ABSTRACT
In this chapter we consider two quasistatic contact problems for elasticviscoplastic materials. The contact is modeled with Signorini’s condition in the first problem, and with normal compliance in the second one. In both problems the adhesion of the contact surfaces, caused by glue, is taken into account. As in Chapters 4 and 5, the evolution of the bonding field is described by a first-order differential equation. We provide the variational formulation for the mechanical problems and prove the existence of the unique weak solution for each model. We also introduce and study a fully discrete scheme for the numerical solutions of the problem and, under suitable assumption on the solution regularity, we derive optimal order error estimates. Moreover, we prove that the solution of the Signorini problem can be obtained as the limit of the solutions of the problem with normal compliance when the stiffness coefficient of the foundation becomes infinite.
We assume that in the reference configuration the body is in adhesive contact with the obstacle over Γ3. The process is quasistatic, and therefore the inertial terms are neglected in the equation of motion. We use (1.11) (page 8) as the constitutive law, (1.38) (page 16) as the tangential boundary condition on the contact surface Γ3 and Equation (1.42) (page 17) to describe the evolution of the bonding field. We denote by u0,σ0, and β0 the initial displacement, stress, and bonding fields, respectively.
