In this seminar I present an explicit expression for the kinetic and elastic energy associated to the dynamic evolution of an elastic material with a prescribed smooth crack, in which the fracture lips open vertically (anti-plane case). This computation has a crucial role in view of the so-called energy-dissipation balance which underlies the dynamics: the energy released during the evolution and the energy dissipated to create the fracture (the latter proportional to the crack surface increment) balance the work done by the external loads. I first characterize the singularity of the displacement near the crack tip; then I derive an explicit expression for the kinetic and elastic energy. Finally I provide necessary and sufficient conditions to guarantee the validity of the energy--dissipation balance.
This is a joint work with I. Lucardesi and E. Tasso.