The Influence of Voids in the Vibrations of Bodies with Dipolar Structure

Abstract

In our study we analyse the vibration of a right cylinder which consists of an elastic material with dipolar structure and has pores. One end of this cylinder is subjected to an excitation, harmonically in time. The other end of the cylinder and its lateral surface are free of loads. We prove that the presence of the voids does not affect the spatial decay of effects away from the excited end, if the harmonic excitation level is below a predetermined threshold.

1. Introduction

Many studies highlight how the presence of voids in a material affects certain thermomechanical properties of some materials. For instance, it is proved that the voids presence in composites, even in small amounts, are detrimental to their thermo-mechanical performance. Voids cause reduction in different mechanical properties including interlaminar shear strength, tensile and flexural strength and modulus, torsional shear, fatigue resistance, and impact. Even at low void occurrence, the mechanical integrity continues to be affected. In addition to reducing elastic properties, voids can have significant effects on micro damage mechanisms. However, this effect of voids should not be generalized, to all properties. In our manuscript highlight a property that is not affected by the presence of voids. We analyse the vibration of a right cylinder which consists of an elastic material with dipolar structure and having pores. One end of this cylinder is subjected to an excitation, harmonically in time. The other end of the cylinder and its lateral surface are free of loads. We prove that the presence of the voids do not affect the spatial decay of effects away from the excited end, if the harmonic excitation level is below a predetermined threshold.

Our study is dedicated to the analysis of the vibrations of a cylindrical body, made of a porous elastic material having a dipolar structure and the results obtained are generalizations of similar results obtained by Flavin and Knops in the paper [1], dedicated to classical elasticity. Our cylinder consists of a homogeneous, anizotropic and porous material with dipolar structure and it is subjected to some vibrations, which vary harmonically in time, only on one of its ends. The lateral surface of the cylinder and other its end are free of loads. It is also assumed that the mass forces, the dipolar mass forces and the intrinsic forces are zero. In a first stage we put down the main equations and conditions of the boundary problem in this context. Then, inspired by the procedure used by Toupin in the work [2], dedicated to classical elastostatics, we consider a measure to be able to evaluate the amplitude of the vibrations. By usual calculations on the solutions of the boundary problem, we prove some estimations on the basis of which we establish a differential inequality of the first order. Assuming that the harmonic excitation level is below a predetermined threshold, we integrate this inequality and obtain a spatial decay estimation from which we deduce that the voids do not influence the fact that the vibration disappears at an appreciable distance from the excited end.

We need to motivate why we used bodies with voids. The initiators of the theory of materials with voids are Cowin and Goodman who conducted pioneering analyses of porous structures (bodies with voids) in the study [3] by Goodman and Cowin. They created what is known as granular theory. Similar studies appear in the paper [4] by Cowin and Nunziato, where the authors proposed the introduction of a new, additional degree of freedom, as Goodman and Cowin anticipated. It was to be used to investigate a new mechanical behaviour of hollow bodies, for which the material of the matrix is an elastic medium and the interstices are the voids in the body.

This new theory was immediately applied in approaching geological materials, such as soils and rocks, or in the composition of human bones. It has also found applications in “artificially” manufactured materials. It is important to highlight the main feature of this theory, namely that a new concept of environment is introduced for which bulk density appears as the product of two functions in which the first represents the density field of the matrix material and the second—the volume fraction field.

Cowin and Nunziato’s work as well as the study of Nunziato and Cowin [5] approach only elastic materials, without considering the effect of the thermal field.

The thermal effect for materials with voids was proposed by Iesan in [6].

Now, we provide some arguments for the use of media with dipolar structure.

The study of dipolar bodies began with the published results by Mindlin in [7], as well as in the paper of Rivlin and Green [8].

These researchers approached in several of their studies the structures of the multipolar type and in concrete situations, the structures of the dipolar type. A domain of influence for this type of bodies was established in [9]. M.E. Gurtin also published a few studies on multipolar structures, see, for instance, [10] by Fried and Gurtin.

In the last period of time, other models of bodies with dipolar structure were introduced, in which the volume fraction field is further used. We list a few of these: [11,12,13,14,15,16,17]. For instance, in [11] the authors consider the mixed problem in the context of bodies with double porosity and prove the existence of the solution, its uniqueness as well as some considerations on the stability of the solution.

The interest and the need to use the combined theory of thermoelasticity of solids with dipolar structure together with the porosity theory have been accentuated since the first works on porosity media. Moreover, in the book [18] by Iesan, the author highlights that the porous structure of a continuous solid can be affected by the displacement of that body. These are the reasons why we, in our work, took into account both the effect related to the dipolar structure and the one related to the porous structure.

The paper [19] by Eringen is dedicated to the elasticity theory of materials with a microstructure which includes rotations of intrinsic type, contractions and some expansions of microstructural type. See also [20,21], where the motivation for combining the two theories reappears.

2. Preliminaries

Let us consider the interior of a right cylinder $\mathcal{C}$, with the length L and where any of its cross section is bounded by a smooth curve. The coordinate axis system is chosen so that its origin is located in one end of the right cylinder and so that the $x_3$-axis is parallel to the generator.

A cross section in the cylinder is denoted by ${\mathsf{\Sigma }}_h$ and it corresponds to $x_3=h,h\ge 0.$ Consider that the boundary of the surface ${\mathsf{\Sigma }}_h$ is a curve denoted by $\partial {\mathsf{\Sigma }}_h$. We will use a comma followed by the indices j to denote the partial derivative with respect to the spatial coordinate $x_j$, and a superposed dot to denote the partial derivative with respect to the temporal variable t.

