Numerical analysis of evolution problems in nonlinear small strains elastoviscoplasticity

Abstract

Summary. The present paper deals with the mathematical and numerical analysis of evolution problems in nonlinear small strains viscoelasticity of Burger's type. After a brief review of the mechanical model, the viscoelastic problem to be solved is written as an abstract evolution problem. The associated operator is proved to be maximal monotone, thus implying existence and uniqueness of solutions. This problem is then solved numerically by a backward Euler discretization in time, a finite element approximation in space and by using a preconditioned conjugate gradient algorithm for solving the resulting nonlinear algebraic systems. Numerical results are finally presented to illustrate the solution procedure.

Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl—Reuss model of elastoplasticity

“…A thorough formulation of the Prandtl-Reuss model in BD(Ω) has recently been provided in [32]. In [13,14] existence and uniqueness of the solution in the framework of Hilbert spaces were proved for a more general constitutive law than (1.2) (see also [61]). The construction via analogical models is the basis of the Prandtl-Ishlinskiȋ models of stop-type, which extend the Prandtl-Reuss model and have been studied in the theory of hysteresis (see, for example, [24,59,60,100]).…”

“…Proof. For p = q = 2 the procedure of [13,14] might be applied. Here we use a different argument, which also provides uniform estimates that will be used in the homogenization procedure.…”

A System of Ordinary and Partial Differential Equations Describing Creep Behaviour of Thin-Walled Shells

“…(These internal variables are, of course, the functions d and EC7.) Mathematical models involving internal variables and pertaining to bulk materials were considered in [12,13,29,31,34,35,37,39,45,47,541 . These references essentially deal with constitutive relations which lead to initial-boundary value problems of the type w 1 + C(w) = 0, where C is a monotone operator.…”

“…Note that if (8,1111 8 ) is a Banach space, then fi . IIB, is a norm, and C°([0,T], 13) equipped with this norm is also a Banach space. for u E WkP(B).…”

Homogenization of nonlinear visco-elastic composites

“…[1,6,23,24,34,39,41,42,[47][48][49]58,64]. A model of visco-elasticity was coupled with the equation of continuum dynamics, e.g., in [13,14,49]. Nonlinear elasticity may appropriately be represented in the framework of the finite-strain theory, see e.g.…”