A SPATIAL DECAY ESTIMATE IN THERMOVISCOELASTIC COMPOSITE CYLINDERS

Transcription

1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LVII, 2011, f.1 A SPATIAL DECAY ESTIMATE IN THERMOVISCOELASTIC COMPOSITE CYLINDERS BY CĂTĂLIN GALEŞ Abstract. This work concerns the study of time harmonic vibrations in a right finite cylinder made of an isotropic and homogeneous mixture consisting of two components: an elastic solid and a Kelvin-Voigt material. The both thermal and viscoelastic effects are used to introduce an adequate measure of the amplitude of vibrations and to establish an exponential decay estimate of Saint-Venant type that holds for every frequency of vibrations and for mixtures for which the constitutive coefficients are supposed to satisfy some mild positive definiteness conditions. Mathematics Subject Classification 2000: 74F20, 74G50, 74H45. Key words: viscoelastic mixtures, harmonic vibrations, spatial behavior, dissipative effects. 1. Introduction The continuum theory of mixtures has been a subject of intensive study in literature. Various theories of mixtures developed in Eulerian or Lagrangian description have been proposed in the last decades in order to describe the thermomechanical behavior of interacting continua. A detailed discussion on the basic formulations and the progress in the field can be found in the review articles by Bowen 5], Atkin and Craine 2, 3], Bedford and Drumheller 4] and the books of Samohyl 25] and Rajagopal and Tao 24]. A great attention has been paid to the theories taking into account the viscoelastic effects. In this connection, Ieşan 19] has developed a continuum theory for a viscoelastic composite as a mixture of a porous elastic solid and a Kelvin-Voigt material. In this theory the independent constitutive variables are: the displacement fields, velocity fields, volume fractions,

2 112 CĂTĂLIN GALEŞ 2 displacement gradients, velocity gradient, volume fraction gradients, temperature and temperature gradient. The theory takes into account the effects of porosity 18, 21], by considering the volume fraction of each constituent as an independent kinematic variable. We note that in the linear dynamic theory some uniqueness and continuous results have been established by Ieşan 19], the spatial behavior problem has been investigated by Galeş 13], some existence and exponential stability results have been derived by Quintanilla 22], while the problem of existence of weak solutions and the asymptotic partition of total energy have been studied by Galeş 14]. The purpose of this paper is to investigate the spatial behavior of solutions describing harmonic vibrations of a right cylinder filled by an isotropic and homogeneous viscoelastic mixture. Our analysis includes thermal effects. But as in 22], we do not consider the effect of porosity. We consider a finite cylinder subject to boundary data varying harmonically in time on one end, while the other end and lateral surface are subjected to null boundary data. The heat supply and the body forces are supposed to be absent. Initial boundary value problems of this type have been treated by Flavin and Knops 10] in the context of the linearly damped wave equation and the linearly elastic damped cylinder. They proved that in both cases the existence of damping gives rise ultimately to a steady-state oscillation, whose amplitude decays exponentially from the excited end provided the exciting frequency is less than a certain critical value. The later case has been investigated under positive definiteness assumption upon elasticity tensor. This work has been followed by further developments (see 1], 6], 7], 11], 12], 16], 20], 23] and references therein). Within the framework of the theory of mixtures, the spatial behavior of solutions describing harmonic vibrations has been investigated previously in 12], where the case of swelling porous elastic soils has been considered, and in 23], where some exponential decay estimates for thermoelastic mixtures have been derived. The results established in all these papers, including those devoted to the theory of mixtures, suggest exponential decay of activity away from the excited end, provided the frequency of vibration is lower than a critical value and the constitutive coefficients satisfy some positive definiteness conditions. In a recent paper 8] we considered a homogeneous and isotropic Kelvin- Voigt material, and we pointed out how the dissipative mechanism may be used to obtain decay estimates valid for every value of the frequency of

3 3 HARMONIC VIBRATIONS IN COMPOSITE CYLINDERS 113 vibrations and without any restriction conditions on the Lamé coefficients. This result was extended in 15] for thermoviscoelastic cylinders consisting of two anisotropic Kelvin-Voigt materials. Here we consider a thermoviscoelastic composite cylinder consisting of an elastic solid and a Kelvin Voigt material. Although one component of the mixture is elastic, we prove that the viscoelastic effect of the second constituent is so strong that for any value of the frequency of vibrations, the activity decays exponentially away from the excited end of the cylinder. In this sense, we construct a measure by combining two type of functions. One function is suggested by the results obtained in the case of classical (or generalized) elastic media (see 1], 6], 7], 10], 11], 12], 16], 20], 23]), while the other one is obtained as in 8]. Then, we follow the well known method, namely we derive a first order differential inequality for this measure, which by an integration leads to a spatial decay estimate of Saint Venant type. This estimate holds for mixtures for which the dissipation energy density is a positive definite quadratic form and some elastic constitutive coefficients are supposed to satisfy mild positive definiteness conditions. 2. Formulation of the problem Throughout this paper, we refer the motion of a continuum to a fixed system of rectangular Cartesian axes 0x k (k = 1, 2, 3). We shall employ the usual summation and differentiation conventions: Latin subscripts are understood to range over the integers (1, 2, 3) whereas Greek subscripts are confined to the range (1, 2), summation over repeated subscripts is implied, subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate, and a superposed dot denotes time differentiation. Let B denote the interior of a right cylinder of length L > 0 whose cross section is bounded by one or more piecewise smooth curves. Choose the Cartesian coordinates such that the origin lies in one end of the cylinder and such that the x 3 -axis is parallel to the generators. Let denote the cross section of the cylinder corresponding to the axial coordinate x 3, and let denote the cross-sectional boundary. We denote by π the lateral surface of the cylinder, that is π = D 0, L]. We assume that a chemically inert mixture consisting of two constituents, a Kelvin Voigt material s 1 and an elastic solid s 2, fills B. According to the linear theory, the fundamental equations that govern the motion of an

