Stress minimization for lattice structures. Part I: Micro-structure design

Lattice structures are periodic porous bodies which are becoming popular since they are a goodcompromise 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. The goal of our two-parts work is to extend lattice optimiza-tion to stress minimization problems in 2-d. The present first part is devoted to the choice of aparametrized periodicity cell that will be used for structural optimization in the second part ofour work. In order to avoid stress concentration, we propose a square cell micro-structure with asuper-ellipsoidal hole instead of the standard rectangular hole often used for compliance minimiza-tion. This type of cell is parametrized in 2-d by one orientation angle, two semi-axis and a cornersmoothing parameter. We first analyse their influence on the stress amplification factor by perform-ing some numerical experiments. Second, we compute the optimal corner smoothing parameter foreach possible micro-structure and macroscopic stress. Then, we average (with specific weights) theoptimal smoothing exponent with respect to the macroscopic stress. Finally, to validate the results,we compare our optimal super-ellipsoidal hole with the Vigdergauz micro-structure which is knownto be optimal for stress minimization in some special cases