Nonlinear composites with hierarchical microgeometries of the self-consistent type

Abstract

Multiphase composites are constructed in such a way that their nonlinear viscoplastic behavior realizes the self-consistent approximation when the behavior of the constituent phases is linearly viscous. Their construction combines sequential laminations of infinite rank with a self-consistent prescription for the homogenized dissipation potential of a uniform reference medium. Specific calculations are carried out assuming that all phases are incompressible, isotropic, exhibit power-law dissipation with the same strain-rate sensitivity, and are arranged with transversely isotropic symmetry about an axis. Predictions for the effective in-plane flow stress are found to satisfy pertinent bounds and to agree identically with exact weak-contrast expansions to second order, suggesting their realizability by some class of hierarchical microgeometries in the strong sense. When one of the phases is void, the effective flow stress displays trends akin to percolative behavior. Interestingly, the critical exponent is identically equal to the strain-rate sensitivity of the phases, and yet bounds are satisfied because the critical threshold varies with the strain-rate sensitivity concomitantly. Possible implications for nonlinear self-consistent approximations based on variational linearizations are contemplated.