Distribution effects in damage mechanics

Key issues pertaining to the development of viable damage evolution equations using a continuum damage mechanics (CDM) approach are addressed. Numerical simulations of evolving crack systems in two-dimensional perfectly brittle solids indicate that certain common damage models incorporating internal state variables (ISVs) may have difficulty in characterizing the effective moduli and damage evolution in brittle microcracked solids when the damage consists of cracks of variable size and/or spatial distributions. Statistical inhomogeneity of evolution may arise when damage interactions become significant or when use of certain low-order damage variables leads to response functions related to evolution that have an intrinsic flaw size dependence. Application of CDM to the statistically inhomogeneous case is problematic, but may minimally require the introduction of additional ISVs that account for the effect of the damage distribution within a representative volume element (RVE) on the macroscale response.;An argument for implementing ISVs that characterize the damage distribution within the RVE used for stiffness determination is presented. A form for higher-order ISVs based upon the gradient of the mesoscale damage distribution is proposed. An approximate method is suggested for calculating the higher-order ISVs using an RVE subvolume averaging procedure; the macroscale constitutive equations, while including terms that depend on multiple varying spatial scales, satisfy the principle of local action.;Numerical calculations suggest the higher-order ISVs may be used to distinguish between different mesoscale damage configurations leading to the same value of low-order damage parameter. Furthermore, the mean norm of the mesoscale gradient of damage appears to be a fairly linear function of the corresponding low-order damage variable. This potentially allows for the formulation of fairly elementary evolution equations for the higher-order ISVs. While still preliminary and requiring further development, use of higher-order ISVs that characterize the sub-RVE damage distribution may provide a framework for extending CDM to the statistically inhomogeneous case. Development of the structure of a nonlocal theory of CDM based on multiple scales of heterogeneity up to the order of the RVE dimension is discussed.