Solving contact forces with the competition between potential energy and entropy in elastic mechanics

Abstract

Energy and entropy are competitors in thermodynamics, as an increase in one usually leads to a decrease in the other. In this article, we show that competition between potential energy and entropy also exists in contact problems, which is revealed in the process of solving contact forces in the finite element model of elastic contact problems based on the principle of minimum potential energy and the principle of maximum entropy. First, we discretize the contact problem with finite elements and model it as a quadratic programming problem with the potential energy as the objective and non-penetration conditions as the constraints. Second, by normalizing the Lagrange multipliers—the potential nodal contact forces—in the Lagrangian duality, we obtain an explicit formulation of the displacements as expressed by the normalized potential nodal contact forces. Third, analogous to the concepts in statistical physics, we consider each normalized potential nodal contact force as the probability of the node occupying a microstate, and with the maximum entropy principle, we obtain an explicit formulation of the normalized potential nodal contact forces as expressed by the displacements. Fourth, we construct an iterative procedure for solving the contact forces based on the previous two formulations. The initial value of each normalized nodal contact force, which is one divided by the number of potential contact nodes, is used to calculate the displacements in the first iteration, the iterations continue until a termination condition that is based on the monotonicity of the potential energy is met. Finally, we provide some examples to verify the iterative algorithm and the competition between the potential energy and entropy in the iterative procedure, which takes the form of a decrease in entropy and an increase in potential energy.