4 114 CĂTĂLIN GALEŞ 4 isotropic and homogeneous mixture, in the absence of the body forces and heat supply, are (see 19, 22]): the equations of motion (2.1) t rl,r p l = ρ 0 1ü l, s rl,r + p l = ρ 0 2ẅ l, where ρ 0 1 and ρ0 2 are the densities at time t = 0 of s 1 and s 2, t rl and s rl are the partial stress tensors, u l and w l are the displacement vector fields and p l is the internal body force characterizing the mechanical interaction between constituents; the energy equation (2.2) ρ 0 T 0 η = q l,l, where ρ 0 = ρ ρ0 2, T 0 is the constant absolute temperature of the body in the reference configuration, q i is the heat flux vector and η is the entropy; the constitutive equations t rl = (λ + ν)e nn δ rl + 2(µ + ζ)e rl + (α + ν)g nn δ rl + (2κ + ζ)g rl + (2γ + ζ)g lr (β (1) + β (2) )T δ rl + λ ė nn δ rl + 2µ ė rl, (2.3) s rl = νe nn δ rl + 2ζe rl + αg nn δ rl + 2κg lr + 2γg rl β (2) T δ rl, p l = ξd l + ξ d l + b T,l, ρ 0 η = β (1) e nn + β (2) g nn + at, q l = k T,l + f d l, where T is the temperature variation from the uniform reference temperature T 0, δ rl is the Kronecker delta, λ, ν, µ, ζ, α, κ, γ, β (1), β (2), ξ, a, λ,µ, ξ, b, k, f are the constitutive coefficients and e rl, g rl and d l are defined by the geometrical equations (2.4) e rl = 1 2 (u r,l + u l,r ), g rl = w r,l + u l,r, d l = u l w l. The Clausius-Duhem inequality implies that the dissipation energy density (2.5) Φ λ ė rr ė ll + 2µ ė rl ė rl + ξ d l d l + 1 ( k T,l T,l + b + 1 f ) d l T,l, T 0 T 0

5 5 HARMONIC VIBRATIONS IN COMPOSITE CYLINDERS 115 is positive. This implies that (2.6) µ 0, 3λ + 2µ 0, k 0, ξ 0, 4ξ k T 0 (b + 1 T 0 f ) 2. We consider the initial boundary value problem defined by the equations of motion (2.1), the energy equation(2.2), the constitutive equations (2.3), the geometrical equations (2.4), the initial conditions (2.7) u l = a (1) l (x 1, x 2, x 3 ), w l = a (2) l (x 1, x 2, x 3 ), T = θ 0 (x 1, x 2, x 3 ), u l = b (1) l (x 1, x 2, x 3 ), ẇ l = b (2) l (x 1, x 2, x 3 ), in B at t = 0, and the lateral boundary conditions (2.8) u l = 0, w l = 0, T l = 0 on π 0, t 0 ) together with the end boundary conditions (2.9) u l = ũ l (x 1, x 2 ) exp( iωt), w l = w l (x 1, x 2 ) exp( iωt), T = T (x 1, x 2 ) exp( iωt), on D(0) 0, t 0 ), and (2.10) u l = 0, w l = 0, T = 0, on D(L) 0, t 0 ), where ω is a positive constant and represents the frequency of vibrations and i = 1. We can consider the decomposition (2.11) u l = U l (x 1, x 2, x 3, t) + v l (x 1, x 2, x 3 ) exp( iωt), w l = W l (x 1, x 2, x 3, t) + ψ l (x 1, x 2, x 3 ) exp( iωt), T = Θ(x 1, x 2, x 3, t) + θ(x 1, x 2, x 3 ) exp( iωt), where (U l, W l, Θ) absorbs the initial conditions and satisfies the null boundary conditions and the equations (2.1),(2.2), (2.3) and (2.4), while (v l, ψ l, θ) satisfies the boundary value problem consisting of (2.12) T rl,r P l = ρ 0 1ω 2 v l, S rl,r + P l = ρ 0 2ω 2 ψ l,

