Minimization of maximal von Mises stress in porous composite microstructures using shape and topology optimization

Strength-oriented optimization of porous periodic microstructures impacts on efficient design of load-bearing lightweight structures avoiding mechanical failure. In this work, the maximal von-Mises stress, predicted by homogenization theory on a planar representative unit-cell domain, is minimized using either shape or topology design changes. Plane stress and linear behaviour are assumed. Two benchmarks problems are revisited, bulk and shear loads. Firstly, a fully stressed design is sought on extremal materials, rank-2 laminates, for comparative purposes. The lamination factors are handled analytically to find a relationship between stress and material volume fraction. Secondly, one numerically minimizes the peak von-Mises stress of single-material unit-cell varying shape or topology. In shear, the optimal topology design tends to approximate its rank-2 counterpart. Under bulk load, the peak stresses are further decreased by allowing an inhomogeneous solid phase. An extra material discrete phase is included in the shape problem while functionally graded material solutions are allowed in the topology problem. The single-material optimal results are consistent with the theoretical ones. This validates not only the proposed shape parameterization problem, based on supershapes, but also the proposed stress-based formulation for microstructural topology optimization, not yet extensively addressed in the literature. The multi-material approaches, to the extent of ideally allowing each spatial point to have a different material property, show that by increasing the material design freedom one achieves lower peak stresses.