To describe the deformation of our dipolar elastic body with voids we will use the following independent variables:

$$v_m(t,x),{\varphi }_{jk}(t,x),\phi (t,x),(t,x)\in [0,t_0)\times \mathcal{C},$$

where $v_m$ represent the components of a vector field called displacement and ${\varphi }_{jk}$ represent the components of a tensor field called dipolar displacement. Additionally, by $\phi$ we denoted a variable to characterize the evolution of the voids, which is known as the change in volume fraction. With the help of the displacement variables, we can define the strain tensors, by means of the following kinematic relations:

$$\begin{array}{c}\hfill e_{mn}=\frac{1}{2}\left(v_{m,n}+v_{n,m}\right),{\epsilon }_{mn}=v_{n,m}-{\varphi }_{mn},{\gamma }_{mnr}={\varphi }_{mn,r}.\end{array}$$

(1)

We denote by E the free energy in the body, which, because we consider only the linear theory, is a quadratic form with respect to all its independent variables, namely:

$$\begin{array}{c}\hfill E=\frac{1}{2}{A}_{klmn}{e}_{kl}{e}_{mn}+G_{klmn}{e}_{kl}{\epsilon }_{mn}+F_{klmnr}{e}_{kl}{\gamma }_{mnr}+\\ \hfill +\frac{1}{2}{B}_{klmn}{\epsilon }_{kl}{\epsilon }_{mn}+D_{klmnr}{\epsilon }_{kl}{\gamma }_{mnr}+\frac{1}{2}{C}_{kljmnr}{\gamma }_{klj}{\gamma }_{mnr}+\\ \hfill +a_{kl}{e}_{kl}\phi +b_{kl}{\epsilon }_{kl}\phi +c_{klj}{\gamma }_{klj}\phi +\frac{1}{2}{d}_{kl}{\phi }_{,k}{\phi }_{,l}+\frac{1}{2}a{\phi }^2.\end{array}$$

(2)

Based on the method introduced by Nunziato and Cowin in [5], we deduce the expressions for the stress tensors $t_{kl}$, ${\tau }_{kl}$, $m_{klj}$:

$$\begin{array}{c}\hfill t_{kl}=\frac{\partial E}{\partial e_{kl}},{\tau }_{kl}=\frac{\partial E}{\partial {\epsilon }_{kl}},m_{klj}=\frac{\partial E}{\partial {\gamma }_{klj}},h_m=\frac{\partial E}{\partial {\phi }_{,m}},g=-\frac{\partial E}{\partial \phi },\end{array}$$

so that, considering the expression of the free energy from (2) we are led to the following constitutive relations:

$$\begin{array}{c}\hfill t_{kl}=A_{klmn}{e}_{mn}+G_{klmn}{\epsilon }_{mn}+F_{klmnr}{\gamma }_{mnr}+a_{kl}\phi ,\\ \hfill {\tau }_{kl}=G_{klmn}{e}_{mn}+B_{klmn}{\epsilon }_{mn}+D_{klmnr}{\gamma }_{mnr}+b_{kl}\phi ,\\ \hfill m_{klj}=F_{kljmn}{e}_{mn}+D_{kljmn}{\epsilon }_{mn}+C_{kljmnr}{\gamma }_{mnr}+c_{klj}\phi ,\\ \hfill h_k=d_{kl}{\phi }_{,l},\\ \hfill g=-a_{kl}{e}_{kl}-b_{kl}{\epsilon }_{kl}-c_{klj}{\gamma }_{klj}.\end{array}$$

(3)

By using the same procedure as Iesan in [18], we obtain the basic equations of elasticity for porous bodies. If mass forces, dipolar body forces and the intrinsic force are missing, then these have the following form:

-

the motion equations:

$$\begin{array}{c}\hfill {\left(t_{kl}+{\tau }_{kl}\right)}_{,l}=\rho {\ddot{v}}_k,\\ \hfill m_{klj,j}+{\tau }_{kl}=I_{km}{\ddot{\varphi }}_{lm};\end{array}$$

(4)

-

the balance of the equilibrated forces:

$$\begin{array}{c}\hfill h_{m,m}+g=\rho \kappa \ddot{\phi }.\end{array}$$

(5)

The new terms which appear in the above equations have the following significations:

• $\varrho$—the mass density;
• $I_{kl}$—the components of the inertia tensor;
• g—the intrinsic force.

$A_{klmn}$, $B_{klmn}$, $C_{kljmnr}$, $D_{kljmn}$, $G_{rsmn}$, $F_{kljmn}$, $a_{kl}$, $b_{kl}$, $c_{klj}$, $d_{kl}$ and a are the characteristic functions to characterize the elastic properties of the material. These satisfy the following symmetric relations:

$$\begin{array}{c}\hfill A_{klmn}=A_{mnkl}=A_{lkmn},B_{klmn}=B_{mnkl},C_{kljmnr}=C_{mnrklj},\\ \hfill F_{kljmn}=F_{kljnm},G_{klmn}=G_{klnm},I_{mn}=I_{nm},\end{array}$$

(6)

where we also included the symmetry of the tensor of inertia.

In order to complete the mixed problem for the cylinder $\mathcal{C}$ we must add the initial and boundary conditions.

We will use the initial conditions in the following form:

$$\begin{array}{c}\hfill v_m(x,0)={\tilde{v}}_m^0\left(x\right),{\dot{v}}_k(x,0)={\tilde{v}}_k^1\left(x\right),{\varphi }_{mn}(x,0)={\tilde{\varphi }}_{mn}^0\left(x\right),\\ \hfill {\dot{\varphi }}_{mn}(x,0)={\tilde{\varphi }}_{mn}^1\left(x\right),\phi (x,0)={\tilde{\phi }}^0\left(x\right),x\in \mathcal{C},\end{array}$$

(7)

where the functions ${\tilde{v}}_m^0,{\tilde{v}}_m^1,{\tilde{\varphi }}_{mn}^0,{\tilde{\varphi }}_{mn}^1$ and ${\tilde{\phi }}^0$ are known on $\mathcal{C}$.

Regarding the boundary conditions, we assume that these are null on all frontiers of the cylinder $\mathcal{C}$, except for the basis (i.e., ${\mathsf{\Sigma }}_0$) where we prescribe an excitation that is harmonic in time.

