Gradient concepts for evolution of damage
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While low-order measures of damage have sufficed to describe the stiffness of bodies with distributed voids or cracks, such as the void volume fraction or the crack density tensor of Vakulenko, A.A., Kachanov, M., 1971., addressing the growth of distributed defects demands a more comprehensive description of the details of defect configuration and size distribution. Moreover, interaction of defects over multiple length scales necessitates a methodology to sort out the change of internal structure associated with these scales. To extend the internal state variable approach to evolution, we introduce the notion of multiple scales at which first and second nearest-neighbor effects of nonlocal character are significant, similar to homogenization theory. Further, we introduce the concept of a cutoff radius for nonlocal action associated with a representative volume element (RVE), which exhibits statistical homogeneity of the evolution, and flux of damage gradients averaged over multiple subvolumes. In this way, we enable a local description at length scales below the RVE. The mean mesoscale gradient is introduced to reflect systematic differences in size distribution and position of damage entities in the evolution process. When such a RVE cannot be defined, the evolution is inherently statistically inhomogeneous at all scales of reasonable dimension, and the concept of macroscale gradients of internal variables is the only recourse besides micromechanics. Based on a series of finite element calculations involving evolution of 2D cracks in brittle elastica arranged in random periodic arrays, we examine the evolution of the mean mesoscale gradients and note some preliminary implications for the utility of such an approach.