6 116 CĂTĂLIN GALEŞ 6 ] (2.13) Q l,l + iωt 0 (β (1) + β (2) )v l,l + β (2) ψ l,l + aθ = 0, where T rl = (λ + α + 2ν iωλ )v n,n δ rl + (µ + 2ζ + 2γ iωµ )v r,l (2.14) + (µ + 2ζ + 2κ iωµ )v l,r + (α + ν)ψ n,n δ rl + (2κ + ζ)ψ r,l + (2γ + ζ)ψ l,r (β (1) + β (2) )θδ rl, S rl = (α + ν)v n,n δ rl + (2κ + ζ)v r,l + (2γ + ζ)v l,r + αψ n,n δ rl + 2κψ l,r + 2γψ r,l β (2) θδ rl, P l = (ξ iωξ )(v l ψ l ) + b θ,l, Q l = k θ,l iωf (v l ψ l ), subjected to the homogeneous boundary conditions (2.15) v l = 0, ψ l = 0, θ = 0 on π and the end boundary conditions (2.16) v l = ũ l (x 1, x 2 ), ψ l = w l (x 1, x 2 ), θ = T (x 1, x 2 ) on D(0) (2.17) v l = 0, ψ l = 0, θ = 0 on D(L). As in 10, 8], it may be established that (U l, W l, Θ) tends to zero as time tends to infinity. So that, (U l, W l, Θ) represents the transient and (v l, ψ l, θ) exp( iωt) the forced oscillation (see 10]). From (2.12), (2.13) and (2.14) we deduce the following system of partial differential equations for the functions v l, ψ l and θ: (2.18) (α 1 iωµ )v l,rr + α 2 iω(λ + µ )]v r,rl + β 1 ψ l,rr + β 2 ψ r,rl (a 1 + b )θ,l (ξ iωξ )(v l ψ l ) + ρ 0 1ω 2 v l = 0, β 1 v l,rr + β 2 v r,rl + γ 1 ψ l,rr + γ 2 ψ r,rl (a 2 b )θ,l + (ξ iωξ )(v l ψ l ) + ρ 0 2ω 2 ψ l = 0, k θ,ll + iω(t 0 a 1 f )v l,l + iω(t 0 a 2 + f )ψ l,l + iωt 0 aθ = 0, where (2.19) α 1 = µ + 2ζ + 2κ, α 2 = λ + µ + α + 2ν + 2γ + 2ζ, β 1 = 2γ + ζ, β 2 = α + ν + 2κ + ζ, γ 1 = 2κ,

7 7 HARMONIC VIBRATIONS IN COMPOSITE CYLINDERS 117 γ 2 = 2γ + α, a 1 = β (1) + β (2), a 2 = β (2). Let us use the notation (P) for the boundary value problem consisting of the field equations (2.18) in B and the boundary conditions (2.15), (2.16) and (2.17). We note that, except the terms µ, λ, ξ, b and f the equations (2.18) are the same as those investigated in 23]. So, it is easy to extend the results established in 23] to the above model. However, based on these new terms, we develop a different technique to obtain more information on the behavior of amplitude of harmonic vibrations. If the estimates given in 23] hold for harmonic vibrations whose frequency is lower than a certain critical value, the result presented here holds for every value of the frequency of vibrations. Moreover, we use milder positive definiteness conditions on the elastic constitutive coefficients. 3. Hypotheses and some preliminary results Throughout this paper we shall assume that the elastic coefficients of the mixture satisfy the following mild positive definiteness conditions: (3.1) γ 1 > 0, γ 1 + 3γ 2 > 0. Moreover, suggested by the dissipation (2.5), we suppose that the dissipation coefficients satisfy the conditions: (3.2) µ > 0, k > 0, 4ξ k > T 0 (b + 1 T 0 f ) 2. Clearly, (2.6) and (3.2) assure that the dissipation potential Φ (see (2.5)) is a positive definite quadratic form in terms of ė ij, d l and T l. Then, there exist the positive constants km and ξm such that the following inequality holds true (3.3) ξ (v l ψ l )(v l ψ l ) T 0 k iθ,l ω ( b + 1 f ) (v l ψ l ) iθ,l T 0 ω + (v l ψ l ) iθ ],l ω iθ,l ω ξ m(v l ψ l )(v l ψ l ) + 1 T 0 ω 2 k mθ,l θ,l, where the superposed bar denotes complex conjugate.

