Dynamic frictional contact of a viscoelastic beam
In this paper, we study the dynamic frictional contact of a viscoelastic beam with a deformable obstacle. The beam is assumed to be situated horizontally and to move, in both horizontal and tangential directions, by the effect of applied forces. The left end of the beam is clamped and the right one is free. Its horizontal displacement is constrained because of the presence of a deformable obstacle, the so-called foundation, which is modelled by a normal compliance contact condition. The effect of the friction is included in the vertical motion of the free end, by using Tresca's law or Coulomb's law. In both cases, the variational formulation leads to a nonlinear variational equation for the horizontal displacement coupled with a nonlinear variational inequality for the vertical displacement. We recall an existence and uniqueness result. Then, by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives, a numerical scheme is proposed. Error estimates on the approximative solutions are derived. Numerical results demonstrate the application of the proposed algorithm.
FRICTIONAL CONTACT PROBLEMS WITH NORMAL COMPLIANCE
Abstract The quasistatic problem of a linear elastic body in unilateral contact with a rigid support subject to Coulomb's friction law and a power law normal compliance is considered. Two approximations are derived. It is proved that the first, the incremental problem, possesses a solution but it is unique only if it is small and the coefficient of friction is small enough. It is suitable for numerical calculations. The second, the rate problem, captures exactly the path dependence of the quasistatic problem, is more suitable for theoretical investigations and possesses a unique solution for “small” coefficients.
Mathematical programming and nonlinear finite element analysis
Abstract The paper is concerned with the application of mathematical programming techniques to a variety of structural plasticity problems. The incremental and finite elastoplastic analysis problems, as well as the plastic limit analysis problem, are formulated as a variety of linear complementarity, quadratic, linear and nonlinear programming problems. A unified computational scheme is presented, and comparisons are made between the mathematical programming techniques and the so-called “direct” methods for nonlinear finite element analysis.