Approximating thermo-viscoelastic heating of largely strained solid rubber components

Mechanically induced viscoelastic dissipation is difficult to compute when the constitutive model is defined by history integrals. The computation of the viscous energy dissipated is in the form of a double convolution integral. In this study, we present a method to approximate the dissipation for constitutive models in history integral form that represent Maxwell-like materials. The dissipation is obtained without directly computing the double convolution integral. The approximation requires that the total stress can be separated into elastic and viscous components, and that the relaxation form of the constitutive law is defined with a Prony series. A numerical approach often taken to approximate a history integral involves interpolating the history integral's kernel across a time step. Integration then yields finite difference equations for the evolution of the viscous stresses in time. In the case when the material is modeled with a Prony series, the form of these finite difference equations is similar to the form of the finite difference equations for a Maxwell solid. Since the dissipation rate in a Maxwell solid can be easily computed from knowledge of its viscous stress and the Prony series constants (spring-dashpot constants), we computationally investigated employing a Maxwell solid's dissipation function to couple thermal and large strain history integral based finite element models of solid rubber components. Numerical data is provided to support this analogy and to help understand its limitations. A rubber cylinder with an imbedded steel disk is dynamically loaded, and the non-uniform heating within the cylinder is computed.