Variational and numerical analysis of the Signorini′s contact problem in viscoplasticity with damage

Abstract

We consider the quasistatic Signorini's contact problem with damage for elastic-viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution of the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

Complete damage in elastic and viscoelastic media and its energetics

“…Note that, like the elastic response, it is natural to assume that also the viscous response depends on the damage f and vanishes in the completely damaged. This substantially differs from previous studies [16,25,43,49] which considered viscosity unchanged even in damaged material. Like in [43,49], we also consider inertia related to the mass density q.…”

“…Combination with viscosity has been addressed in the Maxwellian rheology (even with plasticity) in [16] and in the Kelvin-Voigt rheology in [25,43,49,10].…”

Analysis of a one‐dimensional damage model

“…We note that (4.1) is a variational inequality of the first kind which was solved using a penalty-duality algorithm introduced in [24] and already applied for solving other contact problems ( [16,17,25]). The numerical scheme was implemented on a standard UNIX IBM RISC6000 workstation using FORTRAN 77 and a typical run took less than 5 minutes of CPU time.…”

“…Viscoelastic contact problems with damage were studied in many recent papers (see, e.g., [11][12][13][14], the monograph [15] and references therein); meanwhile the damage problem with viscoplastic materials and different contact conditions were considered in [16,17].…”

Frictional Contact Problem With Wear for Thermo-Viscoelastic Materials With Damage

“…Using the subdifferential of indicator function of the interval [0, 1] guarantees that the damage function ς, which measures the decrease in the load bearing capacity of the material, varies between 0 and 1; when ς = 1 the material has its full capacity; wvhen ς = 0 it is completely damaged, and if ς = 1, the material is partially damaged. Because of the importance of the subject, three-dimensional problems which include this approach to material damage have been investigated recently in [7,12,13] This paper is organized as follows. In Section 2, we present the notation and some preliminaries.…”