INSTABILITY , NONEXISTENCE , AND UNIQUENESS IN ELASTICITY WITH POROUS DISSIPATION

In what follows, we consider a theory for the behaviour of porous solids such that the matrix material is elastic and the interstices are void of material; it is a generalization of the classical theory of elasticity. The theory of porous elastic material has been established by Cowin and Nunziato [2, 11]. In this theory, the bulk density is the product of two scalar fields, the matrix material density and the volume fraction field; it is studied in the book of Ciarletta and Ieşan [1]. Thermal effects were included in the book of Ieşan [4]. Results on linear and nonlinear problems have been obtained recently [5, 14, 15]. The aim of this paper is the study of the qualitative behaviour of the solutions of the elasticity with porous dissipation. It is worth noting that there are very few contributions to this topic. We can recall some contributions to the nonlinear problem in the book of Ciarletta and Ieşan [1], but they are in the case where porous dissipation is not present. Here we prove instability and nonexistence in the nonlinear case when some conditions on the internal energy and the dissipation function are satisfied. We also work in the linear case and we prove exponential growth of solutions whenever the initial data satisfy several conditions. A uniqueness result is also obtained in the case of the backward in time problem for the linear equations. The results of this paper are of interest from the mechanical and from the mathematical point of view: from the mechanical point of view because we obtain some qualitative results in the theory of elastic materials with voids; from the mathematical point of view because we extend some results that were known


Introduction
In what follows, we consider a theory for the behaviour of porous solids such that the matrix material is elastic and the interstices are void of material; it is a generalization of the classical theory of elasticity.The theory of porous elastic material has been established by Cowin and Nunziato [2,11].In this theory, the bulk density is the product of two scalar fields, the matrix material density and the volume fraction field; it is studied in the book of Ciarletta and Ies ¸an [1].Thermal effects were included in the book of Ies ¸an [4].Results on linear and nonlinear problems have been obtained recently [5,14,15].
The aim of this paper is the study of the qualitative behaviour of the solutions of the elasticity with porous dissipation.It is worth noting that there are very few contributions to this topic.We can recall some contributions to the nonlinear problem in the book of Ciarletta and Ies ¸an [1], but they are in the case where porous dissipation is not present.Here we prove instability and nonexistence in the nonlinear case when some conditions on the internal energy and the dissipation function are satisfied.We also work in the linear case and we prove exponential growth of solutions whenever the initial data satisfy several conditions.A uniqueness result is also obtained in the case of the backward in time problem for the linear equations.The results of this paper are of interest from the mechanical and from the mathematical point of view: from the mechanical point of view because we obtain some qualitative results in the theory of elastic materials with voids; from the mathematical point of view because we extend some results that were known 2 Elasticity with porous dissipation in the case of no dissipation to the case that there exists dissipation.Here, we are able to overcome the difficulty proposed by this new term.
It is worth recalling that some of the results are inspired by methods and techniques used in the classical elasticity.For example, Section 3 is inspired by the studies of Knops [6].The methodology of Section 5 has been used by Knops et al. [7], Levine [9], Payne [12], and Knops and Straughan [8].Recently this approach has been also used by Galaktionov and Pohozaev [3] and Quintanilla [13] has applied it to the thermoelasticity without energy dissipation.
The plan of the paper is as follows.In Section 2, we set down the basic equations and the initial and boundary conditions.An instability result is proved in Section 3. Some cases where the instability assumptions hold are presented in Section 4. In Section 5, we set down a nonexistence result and some cases where the assumptions hold are presented in Section 6.An exponential growth of the solutions for the linear case is proved in Section 7. The paper concludes with a uniqueness result for the backward-in-time problem corresponding to the linear equations.

