Existence and exponential decay in the linear theory of viscoelastic mixtures

In this paper we consider a theory of viscoelastic mixtures proposed in Ieşan (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.