Unique solvability of a crack problem with Signorini-type and Tresca friction conditions in a linearized elastodynamic body

Abstract

We consider dynamic motion of a linearized elastic body with a crack subject to a modified contact law, which we call the Signorini contact condition of dynamic type , and to the Tresca friction condition. Whereas the modified contact law involves both displacement and velocity, it formally includes the usual non-penetration condition as a special case. We prove that there exists a unique strong solution to this model. It is remarkable that not only existence but also uniqueness is obtained and that no viscosity term that serves as a parabolic regularization is added in our model. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

Non-smooth variational problems and applications

“…By this, nonlinear boundary conditions are of the first importance for physically consistent modelling. The theme issue studies the unilateral contact appearing for inclusions subject to delamination with cracks [ 2 , 3 ], non-smooth slip [ 9 ], frictional contact [ 4 , 13 ] and degenerating Robin-type transmission conditions for the thin reactive heat-conducting inter-phases [ 11 ]. The rate-independent evolutionary systems driven by non-convex energies have been suggested [ 10 ], which are successful to model properly jump discontinuities in time during quasi-brittle crack propagation.…”

“…The rate-independent evolutionary systems driven by non-convex energies have been suggested [ 10 ], which are successful to model properly jump discontinuities in time during quasi-brittle crack propagation. The elaborated mathematical and mechanical description gives an impact to practise testing methodologies by rigid punch indentation [ 5 ], in fracture mechanics and seismology [ 13 ].…”