Weak and strong singularities reflecting multiscale damage: Micro-boundary conditions for free-free, fixed-fixed and free-fixed constraints

Abstract Analytical modeling of the multiscale character of the material microstructure has been problematic because of the inability of continuum mechanics to consider scale effects. The introduction of length parameters into theories that advocate the vanishing of element size are equally unsatisfying since they hardly contest the main issue of how the constituents at a very small scale can affect the behavior of the same material at a much larger scale. Microscopic effects can sometimes have a significant influence on the macroscopic behavior of material. It is therefore pertinent to have a model that can couple the microscopic effects to those at the macroscopic scale. A new scheme is no doubt needed to overcome the conventional approach adopted in continuum mechanics. The singularity representation scheme will be considered where the local damage at the different scales will be modeled by different orders of the stress singularities. Using the 1/r1/2 stress singularity as the reference, the weak and strong singularities will be used to model the degree of damage at the microscopic scale. The former and latter refer, respectively, to 0.5