2 Preliminaries. Auxiliary results

Transcription

1

2 Preliminaries. Auxiliary results We consider a fixed system of rectangular Cartesian axes Ox i (i =1,, 3). Throughout this paper Latin indices have the range 1,, 3, Greek indices have the range 1,. Typical conventions for differential operations are implied such as a superposed dot or comma followed by a subscript in order to denote the partial derivative with respect to time or to the corresponding cartesian coordinate, respectively. [ Let us consider the region h, h simple C -curve and <h=constant << diam. We call this region plate with the thickness h. We assume that B is the interior of the right cylinder ] of the physical space R 3,where is a domain in R whose boundary is a [ h, h ], and we consider that it is filled by an isotropic and homogeneous thermoelastic material with voids. Let u =(u 1,u,u 3 ) be the displacement field, ϕ the change in volume fraction from the reference volume [1], T the absolute temperature in the reference state and θ the temperature variation from the absolute temperature T over B. In what follows we will assume that the functions u i, ϕ and θ are such that: i) u i and ϕ are of class C on B [, ); ii) θ is of class C,1 on B [, ); iii) u i,ϕand θ are of class C 1 on B [, ). In the bending theory of Mindlin type plates with voids [8, 13] (see also [5 9]), an admissible process is a state of bending on B [, ) provided u α = x 3 v α (x 1,x,t), u 3 = w(x 1,x,t), ϕ = x 3 ψ(x 1,x,t), θ = x 3 T (x 1,x,t) (.1) on B [, ). We [ consider that the average body forces, the average extrinsic forces, the average moments and the average heat source over h, h ] vanish [8]. We assume that the forces, the moments and the heat flux on the faces x 3 = ± h vanish, too. The basic system of partial differential equations from the bending theory of thermoelastic plates with voids [8, 13] is the following M βα,β hn α = ρi v α, N β,β = ρẅ, H α,α + G hγ =ρχi ψ, Q α,α hr = T σ, in (, ), (.) where I = h3, ρ and χ are the bulk mass density and the equilibrated inertia in the reference state. 1 The quantities M βα, N α, H α, G, Γ, Q α, R,andσ are defined [8, 13] by M βα = h/ x 3 t βα dx 3, N α = 1 h h/ t α3 dx 3, H α = Q α = h/ h/ h/ h/ x 3 h α dx 3, G = x 3 q α dx 3, R = 1 h h/ h/ h/ h/ x 3 gdx 3, Γ= 1 h q 3 dx 3, σ = h/ h/ h/ h 3 dx 3, ρx 3 ηdx 3, (.3) h/ h/ h/

3 where t ij, h i, g, q i,andη are quantities from the three dimensional theory [] which represent the components of the stress tensor, the components of the equilibrated stress vector, the intrinsic equilibrated force, the components of the heat flux vector and the specific entropy, respectively. Hence, the constitutive equations [8, 13] are given by M αβ = I[λε γγ δ αβ +με αβ + bψδ αβ βtδ αβ ], N α = μγ α, H β = αiψ,β, Γ = αψ, G = I(bε γγ + ξψ mt ), σ = I(βε γγ + mψ + at ), Q α = kit,α, R = kt, in [, ), (.4) where ε αβ = 1 (v α,β + v β,α ), γ α = v α + w,α. (.5) The quantities γ α (see e.g. [8, 5]) represent the angles of rotation of the cross sections x α = constant about the middle surface, the shear strains, and λ, μ, b, β, α, ξ, k, m, and a are constitutive coefficients. Equations (.) can be expressed in terms of v α, w, ψ,andt. Thus, we obtain I[μΔv α +(λ + μ)v ν,να + bψ,α βt,α ] μh(v α + w,α )=ρi v α, μδw + μv α,α = ρẅ, I[αΔψ bv ν,ν + mt ] (ξi + αh)ψ = ρχi ψ, I[kΔT βt v α,α mt ψ] kht = ait T, in (, ), (.6) where Δ is the Laplace operator. To the field equations (.6) we must adjoin boundary conditions and initial conditions. We consider the following boundary conditions v α =, w =, ψ =, T =, on. (.7) The initial conditions consist in v α (x γ, ) = v α (x γ), v α (x γ, ) = v 1 α (x γ), w(x γ, ) = w (x γ ), ẇ(x γ, ) = w 1 (x γ ), ψ(x γ, ) = ψ (x γ ), ψ(x γ, ) = ψ 1 (x γ ), (.8) T (x γ, ) = T (x γ ), in, where the functions v α,v1 α,w,w 1,ψ,ψ 1,T are given continuous functions. The internal energy density W per unit area of the middle plane, associated with the kinematic fields v α,w and with ψ is defined by W = I [ λε αα ε ββ +με αβ ε αβ + ξψ + αψ,β ψ,β +bψε ββ ] + h(μγα γ α + αψ ). (.9) Without losing the generality, we can assume that all the quantities involved in this paper are dimensionless. If we do not assume this, in view of linearity of the problem, some new notations which involve the constitutive coefficients can be done in order to obtain a dimensionless form of the equations. We denote by P, the boundary initial value problem defined by the relations (.6) (.8).

