Numerical study of a thermo-elasto-viscoplastic contact problem with adhesion using a hybrid method

A dynamic frictionless contact problem between an elasto-visco-plastic body and a deformable obstacle is numerically studied in this paper. The contact is modeled using the normal compliance contact condition involving adhesion effect of contact surfaces and the thermo-elasto-visco-plastic constitutive law is employed to simulate the thermal effects. The adhesion of the contact surfaces is considered and is modeled with a surface variable, the bonding field, whose evolution is described by a first order differential equation. We provide variational formulation which can be put into a general model of system defined by a second order evolution equality, a first order evolution equation and a first order differential equation. The existence and the uniqueness of the weak solution were established. The proofs are based on evolution equations with monotone operators, differential equations and fixed point theorem. The applied approach leads to a system involving variational equalities and we focuse on the study of fully discrete approximation scheme where both the spatial and temporal variables are discretized. The finite difference method and finite element method are popular numerical methods to discretize problems involving ordinary and partial differential equations. In this work, we will use the first-order finite differences to approximate time derivatives. Moreover, our discussion will be limited to the case of uniform partitions of the time interval and we will use the finite elements for spatial discretizations of variational equalities. For a numerical scheme, we prove the existence and uniqueness of the solutions, and derive optimal order error estimates under certain regularity assumption on the solution of the continuous problem. The additional assumptions on the solution regularity are needed for optimal order error estimates and may not always be valid. Nevertheless, weaker solution regularity assumptions can be used to derive error estimates with lower convergence orders.

• thermo-elasto-viscoplasticity
• frictionless contact
• adhesion
• weak solution existence and uniqueness
• fully discrete approximation
• error estimates
• monotone operators
• variational inequalities
• hybrid algorithm

• Contact Mechanics and Variational Inequalities
• Adhesion, Friction, and Surface Interactions
• Advanced Numerical Methods in Computational Mathematics
• thermo-elasto-viscoplasticity
• frictionless contact
• adhesion
• weak solution existence and uniqueness
• fully discrete approximation
• error estimates
• monotone operators
• variational inequalities
• Computational Theory and Mathematics
• Mechanics of Materials
• Computational Mechanics
• Computer Science
• Engineering