Thus, the boundary data have the following form:

$$\begin{array}{c}\hfill v_m=0,{\mathsf{\Phi }}_{mn}=0,\phi =0\mathrm{on}(0,t_0)\times (\partial \mathcal{C}\setminus {\mathsf{\Sigma }}_0),\end{array}$$

(8)

$$\begin{array}{c}\hfill t_{3m}+{\tau }_{3m}=p_m{e}^{i\omega t},m_{3kl}=r_{kl}{e}^{i\omega t},h_3=h{e}^{i\omega t},\mathrm{on}(0,t_0)\times {\mathsf{\Sigma }}_0,\end{array}$$

(9)

where $p_m(x_1,x_2),r_{kl}(x_1,x_2)$ and $q(x_1,x_2)$ are given continuous functions on ${\mathsf{\Sigma }}_0$, $\omega$ is a positive constant and i is the unit of complex numbers, i.e., $i^2=-1$.

Let us denote by ${\mathcal{P}}_0$ the mixed problem consisting of Equations (4) and (5), the initial conditions (7) and the boundary conditions (8) and (9).

Consider the ordered array $\left(w_m,{\mathsf{\Phi }}_{mn},\psi \right)$ as a solution of the above mixed problem ${\mathcal{P}}_0$.

In the following we will use the next decomposition:

$$\begin{array}{c}\hfill u_m=w_m(t,x)+{\tilde{v}}_m(t,x)e^{i\omega t},\\ \hfill {\varphi }_{mn}={\mathsf{\Phi }}_{mn}(t,x)+{\tilde{\varphi }}_m(t,x)e^{i\omega t},\\ \hfill \phi =\psi (t,x)+\tilde{\phi }(t,x)e^{i\omega t}.\end{array}$$

(10)

Thus, we deduce that the ordered array $\left({\tilde{v}}_m,{\tilde{\varphi }}_{mn},\tilde{\phi },\right)$ satisfies the boundary value problem consisting of:

-

the geometric equations:

$$\begin{array}{c}\hfill {\tilde{e}}_{mn}=\frac{1}{2}\left({\tilde{v}}_{m,n}+{\tilde{v}}_{n,m}\right),{\tilde{\epsilon }}_{mn}={\tilde{v}}_{n,m}-{\tilde{\varphi }}_{mn},{\tilde{\gamma }}_{mnr}={\tilde{\varphi }}_{mn,r};\end{array}$$

(11)

-

the constitutive equations:

$$\begin{array}{c}\hfill {\tilde{t}}_{kl}=A_{klmn}{\tilde{e}}_{mn}+G_{klmn}{\tilde{\epsilon }}_{mn}+F_{klmnr}{\tilde{\gamma }}_{mnr}+a_{kl}\tilde{\phi },\\ \hfill {\tilde{\tau }}_{kl}=G_{klmn}{\tilde{e}}_{mn}+B_{klmn}{\tilde{\epsilon }}_{mn}+D_{klmnr}{\tilde{\gamma }}_{mnr}+b_{kl}\tilde{\phi },\\ \hfill {\tilde{m}}_{klj}=F_{kljmn}{\tilde{e}}_{mn}+D_{kljmn}{\tilde{\epsilon }}_{mn}+C_{kljmnr}{\tilde{\gamma }}_{mnr}+c_{klj}\tilde{\phi },\\ \hfill {\tilde{h}}_k=d_{kl}{\tilde{\phi }}_{,l},\\ \hfill \tilde{g}=-a_{kl}{\tilde{e}}_{kl}-b_{kl}{\tilde{\epsilon }}_{kl}-c_{klj}{\tilde{\gamma }}_{klj}.\end{array}$$

(12)

-

the motion equations:

$$\begin{array}{c}\hfill {\left({\tilde{t}}_{mn}+{\tilde{\tau }}_{mn}\right)}_{,n}+\rho {\omega }^2{\tilde{v}}_m=0,\\ \hfill {\tilde{m}}_{ijk,i}+{\tilde{\tau }}_{jk}+{\omega }^2{I}_{kr}{\tilde{\varphi }}_{jr}=0;\end{array}$$

(13)

-

the balance of the equilibrated forces:

$$\begin{array}{c}\hfill {\tilde{h}}_{m,m}+\tilde{g}=\rho \kappa {\omega }^2\tilde{\phi };\end{array}$$

(14)

-

the boundary conditions:

$$\begin{array}{c}\hfill {\tilde{v}}_m=0,{\tilde{\varphi }}_{mn}=0,\tilde{\phi }=0,\mathrm{on}\partial \mathcal{C}\setminus {\mathsf{\Sigma }}_0,\end{array}$$

(15)

$$\begin{array}{c}\hfill t_{3m}+{\tau }_{3m}=p_m,m_{3mn}=r_{mn},h_3=h\mathrm{on}{\mathsf{\Sigma }}_0.\end{array}$$

(16)

3. Spatial Decay Estimates

Let us denote by $\mathcal{P}$ the boundary value problem defined by the Equations (11)–(14) and the boundary conditions (15) and (16).

Using the procedure suggested by Fichera in [22], we can prove the existence of a solution $\left({\tilde{v}}_m,{\tilde{\varphi }}_{mn},\tilde{\phi },\right)$ to the problem $\mathcal{P}$.

In all that follows we consider only problem $\mathcal{P}$ and because there is no danger of confusion and for the sake of simplifying the writing, we will give up the sign $\tilde{}$ for the functions that refer to this problem.

It is known that in classical elasticity the total energy associated with the deformation of any body should be bounded, so we can assume that the elastic tensors are positive definite.

We can also suppose that the components of tensor of inertia $I_{mn}$ and the density $\rho$ are known and are strictly positive and this is both realistic and consistent with the above considerations on the total energy.

Let us make the following notation convention: we will use a superposed bar over a function in order to denote the complex conjugate of that function.

Now we obtain the first spatial decay estimate regarding a solution of the problem $\mathcal{P}$.