Preliminaries
Consider a homogeneous body which occupies a bounded region B that we consider as a reference configuration.The surface ∂B is assumed to have sufficient smoothness to permit applications of the divergence theorem.The particles of the body are identified with their positions in B and we assume that the body is in equilibrium under the action of supply terms and surface tractions.The mechanical motion is described by the displacement u and the volume fraction ν.We assume that they are referred to as the reference configuration and a fixed system of Cartesian axes.B is at rest relative to the considered system of axes.Equilibrium is destroyed at t = 0 by instantaneously imparting to all points of B a given displacement and volume fraction, but keeping the boundary fixed throughout the ensuing mechanical motion.The governing equations of the nonlinear elasticity with voids are the equation of motion the equation and the constitutive equations Here W is the internal energy which depends on the gradient of the displacement u i,K , the gradient of the volume fraction ν ,K , and the volume fraction ν, and D is the dissipation function which also depends on the gradient of the displacement, the gradient of the volume fraction, the volume fraction, and the time derivative of the volume fraction ν.
We assume that D vanishes when ν vanishes and in general we have the condition νD ≤ 0. (2.4) If we introduce the constitutive equations into the evolution equations, we obtain the system of equations ) To the system of field equations we adjoin the initial conditions and the boundary conditions If we multiply (2.5) by ui and (2.6) by ν, integrate over B, and use the boundary conditions, we obtain the energy equation Although we work in the case of bounded domains, it is worth noting that the analysis and results of Sections 3, 5, 7, and 8 can be also used in the case of unbounded domains whenever we assume that the solutions vanish at infinity.

Instability: nonlinear case
The aim of this section is to state and prove the following result.

) be a solution of the initial-boundary-value problem (2.5)-(2.8). Let us assume that
and that where 0 < < 1.Then, the equilibrium state is unstable. Proof.Define In view of the assumptions, we have Then where the last inequality follows from the Schwarz inequality.We see that where δ = ( + 1)/2.So which proves that F δ (t) becomes unbounded whenever dF/dt(0) > 0.

Instability: discussion
The energy criterion is sometimes a condition to determine stability or instability.With the help of Theorem 3.1, it is possible to obtain a result when the equilibrium solution corresponds to a maximum of the energy.If we look for one-dimensional examples, we may consider We have where Arithmetic-geometric mean inequality and | ν| 2 dx . (4.5) In case that r = q = 2(s + 1), condition (3.2) is satisfied whenever there are three positive constants λ < 1, 1 , and 2 such that In particular, if s = 0, the above conditions become In case that r = 2(s + 1), but q = 2(s + 1), condition (3.2) is satisfied whenever there exists a positive constant 1 such that This last condition can be satisfied whenever d 2 < 8ρk(s + 1).If s = 0, the above condition becomes d 2 < 8ρk.When q = 2(s + 1), but r = 2(s + 1), condition (3.2) is satisfied whenever there exists a positive constant 2 such that It is satisfied whenever d 2 μ 2(s+1) 2(s+1) < 8ρk(s + 1).If s = 0, the condition is d 2 μ 2 2 < 8ρk.Another family of examples corresponds to the case when A sufficient condition to guarantee that condition (3.2) is satisfied could be the existence of three positive constants 1 , 2 , and 3 such that 1 2 3 = 1 and In the particular case that s = 1, p = 4, and r = 4, the previous conditions say In case that d 4 < 4 5 ρ 2 k 2 , we can select i satisfying the previous conditions.We consider now s = 0, so r = 2.If there are two positive constants 2 and 3 such that 2 3 = 1 and 2 2 ≤ 4/d, 2 3 < 2ρk/d, then condition (3.2) holds.Thus, we obtain the condition d 2 < 8ρk.We note that it is the same condition as that corresponding to family (4.2) when s = 0, r = 2, and q = 2.

Nonexistence
Here, we analyze the consequence of strengthening conditions (3.1) and (3.2).Thus, we assume that condition (3.1) is satisfied and the inequality is satisfied for ξ > 2. Concerning the dissipation function, we assume that where M. C. Leseduarte and R. Quintanilla 7 We define where F(t) is defined in (3.3).We have Gβ,t0 = F(t) + β. (5.4) In view of the energy equation, we have that (5.5) Schwarz's inequality leads to (5.7) Since ξ * > 2, the last inequality may be put in the form where (5.9) If we choose β such that ξE(0) + (ξ * − 1)β = 0, inequality (5.8) implies that .10)This shows that G −γ β,t0 is a concave function.We have We can always select t 0 sufficiently large to guarantee that the derivative of G β,t0 is positive at zero.This provides a lower bound to G β,t0 which becomes unbounded in finite time.
An integration of the inequality (5.10) also gives the estimate where we assume that the solution exists in 0 ≤ t ≤ T. From (5.12), we may deduce the uniqueness of the null solution, and also continuous dependence upon the initial data.

