Existence of solutions for time-dependent Signorini-type problems in linearised viscoelasticity
Paolo Piersanti
TL;DR
This work addresses the existence of solutions for a time-dependent Signorini-type obstacle problem in linearised viscoelasticity, where a 3D body is confined to a half-space. The authors introduce a novel solution concept that accommodates initial contact with the obstacle and develop a two-layer approximation (penalization of the constraint and Galerkin discretization) to construct solutions. A key technical contribution is the compactness of the time-dependent trace operator, enabling compactness-based passage to the limit via Aubin-Lions-Simon and Minty-Browder arguments. Under additional material assumptions, they show that when body forces are removed the viscoelastic body relaxes back to its rest configuration at exponential rate, consistent with Kelvin-Voigt behavior.
Abstract
In this paper we establish the existence of solutions for a model describing the evolution of a linearly viscoelastic body which is constrained to remain confined in a prescribed half-space. The confinement condition under consideration is of Signorini type, and is given over the boundary of the linearly viscoelastic body under consideration. We show that one such variational problem admits solutions and we coin a novel concept of solution which, differently from the available literature, is valid even in the case where the viscoelastic body starts its motion in contact with the obstacle. Additionally, under additional assumptions on the constituting material, we show that when the applied body force is lifted the deformed linearly viscoelastic body returns to its rest position at an exponential rate of decay.