Simultaneous occurrence of double micro/macro stress singularities for multiscale crack model

Abstract

It is rare to find two distinct stress singularities with different order at the same crack tip while the local displacements remained regular. Such a discovery was made when developing a dual scale micro/macro crack model and validated from a closed form asymptotic analytical solution for a micro/macro crack subjected to mixed boundary conditions. The two different orders of the stress singularities are 0.75 and 0.25 for an incompressible material. The existence of multi-singularities with different orders at the crack point is an unexplored area of research. For a sharp notch wedge, multi-stress singularities at the wedge point can prevail and they depend on the notch angle and Poisson’s ratio. Physical implications of such solutions when viewed at the microscopic scale are particularly significant as they tend to suggest that the multiple combinations of geometries and materials can yield so many different combinations of distortional and dilatational effects. No wonder microscopic irregularities seem to be the rule. Discontinuous numerically solutions, however, may or may not correspond to singularities. In view of the recent research emphases on multiscaling, the multiple singularity solution at a point seems to signify the inherent characteristics of micro-defects. They are irregular and defy self-similarity. The double singularity alone implies that a microcrack can extend in two different directions from the same location, known as branching in the very rapid propagation of a macrocrack. At the microscopic scale, double singularity implies the possibility that branching can occur under macroscopic static load. Such a behavior of static branching of microcracks associated with double singularity is being established. A fundamental aspect of this model is the inherent existence of restraining stress from the material that is essential for the microcrack and only sometimes for the macrocrack. As a rule, the microcrack being smaller would react to the material restraint more readily. Such a feature is necessary in the model in addition to the effects of micro/macro material interaction and the macrocrack characteristic length with reference to the macrocrack length. In normalized form, the minimum of three parameters, say μ ∗, σ ∗ and d ∗, would be necessary for the formulation of the dual scale micro/macro crack model. The combined effects of μ ∗ (micro/macro material), σ ∗ (load and restraint) and d ∗ (characteristic length) cannot be over emphasized in multiscale modeling. In the spirit of classical fracture mechanics, a micro/macro crack intensification factor K micro macro is defined and obtained in closed form. It contains three basic parameters μ ∗, d ∗ and σ ∗ being unique to all microcracks while two parameters ϕ 1 and ϕ 2 can be used to describe additional features of the microstructure. More precisely, the specific material microstructure can be modeled by two or more parameters associated with the eigenvalues selected rather than using the constitutive relations in the traditional manner. The asymptotic form of K micro macro is exact such that the case specific nature of microcracking can be distinguished by two constants. The contribution of this work is to establish the potential use of the dual singularity solution. The basic concept of the method applies to other multiscale problems in mechanics and physics.