Damage of materials: damaging effects of macroscopic vanishing motions

We investigate a mechanical model describing the evolution of damage in elastic and viscoelastic materials. The state variables are macroscopic deformations and the volume fraction of sound material. The equilibrium equations are recovered by refining the principle of virtual power including also microscopic forces. After proving an existence and uniqueness result for a regularized problem, we investigate the behaviour of solutions, in the case when a vanishing sequence of external forces is applied. By use of a rigorous asymptotics analysis, we show that macroscopic deformations can disappear at the limit, but their damaging effect remains in the equation describing the evolution of damage at a microscopic level. Moreover, it is proved that the balance of the energy is satisfied at the limit.