Singular relaxation moduli and smoothing in three-dimensional viscoelasticity

Abstract

Abstract. We develop a semigroup setting for linear viscoelasticity in threedimensional space with tensor-valued relaxation modulus and give a criterion on the relaxation kernel for differentiability and analyticity of the solutions. The method is also extended to a simple problem in thermoviscoelasticity.

Well-Posedness and Stabilizability of a Viscoelastic Equation in Energy Space

“…The functions v and / in ( 1 ) take their values in an infinite-dimensional Hubert space X, and o takes its values in another infinite-dimensional Hubert space Y. The unbounded, closed, and densly denned operator D maps domD c X into Y, and its adjoint D* maps domD* c Y into X.…”

“…J-oo This is the formulation used in [4] (apart from the fact that there the viscosity v throughout was taken to be zero, and £ was throughout nonzero). If one computes an energy balance equation for (14) in the same way as we did for (1), then one gets an extra term Í£||D«(r)||2v-Í£||DM(0)||2v which has an obvious interpretation as the change of potential energy, due to the equilibrium elastic response of the material. The same term is in fact present in the previous computation, too, hidden in the double integral term.…”

“…The formulation simplifies, since there is no need to keep track of u (which represents position; recall that v stands for velocity), and some of the complications in [4] due to the fact that D may have a nontrivial kernel are avoided. The same formulation (1) that we use here is also used in [1] (except that there v = 0, and the kernel is allowed to be tensor-valued instead of scalar-valued).…”

“…In this paper we treat the well-posedness and stabilizability problems for abstract Volterra integrodifferential systems of the type v'(t) = -D'a(t) + f(t), (1) i" a(t) = vDv(t) + / a(t -s)Dv(s)ds, t > 0, J-oo in a Hubert space. These equations arise, for example, in the theory of linear viscoelasticity, in which case v represents velocity, v' acceleration, a stress, -D*o the divergence of the stress, v > 0 pure viscosity, Dv the time derivative of the strain, and a the linear (shear or tensile) stress relaxation modulus of the material.…”

“…The book [16] contains a nice short discussion of linear viscoelastic itv. Additional relevant references are, among others, [1], [2], [3], [8], [10], [11], [12], [14], and [17].…”