# On the Nonlinear Theory of Micromorphic Thermoelastic Solids

Authors: C. Galeş
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

This work aims to study some dynamical problems in the framework of nonlinear
theory of micromorphic thermoelastic solids. First, the continuous dependence of smooth admissible thermodynamic processes upon initial state and supply terms is investigated in the region of state space where the internal energy is a convex function and the elastic material behaves as a definite conductor of heat. Then, a uniqueness result is demonstrated.


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## Body

## 1. Introduction

Motivated by experimental studies, various continuous models of deformable bodies have been proposed in literature in order to describe the thermomechanical behavior of media with microstructure. In the micromorphic theory introduced by Eringen and Şuhubi [1] and Eringen [2], a material body is envisioned as a collection of a large number of deformable particles (subcontinua or microcontinua). Each particle possesses finite size and directions representing its microstructure. The microdeformation gives rise to extra degrees of freedom. Thus, the particle has nine independent degrees of freedom describing both rotations and stretches, in addition to the three classical translational degrees of freedom of its center.Many deformable solids point to the necessity for the incorporation of micromotions into mechanics. Porous solids with deformable grains and pores, composites, polymers with deformable molecules, crystals, solids with microcracks, dislocation and disclinations, and biological tissues (bones and muscles) are just a few examples of deformable solids which require the degrees of freedom given by the micromorphic theory. As a consequence, the micromorphic mechanics is the subject of detailed studies both from theoretical and practical reasons. In the linear context, uniqueness theorems have been proved by Soós [3] and Ieşan [4], variational principles have been established by Maugin [5] and Nappa [6], applications to earthquake problems have been suggested by Teisseyre [7], Dresen et al. [8], and Teisseyre et al. [9], plane harmonic waves have been studied by Eringen [2], reciprocal and existence theorems have been demonstrated by Ieşan [4], and material constants for isotropic materials have been determined by Chen and Lee [10].On the other hand, the theory has been generalized to mixtures of micromorphic materials by Twiss and Eringen [11, 12], to higher-grade materials by Eringen [13], and to electromagnetic micromorphic thermoelastic solids by Eringen [14]. Moreover, the constitutive theory of micromorphic thermoplasticity has been formulated by Lee and Chen [15], the concept of material forces was extended to micromorphic theory by Lee and Chen [16], the problem of heat flow in a micromorphic continua with microtemperatures has been investigated by Ieşan and Nappa [17].This paper deals with the nonlinear micromorphic thermoelasticity. The main purpose is to investigate the stability of smooth thermodynamic processes. In this sense, we use the method developed for nonlinear thermoelastic solids which are nonconductor of heat by Dafermos [18] and updated later by Chiriţă [19] to heat-conducting thermoelastic materials. In the general context of heat-conducting nonlinear micromorphic solids, we prove the continuous dependence of smooth admissible thermodynamic processes upon initial state and supply terms. We present also a uniqueness theorem. Both results are obtained in the region of state space where the internal energy is a convex function and the elastic material behaves as a definite conductor of heat.We recall that the Dafermos method has been utilized recently to prove continuous dependence results for nonconductor-of-heat mixtures [20–22] and for materials with voids [23].The paper is organized as follows. In the next section, we recall the basic equations of the nonlinear theory of micromorphic thermoelasticity. Then, in Section3, we use the consequences of the second law of thermodynamics to prove a uniqueness theorem and the continuous dependence of smooth thermodynamic processes upon initial state and body loads.


