Ángel Rodríguez-Arós

A class of integro-differential variational inequalities with applications to viscoelastic contact

We consider a class of abstract evolutionary variational inequalities arising in the study of frictional contact problems for linear viscoelastic materials with long-term memory. First, we prove an abstract existence and uniqueness result, by using arguments of evolutionary variational inequalities and Banachs fixed-point theorem. Next, we study the dependence of the solution on the memory term and derive a convergence result. Then, we consider a contact problem to which the abstract results apply. The problem models a quasistatic process, the contact is bilateral and the friction is modeled with Trescas law. We prove the existence of a unique weak solution to the model and we provide the mechanical interpretation of the corresponding convergence result. Finally, we extend these results to the study of a number of quasistatic frictional problems for linear viscoelastic materials with long-term memory.

Numerical analysis of a frictional contact problem for viscoelastic materials with long-term memory

We consider a mathematical model which describes the bilateral contact between a deformable body and an obstacle. The process is quasistatic, the material is assumed to be viscoelastic with long memory and the friction is modeled with Tresca’s law. The problem has a unique weak solution. Here we study spatially semi-discrete and fully discrete schemes using finite differences and finite elements. We show the convergence of the schemes under the basic solution regularity and we derive order error estimates. Finally, we present an algorithm for the numerical realization and simulations for a two-dimensional test problem.

A Contact Model for Piezoelectric Beams

We consider a mathematical model which describes the equilibrium of an electro-elastic beam in contact with an electrically conductive foundation. The model is constructed by coupling the beam equation with the one dimensional piezoelectricity system obtained in [13]. We state the unique weak solvability of the model as well as the continuous dependence of the weak solution with respect to the data. We also introduce a discrete scheme for which we perform the numerical analysis, including convergence and error estimates results. Finally, we present numerical simulations in the study of a test problem.