Scaling of volume energy density function reflecting damage by singularities at macro-, meso- and microscopic level

Abstract The 1/r singular field for the strain energy density function dW/dV is the same as that for the intermolecular attraction. In fact, it applies also to the law of gravitation for large bodies at the cosmos scale. The unique character of this function being independent of the specific material type has been used widely in fracture mechanics for analyzing the failure of material and structure when distinctions are made between the local and global behavior. When system homogeneity is preserved the product of dW/dV and the distance r is governed by a perfect hyperbola. This is the situation that prevails ahead of an idealized macro-crack. At the microscopic scale where heterogeneity is the rule rather than the exception, the dW/dV versus r relation will differ and can be denoted by (dW/dV)s · rm = f(x, y) that no longer obeys the hyperbolic relation in general. Here, the superscript “s” refers to the micro-scale. The former appearance of (dW/dV) · r = g(x, y) can still be retained if r is interpreted simply as the distance from the point of interest, say the crack tip. With reference to the stress state, where σ ∼ r−m/2, it can be seen that m = 1 corresponds to the familiar 1/r−1/2 singularity. Using it as a reference, there prevails a group of weak singularities for 0