## 2. Basic Formulation

We consider a micromorphic continuum, and we assume that at timet0 the body occupies the region B of the Euclidean three-dimensional space and is bounded by a piecewise smooth surface ∂B. Following [2], a point C in the body is characterized by its rectangular coordinates X1, X2, X3, in a fixed system of rectangular Cartesian axes OXK, K=1,2,3, and a deformable vector Ξ. One writes C(X,Ξ). Deformation carries C(X,Ξ) to c(x,ξ) in the spatial configuration at time t, where x1, x2, and x3 are the rectangular coordinates with respect to a new coordinate Cartesian system oxi,  i=1,2,3, and ξ is the vector attached to c. The motion can be expressed as(2.1)xi=xi(X,t),ξi=χiK(X,t)ΞK,X∈B,t∈I,
where, and henceforth, the summation convention on repeated indices is understood and I=[t0,t1), t1>t0.The inverse motion can be written as(2.2)XK=XK(x,t),ΞK=XKi(x,t)ξi,
where(2.3)χiKXKj=δij,χiKXLi=δKL.In what followsḟ denotes the material time rate of f. Moreover, we use the notations fK=∂f/∂XK and fi=∂f/∂xi.The balance laws of micromorphic continua can be expressed as follows [1, 2]:(i)
the conservation of mass:(2.4)ρ0=Jρorρ̇+ρvi,i=0,
where ρ0(X) and ρ(x,t) are the mass densities at time t0 and at present time, respectively, J=det(∂xi/∂XK), and vi are the components of velocity vector;(ii)
the conservation of micromorphic inertia:(2.5)ikl=IKLχkKχlL,IKL=iklXKkXLlordikldt=ikmνlm+ilmνkm,
where νkl is the microgyration tensor defined by(2.6)νij=χ̇iKXKj,
and  IKL(X) are the components of the inertia stress tensor at time t0 and ikl(x,t) are the components of the microinertia tensor at time t. Clearly, IKL(X) and ikl(x,t) are symmetric and positive definite;(iii)
the balance of momentum:(2.7)tji,j+ρfi=ρv̇i,(iv)
the balance of moment of momentum:(2.8)mkij,k+tji-sji+ρlij=ρσij,(v)
the balance of energy:(2.9)ρė=tjivi,j+mkijνij,k+(sji-tji)νij+qi,i+ρr,
where t, m, s, σ, e, q are the stress tensor, moment stress tensor, microstress average, spin inertia, internal energy, and heat input, respectively, and f, l, r are the body force, external body moment, and heat source, respectively. The tensor s is symmetric, that is,  sij=sji. The spin inertia σij may be expressed as(2.10)σij=IKLχ̈iKχjL=imj(ν̇im+νinνnm).The second law of thermodynamics is written as(2.11)ρθη̇-qi,i-ρr+1θqiθ,i≥0,
where η is the entropy density and θ is the absolute temperature.The above formulation is described in detail in the book by Eringen [2]. Since we deal with micromorphic solids, we reformulate the basic equations in Lagrangean description. Thus, introducing the Piola-Kirchhoff tensors(2.12)TKi=JXK,jtji,MKiL=JXK,jXLkmjik,QK=JXK,iqi,SKi=JXK,jsji,LiK=XKjlij,
and making the notation(2.13)GiL=XLj[xj,K(SKi-TKi)-χjM,KMKiM],
then, with the help of the relations (2.4), (2.6), and (2.10) and the identity(2.14)(xi,KJ),i=0,
we obtain the equations of motion(2.15)TKi,K+ρ0fi=ρ0ẍi,MKiL,K-GiL+ρ0LiL=ρ0IKLχ̈iK,
the energy equation(2.16)ρ0ė=TKivi,K+MKiLχ̇iL,K+GiKχ̇iK+QK,K+ρ0r,
and the inequality(2.17)ρ0θη̇-QK,K-ρ0r+1θQKθ,K≥0.
Introducing Helmholtz's free energy ψ as(2.18)ψ=e-θη,
then, with the help of (2.15), the energy equation can be expressed as(2.19)ddt[ρ0(ψ+θη+12vivi+12IKLχ̇iKχ̇iL)]=(TKivi+MKiLχ̇iL+QK),K+ρ0(fivi+LiLχ̇iL+r).
Moreover, from (2.16), (2.17), and (2.18), we deduce(2.20)-ρ0(ψ̇+ηθ̇)+TKivi,K+MKiLχ̇iL,K+GiKχ̇iK+1θQKθ,K≥0.The response of a micromorphic thermoelastic solid is characterized by the following constitutive equations:(2.21)ψ=ψ̂(L),TLj=T̂Lj(L),GjL=ĜjL(L),MLjP=M̂LjP(L),η=η̂(L),QL=Q̂L(L),
where(2.22)L={xi,K,χiK,χiK,M,θ,θ,K;XK}
and ψ̂,  T̂Lj,ĜjL,  M̂LjP,  η̂, and Q̂L are sufficient smooth functions.We assume that there is no kinematical constraint. Then, it follows from the inequality (2.20) that(2.23)ψ=ψ̂(xi,K,χiK,χiK,M,θ;XK),QL=Q̂L(L),TLj=ρ0∂ψ̂∂xj,L,GjL=ρ0∂ψ̂∂χjL,MKjL=ρ0∂ψ̂∂χjL,K,η=-∂ψ̂∂θ,(2.24)Q̂L(xi,K,χiK,χiK,M,θ,0;XK)=0.
