Localization in Time of Solutions for Thermoelastic Micropolar Materials with Voids

Marin Marin,Ravi P.Agarwal and Mohamed Othman

1 Introduction

As is already known,the theory of materials with voids or vacuous pores is the simplest extension of the classical theory of elasticity and was first proposed by Nunziato and Cowin(1979).In this theory the authors introduce an additional degree of freedom in order to develope the mechanical behavior of a body in which the skeletalmaterialis elastic and interstices are voids ofmaterial.Itisworth recalling that porous materials have applications in many fields of engineering such as petroleum industry,material science,biology and so on.The intended applications of the theory are to geological materials like rocks and soil and to manufactured porous materials.The linear theory of elastic materials with voids was developed by Cowin and Nunziato(1983).Here the uniqueness and weak stability of solutions are also derived.Iesan(1986)has established the equations of thermoelasticity of materials with voids.

An extension of these results to cover other theories of materials with voids was been made in our studies[Marin(1999,2010a,2010b);Marin,Mahmoud and Al-Basyouni(2013);Munoz-Rivera and Quintanilla(2008)].

In the case ofsystemswhere dissipation mechanism is sufficiently strong,is already proved the localization of solutions in the time variable,that is the decay of the solutions is quite accelerated to ensure that they vanish after a finite time.However,is several cases there is no proof of the impossibility of localization of the solutions.Even if there are cases of thermoelasticity for bodies with voids in which the decay of solutions can be controlled for exponential functions,and,also,for polynomials functions,still not proven the impossibility of localization for the solutions.In previous studies on this subject,the authors demonstrated different upper bounds for the decay of solutions.

In the case of the present study are important the lower bounds for the decay of solutions.It is worth to mention two studies[Casas and Quintanilla(2005);Nunziato(1979)],in which the authors have shown that after a certain period of time,the deformations of the thermoelastic bodies with voids become so small that they can be neglected.However,we are not sure that these deformations are null for every positive time.If we prove the uniqueness of solutions for the backward in time problem of the thermoelasticity of micropolar bodies with voids,then we are sure that the only solution for this problem which vanishes for everyt≥0 is the null solution.In other words,we deduce the impossibility of localization of the solutions of the problem of the thermoelasticity of micropolar bodies with voids.Regarding the uniqueness of solutions for the backward in time problem,have noted the contribution of Ciarletta(2002),but he proved the uniqueness of solutions for the backward in time problem in the context of classical thermoelasticity.

Some results regarding back in time problems have been considered also by Ames and Payne(1991)in orderto obtain stabilizing criteria forsolutionsofthe boundaryfinal value problem.

Quintanilla(2007)improves the uniqueness result obtained by Ciarletta,using more concrete assumptions,in particular considering a strictly positive heat capacity.So,it is proved the impossibility of localization in time of the solutions of the forward in time problem for the linear thermoelasticity of Green and Naghdi’s type.

Other results regarding the backward in time problem,but for classical porous elastic materials has been studied by Iovane and Passarella(2004).

As we can see,such backward in time problem has never been studied for the case of micropolar thermoelastic bodies with voids.

We still use the paper of Ciarletta as a guide in getting results of this study.The main result of our paper is based on Lagrange identities and energy arguments.

2 Organization of a paper

A paper for publication in CMC must contain a title,names and affiliations of the authors,a list of keywords,a brief abstract at the beginning of the main body,a conclusion section at the end of main body,and a list of references that follows the conclusions section.

In the main body of the paper,three different levels of headings(for sections,subsections,and subsubsections)may be used.The typesetting style for these headings is presented in the next section.

3 Basic equations

An anisotropic elastic material is considered.Assume a such body that occupies a properly regular regionBof three-dimensional Euclidian spaceR3bounded by a piecewise smooth surface?Band we denote the closure ofBbyˉB.The boundary?Bis smooth enough to apply the divergence theorem.

We use a fixed system of rectangular Cartesian axesOxi,(i=1,2,3)and adopt Cartesian tensor notation.A superposed dot stands for the material time derivate while a comma followed by a subscript denotes partial derivatives with respect to the spatial coordinates.Einstein summation convention on repeated indices is used.Also,the spatialargumentand the time argumentofa function willbe omitted when there is no likehood of confusion.

