On the Realism of Complementarity Conditions in Rigid Body Collisions

Abstract

When nominally rigid bodies collide, impulse-momentum equations do not contain enough information to predict the outcome of the collision. The missing information is supplied by constitutive relations called impact laws. Since all bodies deform in impacts, 'rigid bodies' are taken as models for sufficiently stiff bodies. A popular modeling approach for simultaneous multiple impacts of 'rigid' bodies uses complementarity conditions at the velocity level. In the frictionless case, to which we restrict our attention, the complementarity conditions are that at each contact location, which is modeled using normal restitution e and which has a pre-collision normal relative approach velocity ViN, the normal contact impulse PN and the post-collision normal relative separation velocity VfN satisfy PN ≥ 0, VfN − eViN ≥ 0, and PN(VfN − eViN) = 0. The physical realism of the complementarity assumptions is investigated by studying a simple but arbitrarily stiff system theoretically for different deformation distribution regimes; a related experiment is also described. Unlike the case of noncollisional dynamics where the complementarity conditions at the acceleration level usually hold true, it is found that the complementarity conditions at the velocity level have no fundamental physical basis for collisions of even extremely stiff objects. Examples of two other well known systems are also given, where the complementarity conditions are clearly violated. These conditions should therefore be viewed as constitutive assumptions which, though algorithmically convenient, are often physically inaccurate.