A micromechanical cracking imperfection model of edge dislocations generated by residual stresses

The nucleation of crystals is a highly non-homogeneous process. It depends on the number of nucleation sites, the rate of cooling, initial imperfections in the molten liquid and other variables. These inhomogeneities not only vary from location to location but they give rise to non-uniform residual stresses throughout the solid. The existence of initial microcracks and dislocations in a polycrystal is the rule rather than the exception. Knowledge of the micro-dislocational initial defects can be important since they can affect the ensuing service stress and strain states, particularly when the size of the system is in microns. The objective of this work is to develop a micro-atomic multiscale damage model for estimating the initial damage during processing of the material. In what follows, an effective analytical solution will be found for the specific case of a microcrack generating edge dislocations under in-plane shear loading. Cross-scale characteristics will be determined although discontinuities cannot be avoided if equilibrium mechanics theories are to be used for the range of microscopic and atomic scales. The unique idea of a "scale multiplier" is used to connect the microcrack and edge dislocations. The transition from microscopic to atomic can be smoothed out by introducing additional meso zones that would further divide the scale ranges into smaller segments. The process involves labor. Since it will not entail new physical insights, elaboration will not be made until the effort can be justified by the need. It is more important to illustrate the method for treating multiscale damage. Numerical results are obtained and displayed graphically for quantities of interest. The sliding displacement attains its maximum value at the end of the segment where a uniform residual stress is assumed to prevail. Edge dislocations are therefore generated. And they reach a maximum at a distance of about one third of the length of the segment from the point at which residual stresses are locked into the system. The location of this peak shifts with the magnitude of the residual stress. The decay and cross scaling effects of the strain energy density functions are also exhibited. They are relevant for discussing potential failure. In addition, these findings are expected to alter if the residual stresses are non-uniform. Such situations will depend on the specific application. The exact shapes of the microcrack and edge dislocation are depicted figuratively; they are not the important issues. Their singular characteristics, represented by the 1/r(0.5) and 1/r, however, are essential to show the difference in strength of local intensification with r being the distance from the singular points. The relevant quantities are the energy level and characteristic length associated with the micro-defect or crack and the atomic-defect or dislocation. Hence, there is no loss in generality from using the line crack-edge dislocation model for considering the effect of residual stresses. Since the corresponding problem for generating screw dislocation has already been considered in previous work, the addition of edge dislocations can be used to generate a more general solution of line dislocations in a plane under the combined effects of in-plane and out-of-plane shear action. Even though this work considers only the case of uniform residual stress trapped ahead of a microcrack, other situations of non-uniform residual stresses can be obtained by superposition.