The inequality (2.20) reduces to(2.25)Q̂Kθ,K≥0.


## 3. Uniqueness and Continuous Dependence

In this section, we establish a uniqueness result and the continuous dependence of smooth thermodynamic processes upon initial state and supply terms.We assume thatB is a bounded region and that ∂B is sufficiently regular to assure the common laws of transformation of surface integrals. We will employ the following notations: NK are the components of the unit outward normal vector to the surface ∂B;  I denotes the time interval [t0,t1), where t1 may be infinity; the symbol |·| denotes a norm, either in Euclidean vector space or in a tensor space, while ||·||L2(B) denotes the L2-norm.Definition 3.1.
By a thermodynamic process onB×I one means an ordered array of functions {xi,χiL,θ,ψ,TKi,GiL,MKiL,η,QK,fi,LiL,r} which satisfy the equations of motion (2.15), the energy equation (2.19), and the constitutive relations (2.21).Definition 3.2.
A thermodynamic process will be called admissible if it also satisfies the Clausius-Duhem inequality (2.17). From the previous section it follows that the Clausius-Duhem inequality (2.17) holds for all admissible processes in B×I if and only if (2.23), (2.24), and (2.25) are satisfied.For admissible thermodynamic processes, one may write the energy equation in reduced form(3.1)ρ0θη̇=QK,K+ρ0r.Definition 3.3.
One will say thatU={xi,χiL,θ} is an admissible state corresponding to the load (fi,LiL,r) if {xi,χiL,θ,ψ,TKi,GiL,MKiL,η,QK,fi,LiL,r} is an admissible thermodynamic process. The admissible state U={xi,χiL,θ} is smooth if xi, χiL, and θ are functions of class C2.LetU={xi,χiL,θ} and U¯={x¯i,χ¯iL,θ¯} be two smooth admissible states on B×I corresponding to the loads (fi,LiL,r) and (f¯i,L¯iL,r¯), respectively. We define the function D(·) on I by(3.2)D=∫B[12ρ0(vi-v¯i)(vi-v¯i)+12ρ0IKL(χ̇iK-χ¯̇iK)(χ̇iL-χ¯̇iL)+ρ0ψ-ρ0ψ¯-T¯Ki(xi,K-x¯i,K)-G¯iK(χiK-χ¯iK)-M¯KiL(χiL,K-χ¯iL,K)+ρ0η(θ-θ¯)]dV,
where(3.3)v¯i=x¯̇i,ψ¯=ψ̂(x¯i,K,χ¯iK,χ¯iK,M,θ¯;XK),Q¯L=Q̂L(x¯i,K,χ¯iK,χ¯iK,M,θ¯,θ¯,K;XK),T¯Lj=ρ0∂ψ¯∂x¯j,L,G¯jL=ρ0∂ψ¯∂χ¯jL,M¯KjL=ρ0∂ψ¯∂χ¯jL,K,η¯=-∂ψ¯∂θ¯.On account of (2.23) and (3.3) it is easy to see that D is of quadratic order in (3.4)‖v-v¯,χ̇-χ¯̇,F-F¯,χ-χ¯,γ-γ¯,θ-θ¯‖L2(B),
where F=(xi,K), χ=(χiK), and γ=(χiL,K). The evolution in time of this function is described by the following.Theorem 3.4.
