The Statistical Property in Finite Element Model of Elastic Contact Problems
When the continuum elastic bodies in a contact problem are discretized by finite elements, we may look some of the nodes as an ensemble to use the concept or method of statistical physics to solve the mechanics problem. Each potential contact node of an elastic structure 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. Finally, an example is examined to verify the correctness and efficiency of the procedure.