Nonexistence: discussion
In this section, we give some examples where conditions (5.1) and (5.2) are satisfied.In fact, we discuss when the examples proposed in Section 4 also satisfy conditions (5.1) and (5.2).First, we see that the family (4.1) with functions of type (4.2) satisfy condition (5.1) whenever p, q, and r are greater than two.Thus, in this section, we consider the function W defined in (4.1) when the parameters p, q, and r satisfy the previous condition.In this case, We can take ξ as near as we want to the minimum of p, q, r.If we assume that r = q = 2(s + 1), the assumptions are satisfied whenever there exist positive numbers λ < 1, ξ * > 2, 2 , and 3 such that In case that r = 2(s + 1), but q = 2(s + 1), condition (3.2) is satisfied whenever there exist two positive constants 2 , ξ * > 2 such that We note that if s = 0, then we have the condition Perhaps, it would be good to give an explicit example.For instance, let us assume that r = 3, s = 1/2, and q, p are greater than three.We can take ξ = 2.5.The condition is the existence of two positive constants 1 and ξ * > 2 such that We have proved the following.
Theorem 7.1 says that whenever the initial conditions are such that E(0) < 0, the solution grows exponentially.In view of the assumptions on constitutive tensors, we can obtain solutions with negative energy.
The quasistatic evolution of microvoids can be considered (see [10]).From a mathematical point of view, it corresponds to the case where we assume that k = 0 in (7.2).Then, we obtain system of (7.1) and the equation τ ν = A i j ν ,i j − B i j u i, j − ην.(7.14)As far as the authors know, this system has not been studied in the literature.However, it is easy to obtain uniqueness and exponential growth of solutions in a similar way as for the system (7.1)-(7.2).To prove uniqueness, we can define the function M. C. Leseduarte and R. Quintanilla 11 When we assume null initial conditions and boundary conditions (2.8), we see that the function J(t) satisfies the inequality (7.8),where ω = 0.This inequality is well known and it implies that (see [6]) for 0 ≤ t ≤ T. This estimate implies that J(t) = 0 whenever we assume null initial condition and then the uniqueness of solutions follows.Also, it is possible to prove the growth exponential of solutions of the problem determined by (7.2) and (7.14).To this end, we consider the function It is worth noting that in this case the energy of the system is given by E 1 (t) = 1 2 B ρ ui ui + C i jrs u i, j u s,r + 2B i j u i, j ν + A i j ν ,i ν , j + ην 2 + 2τ t 0 ( ν) 2 ds dv = E 1 (0). (7.18) The analysis proposed previously shows that J β,t0 (t) satisfies (7.8) and then the exponential growth of solutions is obtained whenever E 1 (0) < 0.

Uniqueness backward in time problem: linear case
In this section, we consider the problem determined by the backward in time problem for a linear porous elastic material.We continue with the assumptions proposed in Section 7 with respect to ρ, k, and τ, but we do not impose any condition neither on the positivity nor on the negativity of the internal energy.Thus, the internal energy has not a defined sign (which is the most general case).Thus we consider a homogeneous and centrosymmetric porous elastic material.In this case, the system of equations can be expressed as ρ üi = C i jrs u j,rs + B i j ν , j , ρkν = A i j ν ,i j − B i j u i, j − ην + τ ν. (8.1)To treat the uniqueness question, let (u i ,ν) be the difference of two solutions that satisfy the same initial and boundary conditions.Thus (u i ,ν) satisfies the system (8.1), with the boundary conditions (2.8) and the homogeneous initial conditions u i (x,0) = ν(x,0) = 0, on B. (8.2) that after a quadrature implies that Z(t) ≤ Z(0)exp(Ct).(8.12) As we assume homogeneous initial conditions, it follows that Z(0) = 0.Then, Z(t) = 0 for all time and we obtain the uniqueness of solutions.

. 1 )
Condition (3.1) is satisfied.To satisfy condition (3.2), let us consider functions of the type