Dual scaling damage model associated with weak singularity for macroscopic crack possessing a micro/mesoscopic notch tip

Multi-scale damage of material has been a concern in recent times owing to emphases of making devices in sizes smaller and smaller until the bulk average of the macroscopic material properties are no longer adequate while particle physics still lacks the expediency of engineering applications. There is the mesoscopic range between the microscopic and macroscopic size scale where the functional well being of engineering products must be addressed in terms of material damage. By in large, the only successful model developed during the early l960s and used up to now has been that of linear elastic fracture mechanics based on the concept of a single dominant macroscopic crack. There are apparent reasons still not understood why the simple model works so well while the refined theories, non-linear or otherwise, of the past half century or more have made insignificant contributions; they have been a disappointment, to say the least. There are signs to revive the atomistic models that have been attempted previously. The efforts were short lasted mainly because of the inability to connect the results with those at the macroscopic scale. It is not so much of the difference in size scale as the contrasting views of the particulate and continuum, a unsettled debate since the days of Aristotle and earlier. What ever it takes, it does appear that there is a need to extend the size scale of applicability for a given formulation where self-consistency is observed. With this in mind, the weak singularity approach is attempted in this work even for no better reasons other than being more satisfying than the piece meal empirical approach. One of the main objectives of this work is to describe the macroscopic and microscopic material damage ahead of a crack covering three to four orders of magnitudes of size effects in a single formulation that satisfies the continuum mechanics axioms with consistency. This includes the continuity of the crack opening displacements from the macro-scale to the micro-scale. More specifically, a micro-notch tip that can vary in shape and hence singularity prevails at the front of the macro-crack. The macro-to-micro geometric continuity provides not only the interacting effects for two scale levels but also a clue of the conditions under which cross scaling effects should be considered. Micro-crack blunting tends to elevate the macroscopic energy density field following a translational shift of the curves while the same effect at the tends to decrease the microscopic energy density field following a rotational shift of the curves. This behavior is shown to hold for both the symmetric and skew-symmetric loadings. The former refers to in-plane extension and latter to in-plane shear. Hence, the way with which material inhomogeneity affects crack tip behavior is not a simple matter of being close or far away from the site of potential failure. The combined effects of load, geometry and material will all contribute. More specifically, the stress state ahead of the main crack is found to be hydrostatic at the macroscopic scale but no so at the microscopic scale where the x- and y-component of the local normal stresses are not the same. This is because the micro-notch tip shape in this model can change shape. When the applied remote stress is in-plane shear, both the macroscopic and microscopic stress state near the main crack is hydrostatic. This implies that micro-cracking ahead of the main crack is more likely to occur under remote applied shear than remote applied normal stress, a result that has not been found or explained previously by analytical means. Hence, it might be useful for the practitioners to establish some rules of thumb in making decisions. The translational and rotational shifts of the energy density levels caused by different degree of micro-defect blunting can also serve a useful purpose for the development of new materials when considering the shapes of micro-defects in fabrication processes. The present approach of singularity representation emphases the need to put more weight on the influence of defect geometry such that non-linearity of constitutive relations can be relaxed as their emphases seem to have exceeded the contribution they are capable of offering.