Thus, it is not difficult to deduce that:

$$\begin{array}{c}\hfill {\nu }_m\left(e_{ij}{\overline{e}}_{ij}+{\epsilon }_{ij}{\overline{\epsilon }}_{ij}+{\gamma }_{ijk}{\overline{\gamma }}_{ijk}\right)\le \\ \hfill \le A_{ijmn}{e}_{ij}{\overline{e}}_{mn}+G_{ijmn}\left(e_{ij}{\overline{\epsilon }}_{mn}+{\overline{e}}_{ij}{\epsilon }_{mn}\right)+F_{mnrij}\left(e_{ij}{\overline{\gamma }}_{mnr}+{\overline{e}}_{ij}{\gamma }_{mnr}\right)+\\ \hfill +B_{ijmn}{\epsilon }_{ij}{\overline{\epsilon }}_{mn}+D_{ijmnr}\left({\epsilon }_{ij}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{ij}{\gamma }_{mnr}\right)+C_{ijkmnr}{\gamma }_{ijk}{\overline{\gamma }}_{mnr}\le \\ \hfill \le {\nu }_M\left(e{\overline{e}}_{ij}+{\epsilon }_{ij}{\overline{\epsilon }}_{ij}+{\gamma }_{ijk}{\overline{\gamma }}_{ijk}\right),\end{array}$$

(17)

where we denote by ${\nu }_M$ and ${\nu }_m$ the maximum and minimum material moduli.

Furthermore, after some calculations we are led to the following double inequality:

$$\begin{array}{c}\hfill {\nu }_m^2\left(e_{ij}{\overline{e}}_{ij}+{\epsilon }_{ij}{\overline{\epsilon }}_{ij}+{\gamma }_{ijk}{\overline{\gamma }}_{ijk}\right)\le \\ \hfill \le \left[A_{ijmn}{e}_{mn}+G_{ijmn}\left(e_{mn}+{\epsilon }_{mn}\right)+B_{ijmn}{\epsilon }_{mn}+\left(F_{mnrij}+D_{ijmnr}\right){\gamma }_{mnr}\right]\times \\ \hfill \times \left[A_{ijkl}{\overline{e}}_{kl}+G_{ijkl}\left({\overline{e}}_{kl}+{\overline{\epsilon }}_{kl}\right)+B_{ijkl}{\overline{\epsilon }}_{kl}+\left(F_{klsij}+D_{ijkls}\right){\overline{\gamma }}_{kls}\right]+\\ \hfill +\left[F_{ijkmn}{e}_{mn}+D_{mnijk}{\epsilon }_{mn}+C_{ijkmnr}{\gamma }_{mnr}\right]\times \\ \hfill \times \left[F_{ijkrs}{\overline{e}}_{rs}+D_{mnijk}{\overline{\epsilon }}_{rs}+C_{ijkprs}{\overline{\gamma }}_{prs}\right]\le \\ \hfill \le {\nu }_M^2\left(e_{ij}{\overline{e}}_{ij}+{\epsilon }_{ij}{\overline{\epsilon }}_{ij}+{\gamma }_{ijk}{\overline{\gamma }}_{ijk}\right).\end{array}$$

(18)

Taking into account the constitutive relations $\left(12\right)$ and the inequalities (17) and (18), we deduce:

$$\begin{array}{c}\hfill t_{ij}{\overline{t}}_{ij}+{\tau }_{ij}{\overline{\tau }}_{ij}+m_{ijk}{\overline{m}}_{ijk}=\\ \hfill \left[A_{ijmn}{e}_{mn}+G_{mnij}{\epsilon }_{mn}+F_{mnrij}{\gamma }_{mnr}+a_{ij}\phi \right]\left[A_{ijkl}{\overline{e}}_{kl}+G_{klij}{\overline{\epsilon }}_{kl}+F_{klrij}{\overline{\gamma }}_{klr}+a_{ij}\overline{\phi }\right]+\\ \hfill +\left[G_{ijmn}{e}_{mn}+B_{ijmn}{\epsilon }_{mn}+D_{ijmnr}{\gamma }_{mnr}+b_{ij}\phi \right]\left[G_{ijkl}{\overline{e}}_{kl}+B_{ijkl}{\overline{\epsilon }}_{kl}+D_{ijklr}{\overline{\gamma }}_{klr}+b_{ij}\overline{\phi }\right]+\\ \hfill +\left[F_{ijkmn}{e}_{mn}+D_{mnijk}{\epsilon }_{mn}+C_{ijkmnr}{\gamma }_{mnr}+c_{ijk}\phi \right]\times \\ \hfill \times \left[F_{ijkls}{\overline{e}}_{ls}+D_{lsijk}{\overline{\epsilon }}_{ls}+C_{ijklsr}{\overline{\gamma }}_{lsr}+c_{ijk}\overline{\phi }\right]\le \\ \hfill \le (1+\epsilon )\left[A_{ijmn}{e}_{mn}+G_{ijmn}\left(e+{\epsilon }_{mn}\right)+B_{ijmn}{\epsilon }_{mn}+\left(F_{mnrij}+D_{ijmnr}\right){\gamma }_{mnr}\right]\times \\ \hfill \times \left[A_{ijkl}{\overline{e}}_{kl}+G_{ijkl}\left({\overline{e}}_{kl}+{\overline{\epsilon }}_{kl}\right)+B_{ijkl}{\overline{\epsilon }}_{kl}+\left(F_{klsij}+D_{ijkls}\right){\overline{\gamma }}_{kls}\right]+\\ \hfill +\left[F_{ijkmn}{e}_{mn}+D_{mnijk}{\epsilon }_{mn}+C_{ijkmnr}{\gamma }_{mnr}\right]\times \\ \hfill \times \left[F_{ijkrs}{\overline{e}}_{rs}+D_{mnijk}{\overline{\epsilon }}_{rs}+A_{ijkprs}{\overline{\gamma }}_{prs}\right]+\frac{1+\epsilon }{\epsilon }\left(a_{ij}{a}_{ij}+b_{ij}{b}_{ij}+c_{ijk}{c}_{ijk}\right)\phi \overline{\phi }\le \\ \hfill \le (1+\epsilon ){\nu }_C^2\left(e_{ij}{\overline{e}}_{ij}+{\epsilon }_{ij}{\overline{\epsilon }}_{ij}+{\gamma }_{ijk}{\overline{\gamma }}_{ijk}\right)+\frac{1+\epsilon }{\epsilon }{C}^2\phi \overline{\phi }\le \\ \hfill \le (1+\epsilon ){\nu }^{*}\left[C_{ijmn}{e}_{ij}{\overline{e}}_{mn}+G_{ijmn}\left(e_{ij}{\overline{\epsilon }}_{mn}+{\overline{e}}_{ij}{\epsilon }_{mn}\right)+F_{mnrij}\left(e_{ij}{\overline{\gamma }}_{mnr}+{\overline{e}}_{ij}{\gamma }_{mnr}\right)+\right.\\ \hfill \left.+B_{ijmn}{\epsilon }_{ij}{\overline{\epsilon }}_{mn}+D_{ijmnr}\left({\epsilon }_{ij}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{ij}{\gamma }_{mnr}\right)+C_{ijkmnr}{\gamma }_{ijk}{\overline{\gamma }}_{mnr}\right]+\frac{1+\epsilon }{\epsilon }{C}^2\phi \overline{\phi },\end{array}$$