8 118 CĂTĂLIN GALEŞ 8 Now, let us consider two constants α1 and α 2, having the same dimension as the constitutive constants α 1, α 2, γ 1, γ 2, such that (3.4) α 1 > β2 1 γ 1, α 1 + 3α 2 > (β 1 + 3β 2 ) 2 γ 1 + 3γ 2. For example, we may set α 1 = β2 1 γ 1 + γ 1 and α 1 + 3α 2 = (β 1 + 3β 2 ) 2 γ 1 + 3γ 2 + γ 1 + 3γ 2. Then, from (3.1) and (3.4) it follows that (3.5) σ = α 1 v r,lv r,l + α 2 v r,rv l,l + β 1 (v r,l ψ r,l + v r,l ψ r,l ) + β 2 (v l,l ψ r,r + v l,l ψ r,r ) + γ 1 ψ r,l ψ r,l + γ 2 ψ l,l ψ r,r, is a positive definite quadratic form in terms of v r,l and ψ r,l. The distinct eigenvalues of the corresponding linear transformation can be calculated rapidly and easily by using a mathematical software. We note that these eigenvalues have been derived in 9] (see the relation (103)). In our notations, they are (3.6) 4β 21 + (α 1 γ 1) 2 ], π 1;2 = 1 α1 2 + γ 1 π 3;4 = 1 α1 2 + γ 1 + 3(α2 + γ 2) 4(β 1 + 3β 2 ) 2 + (α 1 + 3α 2 γ 1 3γ 2 ) 2 ]. Introducing the notations (3.7) π m = min{π 1, π 3 }, π M = max{π 2, π 4 }, then we have (3.8) π m (v r,l v r,l + ψ r,l ψ r,l ) σ π M (v r,l v r,l + ψ r,l ψ r,l ). We associate with the amplitude V = {v l, ψ l, θ} of the steady-state

9 9 HARMONIC VIBRATIONS IN COMPOSITE CYLINDERS 119 vibration the following cross-sectional functionals K(x 3 ) = α 1 (v l,3 v l + v l,3 v l ) + α 2 (v r,r v 3 + v r,r v 3 ) (3.9) and H(x 3 ) = (3.10) + β 1 (ψ l,3 v l + ψ l,3 v l + v l,3 ψ l + v l,3 ψ l ) + β 2 (ψ r,r v 3 + ψ r,r v 3 + v r,r ψ 3 + v r,r ψ 3 ) + γ 1 (ψ l,3 ψ l + ψ l,3 ψ l ) + γ 2 (ψ r,r ψ 3 + ψ r,r ψ 3 ) a 1 (v 3 θ + v 3 θ) a 2 (ψ 3 θ + ψ 3 θ) + f (v 3 ψ 3 )θ + (v 3 ψ T 3 )θ] i k (θθ,3 θθ,3 ) 0 ωt 0 ] iωµ (v l,3 v l v l,3 v l ) iω(λ + µ )(v r,r v 3 v r,r v 3 ) da, iα 1 (v l,3 v l v l,3 v l ) + iα 2 (v r,r v 3 v r,r v 3 ) + iβ 1 (ψ l,3 v l ψ l,3 v l + v l,3 ψ l v l,3 ψ l ) + iβ 2 (ψ r,r v 3 ψ r,r v 3 + v r,r ψ 3 v r,r ψ 3 ) + iγ 1 (ψ l,3 ψ l ψ l,3 ψ l ) + iγ 2 (ψ r,r ψ 3 ψ r,r ψ 3 ) + ia 1 (v 3 θ v 3 θ) + ia 2 (ψ 3 θ ψ 3 θ) if (v 3 ψ 3 )θ (v 3 ψ T 3 )θ] k (θθ,3 + θθ,3 ) + ωµ (v l,3 v l + v l,3 v l ) ωt 0 ] + ω(λ + µ )(v r,r v 3 + v r,r v 3 ) da. The field equations (2.18), the boundary conditions (2.15) and the divergence theorem may be used to deduce the following: Lemma 1. The first derivatives of the functions K( ) and H( ) associated with the solution V = {v l, ψ l, θ} of the boundary value problem (P) are dk (x 3 ) = 2 (α 1 α1 dx )v l,rv l,r + (α 2 α2 )v r,rv l,l + σ 3 (3.11) + ξ(v l ψ l )(v l ψ l ) ρ 0 1ω 2 v l v l ρ 0 2ω 2 ψ l ψ l + 1 (b + f ) θ,l (v l ψ 2 T l ) + θ,l (v l ψ l )] + a θ 2] da, 0

