David E. Stewart

The proof in “Energy balance for viscoelastic bodies in frictionless contact” of an energy balance has a crucial flaw that renders the proof incorrect. At present no correction for this flaw is known.

In this paper it is shown that the change in the energy for a linearly viscoelastic body (with Kelvin–Voigt type viscosity) in frictionless contact with a rigid obstacle can be accounted for by viscous losses and the work done by external forces. Thus there is no change in the energy due to impacts, unlike the case of rigid-body dynamics. The result can be extended to a wide class of dynamic viscoelastic boundary thin obstacle problems of similar type.

Two coupled partial differential equations which describe the motion of a viscoelastic (Kelvin-Voigt type) Timoshenko beam are formulated with the complementarity conditions. This dynamic impact problem is considered a boundary thin obstacle problem. The existence of solutions is proved. A major concern is to pursue an investigation into conservation of energy (or energy balance), which is performed both theoretically and numerically.

Convolution complementarity problems (CCPs) have the following form: given a matrix-valued function k and a vector-valued function q, find a vector-valued function u satisfying 0 ≤ u(t) ⊥ (k*u)(t) + q(t) ≥ 0 for all t. In this paper CCPs are applied to a mechanical impact problem, but they can also be applied to other dynamic problems with hard constraints. CCPs are shown to have solutions provided q(0) ≥ 0 and q is sufficiently regular, k has locally bounded variation and k(0⁺) is a P-matrix. Uniqueness also holds provided, in addition, k(0⁺) is symmetric positive definite. This theory shows that the impact problem studied here has a unique solution, and that energy is conserved. Numerical methods have been devised and implemented for the impact problem, and the results are presented.