(19)

where $\epsilon$ is an positive constant which will be determined later and we used the notation:

$$\begin{array}{c}\hfill {\nu }^{*}=\frac{{\nu }_M^2}{{\nu }_m},C=\sqrt{a_{ij}{a}_{ij}+b_{ij}{b}_{ij}+c_{ijk}{c}_{ijk}}.\end{array}$$

(20)

Now, we integrate the inequality (19) on an arbitrary section ${\mathsf{\Sigma }}_z$ and so we obtain:

$$\begin{array}{c}\hfill {\int }_{{\mathsf{\Sigma }}_z}\left(t_{ij}{\overline{t}}_{ij}+{\tau }_{ij}{\overline{\tau }}_{ij}+m_{ijk}{\overline{m}}_{ijk}\right)dA\le \\ \hfill \le {\int }_{{\mathsf{\Sigma }}_z}\left\{(1+\epsilon ){\nu }^{*}\left[A_{ijmn}{e}_{ij}{\overline{e}}_{mn}+G_{ijmn}\left(e_{ij}{\overline{\epsilon }}_{ij}+{\overline{e}}_{ij}{\epsilon }_{mn}\right)+\right.\right.\\ \hfill +F_{mnrij}\left(e_{ij}{\overline{\gamma }}_{mnr}+{\overline{e}}_{ij}{\chi }_{mnr}\right)+B_{ijmn}{\epsilon }_{ij}{\overline{\epsilon }}_{mn}+D_{ijmnr}\left({\epsilon }_{ij}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{ij}{\gamma }_{mnr}\right)+\\ \hfill \left.\left.+C_{ijkmnr}{\gamma }_{ij}{\overline{\gamma }}_{mnr}\right]+\frac{1+\epsilon }{\epsilon }{C}^2{\phi }_{,r}{\overline{\phi }}_{,s}\right\}dA.\end{array}$$

(21)

In conclusion, we have proven the next result.

Theorem 1.

For any solution $(v_m,{\varphi }_{mn},\phi )$ of the above problem $\mathcal{P}$ we have the estimate (21).

In the following we can work in the hypothesis that $C\ne 0$ because in the case $M=0$ we can separate the thermal effect of the mechanical effect and then obtain two problems which can be treated separately by using, for each of them, a procedure which is similar to the one that will be exposed in the sequel.

In order to obtain another decay estimate, we will use the following functional:

$$\begin{array}{c}\hfill I\left(z\right)=-{\int }_{{\mathsf{\Sigma }}_z}\left[\left(t_{3m}+{\tau }_{3m}\right){\overline{v}}_m+\left({\overline{t}}_{3m}+{\overline{\tau }}_{3m}\right)v_k+m_{3mn}{\overline{\varphi }}_{mn}+{\overline{m}}_{3mn}{\varphi }_{mn}-\delta (h_3\overline{\phi }+{\overline{h}}_3\phi )\right]dA,\end{array}$$

(22)

which is associated with a solution $(v_m,{\varphi }_{mn},\phi )$ of the boundary value problem $\mathcal{P}$. The positive constant $\delta$, used here, will be determined later.

Another form of the functional I is obtained in the following proposition.

Proposition 1.

The functional I can be written in the form:

$$\begin{array}{c}\hfill F\left(z\right)={\int }_{{\mathcal{C}}_z}\left\{2A_{klmn}{e}_{kl}{\overline{e}}_{mn}+G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+2B_{klmn}{\epsilon }_{kl}{\overline{\epsilon }}_{mn}+\right.\\ \hfill +F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+D_{klmnr}\left({\epsilon }_{kl}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+\\ \hfill +2C_{klsmnr}{\gamma }_{kls}{\overline{\gamma }}_{mnr}-2\rho {\omega }^2{v}_m{\overline{v}}_{}-2{\omega }^2{I}_{mn}{\varphi }_{mj}{\overline{\varphi }}_{nj}+2\delta {\phi }_{,m}{\overline{\phi }}_{,m}+\\ \hfill +\left[\sqrt{\alpha {\nu }_m}{e}_{kl}+\frac{1}{\sqrt{\alpha {\nu }_m}}(1+i\delta \omega )a_{kl}\phi \right]\left[\sqrt{\alpha {\nu }_m}{\overline{e}}_{kl}+\frac{1}{\sqrt{\alpha {\nu }_m}}(1-i\delta \omega )a_{kl}\overline{\phi }\right]+\\ \hfill +\left[\sqrt{\alpha {\nu }_m}{\epsilon }_{kl}+\frac{1}{\sqrt{\alpha {\nu }_m}}(1+i\delta \omega )b_{kl}\phi \right]\left[\sqrt{\alpha {\nu }_m}{\overline{\epsilon }}_{kl}+\frac{1}{\sqrt{\alpha {\nu }_m}}(1-i\delta \omega )b_{kl}\overline{\phi }\right]+\\ \hfill +\left[\sqrt{\alpha {\nu }_m}{\gamma }_{prs}+\frac{1}{\sqrt{\alpha {\nu }_m}}(1+i\delta \omega )c_{prs}\phi \right]\left[\sqrt{\alpha {\nu }_m}{\overline{\gamma }}_{prs}+\frac{1}{\sqrt{\alpha {\nu }_m}}(1-i\delta \omega )c_{prs}\overline{\phi }\right]-\\ \hfill \left.-\alpha {\nu }_m\left(e_{kl}{\overline{e}}_{kl}+{\epsilon }_{kl}{\overline{\epsilon }}_{kl}+{\gamma }_{klj}{\overline{\gamma }}_{klj}\right)-\frac{1}{\alpha {\nu }_m}\left(1+{\delta }^2{\omega }^2\right)M^2\phi \overline{\phi }\right\}dV,\end{array}$$

(23)

in which the positive constant α will be determined later.

