Mathematical Justification of the Obstacle Problem in the Case of a Shallow Shell

This paper deals with the asymptotic formulation and justification of a mechanical model for a shallow shell in frictionless unilateral contact with an obstacle. The first three parts of the paper concern the formulation of the equilibrium problem. Special attention is paid to the contact conditions, which are usual within two or three dimensional elasticity, but which are not so usual in shell theories. Lastly the limit problem is formulated in the main part of the paper and a convergence result is presented. Two points are worth stressing here. First, we point out that unlike classical bilateral shell models justifications, the functional framework of the present analysis involves cones. Secondly, while the cones result from a positivity condition on the boundary as long as the thickness parameter is finite, leading to a Signorini problem in the Sobolev space H ¹, the cone results from a positivity condition in the domain, giving rise to a so-called obstacle problem in the Sobolev space H ² at the limit.      

Regularity for the nonlinear Signorini problem

We study the regularity of the solution to a fully nonlinear version of the thin obstacle problem. In particular we prove that the solution is C1,α for some small α>0. This extends a result of Luis Caffarelli of 1979. Our proof relies on new estimates up to the boundary for fully nonlinear equations with Neumann boundary data, developed recently by the authors.     

Unilateral Contact with Coulomb Friction and Uncertain Input Data

Abstract

A quasivariational inequality (QVI) in R ^d , d = 2, 3, with perturbed input data is solved by means of a worst scenario (anti-optimization) approach, using a stability result for the solution set of perturbed QVI-problems. The theory is applied to the dual finite element formulation of the Signorini problem with Coulomb friction and uncertain coefficients of stress-strain law, friction, and loading.     

Asymptotic modeling of Signorini problem with Coulomb friction for a linearly elastostatic shallow shell

In 2002–2003, Paumier studied the Signorini problem with friction in the linear Kirchhoff–Love theory of plates using the convergence method. In 2008, Léger and Miara generalized this study to the case of linearized shallow shell but without friction. The purpose of this paper is to extend those results to the case of linearized shallow shell with a Coulomb friction law. Copyright © 2015 John Wiley & Sons, Ltd.     

An Euler–Bernoulli Beam with Dynamic Frictionless Contact: Penalty Approximation and Existence

In this work, we consider the dynamic frictionless Euler–Bernoulli equation with the Signorini contact conditions along the length of a thin beam. The existence of solutions is proved based on the penalty method. Employing energy functional with the penalty method, we bound integral of contact forces over space and time. Hölder continuity of the fundamental solution plays an important role in the convergence theory.     

Numerical analysis of a quasi-static contact problem for a thermoviscoelastic beam

In this paper we revisit a quasi-static contact problem of a thermoviscoelastic beam between two rigid obstacles which was recently studied in [1]. The variational problem leads to a coupled system, composed of an elliptic variational inequality for the vertical displacement and a linear variational equation for the temperature field. Then, its numerical resolution is considered, based on the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Error estimates are proved from which, under adequate regularity conditions, the linear convergence is derived. Finally, some numerical simulations are presented to show the accuracy of the algorithm and the behavior of the solution.     

Differential variational–hemivariational inequalities with application to contact mechanics

In this paper we study a dynamical system which consists of the Cauchy problem for a nonlinear evolution equation of first order coupled with a nonlinear time-dependent variational–hemivariational inequality with constraint in Banach spaces. The evolution equation is considered in the framework of evolution triple of spaces, and the inequality which involves both the convex and nonconvex potentials. We prove existence of solution by the Kakutani–Ky Fan fixed point theorem combined with the Minty formulation and the theory of hemivariational inequalities. We illustrate our findings by examining a nonlinear quasistatic elastic frictional contact problem for which we provide a result on existence of weak solution.