Stabilization of non homogeneous elastic materials with voids

Description

Niveau: Supérieur, Doctorat, Bac+8
Stabilization of non-homogeneous elastic materials with voids Serge Nicaise?, Julie Valein† May 4, 2011 Abstract We study the asymptotic behavior of the solution of the non-homogeneous elastic system with voids and a thermal effect. We first prove the well-posedness of this system under some realistic assumptions on the coefficients. Since this system suffers of exponential stability (as shown in dimension 1 in [18]), our main results concern strong and polynomial stabilities again under some assumptions on the coefficients. These stabilities are obtained in a closed subspace of the natural Hilbert space. Hence we characterize its orthogonal and further show that in the whole space the energy tends strongly or polynomially to the energy of the projection of the initial datum on this orthogonal space. In this respect we extend and precise former results obtained in one dimension in [18]. 2000 Mathematics Subject Classification: 35L05, 93D15 Keywords: Elasticity, Polynomial stability 1 Introduction and main results There is a large literature devoted to the stabilization of the elasticity systems set in bounded domains of Rd, d ≥ 1 by boundary and/or internal dampings, see [1, 5, 7, 10] and the references cited there. As alternative damping we can couple the elasticity systems with the heat equation (elasticity with thermal effects) and it is well known that the thermal effects provokes the expo- nential decay of the solution [13, 21].

• polynomial stability


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• homogeneous elastic


• ?université de valenciennes et du hainaut cambrésis


• contrac- tions over


• equation


• decay can

Sujets

• Saint Nicaise
• Cartan
• Équation

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 17
• Langue: English

Extrait

Stabilization

of non-homogeneous elastic voids Serge Nicaise ∗ , Julie Valein †

May 4, 2011

Abstract

materials

with

We study the asymptotic behavior of the solution of the non-homogeneous elastic system with voids and a thermal effect. We first prove the well-posedness of this system under some realistic assumptions on the coefficients. Since this system suffers of exponential stability (as shown in dimension 1 in [18]), our main results concern strong and polynomial stabilities again under some assumptions on the coefficients. These stabilities are obtained in a closed subspace of the natural Hilbert space. Hence we characterize its orthogonal and further show that in the whole space the energy tends strongly or polynomially to the energy of the projection of the initial datum on this orthogonal space. In this respect we extend and precise former results obtained in one dimension in [18].

2000 Mathematics Subject Classification:35L05, 93D15 Keywords:Elasticity, Polynomial stability

1 Introduction and main results

There is a large literature devoted to the stabilization of the elasticity systems set in bounded domains ofR d , d≥1by boundary and/or internal dampings, see [1, 5, 7, 10] and the references cited there. As alternative damping we can couple the elasticity systems with the heat equation (elasticity with thermal effects) and it is well known that the thermal effects provokes the expo- nential decay of the solution [13, 21]. In this paper we are interested in porous elastic materials and in that case it was shown in [20] that the porous viscosity was not strong enough to obtain exponential decay of the solutions and that the decay can be very weak. Hence other dissipa- tive mechanisms were considered recently in order to restore such an exponential decay, see for instance [16, 17, 18]. Here we want to consider the thermal and viscoelastic effects on the decay of the multi-dimensional problem (see [8, 11, 12] for the modelisation). Since in dimension 1, this system suffers of exponential stability [18], we concentrate on weak stability results by proving some strong and polynomial stabilities under some realistic conditions on the coefficients. Note that the main aim of this paper is to generalize the results from [18] to the multi-dimensional case and to non constant coefficients. ∗ du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Tech-Université de Valenciennes et niques of Valenciennes, F-59313 - Valenciennes Cedex 9 France, [e-mail] † Institut Elie Cartan Nancy (IECN), Nancy-Université & INRIA (Project-Team CORIDA), B.P. 70239, F-54506 - Vandoeuvre-lès-Nancy Cedex France, [e-mail]

1

Accordingly we consider the stabilization of the following coupled elastic solids with voids set in a bounded domainΩofR d ,d= 1,2or3with a Lipschitz boundaryΓ(for the model, see section 5 of [8], [11] or [12]):

(1)   u c ρ θ t tt d= i i v v( =d k [C r ( θ ( ) u − ) βd +γ iv d  iv ( u u t u t ) − ) − + m ξϕ (b +m ϕ− θ, θβ)Id], inΩ×(0,+∞), J ϕ tt = div(δrϕ)−b  ϕ t with the boundary conditions (nbeing the unit outward normal vector alongΓ)

