Failure Surfaces for Finitely Strained Two-Phase Periodic Solids Under General In-Plane Loading

For ductile solids with periodic microstructures (e.g., honeycombs, fiber-reinforced composites, cellular solids) which are loaded primarily in compression, their ultimate failure is related to the onset of a buckling mode. Consequently, for periodic solids of infinite extent, one can define as the onset of failure the first occurrence of a bifurcation in the fundamental solution, for which all cells deform identically. By following all possible loading paths in strain or stress space, one can construct onset-of-failure surfaces for finitely strained, rate-independent solids with arbitrary microstructures. The calculations required are based on a Bloch wave analysis on the deformed unit cell. The presentation of the general theory is followed by the description of a numerical algorithm which reduces the size of stability matrices by an order of magnitude, thus improving the computational efficiency for the case of continuum unit cells. The theory is subsequently applied to porous and particle-reinforced hyperelastic solids with circular inclusions of variable stiffness. The corresponding failure surfaces in strain-space, the wavelength of the instabilities, and their dependence on micro-geometry and macroscopic loading conditions are presented and discussed.