10 120 and (3.12) CĂTĂLIN GALEŞ 10 dh (x 3 ) = 2ω µ v l,r v l,r + (λ + µ )v r,r v l,l dx 3 + ξ (v l ψ l )(v l ψ l ) + 1 (b + f ) iθ,l 2 T 0 ω (v l ψ l ) + iθ,l ω (v l ψ l )] + 1 k iθ,l T 0 ω iθ ],l da, ω where σ is given by (3.5). Proof. We give the details for the derivative dh/dx 3. The relation (3.11) follows in a similar manner. Differentiating (3.10), we get dh (x 3 ) = iα 2 (v r,r v 3,3 v r,r v 3,3 ) dx 3 + iβ 2 (ψ r,r v 3,3 ψ r,r v 3,3 + v r,r ψ 3,3 v r,r ψ 3,3 ) + iγ 2 (ψ r,r ψ 3,3 ψ r,r ψ 3,3 ) + ia 1 (v 3,3 θ + v 3 θ,3 v 3,3 θ v 3 θ,3 ) + ia 2 (ψ 3,3 θ + ψ 3 θ,3 ψ 3,3 θ ψ 3 θ,3 ) if (v 3,3 ψ 3,3 )θ + (v 3 ψ 3 )θ,3 (v 3,3 ψ T 3,3 )θ (v 3 ψ 3 )θ,3 ] ] (3.13) k θ,3 θ,3 + 2ωµ v l,3 v l,3 + ω(λ + µ )(v r,r v 3,3 + v r,r v 3,3 ) da ωt 0 iα 1 (v l,33 v l v l,33 v l ) + iα 2 (v r,r3 v 3 v r,r3 v 3 ) + iβ 1 (ψ l,33 v l ψ l,33 v l + v l,33 ψ l v l,33 ψ l ) + iβ 2 (ψ r,r3 v 3 ψ r,r3 v 3 + v r,r3 ψ 3 v r,r3 ψ 3 ) + iγ 1 (ψ l,33 ψ l ψ l,33 ψ l ) + iγ 2 (ψ r,r3 ψ 3 ψ r,r3 ψ 3 ) + 1 k (θθ,33 + θθ,33 ) ωt 0 ] + ωµ (v l,33 v l + v l,33 v l ) + ω(λ + µ )(v r,r3 v 3 + v r,r3 v 3 ) da. On the basis of the field equations (2.18), the square bracket of the

11 11 HARMONIC VIBRATIONS IN COMPOSITE CYLINDERS 121 second integral may be written in the form (3.14) iα 1 (v l,33 v l v l,33 v l ) + iα 2 (v r,r3 v 3 v r,r3 v 3 ) + iβ 1 (ψ l,33 v l ψ l,33 v l + v l,33 ψ l v l,33 ψ l ) + iβ 2 (ψ r,r3 v 3 ψ r,r3 v 3 + v r,r3 ψ 3 v r,r3 ψ 3 ) + iγ 1 (ψ l,33 ψ l ψ l,33 ψ l ) + iγ 2 (ψ r,r3 ψ 3 ψ r,r3 ψ 3 ) + 1 ωt 0 k (θθ,33 + θθ,33 ) + ωµ (v l,33 v l + v l,33 v l ) + ω(λ + µ )(v r,r3 v 3 + v r,r3 v 3 ) = iα 1 (v l,ρρ v l v l,ρρ v l ) iα 2 (v r,rρ v ρ v r,rρ v ρ ) iβ 1 (ψ l,ρρ v l ψ l,ρρ v l + v l,ρρ ψ l v l,ρρ ψ l ) iβ 2 (ψ r,rρ v ρ ψ r,rρ v ρ + v r,rρ ψ ρ v r,rρ ψ ρ ) iγ 1 (ψ l,ρρ ψ l ψ l,ρρ ψ l ) iγ 2 (ψ r,rρ ψ ρ ψ r,rρ ψ ρ ) ωµ (v l,ρρ v l + v l,ρρ v l ) ω(λ + µ )(v r,rρ v ρ + v r,rρ v ρ ) + 2ωξ (v l ψ l )(v l ψ l ) 1 ωt 0 k (θ,ρρ θ + θ,ρρ θ) ia 1 (v l,l θ + v l θ,l v l,l θ v l θ,l ) ia 2 (ψ l,l θ + ψ l θ,l ψ l,l θ ψ l θ,l ) + ib θ,l (v l ψ l ) θ,l (v l ψ l )] + if T 0 (v l,l ψ l,l )θ (v l,l ψ l,l )θ]. By using the relation (3.14), the divergence theorem and the boundary conditions (2.15) in the second integral of (3.13), we deduce the relation (3.12). Lemma 2. Let V = {v l, ψ l, θ} be solution of the boundary value problem (P). Then, the functions K( ) and H( ) associated with V = {v l, ψ l, θ} satisfy the estimates: (3.15) K(x 3 ) D 1 v l,r v l,r + D 2 v r,r v l,l + E 1 ψ l,r ψ l,r + E 2 ψ r,r ψ l,l + K θ,l θ,l ]da, (3.16) H(x 3 ) D 1 v l,r v l,r + D 2 v r,r v l,l + E 1 ψ l,r ψ l,r + E 2 ψ r,r ψ l,l + K θ,l θ,l ]da,

