Seminar on Partial Differential Equations

The talk treats mathematical aspects of evolutionary material models for shape-memory alloys at finite-strains. The difficulty of related mathematical analysis consists in the non-linear and non-convex dependence of the energy on the deformation gradient. One possible way, how maintain the analysis tractable, is to suppose that the energy depends also on the second deformation gradient and is convex in it. We relax this assumption by using the recently proposed concept of gradient-polyconvexity. Namely, we consider energies which are convex only in gradients of non-linear minors (i.e. cofactor and determinant in three dimension) of the deformation gradient. As a result, the whole second deformation gradient needs not to be integrable. Yet, at the same time, the obtained compactness is sufficient and, moreover, additional physically desirable properties(e.g. local invertibility) can be shown. We extend the previous result for hyperelastic materials by incorporating a rate-independent... more