Proof. 

By taking into account Equations (12)–(14) and the boundary relations (15), we obtain:

$$\begin{array}{c}\hfill F^{{}^{\prime }}\left(z\right)=-{\int }_{{\mathsf{\Sigma }}_z}\left\{2A_{klmn}{e}_{kl}{\overline{e}}_{mn}+G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+2B_{klmn}{\epsilon }_{kl}{\overline{\epsilon }}_{mn}+\right.\\ \hfill +F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+D_{klmnr}\left({\epsilon }_{kl}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+2C_{klsmnr}{\gamma }_{kls}{\overline{\gamma }}_{mnr}+\\ \hfill +a_{kl}\left(e_{kl}\overline{\phi }+{\overline{e}}_{kl}\phi \right)+b_{kl}\left({\epsilon }_{kl}\overline{\phi }+{\overline{\epsilon }}_{kl}\phi \right)+c_{kls}\left({\gamma }_{kls}\overline{\phi }+{\overline{\gamma }}_{kls}\phi \right)-\\ \hfill -2\rho {\omega }^2{v}_m{\overline{v}}_m-2{\omega }^2{I}_{kl}{\varphi }_{jl}{\overline{\varphi }}_{jk}+2\delta {\phi }_{,m}{\overline{\phi }}_{,m}-\\ \hfill \left.-i\delta \omega \left[a_{kl}\left(e_{kl}\overline{\phi }-{\overline{e}}_{kl}\phi \right)+b_{kl}\left({\epsilon }_{kl}\overline{\phi }-{\overline{\epsilon }}_{kl}\phi \right)+c_{kls}\left({\gamma }_{kls}\overline{\phi }-{\overline{\gamma }}_{kls}\phi \right)\right]\right\}dA.\end{array}$$

(24)

Then, by direct calculations:

$$\begin{array}{c}\hfill F\left(L\right)-F\left(z\right)={\int }_z^L{F}^{{}^{\prime }}\left(w\right)dw=\\ \hfill -{\int }_{{\mathcal{C}}_z}\left\{2A_{klmn}{e}_{kl}{\overline{e}}_{mn}+G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+2B_{klmn}{\epsilon }_{kl}{\overline{\epsilon }}_{mn}+\right.\\ \hfill +F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+D_{klmnr}\left({\epsilon }_{kl}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+\\ \hfill +2C_{klsmnr}{\gamma }_{kls}{\overline{\gamma }}_{mnr}-2\rho {\omega }^2{v}_m{\overline{v}}_m-2{\omega }^2{I}_{kl}{\varphi }_{jl}{\overline{\varphi }}_{jk}+\\ \hfill +2\delta {\phi }_{,m}{\overline{\phi }}_{,m}+\phi (1+i\delta \omega )\left(a_{mn}{\overline{e}}_{mn}+b_{mn}{\overline{\epsilon }}_{mn}+c_{mnr}{\overline{\gamma }}_{mnr}\right)+\\ \hfill +\overline{\phi }(1-i\delta w)\left(a_{mn}{e}_{mn}+b_{mn}{\epsilon }_{mn}+c_{mnr}{\gamma }_{mnr}\right)\}dV.\end{array}$$

(25)

However, according to (15), the boundary data are null so that based on the definition (22) of the functional F, we deduce that $F\left(L\right)=0$, such that (25) becomes (23) and the proof of Proposition 1 is finished. □

We intend to give a new estimate on the solutions of problem $\mathcal{P}$. For this we need the following considerations.

First, by using the notation:

$$\begin{array}{c}\hfill {\mathcal{M}}_1={\int }_{\mathcal{C}}\left(A_{klmn}{e}_{kl}{e}_{mn}+2G_{klmn}{e}_{kl}{\epsilon }_{mn}+B_{klmn}{\epsilon }_{kl}{\gamma }_{mn}+\right.\\ \hfill \left.+2F_{mnrkl}{e}_{kl}{\gamma }_{mnr}+2D_{klmnr}{\epsilon }_{kl}{\gamma }_{mnr}+C_{klsmnr}{\gamma }_{kls}{\gamma }_{mnr}\right)dV,\\ \hfill {\mathcal{M}}_2={\int }_{\mathcal{C}}\left(\rho v_m{v}_m+I_{mn}{\varphi }_{mj}{\varphi }_{nj}+\rho \kappa {\phi }^2\right)dV,\end{array}$$

we can use the suggestion of Flavin and Knops from [1] in order to introduce the size ${\omega }_m(h,L)$ by

$$\begin{array}{c}\hfill {\omega }_m^2(h,L)=\inf \frac{{\mathcal{M}}_1}{{\mathcal{M}}_2}.\end{array}$$

(26)

Theorem 2.

