Application of the material force method to thermo-hyperelasticity

The numerical analysis of material forces in the context of thermo-hyperelasticity constitutes the central topic of the present paper. In contrast to classical spatial forces in the sense of Newton, material forces in the sense of Eshelby indicate the tendency of material inhomogeneities to move relative to their surrounding material. Material forces are thus considered of particular importance in the context of thermo-elasticity where thermal effects can be understood as a potential source of inhomogeneity. The relevant balance equations of thermo-elasticity, i.e. the balance of momentum and energy, which essentially govern the evolution of the deformation and the temperature field are thus illustrated for both, the classical spatial and the material motion context. Guided by arguments of duality, the corresponding weak forms are derived. Next, we carry out the finite element discretization of both problems. While the numerical solution of the spatial motion problem renders the discrete spatial deformation map and the temperature as nodal degrees of freedom, the solution of the material motion problem provides the discrete material node point forces. The former typically relies on the solution of a global system of equations whereas the latter is introduced as a mere post-processing procedure. Since we apply a simultaneous solution of the mechanical and the thermal problem with the deformation and the temperature interpolated in a -continuous way, all the relevant information for the material force method is readily available once the spatial motion problem has been solved. Selected examples from the field of fracture mechanics illustrate the additional insight that is provided by the results of the material force method.