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

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 < λ∗(or γ∗) < 1.0 and 0 < λ∗(or γ∗) < 0.5. Of immediate interest are the lowest minimum eigen-values λ∗ for symmetric loading and γ∗ for skew-symmetric loading. Physically, the different orders of the stress singularities may be related to the different constraint associated with the damage. If the damage is in the form of a micro-notch, the eigen-values will depend on the boundary conditions on the notch edges and notch angles. In contrast to the conventional approach of using complex constitutive relations to reflect nonlinearity at the lower scales, the singularity representation scheme can simplify the description between stress and strain. This is accomplished by shifting the scale of analysis to a lower order such that linearity is retained. The results at the different scales are then connected by the appropriate application of scale invariant criteria. The total energy and force quantities can be used. To illustrate the scheme of the method, a multiscale model involving macro-, meso- and micro-damage will be presented where the degree of damage will be reflected by the orders of the stress singularities. They are determined from the stress and displacement conditions on the boundary of the defect at the macro-, meso- and micro-scale which are coupled. The sizes of damage at the different scales are determined by solving a system of nonlinear equations. Additional damage size, say the nano-scale, may also be added to the model, when relevant, but this effort will be left for future consideration.