12 122 CĂTĂLIN GALEŞ 12 where D 1 = α 1 + α 2 + β 1 + β 2 + ωµ + ω(λ + µ ) + T 0 a 1 + f χ1, D 2 = α 2 + β 2 + ω(λ + µ ) χ1, (3.17) E 1 = γ 1 + γ 2 + β 1 + β 2 + T 0 a 2 + f χ1, E 2 = γ 2 + β 2 χ1, K = a 1 + a 2 + 2T 1 0 f + k χ 1 ω 1 T 0 χ 1 χ1. Proof. Let us note that, on the basis of the boundary conditions (2.15), the following inequalities hold: (3.18) v l,ρ v l,ρ da χ 1 ψ l,ρ ψl,ρ da χ 1 θ,ρ θ,ρ da χ 1 v l v l da, ψ l ψl da, θ θda, where χ 1 is the lowest eigenvalue in the two-dimensional clamped membrane problem for the cross section D. It should be mentioned that the dimension of χ 1 is χ 1 ] = l 2 ] (l denotes length unit). Then, the arithmetic-geometric and Schwarz s inequalities, and the inequalities (3.18) may be used to estimate the terms in the right hand of (3.9) and (3.10). We give the details for four of them. Thus, we have (3.19) α 1 (v l,3 v l + v l,3 v l )da ( 2 α 1 α 1 χ1 v l,3 v l,3 da v l,r v l,r da ; ) 1/2 ( 1 χ 1 ) 1/2 v l,ρ v l,ρ da

13 13 HARMONIC VIBRATIONS IN COMPOSITE CYLINDERS 123 iω(λ + µ )(v r,r v 3 v r,r v 3 )da (3.20) ω(λ + µ ) (v r,r v l,l + v 3,ρ v 3,ρ )da; χ1 (3.21) (3.22) f (v 3 ψ 3 ) θ + ( v 3 T ψ 3 )θ]da 0 f T 0 (v 3,ρ v 3,ρ + ψ 3,ρ ψ3,ρ )da T 0 χ1 + 2 T 0 χ 1 ] θ,ρ θ,ρ da ; ik (θ θ,3 ωt θθ,3 )da 0 ( ) 1/2 ( 2k θ,3 θ,3 da θ,ρ θ,ρ da ωt 0 χ1 k θ,l θ,l da. ωt 0 χ1 ) 1/2 The others estimates can be obtained in a similar manner. Collecting the results, we deduce (3.15) and (3.16). 4. A spatial decay estimate for the amplitude of the steadystate vibration In this section, by using the properties of the functions K( ) and H( ), we construct a measure for the solution V of the boundary value problem (P) that decays more rapidly than an exponential function of the distance from the excited end D(0) of the cylinder. In this sense, we consider the function I( ) K( ) + κh( ), where κ is chosen in such a way that di dx 3 (x 3 ) 0 for all x 3 0, L]. Then, we prove that I( ) is an acceptable measure (namely it is positive for all v r, ψ r and θ and it vanishes only when v r = ψ r = 0 and θ = 0) that satisfies a first order differential inequality. This differential inequality leads to an estimate which holds for every frequency ω of vibrations. In order to introduce the parameter κ we need first to give some estimates for the derivatives dk/dx 3 and dh/dx 3. Thus, using the arithmetic

14 124 CĂTĂLIN GALEŞ 14 geometric inequality (4.1) z 1 z 2 + z 1 z 2 ϵz 1 z ϵ z2 z 2, z 1, z 2 C, ϵ (0, ), we have (4.2) ξ(v l ψ l )(v l ψ l ) = ξv l v l + ξψ l ψ l ξ(v l ψ l + v l ψ l ) (ξ ξ )(v l v l + ψ l ψ l ), (4.3) 1 (b + f )(θ,l v l + θ,l v l ) 1 b + f ( T 0 χ 1 v l v l + 1 ) θ,l θ,l, 2 T 0 2 T 0 T 0 χ 1 (4.4) 1 (b + f )(θ,l ψ 2 T l + θ,l ψ l ) 1 b + f ( T 0 χ 1 ψ l ψ 0 2 T l + 1 ) θ,l θ,l. 0 T 0 χ 1 Moreover, the inequality (3.18) yields (4.5) aθθda a χ 1 θ,ρ θ,ρ da a θ,l θ,l da. χ 1 From (3.11) and the estimates (4.2) (4.5) we obtain dk 2 (α 1 α1 dx )v l,rv l,r + (α 2 α2 )v r,rv l,l + σ (4.6) 3 Λv l v l Ωψ l ψ l Υθ,l θ,l ]da, where (4.7) Λ = ρ 0 1ω 2 + ξ ξ + 1 b + f T0 χ 1, 2 T 0 Ω = ρ 0 2ω 2 + ξ ξ + 1 b + f T0 χ 1, 2 T 0 Υ = a + 1 b + f. T 0 T 0 χ 1 T 0 Clearly, Λ and Ω are positive (since ρ 0 1 > 0 and ρ0 2 > 0) and Υ is non negative. As regards the function H( ), we note that the inequality ( (4.8) (v l ψ l )(v l ψ l ) 1 1 ) v l v l + (1 ε)ψ l ψ ε l,