4 3 Auxiliary results Let us consider the following bilinear form F(ζ (1), ζ () ) = 1 { λζ αα (1) ζ() ββ +μζ(1) αβ ζ() αβ + ξφ(1) φ () + αφ (1) β φ() β + b(φ(1) ζ () ββ + φ() ζ (1) ββ ) +μϑ (1) α ϑ() α + ατ (1) τ }, () { } for all vectors ζ (α) = ζ (α) 11,ζ(α),ζ(α) 1,ζ(α) 1,φ(α) 1,φ (α),φ(α),ϑ (α) 1,ϑ(α),τ(α) R 1. We associate with the solution of the problem P, the kinetic energy K(t) = 1 ρ(i v α (t) v α (t)+hẇ (t)+χi ψ (t))da, (3.) the internal energy U(t) = the dissipation energy D(t) = the thermal energy S(t) = 1 and the total energy t W(t)da, (3.1) (3.3) k T (IT,α (τ)t,α (τ)+ht (τ))dadτ, (3.4) ait (t)da, E(t) =K(t)+U(t)+D(t)+S(t). (3.6) From the work [8] we have the following energy conservation law Lemma 3.1. Suppose that vα,w,ψ W 1 (), vα 1,w1,ψ 1 W () and T W ().Let(v α,w,ψ,t) be a solution of the boundary initial value problem P. Then the following energy conservation holds: E(t) =E(), for all t [, ). (3.7) If f is a continuous function on [, ), then we denote by f the function defined by f(x,t)= t (3.5) f(x,τ)dτ, (x,t) [, ). (3.8) Let us introduce the function J(t) = 1 ρ(iv α (t)v α (t)+hw (t)+χiψ (t))da + 1 k t [ ] h T (τ)+i T,α (τ) T,α (τ) dadτ. T In this section we establish some identities of Lagrange-Brun type [3], in terms of the function J( ), that are essential in investigating the problem of asymptotic partition of total energy. Lemma 3.. Suppose that vα,w,ψ W 1 (), vα 1,w1,ψ 1 W () and T W ().Let(v α,w,ψ,t) be a solution of the boundary initial value problem P. Then the following identity holds: dj dj (t) = dt dt () + t [K(τ) W (τ) S(τ)]dτ + t (3.9) σ()t (τ)dadτ, for all t [, ). (3.1)

5 P r o o f. We introduce the following notation H(t) =M βα (t)ε αβ (t)+hn α (t)γ α (t)+h α (t)ψ,α (t) G(t)ψ(t)+hΓ(t)ψ(t)+σ(t)T (t). (3.11) In view of relations (.4), we deduce that H(t) =W(t)+aIT (t). (3.1) Let us remark that the energy equation can be written in the following form Q α,α h R = T σ T σ(). (3.13) By using the equations of motion (.) and the Eq. (3.13), we obtain H(t) =[M βα (t)v α (t)+hn β (t)w(t)+h β (t)ψ(t)+ 1 Qβ (t)t (t)],β (3.14) T ρ(i v α (t)v α (t)+hẅ(t)w(t)+χi ψ(t)ψ(t)) 1 T Q α (t)t,α (t) h T R(t)T (t)+σ()t (t). Thus, in view of the relations (3.1), (3.14), and (.4), using the divergence theorem we have d ρ [ I v α (t)v α (t)+hẇ(t)w(t)+χi dt ψ(t)ψ(t) ] da [ k + I T T,α (t)t,α (t)+ k ] h T T (t)t (t) da =(K(t) W (t) S(t)) + σ()t (t)da (3.15) and the proof is complete. Lemma 