For those solutions $\left(v_m,{\varphi }_{mn},\phi \right)$ of the problem $\mathcal{P}$ for which the components are continuous differentiable real functions and satisfy the conditions $v_m=0$, ${\varphi }_{mn}=0$ and $\phi =0$ on the boundary, we have the following estimate:

$$\begin{array}{c}\hfill m_0\left(1-\frac{{\omega }^2}{{\omega }_{*}^2}\right){\int }_{R_z}\left[A_{klmn}{e}_{kl}{\overline{e}}_{mn}+G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+\right.\\ \hfill +B_{klmn}{\gamma }_{kl}{\overline{\gamma }}_{mn}+F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+\\ \hfill \left.+D_{klmnr}\left({\epsilon }_{kl}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+A_{klsmnr}{\chi }_{kls}{\overline{\chi }}_{mnr}+\delta {\phi }_{,m}{\overline{\phi }}_{,m}\right]dV\le \\ \hfill \le \gamma (1+\epsilon ){\nu }^{*}{\int }_{{\mathsf{\Sigma }}_z}\left[A_{klmn}{e}_{kl}{\overline{e}}_{mn}+G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+\right.\\ \hfill +B_{klmn}{\epsilon }_{kl}{\overline{\epsilon }}_{mn}+F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+\\ \hfill \left.+D_{klmnr}\left({\epsilon }_{kl}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+C_{klsmnr}{\gamma }_{kls}{\overline{\gamma }}_{mnr}\right]dA+\\ \hfill +\frac{1}{\gamma }{\int }_{{\mathsf{\Sigma }}_z}\left(v_m{\overline{v}}_m+{\varphi }_{mn}{\overline{\varphi }}_{mn}+{\phi }^2\right)dA,\end{array}$$

(27)

where the positive constants $m_0,\delta$ and γ will be determined later.

Proof. 

If we take into account the relations (17) and (26), then the equality (23) leads to the following inequality:

$$\begin{array}{c}\hfill F\left(z\right)\ge {\int }_{{\mathcal{C}}_z}\left\{\left(2-\alpha -2\frac{{\omega }^2}{{\omega }_m^2}\right)\left[A_{klmn}{e}_{kl}{\overline{e}}_{mn}+G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+\right.\right.\\ \hfill +B_{klmn}{\epsilon }_{kl}{\overline{\epsilon }}_{mn}+F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+D_{klmnr}\left({\epsilon }_{kl}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+\\ \hfill \left.\left.+C_{klsmnr}{\gamma }_{kls}{\overline{\gamma }}_{mnr}\right]-\frac{1}{\alpha {\nu }_m}\left(1+{\delta }^2{\omega }^2\right)C^2\phi \overline{\phi }\right\}dV.\end{array}$$

(28)

As we anticipated, we will choose the positive constants $\alpha$ and $\delta$ so that:

$$\begin{array}{c}\hfill 0<\alpha <2,\delta >\frac{C^2}{2\delta \alpha {\lambda }_1{\nu }_m},\end{array}$$

(29)

and suppose that the exciting frequencies are subject to the condition:

$$\begin{array}{c}\hfill \omega <{\omega }_{*}(h,L,\alpha ,\delta ),\end{array}$$

where the critical frequency ${\omega }_{*}(h,L,\alpha ,\delta )$ is defined by:

$$\begin{array}{c}\hfill {\omega }_{*}^2(h,L,\alpha ,\delta )=\min \left\{\left(1-\frac{\alpha }{2}\right){\omega }_m^2(h,L),\frac{2\alpha {\lambda }_1{\nu }_m}{\delta C^2}\right\}.\end{array}$$

(30)

Taking into account these specifications, the inequality (28) receives the form:

$$\begin{array}{c}\hfill F\left(z\right)\ge m_0\left(1-\frac{{\omega }^2}{{\omega }_{*}^2}\right){\int }_{{\mathcal{C}}_z}\left[A_{klmn}{e}_{kl}{\overline{e}}_{mn}+G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+\right.\\ \hfill +B_{klmn}{\epsilon }_{kl}{\overline{\epsilon }}_{mn}+F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+D_{klmnr}\left({\epsilon }_{kl}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+\\ \hfill \left.+C_{klsmnr}{\gamma }_{kls}{\overline{\gamma }}_{mnr}+\delta {\phi }_{,m}{\overline{\phi }}_{,m}\right]dV,\end{array}$$

(31)

where we used the notation:

$$\begin{array}{c}\hfill m_0(\alpha ,\delta )=\min \left\{2-\alpha ,2-\frac{C^2}{\delta \alpha {\lambda }_1{\mu }_m}\right\}.\end{array}$$

(32)

We will use an extension of the arithmetic-geometric means inequality, which for a non-null parameter p, has the form:

$$\begin{array}{c}\hfill ab\le \frac{1}{2}\left(\frac{a^2}{p^2}+b^2{p}^2\right).\end{array}$$

(33)

Now, we will use the Schwarz inequality and the means inequality (33) for the terms under the integral from the right-hand side of (21) so that considering (22) we deduce:

$$\begin{array}{c}\hfill F\left(z\right)\le \gamma (1+\epsilon ){\nu }^{*}{\int }_{{\mathsf{\Sigma }}_z}\left[C_{klmn}{e}_{kl}{\overline{e}}_{mn}+G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+\right.\\ \hfill +B_{klmn}{\epsilon }_{kl}{\overline{\epsilon }}_{mn}+F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+\\ \hfill \left.+D_{klmnr}\left({\epsilon }_{kl}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+C_{klsmnr}{\gamma }_{kls}{\overline{\gamma }}_{mnr}\right]dA+\\ \hfill +\frac{1}{\gamma }{\int }_{{\mathsf{\Sigma }}_z}\left(v_m{\overline{v}}_m+{\varphi }_{mn}{\overline{\varphi }}_{mn}+{\phi }^2\right)dA,\end{array}$$

(34)

where $\gamma$ has the expression:

$$\begin{array}{c}\hfill \gamma ={\left[(1+\epsilon ){\nu }^{*}\rho {\omega }_0^2\left(h\right)\right]}^{-\frac{1}{2}}.\end{array}$$

(35)

Finally, by using the inequalities (31) and (34), we obtain the estimate (27) and the proof of Theorem 2 is finished. □

Based on a suggestion given by Toupin in [2], in order to obtain another estimate of the solutions to problem $\mathcal{P}$, we will use the following two auxiliary functions:

$$\begin{array}{c}\hfill W\left(z\right)={\int }_{{\mathcal{C}}_z}\left[A_{klmn}{e}_{kl}{\overline{e}}_{mn}+G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+\right.\end{array}$$

$$\begin{array}{c}\hfill +F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+D_{klmnr}\left({\epsilon }_{kl}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+\\ \hfill \left.+B_{klmn}{\epsilon }_{kl}{\overline{\epsilon }}_{mn}+C_{klsmnr}{\gamma }_{kls}{\overline{\gamma }}_{mnr}+\delta {\phi }_{,m}{\overline{\phi }}_{,m}\right]dV,\end{array}$$

