

fp = f

(cf. (2.6)).

Otherwise, let ~ # 0 on 0 H be an element of the fractional Sobolev space HI(OH), and let g ~ H a be such that its trace on the boundary of H equals ~ :

We may formally restate the latter results in the following T h e o r e m . Let V c H 1, Ev and fp be as above. Then, there exists one and only one solution up ~ V of the distributional equation E~u = fv.

Or{U) = y

aT(u, u) - - (fp, u).

(3.8)

8 Henceforth we drop the superposed bars to distinguish I~tranforms. 196

U----u--g,

(3.7)

According to an interpretation that has by now become almost standard, we may also think of (3.7) as the Euler equation (in the weak sense) o f the following functional on Ht:

1

let further u be a solution of the nonhomogeneous problem. We then set

and again deal with the problem of finding one and only one U ~ V = H01 which solves (3.4) with

(f., v) = (f, v) + a~(g, % ii) Traction problem. In the absence of any Dirichlet datum, no kinematical conditions intervene in the definition of V. However, V cannot be identified with H a, otherwise we are unable to MECCANICA

guarantee the coerciveness of the form b ( . , . ) . homogeneous case we therefore chose

In the

V= (u eH~l fau-- San(x--xo) × u=o), and for consistency impose the following conditions of

compatibiliO~ on f:

j'f = j'jx-

×

On the contrary, when the Neumann datum ~ does not vanish, and we have ~ E L~(B~), 4 we simply put in equation (3.4)

REFERENCES 1 LIONS J. L. & ~L~GENES E., "Probl6mes aux limites non homog~.nes et applications." Vol. 1, Dunod, 1968. 2 CAMVANATOS., "La diseguagliartza di Korn in elasticitY," Symp. on "Existence and Stability in Elasticity," Udine, June 18-23, 1971. SEPTEMBER 1975

(f,, V) = (f, V) + S 0 ~ °v. iii) Mixed problem. In the homogeneous case, if