Cyclic approximation of the heat equation in finite strains for the heat build-up problem of rubber

It is well-known that rubber exhibits hysteretic mechanical behavior and has a low thermal conductivity. The main consequences are the heat generation and heat build-up phenomena which occur in a rubber component when subjected to repeated deformations. Estimating the heat build-up temperature implies the solution of a coupled thermomechanical problem. Due to the difference between the mechanical and the thermal diffusion characteristic times, a cyclic uncoupled approach is often used to solve the heat build-up problem. In the uncoupled approach, the heat sources are first determined with a mechanical analysis, and the heat equation is then solved on a fixed geometry. At finite strains, the geometry of the body varies with the deformation but the foregoing method does not account for such changes in geometry. The exact solution would require describing the body deformation while solving the thermal problem, but this does not take advantage of the difference between the characteristic times of the thermal diffusion and the mechanical behaviour, respectively, and the exact numerical resolution is therefore unnecessarily time-consuming. The purpose of the current work is to take into account kinematics in the thermal problem when using a cyclic uncoupled approach. The heat problem is written in the reference configuration. That implies that the problem is defined on a fixed domain: the initial configuration of the body. The changes in geometry in the reference heat equation are thus described by mechanical time-dependent variables. The cyclic assumption allows mean variables to be defined, for example the mean temperature. A time- integration method and an approximation of the heat equation are developed, leading to a simplified formulation with mechanical time-independent terms. This simplified heat problem is based on the mean variables.