Phase‐Field Optimization of Elasto‐Plastic Contact Structures

ABSTRACT The paper is concerned with the analysis as well as the numerical solution of the topology optimization problems for elasto‐plastic rather than elastic structures in unilateral frictional contact with a rigid foundation. The contact phenomenon is governed by the system of two coupled variational inequalities. The small strain plasticity model with linear hardening and a von Mises effective stress is used. The material density function is chosen as a design variable. The topology optimization problem consists of finding such material distribution of the domain occupied by the body in contact to minimize the contact stress and to ensure the uniform distribution of this stress. Using the regularization of the stress projection operator on the set of admissible generalized stresses as well as of the friction functional, the original contact problem is formulated in terms of two coupled nonlinear elliptic PDEs depending on the regularization parameter. The existence of solutions is shown. The topology optimization problem for the structure in contact is formulated in terms of the phase field approach. The relation between sharp‐interface and phase‐field optimization problems is investigated. The cost functional derivative with respect to the design variable is calculated. The Lagrange multiplier technique is used to formulate the set of necessary optimality conditions. Gradient flow approach in the form of modified Cahn–Hilliard boundary value problem is used to compute the optimal topology domain. Mixed finite element approximation of modified Cahn–Hilliard problem is used. The examples of minimal contact stress topologies are provided and discussed.