Department of Engineering

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 in macroscopic strain or stress space a microscopic (local) failure surface, which corresponds to the onset of the first bifurcation instability in the fundamental solution, for which all cells deform identically. One also defines a macroscopic (global) failure surface, which corresponds to the onset of the first long wavelength instability in the solid and whose determination requires the calculation of the homogenized macroscopic moduli tensor. By following all possible loading paths in strain or stress space, one can thus construct microscopic and macroscopic onset of failure surfaces for finitely strained, rate independent solids with arbitrary microstructures. The calculations required for the microscopic onset of failure surfaces 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 microgeometry and macroscopic loading conditions are presented and discussed.

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