(2)

u= 0, δrϕ∙n= 0, krθ∙n= 0onΓ×(0,+∞),

and, finally, the initial conditions (3)  u u t ( ( ,x ,x )0= =)0 u u 1 0 ( ( x x ) ) ,  ϕ ϕ t ( ( x x, , 0) ) = =0ϕ ϕ 0 1 ( (x x) ) , θ(x,0) =θ 0 (x)inΩ. Here the variablesu= (u i ) id =1 ,ϕandθare the (vectorial) displacement of the solid elastic material, the volume fraction and the temperature respectively. The coefficientsρ,b,β,γ,J,ξ,mandc belongs toL ∞ (Ω)and are related to the constitutive material. Similarlykandδared×dsymmetric matrices and are assumed to belong toL ∞ (Ω) d×d . FinallyC= (c ijk` )is a tensor such that c ijk` =c ji`k =c k`ij ∈L ∞ (Ω), all indices running over the integers1,∙ ∙ ∙, d usual for. Asu= (u i ) id =1 ,(u)is the linear strain tensor defined by (u) = ( ij (u)) ,ij d =1 with ij (u=)( 2 1 ∂ i u j +∂ j u i ). For ad×dmatrix= ( ij ) ,ji d =1 the productC= ((C) ij ) j,i d =1 is thed×dmatrix given by d (C) ij = X c ijk`  k` . k,`=1 Finally for a (smooth enough) vector valued functionv: Ω→R d ,divvis its standard divergence, namely d divv= X ∂ j v j , j=1 while for a (smooth enough) matrix-valued functionw= (w ij ) : Ω→R d×d ,divwis its divergence line by line, i.e., d divw= ( X ∂ j w ij ) id =1 . j=1 For well-posedness reason we assume that the first two equations of our system is of hyperbolic type while the third one is of parabolic type. Hence we require that there exist a positive function µand positive real numbersk 0 , δ 0 , ρ 0 , J 0 , c 0 , ξ 0 andµ 0 such that for almost allx∈Ω (4)ρ(x)≥ρ 0 , J(x)≥J 0 , c(x)≥c 0 , ξ(x)≥ξ 0 ,

2

(5)

and

k(x)X∙X≥k 0 |X| 2 ,

δ(x)X∙X≥δ 0 |X| 2 ,∀X∈R d ,

d×d (6)C(x):≥µ(x)|| 2 ≥µ 0 || 2 ,∀∈R, where|| 2 = P d j,i=1 | ij | 2 for all∈R d×d and:τdenotes the contraction of the two matrices, i.e., d :τ= X  ij τ ij , i,j=1

and finally

(7)

γ(x)≥0.

This paper is organized as follows. In Section 2 assuming (8) Z Ω (cθ 0 +mϕ 0 +βdivu 0 )dx= 0,

we will prove that the system (1)-(3) is well-posed under some assumptions on the coefficients. We then find in Section 3 sufficient conditions that garantee the strong stability of the system, these conditions are mainly based on some spectral properties of a system coupling the elasticity system with a diffusion equation. In Section 4, we prove some polynomial stability by using a frequency domain approach and by taking the initial data in an appropriate subspaceH 0 of the natural space H. Ifγis positive definite andm6≡0, the orthogonal of the spaceH 0 is at most of dimension 2, on the contrary the situation is more delicate as seen in Section 5, where we characterize this spaceH 0 when all the coefficients are constants and whenγ= 0. Let us finish this introduction with some notation used in the remainder of the paper: The L 2 (Ω)-inner product (resp. norm) will be denoted by(∙,∙)(resp.k ∙ k). The usual norm and semi-norm ofH s (Ω)(s >0) are denoted byk ∙ k s,Ω and| ∙ | s,Ω  shortness, we will, respectively. For use the same notation inH s (Ω) d .

2 Well-posedness of the system

We consider the Hilbert space H={(u, v, ϕ, φ, θ)∈H 01 (Ω) d ×L 2 (Ω) d ×H 1 (Ω)×L 2 (Ω)×L 2 (Ω)satisfying (9) below} (9) Z (cθ+mϕ+βdivu)dx= 0. Ω

OnH, we introduce the sesquilinear form hU, U ∗ Z Ω (u) :(¯u ∗ ) +ρv∙v¯ ∗ +δrϕ∙ rϕ¯ ∗ +ξϕϕ¯ ∗ +J φφ ¯ ∗ +cθθ ¯ ∗ +b(divu ϕ¯ ∗ + div ¯u ∗ ϕ)  dx i H =  C withU= (u, v, ϕ, φ, θ) > ,U ∗ = (u ∗ , v ∗ , ϕ ∗ , φ ∗ , θ ∗ ) > ∈ H.

3