Publisher Summary Multifield theories indicate the wide range of models in which some graphic fields must be introduced to describe the influence of material substructures on the gross mechanical behavior of solids. Solidification of metal alloys and their possible shape memory, damage states and evolution, and the influence of long chains of macromolecules on the behavior of polymers can be successfully analyzed with the help of multifield theories. Models with internal variables can be derived from multifield theories by appropriate internal constraints. The chapter focuses on the multifield description of continua that is a flexible framework to study physical situations in which the analysis of substructures is important for both practical and theoretical reasons. The chapter discusses the configurations and deduction of balance equations for bodies lacking in discontinuity surfaces, the behavior of the substructure, the influence of the substructure on the axial decay of energy in linear elastic cylinders, the derivation of thermomechanical balance at discontinuity surfaces, constitutive restrictions arising from a mechanical version, the second law of thermodynamics and the measures of substructural interactions, the analysis of the influence of substructures on configurational forces that drive the evolution of interfaces, the evaluation of the influence of material substructures on macrocrack propagation, the case where the substructure becomes latent in presence of appropriate internal constraints, and the application of the general theory to special cases.

Multifield theories in mechanics of solids

In this paper we consider a theory of viscoelastic mixtures proposed in Iesan (2004). In this theory the dissipation effects are determined by the viscosity of rate type of a constituent and the relative velocity. We state the linear equations of the thermomechanical deformations. Then, we obtain the suitable framework where the linear problem of thermomechanical deformations of viscoelastic mixtures is well posed. Then, we study several suitable conditions to guarantee the exponential stability of solutions. As we want to emphasize where the mechanical damping is sufficient to guarantee the exponential decay, we restrict our attention to the isothermal question. First, we prove that when both dissipative mechanisms are present the solutions decay exponentially. Later, we concentrate on the case where only one of the dissipative mechanisms is present. As this question seems a little more cumbersome we restrict our attention to an easier problem. As anti-plane shear deformations are among the simplest strains that a material can undergo, we concentrate our study on this kind of deformations. We prove that generically the solutions decay exponentially, but we note that there exist some examples where purely elastic (not damped) solutions can exist.

Existence and exponential decay in the linear theory of viscoelastic mixtures

A general model of the equations of generalized thermo–microstretch for a homogeneous isotropic elastic half–space is given The modulus of elasticity is taken as a linear function of reference temperature The formulation is applied to generalized thermoelasticity theories, the Lord–Shulman and Green–Lindsay theories, as well as the classical dynamical coupled theory The normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses, temperature, couple stresses and microstress distribution The variations of the considered variables through the horizontal distance are illustrated graphically A comparison is made with the results predicted by the three theories in case of temperature–independent modulus of elasticity

Generalized thermo-microstretch elastic medium with temperature dependent properties for different theories

Purpose – The present paper aims to study the effect of rotation on the general model of the equations of generalized thermo‐microstretch for a homogeneous isotropic elastic half‐space solid whose surface is subjected to a thermal shock is considered. The problem is in the context of the generalized thermoelasticity Lord‐Shulman's theory with one relaxation time, as well as the classical dynamical coupled theory.Design/methodology/approach – The normal mode analysis is used to obtain the exact expressions.Findings – For the displacement components, force stresses, temperature, couple stresses and microstress distribution.Originality/value – The variations of the considered variables through the horizontal distance are illustrated graphically. Comparisons are made with the results in the presence and absence of rotation and in the presence and absence of microstretch constants between the two theories.

Effect of rotation on plane waves in generalized thermo‐microstretch elastic solid with one relaxation time

Reflection and transmission of waves from a plane interface between two microstretch solid half-spaces

Anti-plane shear deformations of swelling porous elastic soils

In this paper, we investigate the spatial decay of some diffusion-reaction equations. In the stationary case we obtain results of exponential decay. In the transient case, we find a decay which is as fast as the exponential of a quadratic polynomial. The main tool of the paper is the maximum principle together with the usual methods of decay estimates for linear problems. We also obtain an estimate for the amplitude term by means of the boundary conditions.

On the spatial decay of solutions for a class of diffusion-reaction equations

Existence, uniqueness, and continuous dependence of solutions to the evolution equations that govern small thermoelastic deformations superposed on a general nonlinear thermomechanical deformation with nonuniform temperature is obtained. In our case, the domain occupied by the body may be unbounded and the elasticity tensor may be strongly elliptic. The problem is solved by means of the semigroup theory.

Existence of solutions of the equations of incremental thermoelasticity for unbounded domains

We prove continuous dependence on the intensity coefficient and
continuous dependence on the external data in the theory of
magneto-elasticity. We do not require the Lame coefficients to
be positive. We use logarithmic convexity arguments similar to
those of Ames and Straughan (1992) in classical thermoelasticity.

/pdf/continuous-dependence-of-solutions-in-magneto-elasticity-pe8rif3a31.pdf

Continuous dependence of solutions in magneto-elasticity theory

Some qualitative results for the linear theory of thermo-microstretch elastic solids

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