An entropy-based evaluation of contact forces in continuum mechanics of elastic structures

• We extend entropy and its relevance in statistical physics, e.g. microsate, ensemble, and information, to evaluate the contact forces of elastic bodies, which are discretized with finite elements, some concepts in contact mechanics are considered as the same or similar meaning of that in statistical physics.
• We obtain an explicit probability distribution for the normalized contact forces by maximizing entropy subject to an expectation value and the principle of minimum potential energy.
• We construct an efficient iterative procedure by solving a series of isolated systems to obtain the contact forces, and give a novelty termination criteria to terminate the iteration.
• We also examine the iterative procedure by some examples to show its feasibility, efficiency and robustness.

The maximum entropy principle has proved to be a versatile tool for solving problems in many fields. In this paper, we extend entropy and its use in statistical physics to evaluate contact forces in the continuum mechanics of elastic structures. Each potential contact node of an elastic structure discretized by finite elements along with the normalized contact force on the node is considered as a system and all potential contact nodes together with their normalized contact forces are considered as a canonical ensemble, with the normalized contact force of each node representing the microstate of the node. The product of non-penetration conditions for potential contact nodes and the normalized nodal contact forces then act as an expectation that its value will be zero, and maximizing the entropy under the constraints of the expectation and the minimum potential energy principle results in an explicit probability distribution for the normalized contact forces that shows the relation between contact forces and displacements in a formulation similar to the formulation for particles occupying microstates in statistical physics. Moreover, an iterative procedure that solves a series of isolated systems to find the contact forces is presented, with a novel termination condition. Finally, some examples are examined to verify the correctness and efficiency of the procedure.