IfU={xi,χiL,θ} and U¯={x¯i,χ¯iL,θ¯} are two smooth admissible states on B×I corresponding to the loads (fi,LiL,r) and (f¯i,L¯iL,r¯) in L∞(B×I), then
(3.5)Ḋ=∫∂BΓdA+∫B[Λ+Z+ρ0(fi-f¯i)(vi-v¯i)+ρ0(LiL-L¯iL)(χ̇iL-χ¯̇iL)+ρ0θ(r-r¯)(θ-θ¯)-1θθ¯(Q¯K,K+ρ0r¯)(θ-θ¯)2]dV,
where
(3.6)Γ=[(TKi-T¯Ki)(vi-v¯i)+(MKiL-M¯KiL)(χ̇iL-χ¯̇iL)+1θ(QK-Q¯K)(θ-θ¯)]NK,(3.7)Λ=x¯̇i,K[TKi-T¯Ki-∂T¯Ki∂x¯j,L(xj,L-x¯j,L)-∂T¯Ki∂χ¯jL(χjL-χ¯jL)-∂T¯Ki∂χ¯jL,M(χjL,M-χ¯jL,M)-∂T¯Ki∂θ¯(θ-θ¯)]+χ¯̇iK[GiK-G¯iK-∂G¯iK∂x¯j,L(xj,L-x¯j,L)-∂G¯iK∂χ¯jL(χjL-χ¯jL)-∂G¯iK∂χ¯jL,M(χjL,M-χ¯jL,M)-∂G¯iK∂θ¯(θ-θ¯)]+χ¯̇iL,K[MKiL-M¯KiL-∂M¯KiL∂x¯j,M(xj,M-x¯j,M)-∂M¯KiL∂χ¯jM(χjM-χ¯jM)-∂M¯KiL∂χ¯jM,N(χjM,N-χ¯jM,N)-∂M¯KiL∂θ¯(θ-θ¯)]-ρ0θ¯̇[η-η¯-∂η¯∂x¯i,K(xi,K-x¯i,K)-∂η¯∂χ¯iK(χiK-χ¯iK)-∂η¯∂χ¯iL,K(χiL,K-χ¯iL,K)-∂η¯∂θ¯(θ-θ¯)],(3.8)Z=-(QK-Q¯K)(θ-θ¯θ),K.Proof.
From (3.2) we obtain
(3.9)Ḋ=∫B{ddt[ρ0(ψ+θη+12vivi+12IKLχ̇iKχ̇iL)]-ddt[ρ0(ψ¯+θ¯η¯+12v¯iv¯i+12IKLχ¯̇iKχ¯̇iL)]-ρ0(viv¯̇i+v̇iv¯i-2v¯iv¯̇i)-ρ0IKL(χ̇iKχ¯̈iL+χ̈iKχ¯̇iL-2χ¯̇iKχ¯̈iL)-T¯̇Ki(xi,K-x¯i,K)-G¯̇iK(χiK-χ¯iK)-M¯̇KiL(χiL,K-χ¯iL,K)-T¯Ki(ẋi,K-x¯̇i,K)-G¯iK(χ̇iK-χ¯̇iK)-M¯KiL(χ̇iL,K-χ¯̇iL,K)-ρ0θ¯̇(η-η¯)-ρ0θ¯(η̇-η¯̇)}dV.
Using the balance laws (2.15) and (2.19), we may write (3.9) in the form
(3.10)Ḋ=∫B{[(TKi-T¯Ki)(vi-v¯i)+(MKiL-M¯KiL)(χ̇iL-χ¯̇iL)],k+ρ0(fi-f¯i)(vi-v¯i)+ρ0(LiL-L¯iL)(χ̇iL-χ¯̇iL)+R+P}dV,
where
(3.11)R=-T¯̇Ki(xi,K-x¯i,K)+x¯̇i,K(TKi-T¯Ki)-M¯̇KiL(χiL,K-χ¯iL,K)+χ¯̇iL,K(MKiL-M¯KiL)-G¯̇iK(χiK-χ¯iK)+χ¯̇iK(GiK-G¯iK)-ρ0θ¯̇(η-η¯)+ρ0η¯̇(θ-θ¯),P=ρ0r+QK,K-ρ0r¯-Q¯K,K-ρ0(η¯̇θ+η̇θ¯-2η¯̇θ¯).
It follows from (3.3) that
(3.12)T¯̇Ki=∂T¯Ki∂x¯j,Lx¯̇j,L+∂T¯Ki∂χ¯jLχ¯̇jL+∂T¯Ki∂χ¯jL,Mχ¯̇jL,M+∂T¯Ki∂θ¯θ¯̇,G¯̇iK=∂G¯iK∂x¯j,Lx¯̇j,L+∂G¯iK∂χ¯jLχ¯̇jL+∂G¯iK∂χ¯jL,Mχ¯̇jL,M+∂G¯iK∂θ¯θ¯̇,M¯̇KiL=∂M¯KiL∂x¯j,Mx¯̇j,M+∂M¯KiL∂χ¯jMχ¯̇jM+∂M¯KiL∂χ¯jM,Nχ¯̇jM,N+∂M¯KiL∂θ¯θ¯̇,η¯̇=∂η¯∂x¯i,Kx¯̇i,K+∂η¯∂χ¯iKχ¯̇iK+∂η¯∂χ¯iK,Lχ¯̇iK,L+∂η¯∂θ¯θ¯̇.
