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

Abstract

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. The goal of our two-part work is to extend lattice optimization to stress minimization problems two-dimensionally. The present first part is devoted to the choice of a parametrized periodicity cell that will be used for structural optimization in the second part of our work. In order to avoid stress concentration, we propose a square cell microstructure with a super-ellipsoidal hole instead of the standard rectangular hole often used for compliance minimization. This type of cell is parametrized two-dimensionally by one orientation angle, two semi-axis and a corner smoothing parameter. We first analyse their influence on the stress amplification factor by performing some numerical experiments. Second, we compute the optimal corner smoothing parameter for each possible microstructure and macroscopic stress. Then, we average (with specific weights) the optimal smoothing exponent with respect to the macroscopic stress. Finally, to validate the results, we compare our optimal super-ellipsoidal hole with the Vigdergauz microstructure which is known to be optimal for stress minimization in some special cases. This article is part of the theme issue ‘Topics in mathematical design of complex materials’.

Two-scale optimization of graded lattice structures respecting buckling on micro- and macroscale

“…The microscopic initial stress tensor e describes how the macroscopic strain ε is distributed in unit cell Y. We note that (18) already takes stress amplification (see Ferrer et al 2021) into account. The macroscopic strain is given as solution of the macroscopic constitutive equation…”

Inverse homogenization using the topological derivative

“…The inconvenience is that this process is computationally expensive since the cell problem should be solved for as many microstructures to characterize the parametric domain. See for example, Ferrer et al (2021).…”

Architecting materials for extremal stiffness, yield, and buckling strength

“…A few studies have focused on improving microstructure strength, including stress minimisation and buckling strength optimisation. It has been shown that the maximum von Mises stress can be reduced when taking stress into account during the microstructure optimisation procedure either as constraints or objectives (Collet et al, 2018;Coelho et al, 2019;Alacoque et al, 2021;Ferrer et al, 2021). The stress singularity issue, where the stress at a point approaches infinity as the density at that point approaches zero, was handled by stress relaxation methods (Cheng and Guo, 1997;Duysinx and Bendsøe, 1998;Bruggi, 2008;Le et al, 2010).…”