3.3. Suppose that vα,w,ψ W 1 (), vα,w 1 1,ψ 1 W () and T W ().Let(v α,w,ψ,t) be a solution of the boundary initial value problem P. Then the following identity holds: dj dt (t) = 1 { { ρi[ vα (t)v α () + v α ()v α (t)] +hρ[ẇ(t)w() + ẇ()w(t)] + ρχ[ ψ(t)ψ() + ψ()ψ(t)] } da (3.16) t } + σ()[t (t τ) T (t + τ)]dadτ, for all t [, ). Proof. Letusdefine L(t, τ) =M βα (t)ε βα (τ)+hn α (t)γ α (τ) G(t)ψ(τ)+H β (t)ψ,β (τ)+hγ(t)ψ(τ) σ(t)t (τ). (3.17) In view of (.) and (3.1), we have L(t, τ) =F(ζ(t), ζ(τ)) I[m(T (t)ψ(τ)+t(τ)ψ(t)) + β(t (t)ε ββ (τ)+t(τ)ε ββ (t)) + at (t)t (τ)] (3.18) where ζ(s) ={ Iε11 (s), Iε (s), Iε 1 (s), Iε 1 (s), Iχψ,1 (s), Iχψ, (s), Iψ(s), hγ 1 (s), hγ (s), } hψ(s). Let us remark that L(t, τ) =L(τ,t). (3.19)

6 On the other hand, using the relations (.), (.4), and (3.13), the divergence theorem and the boundary conditions, we deduce that { L(t, τ)da = ρi v α (t)v α (τ)+ρhẅ(t)w(τ)+ρχ ψ(t)ψ(τ) + k I T T,α (t)t,α (τ)+ k } h T T (t)t (τ)+σ()t (τ) da. (3.) Using the above relation we have [L(t + τ,t τ) L(t τ,t + τ)]da = { ρi[ v α (t + τ)v α (t τ) v α (t τ)v α (t + τ)] + ρh[ẅ(t + τ)w(t τ) ẅ(t τ)w(t + τ)] + ρχ[ ψ(t + τ)ψ(t τ) ψ(t τ)ψ(t + τ)] + k I[ T T,α (t + τ)t,α (t τ) T,α (t τ)t,α (t + τ)] + k } h[ T T (t + τ)t (t τ) T (t τ)t (t + τ)] + σ()[t (t τ) T (t + τ)] da. (3.1) For any two functions f, h C ([, )), the following identities hold t t f(t + τ)h(t τ)dτ = f(t)h() f(t)h(t)+ f(t + τ)ḣ(t τ)dτ = t ḣ(t τ) f(t + τ)dτ, t h()f(t)+f(t)h(t)+ f(t + τ)h(t τ)dτ. (3.) We integrate the relation (3.1) over [,t] and then we use the relations (3.). This direct integration leads to the relation (3.16) and the proof is complete. 4 Asymptotic partition of energy in the Cesàro sense In this section we study the asymptotic partition of energy for the solution of the problem P. Throughout this section, we suppose that the internal energy density W is a positive definite quadratic form in terms of the variables { ε 11,ε,ε 1,γ 1,γ,ψ,ψ,1,ψ, }. This is true if and only if the constitutive coefficients satisfy the conditions We have that μ>, α>, λ+ μ>, ξ(λ + μ) >b. (4.1) F(e, e) =W, (4.) for { e = Iε11, Iε, Iε 1, Iε 1, Iχψ,1, Iχψ,, Iψ, hγ 1, hγ, } hψ. (4.3) In view of the assumptions (4.1) we have that there are two positive constants ν m and ν M such that ν m (ζ αβ ζ αβ + φ + φ β φ β )+μϑ α ϑ α + ατ F(ζ, ζ) ν M (ζ αβ ζ αβ + φ + φ β φ β )+μϑ α ϑ α + ατ, (4.4) for all vectors ζ = {ζ 11,ζ,ζ 1,ζ 1,φ 1,φ,φ,ϑ 1,ϑ,τ} R 1.