With the help of (3.12) and (3.3), we find that
(3.13)R=Λ.
On the other hand, using (3.1), we have
(3.14)P=ρ0r+QK,K-ρ0r¯-Q¯K,K+θ¯θ(Q¯K,K+ρ0r¯)-θ¯θ(QK,K+ρ0r)-(Q¯K,K+ρ0r¯)(θ¯θ-2+θθ¯)=ρ0θ(r-r¯)(θ-θ¯)+[(QK-Q¯K)θ-θ¯θ],K-(QK-Q¯K)(θ-θ¯θ),K-1θθ¯(Q¯K,K+ρ0r¯)(θ-θ¯)2.
Collecting (3.10), (3.13), and (3.14) and using (3.6), (3.7), and (3.8) and the divergence theorem we conclude that (3.5) holds. The proof is complete.From (2.21), it follows that(3.15)QK=Q¯K+K¯KL(θ,L-θ¯,L)+QK+QK0,
where(3.16)K¯KL=∂Q̂K∂θ,L(L¯),L¯=(x¯i,M,χ¯iM,χ¯iM,N,θ¯,θ¯,M;XM),
and 𝔔K is given by(3.17)QK=B¯KiL(xi,L-x¯i,L)+C¯KiL(χiL-χ¯iL)+E¯KiLM(χiL,M-χ¯iL,M)+α¯K(θ-θ̅),(3.18)B¯KiL=∂Q̂K∂xi,L(L¯),C¯KiL=∂Q̂K∂χiL(L¯),E¯KiLM=∂Q̂K∂χiL,M(L¯),α¯K=∂Q̂K∂θ(L¯).
In (3.15), QK0 is a function of order o(ϱ), ϱ being defined by(3.19)ϱ=|F-F¯|+|χ-χ¯|+|γ-γ¯|+|θ-θ¯|+|g-g¯|,
where F=(xi,K), χ=(χiK), γ=(χiL,K), and g=(θ,K).Let us introduce the following definition for a definite heat conductor material (see Chiriţă [19]).Definition 3.5.
One says that the admissible stateU¯ resides in the region where the material is a definite heat conductor if
(3.20)k¯KLζKζL>0,
for any nonzero ζK, where k¯KL is given by
(3.21)k¯KL=12(K¯KL+K¯LK).
We introduce the following notation:(3.22)y(t)=∥(F-F¯,χ-χ¯,γ-γ¯,θ-θ¯)(⋅,t)∥L2(B),t∈I.Theorem 3.6.
LetU¯={x¯i,χ¯iL,θ¯} be a smooth admissible state residing in the region where the material is a definite conductor of heat. Then there exist the positive constants δ, m1, and m2 with the following property: if U={xi,χiL,θ} is any smooth admissible process defined on B×I, such that
(3.23)ϱ=|F-F¯|+|χ-χ¯|+|γ-γ¯|+|θ-θ¯|+|g-g¯|<δ,
on B×I, then
(3.24)∫BZdV≤-m1∥(g-g¯)(⋅,t)∥L2(B)2+m2y2(t),t∈I.Proof.
In view of (3.8), (3.15), and (3.21), it follows that
(3.25)Z=-1θ(QK-Q¯K)(θ,K-θ¯,K)+1θ2θ,K(QK-Q¯K)(θ-θ¯)=-1θk¯KL(θ,K-θ¯,K)(θ,L-θ¯,L)+K+o(ϱ2),
where
(3.26)K=-1θQK(θ,K-θ¯,K)+1θ2θ,K[K¯KL(θ,L-θ¯,L)+QK](θ-θ¯).
