A mathematical model of damage accumulation taking into account microstructural effects

It is shown that the influence of microstructure in the damage accumulation process leads to a nonlinear diffusion effect, with a strongly stress-dependent diffusion coefficient. A nonlinear parabolic equation with a source term is obtained for the damage parameter. This equation is relevant to blow-up and quenching problems well known to mathematicians with rupture corresponding to blow-up or quenching. However, the damage accumulation equation possesses an additional nonlinearity due to the non-healing of damage. Depending on the value of a dimensionless constant parameter (the ratio of a properly defined microstructural length-size to a characteristic length-size of the initial damage distribution), two essentially different types of damage accumulation process appear to be possible for a given initial damage distribution over the bar length. In processes of the first type, the damage accumulation remains non-homogeneous over the length of the bar, so that the lifetime for the whole specimen is determined by the maximal initial damage within the bar. For processes of the second type the damage distribution over the specimen at first becomes homogeneous (at least in a considerable part of specimen), and then the damage accumulation proceeds uniformly over all or part of the specimen. The lifetime for processes of the second type is essentially longer than the first. Results of a numerical experiment based on the proposed model are presented. In particular, the origin and development of damage propagation waves is demonstrated. Also, it is demonstrated that when there is substantial damage transfer, the ultimate value of the damage parameter in the life-time calculation is of no significance.