15 15 HARMONIC VIBRATIONS IN COMPOSITE CYLINDERS 125 where ε is a positive constant that will be chosen later, the hypotheses (3.2) and the inequality (3.3) imply that dh/dx 3 (see the Lemma 1), satisfies the following inequality: (4.9) dh dx 3 (x 3 ) 2ω µ v l,r v l,r + (λ + µ )v r,r v l,l ( + ξm 1 1 ) v l v l + ξ ε m(1 ε)ψ l ψ l + 1 ] T 0 ω 2 k mθ,l θ,l da. Now, since ξm(1 1/ε) and ξm(1 ε) are not both positive, following an idea introduced in 17], we use the first term in (4.9) and the inequality (3.18) in order to obtain dh { µ (4.10) (x 3 ) 2ω dx 3 µ χ ( ξm ε We choose ε such that (4.11) 2 v l,rv l,r + (λ + µ )v r,r v l,l )] v l v l + ξm(1 ε)ψ l ψ l + 1 } T 0 ω 2 k mθ,l θ,l da. 2ξ m 2ξ m + µ χ 1 < ε < 1. All the preliminaries are set to introduce the function (4.12) I(x 3 ) = K(x 3 ) + κh(x 3 ), where κ is sufficiently large such that the following inequalities hold: (4.13) α 1 α1 + κωµ 0, α 2 α2 2 + κω(λ + µ ) 0, µ χ ( 1 Λ + κω + ξm 1 1 )] 0, Ω + κωξ 2 ε m(1 ε) 0, Υ + κk m T 0 ω > 0. In the following, for simplicity we set { 2(α κ = max 1 α 1 ) α2 ωµ, α 2 ω(λ + µ ), (4.14) Ω ωξm(1 ε), 2ΥT } 0ω km. We can state the following: 2Λ ωµ χ 1 + 2ωξ m(1 1 ε ),

16 126 CĂTĂLIN GALEŞ 16 Lemma 3. If ρ 0 1 and ρ0 2 are positive and the conditions (3.1) and (3.2) hold true, then the function I( ) associated with the solution V = {v l, ψ l, θ} of the problem (P) is a non increasing function on 0, L]. Moreover, we have di (4.15) (x 3 ) 2 π m (v r,l v r,l + ψ r,l ψ dx r,l ) + κk ] m 3 2T 0 ω 2 θ,lθ,l da, where π m is defined by (3.7). Proof. From (4.6), (4.10), (4.12) and (4.14) we obtain di (4.16) (x 3 ) 2 σ + κk ] m dx 3 2T 0 ω 2 θ,lθ,l da. (3.8) and (4.16) lead to (4.15) and the proof is complete. From Lemma 2 we obtain Lemma 4. Let V = {v l, ψ l, θ} be solution of the boundary value problem (P). Then, the functions I( ) associated with V = {v l, ψ l, θ} satisfies the estimate: ] (4.17) I(x 3 ) (1 + κ) M(v l,r v l,r + ψ l,r ψ l,r ) + K θ,l θ,l da, where (4.18) M = max{(d 1 + 3D 2 ), (E 1 + 3E 2 )}. Proof. The lemma follows from the inequalities (3.15), (3.16) and the fact that the distinct eigenvalues of the linear transformations associated with the positive definite quadratic forms (4.19) σ (1) = D 1 v l,r v l,r + D 2 v l,l v r,r, σ (2) = E 1 ψ l,r ψl,r + E 2 ψ l,l ψr,r, are (4.20) π (1) 1 = D 1, π (1) 2 = D 1 + 3D 2, and (4.21) π (2) 1 = E 1, π (2) 2 = E 1 + 3E 2, respectively. We are ready now to prove the main result of the paper.

17 17 HARMONIC VIBRATIONS IN COMPOSITE CYLINDERS 127 Theorem 1. In the context of a finite composite cylinder made of a mixture consisting of an elastic solid and a Kelvin Voigt material, whose constitutive coefficients satisfies the mild positive definiteness conditions (3.1) and (3.2), the cross sectional functional I( ) represents an acceptable measure of the solution V = {v r, ψ r, θ} of the problem (P). Moreover, it satisfies the following exponential decay estimate ( (4.22) 0 I(x 3 ) I(0) exp x 3 C where { M (4.23) C = (1 + κ) max ), x 3 0, L], 2π m, K T 0 ω 2 κk m Proof. It follows immediately from (4.15) and (4.17) that }. (4.24) I(x 3 ) C di dx 3 (x 3 ), x 3 0, L]. Since I( ) is a non increasing function and I(L) = 0 (see (2.17), (3.9), (3.10) and (4.12)), it follows that I(x 3 ) 0 for every x 3 0, L]. Moreover, integrating (4.15) on x 3, L], it follows that (4.25) I(x 3 ) 2 B(x 3,L) π m (v r,l v r,l + ψ r,l ψ r,l ) + κk ] m 2T 0 ω 2 θ,lθ,l dv, where B(x 3, L) = x 3, L]. This relation together with the boundary conditions prove that I(x 3 ) = 0 implies v r = ψ r = 0 and θ = 0 on B(x 3, L), so that I( ) is an acceptable measure of the amplitude of the steady state vibration. Therefore, (4.24) became (4.26) 1 C I(x 3) + di (x 3 ) 0, dx 3 x 3 0, L]. By integration one obtains the estimate (4.22) and the proof is complete. 5. Conclusions In this paper we have studied the time harmonic vibrations in a cylinder filled by a binary mixture consisting of an elastic solid and a Kelvin Voigt material. The dissipative mechanism has been used to derive a measure which leads to the decay estimate of Saint Venant type (4.22).

