Optimal long-time $L_p}(0,T)$ stability and semidiscrete error estimates for the Volterra formulation of the linear quasistatic viscoelasticity problem

Abstract

The purpose of this article is to show how the solution of the linear quasistatic (compressible) viscoelasticity problem, written in Volterra form with fading memory, may be sharply bounded in terms of the data if certain physically reasonable assumptions are satisfied. The bounds are derived by making precise assumptions on the memory term which then make it possible to avoid the Gronwall inequality, and use instead a comparison theorem which is more sensitive to the physics of the problem. Once the data-stability estimates are established we apply the technique also to deriving a priori error bounds for semidiscrete finite element approximations. Our bounds are derived for viscoelastic solids and fluids under the small strain assumption in terms of the eigenvalues of a certain matrix derived from the stress relaxation tensor. For isotropic materials we can be explicit about the form of these bounds, while for the general case we give a formula for their computation.

Negative norm error control for second-kind convolution Volterra equations

“…Kernel functions of this type are appropriate to viscoelasticity problems, and using the methods described in [11] we have,…”

“…In general a priori estimates for S will derive from Gronwall's inequality and will be exponentially large, but for viscoelasticity problems with fading memory optimal estimates have been given in [11]. We will exploit this sharper result later when we come to the numerical examples.…”

“…In these problems a convolution kernel with fading memory is appropriate, and sharp stability estimates can be derived by exploiting the fading memory. In this direction see [11].…”

Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity

“…This assumption is realistic and, in particular, allows for a much sharper analysis for the time dependence than does the usual Gronwall lemma. We refer to [18] for further details, and simply note here that a prototype for ϕ is,…”

“…where we assume from Korn's inequality (see for example Friedrichs [4] or Horgan [6]) that A is a (self-adjoint) V -elliptic operator, with B(t, s) similar (see [18]). If each D ij kl ∈ W 1 1 (J ; L ∞ ( )) as well as, for example, f ∈ L ∞ (J ; V ) and g ∈ L ∞ (J ; ∂V ) (where ∂V is an appropriate space of traces), then the existence and uniqueness of a solution u ∈ L ∞ (J ; V ) follows from the Riesz representation theorem and the theory of Volterra equations (e.g.…”