7 If (v α,w,ψ,t) is the solution of the problem P, for the different energies, we introduce the following Cesàro means K c (t) = 1 t W c (t) = 1 t S c (t) = 1 t D c (t) = 1 t t t t t K(s)ds, W (s)ds, S(s)ds, D(s)ds. The Cesàro mean values of the kinetic and internal energy have been introduced by Day [31] while the Cesàro mean values of the thermal and dissipation energy have been introduced by Chiriţă [14]. Using the identities (3.7), (3.1), and (3.16), we establish the relations that describe the asymptotic behaviour of the mean energies (4.5). The following theorem proves that the mean thermal energy tends to infinity and the asymptotic equipartition occurs between the Cesàro means of the kinetic and internal energies. Moreover, a relation describing the asymptotic behaviour of the mean dissipation energy is given by this theorem. Theorem 4.1. Let (v α,w,ψ,t) be a solution of the boundary initial value problem P. If the internal energy density is positive definite, then for all choices of the initial data so that vα 1,w,ψ W 1 (), vα,w1,ψ 1 W () and T W (), we have (4.5) P r o o f. lim S c(t) =, t lim K c(t) = lim W c (t), t t lim D c(t) =E() lim K c (t) t t = E() lim W c (t). t Because (v α,w,ψ,t) is a solution of the boundary initial value problem P, from (3.7) we have (4.6) (4.7) (4.8) K c (t)+w c (t)+d c (t)+s c (t) =E(), for all t (, ). (4.9) On the other hand, the relations (3.1) and (3.16) imply K c (t) W c (t) S c (t) = 1 di t dt () + 1 { ρi[ v α (t)v α () + v α ()v α (t)] 4t + hρ[ẇ(t)w() + ẇ()w(t)] + ρχ[ ψ(t)ψ() + ψ()ψ(t)]da t } + σ()[t (t τ) T (t + τ)]dadτ 1 t σ()t (τ)dadτ, for all t (, ). (4.1) t On the basis of the Poincaré inequality and the relation (3.7), we deduce that S c (t) at { C 4tk max I, 1 } t ( k IT,α (τ)t,α (τ)+ k ht (τ)t (τ) h T T at 4tk max { C I, 1 h ) dadτ } E(), for all t (, ), (4.11) where C is a positive constant. By letting t tend to infinity and using that vα,w,ψ W 1 (), vα 1,w1,ψ 1 W (), andt W () we get the relation (4.6). Further, by means of the Schwarz inequality and the Poincaré inequality we deduce ( ) 1/ ρi v α ()v α (t)da [K()] 1/ ρiv α (t)v α (t)da ( 1/ (4.1) [E()] 1/ (ρic) 1/ v α,β (t)v α,β (t)da).

