Simultaneity of multiscaling for macro–meso–micro damage model represented by strong singularities

Abstract

The state of affairs near a crack tip is very different when viewed at the different scales, say from the macroscopic to the atomic. These different appearances have intrigued solid state physicists, mechanists and material scientists for decades. Although they are physically connected, it is not obvious what roles would their mathematical relationships play? Investigations associated with scaling in size and time, evolution of material damage, multiscaling of macro-, meso- and micro-mechanics theories are presumably topics that could provide insights to the behavior of the internal structures of matters and their interactions at the different scales. When seeking for analytical models that can address multiscaling, continuum mechanics seems to give way to the particle approach when the scale reduces to that for the atoms. It is also not clear whether the particle and continuum view points should be regarded as separate disciplines for their differences have generated much discussion with little to gain. There is the general feeling that a re-examination of what has been taken for granted may be in order. Based on the comments made earlier, a multiscale crack damage model is developed by duplicating the physics much like the capability of an optical or electronic microscope. The model contains the macroscopic, mesoscopic and microscopic damage represented by singularities. The results at the different scale levels are connected via stress and displacement compatibility conditions in continuum mechanics and they can switch from one scale to another in a discrete manner where equilibrium is assumed within the range of an arbitrarily defined scale. The sizes involved at the different scales depend on the material, load and geometry and they are determined by solving two highly non-linear equations. Simplicity is gained by using singularities to represent the geometrical discontinuities such as intergranular or transgranular failure. This discussion is limited to a particular group of material damage that can be modelled by the so-called strong singularities as compared to the r −0.5 singularity at the continuum level. The strong singularities would correspond to 0.25 < λ* < 0.5 where λ* = 0.5 corresponds to the singular stress behavior for a line crack in an elastic medium. The lowest real part of the eigenvalue λ is denoted by λ*. In passing, it should be pointed out that the 1/ r singularity (i.e. λ* = 0) corresponds to an edge dislocation which will not be included in this discussion. In order not to cluster the presentation with unnecessarily complex notations of distinguishing the difference between the macroscopic and microscopic constants, the same set of constants will be assumed. Besides, there are fundamental issues with reference to the definition of microscopic material properties whether they should be obtained from testing micro-size specimens or considered as local material properties as part of the bulk. These two contrasting views are not the same. The choice to use one or the other remains debatable. The emphasis here is placed on relating strong singularity group to the geometry of a micro-notch where one notch edge is free and the other is fixed. It is connected to a transition or mesoscopic zone. The notch angle can be varied to model different degrees of intergranular opening. The microscopic non-homogeneous features are reflected by the rotations of the lines with different notch angles in a log–log plot of the strain energy density function with distance. Keep in mind that when the same near crack tip region is viewed at the macroscopic scale, the state of affairs are homogeneous. This in essence is what a microscope does. The objective of this work is to demonstrate that such a capability can be built into an analytical model with all of the complexities of the microstructure, the description of which are made possible by the singularity representation approach.