Using the arithmetic-geometric inequality(3.27)2ab≤a2β2+b2β-2,forallnonzeroconstantβ,
Schwarz's inequality, and (3.17), we deduce
(3.28)2∫BKdV≤(β12+β22+β32+β42)∥(g-g¯)(⋅,t)∥L2(B)2+(M12β1-2+γ12)∥(F-F¯)(⋅,t)∥L2(B)2+(M22β2-2+γ22)∥(χ-χ¯)(⋅,t)∥L2(B)2+(M32β3-2+γ32)∥(γ-γ¯)(⋅,t)∥L2(B)2+(M42β4-2+N12γ1-2+N22γ2-2+N32γ3-2+N2)∥(θ-θ¯)(⋅,t)∥L2(B)2,
where βp, (p=1,2,3,4) and γs, (s=1,2,3) are arbitrary nonzero constants and
(3.29)M1=max⁡|θ-1B¯|,M2=max⁡|θ-1C¯|,M3=max⁡|θ-1E¯|,M4=max⁡|θ-1α¯+θ-2K¯Tg|,N1=max⁡|θ-2B¯Tg|,N2=max⁡|θ-2C¯Tg|,N3=max⁡|θ-2E¯Tg|,N2=2max⁡|θ-2α¯g|.
In view of (3.20), we conclude that there exists a positive constant λ such that
(3.30)-∫B1θk¯KL(θ,K-θ¯,K)(θ,L-θ¯,L)dV≤-λ∥(g-g¯)(⋅,t)∥L2(B)2,t∈I.
Collecting (3.25), (3.28), and (3.30), we conclude that there exists a positive constant δ with the property that whenever (3.23) holds, we have
(3.31)∫BZdV≤-m1∥(g-g¯)(⋅,t)∥L2(B)2+c1∥(F-F¯)(⋅,t)∥L2(B)2+c2∥(χ-χ¯)(⋅,t)∥L2(B)2+c3∥(γ-γ¯)(⋅,t)∥L2(B)2+c4∥(θ-θ¯)(⋅,t)∥L2(B)2,
where
(3.32)m1=λ-12(β12+β22+β32+β42),2c1=M12β1-2+γ12,2c2=M22β2-2+γ22,2c3=M32β3-2+γ32,2c4=M42β4-2+N12γ1-2+N22γ2-2+N32γ3-2+N2.
Now, taking the constants βp, (p=1,2,3,4) such that m1>0 and setting
(3.33)m2=max⁡{c1,c2,c3,c4},
from (3.22) and (3.31), we obtain (3.24), and the proof is complete.Following [18], we introduce the following.Definition 3.7.
One says that the admissible stateU¯={x¯i,χ¯iL,θ̅} resides in the convexity region of the internal energy if the following two conditions are satisfied:(i)
for each(X,t)∈B¯×I, there exists a positive constant μ such that
(3.34)∂ψ¯∂x¯i,Kx¯j,LΥiKΥjL+∂ψ¯∂χ¯iKχ¯jLΩiKΩjL+∂ψ¯∂χ¯iK,Mχ¯jL,NΦiKMΦjLN+2∂ψ¯∂x¯i,Kχ¯jLΥiKΩjL+2∂ψ¯∂x¯i,Kχ¯jL,MΥiKΦjLM+2∂ψ¯∂χ¯iKχ¯jL,MΩiKΦjLM≥μ(ΥiKΥiK+ΩiKΩiK+ΦiKLΦiKL),
for all ΥiK, ΩiK, and ΦiKL(ii)
and(3.35)∂η¯∂θ¯>0.Our study on stability and uniqueness is based on the following Gronwall-type inequality [18].Lemma 3.8.
Assume that the nonnegative functionsz(t)∈L∞[0,s] and g(t)∈L1[0,s] satisfy the inequality
(3.36)z2(τ)≤M2z2(0)+2∫0τ[(α+2βτ)z2(t)+Ng(t)z(t)]dt,τ∈[0,s]
with α, β, M, and N  being nonnegative constants. Then
(3.37)z(s)≤[Mz(0)+N∫0sg(t)dt]exp⁡(σs+βs2),
where σ=α+β/α.Now, we are ready to state the following stability result.Theorem 3.9.
LetU¯={x¯i,χ¯iK,θ¯} be a smooth admissible state on B×I corresponding to the loading (f¯i,L¯iL,r¯)∈L∞(B×I) and residing in the region where the internal energy is a convex function and the material is a definite conductor of heat. Then there exist the positive constants δ̃0, α0, M0, and N0 with the following property: if U={xi,χiK,θ} is any smooth admissible state on B×I corresponding to the loading (fi,LiK,r)∈L∞(B×I), such that
(3.38)ϱ=|F-F¯|+|χ-χ¯|+|γ-γ¯|+|θ-θ¯|+|g-g¯|<δ̃0,
on B×I, and
(3.39)[(TKi-T¯Ki)(vi-v¯i)+(MKiL-M¯KiL)(χ̇iL-χ¯̇iL)+1θ(QK-Q¯K)(θ-θ¯)]NK=0,on∂B×I,
then for any s∈I, one has
(3.40)z0(s)≤[M0z0(0)+N0∫0sg0(t)dt]exp⁡(α0s),
where
(3.41)z0(s)=‖(v-v¯,F-F,¯χ-χ¯,χ̇-χ¯̇,γ-γ¯,θ-θ¯)(⋅,s)‖L2(B),g0(s)=‖(f-f¯,L-L,¯r-r¯)(⋅,s)‖L2(B).Proof.
