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Optimal long-time $$L_p(0,T)$$ stability and semidiscrete error estimates for the Volterra formulation of the linear quasistatic viscoelasticity problem. (English) Zbl 1002.74020
Authors’ abstract: We show how the solution of 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 makes it possible to avoid the Gronwall inequality and to use a comparison theorem which is more sensitive to the physics of the problem. Once data-stability estimates are established, we apply our technique also to derive a priori error bounds for semidiscrete finite element approximations. Our bounds are derived for viscoelastic solids and fluids under small strain assumption in terms of eigenvalues of a certain matrix derived from the stress relaxation tensor. For isotropic materials we give the form of these bounds explicitly, while in the general case we give a formula for their computation.

##### MSC:
 74D05 Linear constitutive equations for materials with memory 74H40 Long-time behavior of solutions for dynamical problems in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics
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