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188.8.131.52 Damage Measures
For elastic damage, we may use the same functional measure as proposed for plastic yielding (e.g., the von Mises stress); this implies that the functions and have the same form. For plastic damage, experimental results show that during plastic flow damage is coupled with measures of plastic strain , necessitating a (pseudo-)strain-based plastic damage measure . For yielded bonds in a bond family , we can use the constitutively-determined mapping to define plastic right Cauchy-Green and Lagrange strain tensors through One possible constitutive relation for , which remains valid for general deformations, is to set it equal to the effective plastic strain for the various bond families , In a numerical implementation, the effective plastic strain of the first bond family to yield may be reported as the effective plastic strain in the entire material, for consistency with plastic strain measures in classical models of plasticity.
Quantities in this section do not represent plastic strains or plastic strain tensors, though we adopt the terminology due to similarities. Recall that the non-observable function of state is a time-invariant mapping providing the reference configuration of a yielded bond with respect to the reference configuration of the master constituent , for family . The quantities and then also represent non-observable functions of state calculated as strain tensors. Consequently, is a measure of the relative motion of the reference configuration of bond family , expressed as a scalar “strain”. Physically, this amounts to the modeling assumption that once the breaking-and-reforming process takes a bond family out of a local neighborhood centered about its original position, the bond begins to degrade with further breaking-and-reforming processes. That each of these quantities exists for every bond family emphasizes the lack of any true or unique plastic strain measure in this framework.