Collisions and fractures: a predictive theory

We investigate collisions of solids which can fracture. Equations of motion and constitutive laws provide a predictive theory. Assuming the collision as instantaneous, the equations of motion are derived from the principle of virtual work introducing new interior forces which describe the very large stresses and the very large contact forces resulting from the kinematic incompatibilities. They are interior volume percussion stresses and interior surface percussions both on the unknown fractures and on the colliding surface. In order to approximate these equations, we assume solids are damageable. In this point of view, it results that velocity is continuous with respect to space but its strain rate is very large in a thin region where the material is completely damaged, so approximating a fracture. When the velocity before collision is very large, the damaged zone may be large accounting for parts of the solid completely transformed into powder. The constitutive laws result from dissipative functions satisfying the second law of thermodynamics and able to model the fracturation phenomenon at the macroscopic engineering level. Representative numerical examples confirm that the model accounts for the fracturation qualitative properties.