8 Using the divergence theorem and the boundary conditions, we obtain ε αβ (t)ε αβ (t)da 1 [v α,β (t)v α,β (t)+v β,α (t)v α,β (t)]da 1 [v α,β (t)v α,β (t)+v α,α (t)v β,β (t)]da 1 v α,β (t)v α,β (t)da. (4.13) Hence, we have ( 1/ ρi v α ()v α (t)da [E()] 1/ (ρc) 1/ Iε αβ (t)ε αβ (t)da). (4.14) Because the internal energy is positive definite, from the relations (4.4) and (3.7) we obtain ( ) 1/ ( ) 1/ ρc ρi v α ()v α (t)da [E()] 1/ 4W(t)da ν m ( ) 1/ ρc E(). ν m We also have ( ρi v α (t)v α ()da [K(t)] 1/ [E()] 1/ ( ) 1/ ρiv α ()v α ()da ρiv α ()v α ()da) 1/. (4.15) (4.16) Thus, we have 1 lim ρi[ v α ()v α (t)+ v α (t)v α ()]da =. (4.17) t t The Schwarz inequality and the relation (3.7) give us ( 1/ ρhẇ(t)w()da [E()] ρhw()w()da) 1/. (4.18) On the other hand, the Schwarz inequality, the Poincaré inequality and the relation (3.7) give us ρhẇ()w(t)da [E()] 1/( ρc ) ( 1/ 1/ hw,α (t)w,α (t)da). (4.19) We have that hγ α (t)γ α (t)da {( h 1 1 ) v α (t)v α (t)+ ( 1 δ ) } w,α (t)w,α (t) da, for all <δ<1, (4.) δ which imply 1 hγ α (t)γ α (t)da + 1 hv α (t)v α (t)da hw,α (t)w,α (t)da. (4.1) 1 δ δ In view of (4.1), (4.4), and (4.13) we have hv α (t)v α (t)da hc I Iε αβ (t)ε αβ (t)da. (4.)

9 Moreover, using (4.4) and (4.1) we have hγ α (t)γ α (t)da 1 W(t)da. (4.3) μ Thus, in view of (3.7), (4.), and (4.3) we have hγ α (t)γ α (t)da μ E(), hv α (t)v α (t)da 4hC (4.4) E(). Iν m Using the relations (4.1) and (4.4) we have [ ρhw,α w,α da ρe() 1 μ(1 δ) + hc Iν m δ ], for all <δ<1. (4.5) The relations (4.19), (4.1), and (4.4) lead to [ ρhẇ()w(t)da E()(ρC) 1/ 1 μ(1 δ) + hc ] 1/, for all <δ<1. (4.6) Iν m δ Moreover, using (3.7), (4.1), and (4.4), we deduce ( 1/ ρχ ψ(t)ψ()da [E()] 1/ ρχψ ()da), (4.7) and ρχ ψ()ψ(t)da Using the inequality (3.7) we get t t ( K() ρχ αh ( K() ρχ αh ( ) 1/ ρχ E(). αh ) 1/ ( αhψ (t)da ) 1/ [W (t)] 1/ ) 1/ σ()[t (t τ) T (τ)]dadτ 3t 1/ [area()] 1/ [ T k max { C I, 1 } h (4.8) 1/ E()] sup σ() (4.9) x [ { σ()t (t + τ)dadτ (t) 1/ [area()] 1/ T C k max I, 1 } 1/ E()] sup σ(). (4.3) h x The method used above gives us that di () is bounded. We make t in (4.1) and we use the relation (4.11), dt (4.17), (4.18), (4.6) (4.3) to obtain the desired relation (4.7). Relation (4.8) is a direct consequence of the relations (4.6), (4.7) and (4.9). 5 Some extensions In this section we prove that the results established in the previous section hold true for a larger class of materials with voids. We consider the class of materials with voids for which the constitutive coefficients satisfy the following inequalities { } λ +μ λ +3μ α>, μ>, ξ>, λ+μ>, ξmax, >b. (5.1) This class of materials is included in the class of strongly elliptic materials [1] and includes the class of materials for which the internal energy density W is a positive definite quadratic form. To this aim we consider the following cases: i) α>,μ>, λ +3μ>,ξ(λ +3μ) > b, ii) α>,μ>,λ<,λ+μ>,ξ(λ +μ) > b.