We consider the mixed problem associated with the theory of thermoelasticity of micropolar bodies with voids on the time intervalI.So,in the absence of supply terms,it is known that the basic equations onB×Iare,(see,for instance[Marin,Mahmoud and Al-Basyouni(2013)])

The equations(1)are the motion equations,(2)is the balance of the equilibrated forces and(3)is the energy equation.

In the following,we restricte ourconsiderationsonly to the case where the materials have a centerofsymmetry.Consequently,the constitutive tensorsofodd ordermust vanish and the constitutive equations become

where the strain tensorsεij,γijand the temperature are defined by means of the kinetic relations

In the above equations we have used the following notations:ρ-the constant mass density;η-the specific entropy;T0-the constant absolute temperature of the body in its reference state;Iij-coefficients of microinertia;J-a positive function and it is the product of the mass density and the equilibrated inertia;ui-the components of displacement vector;?i-the components of microrotation vector;φ-the volume distribution function which in the reference state isφ0;θ-the temperature variation measured from the reference temperatureT0;εij,γij-kinematic characteristics of the strain;tij-the components of the stress tensor;mij-the components of the couple stress tensor;hi-the components of the equlibrated stress vector;qi-the components of the heat flux vector;g-the intrinsic equilibrated force;Kij-the heat conductivity tensor;c=the heat capacity which is assumed positive;Aijmn,Bijmn,...,αijfrom the constitutive equations are the characteristic functions of the material,and they obey the symmetry relations

and the following null boundary conditions

whereniare the components of the outward unit normal to the boundary surface.Also,?B1,?B2,?B3and?B4with respective complements?Bc1,?Bc2,?Bc3and?Bc4are the subsets of the surface?Bsuch that

Introducing(5)and(4)into equations(1),(2)and(3),we obtine the following system of equations

By a solution of the mixed boundary-final value problem of the theory of thermoelasticity of micropolar bodies with voids in the cylinder ?0=B×(-∞,0]we mean an ordered array(ui,?i,θ,σ)which satisfies the system of equations(9)for all(x,t)∈?0,the boundary conditions(8)and the final conditions(7).

4 Basic relations

We need to impose the positivity of several tensors.Also,in all what follows we shall use the following assumptions on the material properties

These assumptions are in agreement with the usual restrictions imposed in the mechanics of continua.While the interpretation of conditions(i)is obvious,the assumption(ii)represent a considerable strenghtening of the entropy production inequality.

Conditions(iii)and(iv)ensure that the internal energy is positive and the best of their interpretation finds its place in the theory of mechanical stability.

By using an appropiate change of variables and notations suitably chosen,we can transform the mixed boundary-final value problemPinto the mixed initial boundary value problemP?.In other words,we set,generally speaking,f?(t?)=f(t),witht?=-t.For the sake of simplicity,we remove the sign?from the notations such that we obtain the problemP?,called the backward in time problem corresponding to the problem of thermoelasticity of micropolar bodies with voids.The problemP?is defined by(see also Ciarletta(2002)):-the system of equations

that occur for any(x,t)∈B×[0,∞);

-the constitutive equations(4)inˉB×[0,∞);

-the geometric equations(4)onˉB×[0,∞);

-the following homogeneous boundary conditions

where?Biand?Bciare defined in Section 2;

-the initial conditions(7)whereu0i,u1i,?1i,?0i,φ0,φ1andθ0are prescribed continuous functions onB.

We now state and prove some basic relations which qbe used to prove the main result of our study.The first of these is an energy relation.

Proposition 1.If(ui,?i,ψ,θ)is a solution of the problem consists of system of equations(10),initial conditions(7)and the boundary conditions(11),then we have the following equality

Proof.First,we multiply(10)1by˙uiand after simple calculations we obtain

If we multiply(10)2by˙?ithen using the product rule derivation we deduce

Now,we multiply(10)3by˙φand after simple calculations we lead to the result

Finally,we multiply(10)4byθand obtain

If we add the equalities(13)to(16),term by term,and use geometric equations(5),we find the relation

Equality(17)can be rewritten in the form

Now we integrate in(18)over[0,t]×Band apply the divergence theorem.Given the boundary and the initial conditions,we are led to the desired result(12).

