Stabilization in elastic solids with voids

In this paper we study the asymptotic behavior and the analyticity of the solutions of the one-dimensional porous-elasticity problem with thermal effect. Our main result is to prove the lack of exponential stability in case of the porous-elasticity with thermal effect when viscoelasticity is present. We prove the analyticity of the problem when a porous viscosity is present. We conclude by showing the impossibility of localization in time of the solutions in the isothermal case.     

On the time polynomial decay in elastic solids with voids

In this paper we investigate the temporal decay behavior of the solutions of the one-dimensional problem in various theories of continua with voids. It has been proved that the coupling of the elastic structure with porous microstructure is weak in the sense that in many situations the temporal decay of solutions is slow. We have considered some theories of porous continua when the deformation-rate tensor or time-rate or porosity function or thermal effects is present. We have proved that the decay cannot be controlled by a negative exponential. The natural question now is whether there exist or not a polynomial rate of decay of the solution in some appropriate norms. In this paper we consider some cases where the decay is slow and we obtain polynomial decay estimates. In concrete we consider the case when only the viscoelastic effect is present, the case when the motion of voids is assumed to be quasi-static and the porous viscosity is present and we finish with the case of the porous-elasticity when thermal effect is coupled.     

On the time decay of solutions in porous-elasticity with quasi-static microvoids

In this paper we investigate the temporal asymptotic behavior of the solutions of the one-dimensional porous-elasticity problem with porous dissipation when the motion of microvoids is assumed to be quasi-static. This question has been recently studied in the general dynamical case. Thus, the natural question is to know if the assumption of quasi-static motion for the microvoids implies significant differences in the behavior of the solutions from the results obtained in the general dynamical case. It is worth noting that this assumption involves a qualitative change in the system of equations to be analyzed because it arises from the combination of a parabolic equation with an hyperbolic one, rather different from the well-known system of the thermo-elastic problem. First, we study the coupling of elasticity with porosity and we show that if only porous dissipation is present, the decay of solutions is slow, but if viscoelasticity is added, then the solutions decay exponentially. After that, we introduce thermal effects in the system and we show that while temperature brings exponential stability to the solutions, microtemperature does not.     

On the rate of decay in interacting continua with memory

This paper is concerning the linear theory of isothermal interacting continua with memory. We consider anti-plane shear deformations in a mixture of two elastic solids where the dissipation mechanisms can be the viscosity in one of the components and the viscosity with respect to the relative displacement. We have seen that when the only dissipation mechanism applies on the relative displacement we cannot expect the exponential decay for the solutions. We have also analyzed the case when the viscosity mechanism applies on a constituent. We have seen that generically the decay is of exponential type. However if the coupling constitutive parameter vanishes the decay is slow.     

On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary

We study the asymptotic behavior of the solution of a 3D hyperbolic system arising in the Green-Naghdi models of thermoelasticity of type II and III with a dissipative boundary condition for the displacement and prove that the energy exponentially decays in time.     

Exponential decay in nonsimple thermoelasticity of type III

This paper deals with the model proposed for nonsimple materials with heat conduction of type III. We analyze first the general system of equations, determine the behavior of its solutions with respect to the time, and show that the semigroup associated with the system is not analytic. Two limiting cases of the model are studied later. Copyright © 2015 John Wiley & Sons, Ltd.     

General decay of solutions in one-dimensional porous-elastic system with memory

In this work we consider a one-dimensional porous-elastic system with memory effects. It is well-known that porous-elastic system with a single dissipation mechanism lacks exponential decay. In contrary, we prove that the unique dissipation given by the memory term is strong enough to exponentially stabilize the system, depending on the kernel of the memory term and the wave speeds of the system. In fact, we prove a general decay result, for which exponential and polynomial decay results are special cases. Our result is new and improves previous results in the literature.     

On the decay of solutions for porous-elastic systems with history

In this paper we study the asymptotic behavior to an one-dimensional porous-elasticity problem with history. We show the lack of exponential stability when the porous dissipation or the elastic dissipation is absent. And we show the lack of analyticity and exponential stability when the porous viscosity and the elastic dissipation are present.     

Asymptotic stability of semigroups associated with linear weak dissipative systems with memory

We study the asymptotic behavior of the solutions of a class of linear dissipative integral differential equations. We show in the abstract setting a necessary and sufficient condition to get an exponential decay of the solution. In the case of the lack of exponential decay, we find the polynomial rate of decay of the solution. Some examples are given.     

