Collisions and fractures: a model in SBD (Q2568760)

This paper investigates collisions, assumed to be instantaneous, and fractures of 3D solids. A collision is characterized by a time discontinuity of the velocity field. A fracture, resulting from the collision, is found by a spatial discontinuity of the velocity field after the collision. The equations of motion are derived by the principle of virtual work at collision time, which includes interior percussions, accounting for very large stresses and acceleration forces related to the kinematic incompatibilities. The constitutive laws are derived by use of dissipative potentials in coherence with the second law of thermodynamics, i. e. to satisfy the Clausius-Duhem inequality. The equations for the velocity after collision are derived by the principle of virtual work combined with the constitutive relations and the expression of dissipative potentials. A variational formulation of this problem is considered, written in a suitable space of discontinuous and sufficiently smooth velocities. By this, the weak formulation is stated of a minimization problem associated with the functional corresponding to the above problem. The variational problem is solved in the space of special functions of bounded deformations (SBD). Then, the authors state a unified weak formulation of the impenetrability condition on the fractures and on the contact boundary of the solid, colliding with a fixed obstacle. Finally, the minimization problem is stated in the convex subset of smooth kinematically admissible velocities (SCV). Then, a theorem is formulated which states existence of a solution for the minimization problem. To prove this theorem, the authors use the direct method of calculus of variations, which is based on compactness and lower semicontinuity arguments. A minimization sequence is considered for a functional in SCV. At the first step it is shown that the functional is coercive in SBD. Then, authors prove a weak compactness result holding in SBD and allowing to pass to the limit in a suitable weak sense in such a way that considered functional turns out to be lower semicontinuous. The compactnes and lower semicontinuity results exploit two theorems for rather general dissipative potentials. The main idea of the proof is to reduce the formulation to 1D sections of SBD functions (i. e. investigate the velocity field restricted to fixed directions in space), then apply the results known for SBD in 1D domains, and finally integrate these informations on all directions. To complete the proof of the existence theorem, it is shown that the velocity field satisfies the impenetrability condition in the extended domain.