From (3.6) and (3.39), we have Γ=0. In view of (3.5), (3.7), (3.24), and Schwarz inequality, it follows that there exist the positive constants δ, ν1, and ν2 such that whenever (3.23) holds, we have
(3.42)Ḋ(t)≤ν1y2(t)+ν2g0(t)w(t),t∈I,
where y(t) is defined in (3.22) and
(3.43)w(t)=∥(v-v¯,χ̇-χ¯̇,θ-θ¯)(⋅,t)∥L2(B).
Let us fix s∈I and integrate (3.42) over [0,τ], with τ∈[0,s]. Then, we have
(3.44)D(τ)≤D(0)+ν1∫0τz02(t)dt+ν2∫0τg0(t)z0(t)dt.
Here we used the inequalities y(t)≤z0(t) and w(t)≤z0(t), t∈I.
On the other hand, in view of (2.23) and (3.3), we obtain
(3.45)ρ0ψ-ρ0ψ¯-T¯Ki(xi,K-x¯i,K)-G¯iK(χiK-χ¯iK)-M¯KiL(χiL,K-χ¯iL,K)+ρ0η(θ-θ¯)=ρ02[∂ψ¯∂x¯i,K∂x¯j,L(xi,K-x¯i,K)(xj,L-x¯j,L)+∂ψ¯∂χ¯iK∂χ¯jL(χiK-χ¯iK)(χjL-χ¯jL)+∂ψ¯∂χ¯iK,M∂χ¯jL,N(χiK,M-χ¯iK,M)(χjL,N-χ¯jL,N)+2∂ψ¯∂x¯i,K∂χ¯jL(xi,K-x¯i,K)(χjL-χ¯jL)+2∂ψ¯∂x¯i,K∂χ¯jL,M(xi,K-x¯i,K)(χjL,M-χ¯jL,M)+2∂ψ¯∂χ¯iK∂χ¯jL,M(χiK-χ¯iK)(χjL,M-χ¯jL,M)]+12∂η¯∂θ¯(θ-θ¯)2+o(|F-F¯|2+|χ-χ¯|2+|γ-γ¯|2+|θ-θ¯|2).
It follows from (3.2), (3.34), (3.35), and (3.45) that there exist the positive constants δ0 and ν3 such that whenever
(3.46)|F-F¯|+|χ-χ¯|+|γ-γ¯|+|θ-θ¯|<δ0,
we have
(3.47)ν3z0(t)≤2D(t),t∈I.
Setting δ̃0=min(δ,δ0) in (3.38), from (3.44) and (3.47), we obtain
(3.48)ν3z02(τ)≤2D(0)+2ν1∫0τz02(t)dt+2ν2∫0τg0(t)z0(t)dt.
Using the estimate
(3.49)D(0)≤ν4z0(0),ν4>0,
and the notations
(3.50)M02=2ν4ν3,α0=ν1ν3,N0=ν2ν3,
then (3.48) implies that
(3.51)z02(τ)≤M02z0(0)+2∫0τ[α0z02(t)dt+N0g0(t)z0(t)]dt.
An application of Lemma 3.8 completes the proof.A direct consequence of the above theorem is the following uniqueness result.Theorem 3.10.
LetU and U¯ be as in Theorem 3.9. Assume that the corresponding body loads coincide on B×I and U and U¯ originate from the same state, namely,
(3.52)xi(X,0)=x¯i(X,0),vi(X,0)=v¯i(X,0),χiL(X,0)=χ¯iL(X,0),χ̇iL(X,0)=χ¯̇iL(X,0),θ(X,0)=θ¯(X,0),X∈B.
Then,  U and U¯ coincide on B×I.



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*Source: 415304-2010-12-22.xml*