Strong solvability of 3-D Cahn–Hilliard system in elastic solids

In this paper we prove the existence and uniqueness of a strong solution to a 3-D model of phase separation in elastic solids. The model has the form of an initial-boundary-value problem for a nonlinear coupled system of hyperbolic–parabolic type. The key idea of the proof is based on the analysis of the system once- and twice differentiated with respect to time variable. The paper develops results of the previous work (Top. Meth. Nonlin. Anal., in press). Copyright © 2007 John Wiley & Sons, Ltd.     

Harmonic decomposition analysis of contact mechanics of bonded layered elastic solids

The paper presents an iterative method for obtaining footprint, pressure distribution, local deformation and sub-surface stress field for the contact between a rigid cylindrical indenter and an elastic flat substrate. The methodology is applicable for semi-infinite, as well as for thin or thick bonded elastic layered solids with high or low elastic moduli. All findings are in accord with the observed behaviour of hard wear resistant and soft solid lubricating coatings. It is shown that the decomposed contact pressure distribution into a series of harmonic waves induces sub-surface stress fields that decay into the depth of the solid according to their wavelengths. Consequently, conditions vis-à-vis fatigue spalling and adhesion performance may be predicted for given thickness of layered bonded elastic solids.     

On pathwise super-exponential decay rates of solutions of scalar nonlinear stochastic differential equations

This paper studies the pathwise asymptotic stability of the zero solution of scalar stochastic differential equation of Itô type. In particular, we provide conditions for solutions to converge to zero at a given rate, which is faster than any exponential rate of decay. The results completely classify the rates of decay of many parameterised families of stochastic differential equations.     

Effective elastic moduli of two-dimensional solids with multiple cracks

An accurate method which directly accounts for the interactions between different microcracks is used for analyzing the elastic problem of multiple cracks solids. The effective elastic moduli for randomly oriented cracks and parallel cracks are evaluated for the representative volume element (RVE) with microcracks in infinite media. The numerical results are compared with those from various micromechanics models and experimental data. These results show that the present method is simple and provides a direct and efficient approach to dealing with elastic solids containing multiple cracks. Project supported by the National Natural Science Foundation of China (Grant No. 19704100), and the National Natural Science Foundation of Chinese Academy of Sciences (Project KJ951-1-201).     

On interface waves in misoriented pre-stressed incompressible elastic solids

Some relationships, fundamental to the resolution of interfacewave problems, are presented. These equations allow for thederivation of explicit secular equations for problems involvingwaves localized near the plane boundary of anisotropic elastichalf-spaces, such as Rayleigh, Scholte, or Stoneley waves. Theyare obtained rapidly, without recourse to the Stroh formalism.As an application, the problems of Stoneley wave propagationand of interface stability for misaligned predeformed incompressiblehalfspaces are treated. The upper and lower half-spaces aremade of the same material, subject to the same prestress, andare rigidly bonded along a common principal plane. The principalaxes in this plane do not, however, coincide, and the wave propagationis studied in the direction of the bisectrix of the angle betweena principal axis of the upper half-space and a principal axisof the lower half-space.     

Exponential decay of the solutions for nonlinear multiharmonic equations

The purpose of this paper is to establish the exponential decay properties of the solutions for the nonlinear multiharmonic equation

where the condition plays the role of a boundary value condition.     

A spatial decay in the linear theory of microstretch piezoelectricity

The electromagnetic theory of microstretch elasticity is an adequate tool to describe the behavior of porous bodies, animal bones and solids with deformable microstructures. In this paper we study the linear theory of microstretch piezoelectricity. First, we establish a spatial decay estimate. Then, we obtain an upper bound for the amplitude term.     

广义静态梁方程非负解的存在性

运用Leray—Schauder拓扑理论,证明了广义静态梁方程和静态梁方程非负解的存在性,仅要求非线性项f在原点的某个邻域满足一定的符号条件,突破了以往对非线性项f的增长性限制．所获结果对工程设计具有重要的理论意义和实用价值．     

General decay of solutions of a wave equation with memory term and acoustic boundary condition

In this paper, the global solvability to the mixed problem involving the wave equation with memory term and acoustic boundary conditions for non‐locally reacting boundary is considered. Moreover, the general decay of the energy functionality is established by the techniques of Messaoudi. Copyright © 2016 John Wiley & Sons, Ltd.     

Energy decay of solutions to Maxwell's equations with conductivity and polarization

In this paper, Maxwell's equations involving generally nonlinear polarization and field-dependent currents are studied. The main objective is the asymptotic behavior of the solution for t→∞ if no damping term occurs in the equation governing the polarization field.