10 5.1 The case α>,μ >, λ +3μ>,ξ(λ +3μ) > b For thermoelastic materials with voids characterized by inequalities i) we have the following result: Theorem 5.1. Let (v α,w,ψ,t) be a solution of the boundary initial value problem P. If the constitutive coefficients satisfy the assumptions i), then for all choices of the initial data so that vα,w,ψ W 1 (), vα,w 1 1,ψ 1 W (), and T W (), thecésaro mean energies (4.5) have the asymptotic behaviour given by (4.6), (4.7), and (4.8). P r o o f. In order to treat the temporal behaviour under hypothesis i) upon the constitutive constants of the material, we write the basic equations (.6) 1 in the form where M (1) βα,β hn α = ρi v α, M (1) βα = I[μv α,β +(λ + μ)v γ,γ δ αβ + bψδ αβ βtδ αβ ]. (5.3) Let us define W 1 (t) = M (1) βα (t)v α,β(t)+hn α (t)γ α (t)+h α (t)ψ,α (t) G(t)ψ(t)+hΓ(t)ψ(t) (5.4) +I(βv γ,γ (t)+mψ(t))t (t). In view of the boundary conditions and using the relations (5.) and (.), it is easy to deduce M (1) βα (t)v α,β(t)da = Hence, we obtain W 1 (t)da = Let us consider the quadratic forms W(t)da. [hn α (t)+ρi v α (t)]da = (5.) M βα (t)v α,β (t)da. (5.5) (5.6) F 1 = I[(λ +μ)(v1,1 + v, )+(λ + μ)v 1,1v, + ξψ +bψ(v 1,1 + v, )], F = μi(v1, + v,1), (5.7) in the variables v 1,1,v,,ψand v 1,,v,1, respectively. In view of the assumptions i), we can say that the quadratic forms F 1 and F are positive definite quadratic forms. Clearly, we can find two positive constants k m and k M such that Moreover, we have Hence k m I(v α,β v α,β + ψ ) F 1 + F k M I(v α,β v α,β + ψ ). (5.8) W 1 = F 1 + F + h(μγ α γ α + αψ ). (5.9) k m I(v α,β v α,β + ψ )+h(μγ α γ α + αψ ) W 1 k M I(v α,β v α,β + ψ )+h(μγ α γ α + αψ ). (5.1) The Lagrange-Brun inequalities established in Sect. 3 were proved without any assumption upon the constitutive coefficients. In the proof of Theorem 4.1 we have used the assumption that the internal energy is positive definite in the deduction of the relations (4.15), (4.6), and (4.8). In the following, we show that similar inequalities can be obtained under the conditions i) upon the constitutive coefficients. To this aim let first remark that the inequalities (4.13) and (4.) are satisfied without any assumption upon the constitutive coefficients. Using the inequalities (5.1), (4.13), and (4.4) we deduce W 1 (t)da, hγ α (t)γ α (t)da 1 μ hv α (t)v α (t)da hc I Iε αβ (t)ε αβ (t)da hc W 1 (t)da. Ik m (5.11)

11 Using the relations (4.1) and (5.6) we have [ ρhẇ()w(t)da E()(ρC) 1/ 1 μ(1 δ) + hc ] 1/, for all <δ<1. (5.1) Ik m δ Similar arguments lead to the inequality ( ρχ ψ()ψ(t)da K() ρχ ) 1/ [ ] 1/ W 1 (t) αh ( ) 1/ (5.13) ρχ E() αh Thus, relations similar with (4.6) and (4.8) hold true under the assumption i). Moreover, the relations (4.14) and (5.1) lead to the inequality ( ) 1/ ( 1/ ρc ρi v α ()v α (t)da [E()] 1/ W 1 (t)da). (5.14) k m So, in view of (5.6) we have ( ) 1/ ρc ρi v α ()v α (t)da E(). (5.15) k m The conclusions of the present theorem follow from the relations (4.9) (4.1), (4.16), (4.18), (4.7), (4.9), (4.3), (5.1), (5.13), (5.15) and the boundedness of di dt (). 5. The case α>, μ>, λ<, λ +μ>, ξ(λ +μ) > b In this section we give a result which describe the thermal behaviour of the mean energies (4.5) for the thermoelastic materials with voids characterized by the assumptions ii). Theorem 5.. Let (v α,w,ψ,t) be a solution of the boundary initial value problem P. If the constitutive coefficients satisfy the assumptions ii), then for all choices of the initial data so that vα,w,ψ W 1 (), vα,w1,ψ 1 W (), and T W (), thecésaro mean energies (4.5) have the asymptotic behaviour given by (4.6), (4.7), and (4.8). P r o o f. In order to treat the temporal behaviour under the hypotheses ii) upon the constitutive constants of the material we write the basic equations (.6) 1 in the form M () βα,β hn α = ρi v α, (5.16) where We consider the energy M () βα = I[μv α,β +(λ + μ)v β,α + bψδ αβ βtδ αβ ]. (5.17) W (t) = M () βα (t)v α,β(t)+hn α (t)γ α (t)+h α (t)ψ,α (t) G(t)ψ(t)+hΓ(t)ψ(t) +I(βv γ,γ (t)+mψ(t))t (t). In view of the boundary conditions and using the relations (5.16) and (.), we have W (t)da = W(t)da. Let us consider the quadratic form Q 1 = I[(λ +μ)(v1,1 + v,)+ξψ +bψ(v 1,1 + v, )], Q = I[μ(v1, + v,1 )+(λ + μ)v 1,v,1 ], (5.18) (5.19) (5.)

