Asymptotic micro-stress field dependency on mixed boundary conditions dictated by micro-structural asymmetry: Mode I macro-stress loading

Abstract

Fine line cracks are defined as closed with varying degrees of tightness in surface contact. Their profiles can be irregular depending on the material microstructure and load orientation. They will be referred to as microcracks that cannot be seen in contrast to macrocracks that are visible. There exists no strict definition of what is microscopic and what is macroscopic in view of the transitional region where one scale blends into another which can be referred to as the mesoscopic zone. Under alternating loads, microcracks can open and become macrocracks that can also close and revert back to microcracks. Allowance for switching scale size of cracks is a possibility that should not be excluded in the general development of crack models. Without limiting the discussion to a specific material microstructure, three normalized parameters μ micro/ μ macro, d/ a and σ o / σ ∞ are considered as essential. The shear moduli ratio μ micro/ μ macro is introduced to denote the micro/macro material stiffness difference while d represents the characteristic length distinguishing micro/macro effects. The restraining stress σ o normalized by the applied stress σ ∞ serves as an adjustment for the tightness of the closed microcrack. In the spirit of traditional fracture mechanics, a micro-stress intensity factor K I micro is used although its physical meanings differs fundamentally from the classical mode I macro-stress intensity factor K I macro . The former has been generalized to closed or open cracks. Hence, it can be microscopic or macroscopic. The same functional form of K I [ t ] has been found with [t] standing for the transitional process of micro → meso → macro. One of the objectives is to free the factor K I micro or K I macro from the bondage of being tied to a specific scale as a priori. Without loss in generality, a special microcrack model will be used for illustration. This corresponds to a micro-stress crack tip singularity of order r −0.75 which as a rule is stronger than the macro-stress singularity of r −0.5. The simple model has the ability to simulate more complex conditions and to show that microcracks have the inherent character of being non-self similar. Moreover, the direction of microcrack initiation can be shown to deviate from self-similarity and to take place off to the side even under mode I applied macro-stress. Hence, the model of a microcrack with a notch tip is also discussed since the present scheme can also be used to obtain closed form asymptotic solutions for the notch geometry under mixed boundary conditions. It should be emphasized that the interpretation of linearity and non-linearity will differ from that in the traditional theory of continuum mechanics. Under multiscaling where size and time scale can change, the range of each scale can be made sufficiently small to retain linearity. This will be equivalent to and consistent with the assumption of system homogeneity. Therefore, the use of linear elasticity can be justified at all scales as it has been used in atomic crystal lattice studies. Material non-linearity is interpreted as the reflection of lower scale effects in the current state being one scale level higher. If the current state is elastic then plasticity would be regarded as effect from a lower scale. The use of linear analyses in conjunction with mutliscaling, however, is achieved at the expense of segmentation. Connection of the results for the different segments is made by applying the concept of scale multiplier developed in early works.