Triple scale segmentation of non-equilibrium system simulated by macro–micro-atomic line model with mesoscopic transitions

Abstract

The concept of restraining stress ahead of a macro-crack was first applied to the development of a dual scale line crack model. The local stress intensity is said to be lowered due to the restraint represented by the opposing intensity of the strip zone, the size of which determines the amount of restraint. The description used for this effect may have been referred to as plastic flow, energy dissipation region, or cohesive stress zone. They, however, encounter conceptual difficulties when scales are gradually reduced. A general interpretation is needed that can be applied to both the macroscopic, microscopic and lower size scales. This restraining stress is passive. Its exact nature is not known except it decreases with increasing degradation of the material quantified by its zone size. While the foregoing assumption is clear, the resulting crack behavior still depends on available energy ahead of the crack that is loading specific. This energy can decrease or increase with damage and react with the restraint in a complicated manner. In macroscopic terms, the combined reaction of load with material has escaped the attention of the analysts except when the size is enlarged to that of a full scale structure. This is a particular concern for the behavioral properties of matter at the atomic scale. Experimental evidence of this behavior can be seen from the different dislocation patterns emitted from a macro-crack tip when the macroscopic loading is altered in magnitude and/or type. When the energy input is localized to the crack tip, less and less energy would reach the crack front as the damage extends. This situation may represent the release of initial or residual stresses trapped in the microstructure. The local damage would settle to an equilibrium position once dissipated. On the other hand, if a crack is engulfed in a far remote applied stress field, it would experience continuous input energy and damage. The restraint in the material would be overshadowed. These features will be reflected by the results of the present triple scale line segment crack/dislocation model. Similar trends of the local stress intensity may be caused by different physical mechanisms. Decrease of stress intensification at the macro-crack due to microscopic and atomic effects are shown to be second and third order effects. The influence of dislocations to the micro-crack can be first/second order. All imperfections considered take the configuration of lines. They can have different sizes stretching from the atomic or smaller scale to that of the macroscopic scale. More precisely, three sections are considered. They consist of the macro-, micro- and dislocation-segments. Two mesoscopic zones are introduced to smooth out the transitions where the scale range is shifted. Each scale segment are defined arbitrarily as sub-atomic to nano from 10 −11 to 10 −7 cm, nano to micro from 10 −7 to 10 −4 cm and micro to macro from 10 −4 to 10 −1 cm. These segments as they stand are too coarse. Mathematically, they can be made arbitrarily small to permit the application of equilibrium mechanics with sufficient accuracy. For the simplicity of illustration, discontinuities on the average are assumed to occur at 10 −7 and 10 −4 cm in lineal dimension along the line segment for a fixed half length of the macro-crack a = 10 mm and micro-crack g = 1 mm. The line size of dislocation depends on the number of dislocation generated. It is an unknown to be determined. The length scale is sufficiently long for testing the sensitivity of the model. The properties of the material at the different scales are distinguished although the terminologies adopted for those at lower scales leave much to be desired. Under anti-plane shear, only the shear modulus shows up. Judicious values of the ratios of μ macro/ μ micro and μ micro/ μ disl are selected from a knowledge of previous atomic simulation calculations. In dependent checks, however, are implemented from the positiveness requirements imposed on the stress intensity factors and the scale multipliers that connect scale sensitive quantities from one scale to another. These conditions are used to determine the range of the free parameters. Indeed, the choice is limited. In retrospect, the numerical results represent the direct consequence of the underlying physical assumptions. The objective is to achieve “consistency” in multiscaling such that the sensitivity of parameters at the atomic level for affecting macroscopic behavior can be checked. The basic idea used in the cut-and-patch procedure of line segment model can be extended to imperfections in the forms of points, areas and volumes. Patching of the segmented scales can appeal to the use of a scale invariant criterion, a topic for future considerations.