12 in the variables v 1,1,v,,ψand v 1,,v,1, respectively, We have W = Q 1 + Q + h(μγ α γ α + αψ ). (5.1) The quadratic forms Q 1 and Q are positive definite quadratic forms if and only if the inequalities ii) are satisfied. Thus, we can find two positive constants c m and c M such that c m I(w α,β w α,β + ψ )+h(μγ α γ α + αψ ) W c M I(w α,β w α,β + ψ )+h(μγ α γ α + αψ ). (5.) In the following, the proof of the present theorem becomes similar as the proof of Theorem 5.1. Using (4.13), (4.1), (4.6), and (5.19) we will have [ ρhẇ()w(t)da E()(ρC) 1/ 1 μ(1 δ) + hc ] 1/, for all 1 <δ<1 Ic m δ ( ) 1/ ρc ρi v α ()v α (t)da E(), (5.3) c m ( ) 1/ ρχ ψ()ψ(t)da ρχ E() αh Thus, the conclusions of the present theorem follow from the relations (4.9) (4.1), (4.16), (4.18), (4.7), (4.9), (4.3), (5.3) and the boundedness of di dt (). 5.3 Concluding result We have the following main result: Theorem 5.3. Let (v α,w,ψ,t) be a solution of the boundary initial value problem P. If the constitutive coefficients satisfy the assumptions (5.1), then for all choices of the initial data so that vα,w,ψ W 1 (), vα,w 1 1,ψ 1 W () and T W (), thecésaro mean energies (4.5) have the asymptotic behaviour given by (4.6), (4.7) and (4.8). P r o o f. The proof is a direct consequence of the Theorems 5.1 and 5.. To prove this fact, let us remark that { λ +μ max, } λ +3μ = λ +μ λ +3μ if λ + μ, if λ + μ>. We split our analysis in two cases: a) λ + μ>and b) λ + μ. a) If λ + μ >, then the inequalities (5.1) imply (5.4) α>, μ >, ξ >, λ+ μ>, ξ(λ +3μ) > b. (5.5) The class of materials for which the constitutive coefficients satisfy the inequalities (5.5) is included in the class of materials considered in the Sect In consequence, if λ + μ>, then the Césaro mean energies (4.5) have the asymptotic behaviour given by (4.6), (4.7), and (4.8). b) If λ + μ, then the inequalities (5.1) imply α>, μ >, ξ >, μ λ> μ, ξ(λ +μ) > b. (5.6) In this case, the class of materials for which the constitutive coefficients satisfy the inequalities (5.6) is included in the class of materials considered in the Sect. 5.. Hence, if λ + μ, then the Césaro mean energies (4.5) have the asymptotic behaviour given by (4.6), (4.7), and (4.8), too. In conclusion, the Césaro mean energies (4.5) have the asymptotic behaviour given by (4.6), (4.7), and (4.8) for the class of materials with voids characterized by the inequalities (5.1).