18 128 CĂTĂLIN GALEŞ 18 The estimate (4.22) holds for every value of the frequency of vibrations. Moreover, the assumptions (3.2) represent a natural extension of the consequences (2.6) assured by the dissipation inequality, while the conditions (3.1) are milder than those utilized in 23]. The result obtained here may be extended to a semi infinite cylinder to obtain an appropriate alternative of Phragmèn Lindelöf type. Acknowledgments The author was supported by the Romanian Ministry of Education and Research, CNCSIS Grant code ID-401, Contract no. 15/ REFERENCES 1. Aron, M.; Chiriţă, S. Decay and continuous dependence estimates for harmonic vibrations of micropolar elastic cylinders, Arch. Mech. (Arch. Mech. Stos.), 49 (1997), Atkin, R.J.; Craine, R.E. Continuum theories of mixtures: basic theory and historical development, Quart. J. Mech. Appl. Math., 29 (1976), Atkin, R.J.; Craine, R.E. Continuum theories of mixtures: applications, J. Inst. Math. Appl., 17 (1976), Bedford, A.; Drumheller, D.S. Theories of immiscible and structured mixtures, Internat. J. Engrg. Sci., 21 (1983), Bowen, R.M. Theory of Mixtures, in A.C. Eringen (ed.), Continuum Physics, vol.iii, Academic Press, New York, Chiriţă, S. Spatial decay estimates for solutions describing harmonic vibrations in a thermoelastic cylinder, J. Thermal Stresses, 18 (1995), Chiriţă, S. Further results on the spatial behaviour in linear elastodynamics, An. Ştiinţ. Univ. Al.I. Cuza Iaşi, Mat. (N.S.), 50 (2004), Chiriţă, S.; Galeş, C.; Ghiba, I.-D. On spatial behavior of the harmonic vibrations in Kelvin-Voigt materials, J. Elasticity, 93 (2008), D Apice, C.; Tibullo, V.; Chiriţă, S. On the spatial behavior in the dynamic theory of mixtures of thermoelastic solids, J. Thermal Stresses, 28 (2005), Flavin, J.N.; Knops, R.J. Some spatial decay estimates in continuum dynamics, J. Elasticity, 17 (1987), Flavin, J.N.; Knops, R.J.; Payne, L.E. Decay estimates for the constrained elastic cylinder of variable cross section, Quart. Appl. Math., 47 (1989),

19 19 HARMONIC VIBRATIONS IN COMPOSITE CYLINDERS Galeş, C. Spatial decay estimates for solutions describing harmonic vibrations in the theory of swelling porous elastic soils, Acta Mechanica, 161 (2003), Galeş, C. On spatial behavior in the theory of viscoelastic mixtures, J. Thermal Stresses, 30 (2007), Galeş, C. Some results in the dynamics of viscoelastic mixtures, Math. Mech. Solids, 13 (2008), Galeş, C. On spatial behavior of the harmonic vibrations in thermoviscoelastic mixtures, J. Thermal Stresses, 32 (2009), Ghiba, I.-D. On the spatial behaviour of harmonic vibrations in an elastic cylinder, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Mat. (N.S.), 52 (2006), Ghiba, I.-D. Some uniqueness and stability results in the theory of micropolar solid-fluid mixture, J. Math. Anal. Appl., 355 (2009), Goodman, M.A.; Cowin, S.C. A continuum theory for granular materials, Arch. Rational Mech. Anal., 44 (1972), Ieşan, D. On the theory of viscoelastic mixtures, J. Thermal Stresses, 27 (2004), Knops, R.J. Spatial Decay Estimates in the Vibrating Anisotropic Elastic Beam, Waves and stability in continuous media (Sorrento, 1989), , Ser. Adv. Math. Appl. Sci., 4, World Sci. Publ., River Edge, NJ, Nunziato, J.W.; Cowin, S.C. A nonlinear theory of elastic materials with voids, Arch. Rational Mech. Anal., 72 (1979/80), Quintanilla, R. Existence and exponential decay in the linear theory of viscoelastic mixtures, Eur. J. Mech. A Solids, 24 (2005), Passarella, F.; Zampoli, V. Some exponential decay estimates for thermoelastic mixtures, J. Thermal Stresses, 30 (2007), Rajagopal, K.R.; Tao, L. Mechanics of Mixtures, Series on Advances in Mathematics for Applied Sciences, 35, World Scientific Publishing Co., Inc., River Edge, NJ, Samohyl, I. Thermodynamics of Irreversible Processes in Fluid Mixtures, Approached by rational thermodynamics, Teubner-Texte zur Physik, 12, Teubner Verlagsgesellschaft, Leipzig, Received: 17.IX.2009 Revised: 5.X.2009 Faculty of Mathematics, University Al.I. Cuza, 11, Bd. Carol I, , Iaşi, ROMANIA cgales@uaic.ro

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