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

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 < m < 1 and another group of strong singularities for 1 < m < 1.5 while the displacements at the crack tip r = 0 are still kept finite, a requirement for a regular solution within the context of elastic continuum mechanics. It is one of the objectives of this work to explore the dW/dV behavior for these micro-scale singularities arising from a micro-notch with different bluntness in conjunction with the effects of varying the applied load and boundary constraint on the micro-notch edges. The macro- and micro-effects are connected by application of the stress and displacement compatibility conditions. The former is in fact related to dimensionality consideration while the latter to connecting the macro-crack tip region to that of the micro-crack via a mesoscopic zone. Two distinct behavior of dW/dV were found at the macro- and micro-scale. As the micro-notch bluntness changes, the macro-dW/dV curves tend to translate while the micro-dW/dV curves tend to rotate. The amount of translation and rotation depends on the type of loading and the ways with which the micro-notch edges are constrained. The macro-meso-micro-interactions are coupled. Discontinuities of the dW/dV are seen via the meso-zone which could have been more dramatic without the introduction of the restraining stress or meso-zone. Elaboration on smoothing out the connection is of secondary concern in comparison with the basic idea of the multiscale model. Further improvement of the model is believed to lie in applying the scale invariant criterion to results for two successive scale ranges. This will also involve defining the degree of system homogeneity, a fundamental issue in scale shifting.