Some results in a semi-linear thermoelasticity for a body with Cosserat structure

Purpose The purpose of this study is to approach thermoelastic media with Cosserat structure. The mixed initial-boundary value problems that the authors formulate in this context includes both nonlinear equations, with respect to temperature and the gradients of the displacement, and linear equations, with respect to thermal movements. Design/methodology/approach The authors designated a mathematical model based on motion’s equations, equation of energy by taking into account the mechanical and thermal effects. The authors approach a conservation law of the energy attached for any solution for the mixed problem formulated in this context. Another approach is to find the conditions in which the energy can no longer be bounded, in other words, the authors have an unstable equilibrium state. Then, the authors obtain the circumstances in which there is no solution to mixed problem which is well defined over a finite time interval. Findings In this paper the authors have the following findings. First, the authors formulate and prove a result regarding the conservation of energy. Then, the authors find the conditions in which they have an unstable equilibrium state, that is, the energy can no longer be bounded. Other finding of this study is to establish the conditions in which the mixed problem does not have a solution that can be defined over a finite time interval. Originality/value The novelty of this work lies in the consideration of semi-linear mixed problems in the context of thermoelastic media with the Cosserat structure. More exactly, in the mixed problems the considered equations are non-linear with respect to temperature and gradients of displacements and remain linear only with respect to the thermal displacement. Also, the authors consider the case where only the thermal effect is taking into account. In other words, the authors will neglect mechanical effects. In this situation, the authors are dealing with a nonlinear problem. For certain mixed problems, in this theory formulated, the authors state and demonstrate for such king of problems certain results with regard to the non-existence of solutions. Furthermore, for these problems the authors approach the problem of instability of solutions.