Incompatible deformations in solids with hyperbolic thermoelasticity

The present study aims to develop a theory of incompatible deformations and evolutionary problems for media with hyperbolic heat conductivity. The specificity of the problem lies in the fact that, although it seems preferable toadopt Hamilton’s principle of least action as the fundamental postulate, this cannot be done in the present work, since thermal effects are taken into account, and therefore dissipation is present. For this reason, we are forced to immediately postulate a series of balance equations, which themselves are in many ways already Noether symmetries, but with some extra terms corresponding to thermal effects. At the same time, it remains possible to use methods of differential geometry to model incompatible deformations. A local discharging hypothesis is formulated in this paper, which intuitively generalizes the procedure for partitioning a self-stressed body into infinitesimal fragments. These fragments then unload independently to a stress-free state. The globally self-stressed state is modeled by using Riemannian geometry. An evolution equation is proposed that characterizes the relationship between the rate of change of the Ricci tensor and the action of the temperature field, its primitive, and gradients.