In next proposition we prove a second useful relation which is also an energy relation.

Proposition 2.If(ui,?i,ψ,θ)is a solution of the problem consists of system of equations(10),initial conditions(7)and the boundary conditions(11),then we have the following equality

Proof.First,we multiply(10)1by˙uiand obtain the equality(13).If we multiply(10)2by˙?iwe deduce the equality(14).Now,we multiply(10)3by˙φand we lead to the equality(15).Finally,we multiply(10)4by-θand obtain

If we add the equalities(13)to(15)and(20),term by term,and use geometric equations(5),we find the relation

Now we integrate in(21)over[0,t]×Band apply the divergence theorem.Given the boundary and the initial conditions,we are led to the desired result(19).

Third relation is obtained using the Lagrange identity method.

Proposition 3.If(ui,?i,ψ,θ)is a solution of the problem consists of system of equations(10),initial conditions(7)and the boundary conditions(11),then we have the following equality

Proof.Using the equation(10)1we have

With the help of the equation(10)2we deduce

Using the equation(10)3we have

Similarly,with the help of the equation(10)4we obtain

Using the equalities(22)-(25)and the geometric equations(5)we are lead to

Now we integrate in(27)over[0,t]×Band apply the divergence theorem.Given the boundary and the initial conditions,we are led to the desired result(22)such that Proposition 3 is now concluded.

5 Main result

In this section we will prove the main result of our paper,that is,it is not possible the localization of the solutions of the mixed initial boundary value problem in thermoelasticity of micropolar osies with voids.For this we use relations(12),(19)and(22)and,also,the result of the following proposition.

Proposition 4.Let(ui,?i,ψ,θ)be a solution of the mixed problem consists of system of equations(10),initial conditions(7)and the boundary conditions(11).If the standard assumptions(i)-(iv)are satisfied,then we have

Closely related to relations(12)and(28)we define the functions

whereεis a small positive constant.

It is easy to deduce that

Since we can writeE(t)in the form

If we chooseε1sufficiently small,we can deduce

where the positive constantK1can be calculated in termsof constitutive coefficients andε1.

Also,we can compute a positive constantK2such that

With the help of the relations(31)to(33)we deduce that if we chooseε1≤1-ε,there exists a positive constantCsuch that

If we take into account the inequalities(32)to(34)we obtain the estimate

which is satisfied for everyt≥0.

In the inequality(35)C?is a positive constant which can be calculated taking into account the above inequalities.

Of course,from(35)we deduce

But the initial conditions,(7),were presumed null,therefore we have

Taking into account the definition(29)of functionE(t),it follows that

Considering again the null initial conditions,we deduce that our mixed problem has only null solution and the proof of Proposition 4 is concluded.

Let us consider the mixed initial boundary value problem consists of the equations(1)to(5),the homogeneous boundary conditions(11)and the initial conditions(7).We want to show the impossibility of localization in time of solutions of this problem.This is equivalent to show that the only solution for this problem that vanishes after a finite time is the null solution.

Theorem 1.Suppose the standard assumptions(i)-(iv)and the symmetry relations(6)are satisfied.Consider(ui,?i,φ,θ)a solution of the mixed problem,previous defined,that vanishes after a finite time t1≥0,that is

Proof.To this aim we consider the corresponding backward in time problem in the time interval(-∞,t1],consists of the equations(1)to(5),the homogeneous boundary conditions(11)and the following null final conditions

Taking into account Proposition 4,this problem has only the null solution.

6 Conclusion

The main result of our study prove the uniqueness of the solution for the backward in time problem in the context of thermoelasticity of micropolar bodies with voids.This means that the only solution to the backward in time problem that vanishes for everyt≥t1＞0 is the null solution.In other words,we have shown the impossibility of localization in time of the solutions of the mixed intial boundary value problem for thermoelastic micropolar bodies.

Thus,the combination of the micropolar structure with the thermal and porous dissipation is not so strong to ensure that the mechanical deformations vanish a finite time.

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