The onset of cavitation for the equations of polyconvex elasticity
Athanasios Tzavaras (University of Crete)
Crack initiation and growth: methods, applications, and challenges
Tue 2:40 - 4:00
Sayles Auditorium
In this talk I will review how the method of relative entropy may be used in order to show how certain spring-mass lattice systems approximate the equations of elasticity in the regime before the formation of shocks. In the second part we will review certain recent results on the onset o cavitation in dynamic elasticity. Cavitation refers to the opening of holes during the dynamic deformation of an elastic material and is a phenomenon that lies at the boundary of continuum modeling. Nevertheless, it has been successfully predicted at the level of continuum elasticity models both in the static case (by J. Ball) as well as in the dynamic case (in works of Pericak-Spector and Spector). The latter work on the one hand provides a solution of a dynamically opening cavity originating from a homogeneously deformed state, and on the other hand it poses challenges to the existence theory of multi-d conservation laws, as the constructed cavitating solution turns out to decrease the mechanical energy. We will analyze this example from the perspective of a theory that accounts for the singular layers of the cavitating solution (the resulting notion of solution is called singular limiting induced from continuum solution). This perspective accounts for the cost of energy associated with producing the cavity, and the energy of the cavitating solution is larger than that of a homogeneously deformed state.