Stress minimization for lattice structures. Part I: Micro-structure design
Résumé
Lattice structures are periodic porous bodies which are becoming popular since they are a good compromise between rigidity and weight and can be built by additive manufacturing techniques. Their optimization has recently attracted some attention, based on the homogenization method, mostly for compliance minimization [1], [2], [3]. The goal of the present two-part work is to extend this lattice optimization to an objective function involving stress minimization. As is well known in structural optimization, stress optimization is a very dicult problem. While the second part of our work will be devoted to the macroscopic optimization process itself, the present rst part is devoted to the choice of a parametrized periodicity cell that will be optimally selected in the second part of this work. Designing the right periodicity cell is of paramount importance for the success of the optimization process. For manufacturability reasons it is crucial that this cell is parametrized by just a few parameters. According to homogenization theory, one has to compute the eective elasticity tensor, as well as the corrector terms accounting for possible stress concentrations at the cell or microscopic scale. For compliance minimization a standard choice in 2-d is a square cell with a rectangular hole, or a rank-2 laminate. However, since these microstructures feature corners, they are not optimal for stress minimization. Therefore we propose a square cell with a super-ellipsoidal hole which exhibits no corners. This type of cell is parametrized in 2-d by one orientation angle, two semi-axis and an exponent in its dening equation which can be interpreted as a corner smoothing parameter. We rst analyse the inuence of these parameters on the stress norm by performing some numerical experiments. Second, the optimal corner smoothing parameter is found for each possible micro-structure and macroscopic stress. In order to obtain an optimal micro-structure that depends only on geometrical parameters and not on the stress value, we further average (with specic weights) the optimal smoothing exponent with respect to the macroscopic stress. For simplicity, the optimal values of the corner smoothing parameter are interpolated by an analytical approximated formula. Finally, to validate the results, we compare our optimal super-ellipsoidal hole with the Vigdergauz micro-structure which is known to be optimal for stress minimization in some special cases.
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