(36)

$$\begin{array}{c}\hfill V(z,h)=\frac{1}{h}{\int }_z^{z+h}W\left(\zeta \right)d\zeta .\end{array}$$

(37)

Furthermore, on a cylinder having the plane ends $x_3=z$ and $x_3=z+h$, denoted by $C(z,h)$, we will use the Poincaré type inequality, in the following form:

$$\begin{array}{c}\hfill {\int }_{C(z,h)}\left[A_{klmn}{e}_{kl}{\overline{e}}_{mn}+G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+\right.\\ \hfill +F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+B_{klmn}{\epsilon }_{kl}{\overline{\epsilon }}_{mn}+\\ \hfill \left.+D_{klmnr}\left({\epsilon }_{kl}{\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+C_{klsmnr}{\gamma }_{kls}{\overline{\gamma }}_{mnr}\right]dV\ge \\ \hfill \ge \rho {\omega }_0^2{\int }_{C(z,h)}\left(v_m{\overline{v}}_m+{\varphi }_{mn}{\overline{\varphi }}_{mn}+{\phi }^2\right)dV,\end{array}$$

(38)

where ${\omega }_0\left(h\right)/2\pi$ is the notation for the lowest level of vibration frequency of a cylinder of thickness h whose lateral surface is clamped and whose ends are free of loads.

Based on the above considerations, we can give a new estimate on the solutions of problem $\mathcal{P}$.

Theorem 3.

Under the conditions specified above, the following spatial decay estimation takes place:

$$\begin{array}{c}\hfill W\left(z\right)\le W\left(0\right)\exp \left\{-(z-h)\frac{1-\frac{{\omega }^2}{{\omega }_{*}^2}}{s_c}\right\},h\le z\le L,\end{array}$$

(39)

if the following condition is met:

$$\begin{array}{c}\hfill \omega <{\omega }_{*}(h,L,\alpha ,\delta ),\end{array}$$

(40)

where ${\omega }_{*}(h,L,\alpha ,\delta )$ is defined in (30) and $s_c(h,\alpha ,\delta )$ has the expression:

$$\begin{array}{c}\hfill s_c(h,\alpha ,\delta )=\frac{2}{m_0(\alpha ,\delta )}\sqrt{\frac{(1+\epsilon ){\nu }^2}{\rho {\omega }_0^2\left(h\right)}}.\end{array}$$

(41)

Proof. 

We start by integrating the inequality (27) over the interval $[z,z+h]$. Therefore, if we use the expressions of the functions W in (36) and V in (37), with the help of the Poincaré inequality (38), we arrive at the following estimate:

$$\begin{array}{c}\hfill m_0\left(1-\frac{{\omega }^2}{{\omega }_{*}^2}\right)V(z,h)\le \frac{1}{h}\left[\gamma (1+\epsilon ){\nu }^{*}+\frac{1}{\gamma \rho {\omega }_0^2\left(h\right)}\right]\times \\ \hfill \times {\int }_{C(z,h)}\left[A_{klmn}{e}_{kl}{\overline{e}}_{mn}+G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+\right.\\ \hfill +F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+B_{klmn}{\epsilon }_{kl}{\overline{\epsilon }}_{mn}+\\ \hfill \left.+D_{klmnr}\left(\epsilon {\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+C_{klsmnr}{\gamma }_{kls}{\overline{\gamma }}_{mnr}\right]dV.\end{array}$$

(42)

Taking into account the expression of $\gamma$ from (35), the inequality (42) can be rewritten in the following form:

$$\begin{array}{c}\hfill m_0\left(1-\frac{{\omega }^2}{{\omega }_{*}^2}\right)Q(z,h)\le \frac{2}{h}{\int }_{C(z,h)}\left[A_{klmn}{e}_{kl}{\overline{e}}_{mn}+B_{klmn}{\epsilon }_{kl}{\overline{\epsilon }}_{mn}+\right.\\ \hfill +G_{klmn}\left(e_{kl}{\overline{\epsilon }}_{mn}+{\overline{e}}_{kl}{\epsilon }_{mn}\right)+F_{mnrkl}\left(e_{kl}{\overline{\gamma }}_{mnr}+{\overline{e}}_{kl}{\gamma }_{mnr}\right)+\\ \hfill \left.+D_{klmnr}\left(\epsilon {\overline{\gamma }}_{mnr}+{\overline{\epsilon }}_{kl}{\gamma }_{mnr}\right)+C_{klsmnr}{\gamma }_{kls}{\overline{\gamma }}_{mnr}\right]dV.\end{array}$$

(43)

Now, we consider the expressions of W, V and $s_c$ from (36), (37), (41), respectively, so that the inequality (43) acquires the simpler expression:

$$\begin{array}{c}\hfill s_c(h,\alpha ,\delta )\frac{dV(z,h)}{dz}+\left(1-\frac{{\omega }^2}{{\omega }_{*}^2}\right)V(z,h)\le 0,\end{array}$$

(44)

and this first-order differential inequality is integrated on $[h,L]$ to obtain the desired estimate (39), which concludes the proof of Theorem 3. □

4. Conclusions

It is appropriate to give a more explicit interpretation of the estimation (39).

If on the end of the vibrating cylinder the harmonic excitation level is below a predetermined threshold, then the presence of the voids does not affect the spatial decay of vibration in all points away from the end which is harmonically excited. In our case, the predetermined threshold is given in (30), and it is smaller than the one found in classical elasticity (see Flavin and Knops, [1]).

Obviously, if an upper bound can be obtained for $W\left(0\right)$, then the efficiency of the estimation (39) increases. Indeed, if the frequency $\omega$ is small enough and satisfies (40), an upper bound for $W\left(0\right)$ can be obtained using the boundary data on the excited end. Considering this bound for $W\left(0\right)$ in estimation (39) we obtain a real spatial decay of vibration, at an appreciable distance from the excited end.