13 Acknowledgements The author acknowledges support from the Romanian Ministry of Education and Research, CNCSIS Grant code PN II-ID-PCE References [1] M. A. Goodman and S. C. Cowin, A continuum theory for granular materials, Arch. Ration. Mech. Anal. 44, (197). [] J. W. Nunziato and S. C. Cowin, A non-linear theory of elastic materials with voids, Arch. Ration. Mech. Anal. 7, (1979). [3] D. Ieşan, A Theory of thermoelastic materials with voids, Acta Mech. 6, (1986). [4] S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids, J. Elast. 13, (1983). [5] G. Capriz and P. Podio-Guidugli, Materials with spherical structure, Arch. Ration. Mech. Anal. 75, (1981). [6] G. Capriz, Continua with Microstructure, in: Springer Tracts in Natural Philosophy, Vol. 35, edited by C. Truesdell (Springer- Verlag, New York, 1989). [7] D. Ieşan, Thermoelastic Models of Continua (Kluwer Academic Publishers, Boston/Dordrecht/London, 4). [8] M. Bîrsan, A bending theory of porous thermoelastic plates, J. Thermal Stresses 6, 67 9 (3). [9] M. Bîrsan and H. Altenbach, On the dynamical theory of thermoelastic simple shells, Z. Angew. Math. Mech. (ZAMM) 91, (11). [1] M. Bîrsan and H. Altenbach, On the theory of porous elastic rods, Int. J. Solids Struct. 48, (11). [11] J. Altenbach, H. Altenbach and V. A. Eremeyev, On generalized cosserat-type theories of plates and shells: a short review and bibliography, Arch. Appl. Mech. 8, (1). [1] M. Bîrsan and H. Altenbach, Theory of thin thermoelastic rods made of porous materials, Arch. Appl. Mech. 81, (11). [13] E. Scarpetta, Minimum principle for the bending problem of elastic plates with voids, Int. J. Eng. Sci. 4, (). [14] S. Chiriţă, On the asymptotic partition of energy in linear thermoelasticity, Q. Appl. Math. 14, (1987). [15] H. A. Levine, An equipartition of energy theorem for weak solutions of evolutionary equations in Hilbert space: the lagrange identity method, J. Differ. Equ. 4, (1977). [16] R. Lakes, Foam structure with negative Poisson s ration, Science 35, (1987). [17] R. S. Lakes, No contractile obligations, Nature 358, (199). [18] R. Lakes, A broader view of membranes, Nature 414, (1). [19] S. Chiriţă, Further results on the spatial behavior in linear elastodynamics, An. Şt.Univ. Iaşi,Ser.Mat.a5, (4). [] S. Chiriţă, On some exponential decay estimates for porous elastic cylinders, Arch. Mech. 56, (4). [1] S. Chiriţă and I. D. Ghiba, Inhomogeneous plane waves in elastic materials with voids, Wave Motion 47, (1). [] S. Chiriţă and I. D. Ghiba, Strong ellipticity and progressive waves in elastic materials with voids, Proc. R. Soc. Lond. A 466, (1). [3] S. Chiriţă, Rayleigh waves on an exponentially graded poroelastic half space, J. Elast., DOI: 1.17/s z (1). [4] S. Chiriţă and A. Scalia, On the spatial and temporal behaviour in linear thermoelasticity of materials with voids, J. Thermal Stresses, 4, (1). [5] C. Constanda, A Mathematical Analysis of Bending of Plates with Transverse Shear Deformation (Longman Scientific & Technical, Harlow, 199). [6] R. D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, J. Appl. Mech. 18, (1951). [7] E. Reissner, On the theory of bending of elastic plates, J. Math. Phys. 3, (1944). [8] P. Schiavone and C. Constanda, Oscillation problems in thin plates with transverse shear deformation, SIAM J. Appl. Math. 53, (1993). [9] P.Schiavone and R.J. Tait,Steady time-harmonic oscillations in a linear thermoelastic plate model, Q. Appl. Math. LIII, 15 3 (1995). [3] L. Brun, Méthodes Énergétique dans les Systèmes Évolutifs Linéaires, J. Mec. 8, (1969). [31] W. A. Day, Means and autocorrelations in elastodynamics, Arch. Ration. Mech. Anal. 73, (198).