Evolution of growth of interacting near-by micro-cavities and macroscopic damage amplification

Abstract

The elastic stress in the ligament between two near-by holes (two-dimensional cylindrical cavities) depends singularly on the ligament thickness small parameter ζ = δ/R (where δ is the ligament thickness and R the hole radius), as ζ becomes vanishingly small, with the exponent a of ζ depending on the loading and the geometry. The damage of a solid containing a periodic distribution of pairs of near-by holes is treated within the theory of asymptotic homogenization in a two-scale model. By renormalization of the unit cell with the scale y = x/ε, where ε is the periodicity length between the pairs of near-by holes (a characteristic scale of the microstructure), the leading term in the singular amplification of the stress in the ligament between the holes is carried analytically to the macro-scale. The evolution of the growth of the holes (in the micro-scale) is governed by Eshelby mechanics (in this case the Budiansky and Rice path-independent M integral), which can be considered as a dissipation mechanism as the defects evolve. In the unit cell, the rate of change of energy in the (self-similar) growth of the holes balances the rate of change of the strain energy in the volume as the ligament thickness decreases, and this, by the renormalization, is carried to the macroscopic level and defined as the energy-release rate of “damage” at that point. In the strain hardening regime, the amplification of the damage is determined for two cases of distribution of pairs of interacting near-by small holes near large ones, with the larger (singular as ~ζ^−1/2) damage amplification occurring for a pressurized hole near a large unpressurized one, and being larger than the one when the loading is tension at infinity, for the same geometry.