Optimization of periodic truss materials including constitutive symmetry constraints

Abstract

Periodic truss materials (PTM) -or Lattice Block Materials (LBM) -belong to the family of the so called ultralight cellular materials, which have attractive engineering properties such as high stiffness/weight ratio and high energy absorption capability. These materials are obtained by the periodic repetition of a unit cell given by a truss structure, so that the orientation of the bars and the cross sectional areas define the material9s macroscopic behavior. In recent years, the development of new process technologies for manufacturing PTMs have brought increased interest in modeling their structural and thermal behavior. In addition to usual analytical and numerical approaches for analyzing the effect of different unit cell configurations on macroscopic properties, optimization techniques have been applied in order to tailor unit cells for achieving desired macroscopic constitutive properties. The present contribution uses mathematical programming for tailoring PTMs through a number of optimization problems in which selected homogenized elastic or thermal constitutive properties are maximized, at the same time that isotropy tries to be enforced by the inclusion of adequate material symmetry constraints. Cross sectional areas and nodal coordinates of the bars are adopted as design variables. 2D examples are presented showing qualitative agreement with results available for 2D elasticity.

An Extended Multiscale Finite Element Method (EMsFEM) Analysis of Periodic Truss Metamaterials (PTMM) Designed by Asymptotic Homogenization

“…In this work, we use the EMsFEM to study structures made up of PTMMs that were optimized for maximum and minimum Poisson ratios using asymptotic homogenization (Guth et al, 2012). We verify that as asymptotic homogenization assumes an infinitesimal microscale (unit cell dimension) and is strain-driven, it is uncapable to detect important phenomena that can be naturally captured with EMsFEM.…”

“…Figures 1 (a) and (b) display, respectively, an example of a 3D unit PTM cell and the corresponding porous material obtained by its periodic repetition. (Muñoz-Rojas et al, 2010); (b) Corresponding material (Guth et al, 2012).…”

“…Muñoz-Rojas et al (2010) proposed the layout optimization (simultaneous cross sectional areas and coordinates as design variables) of truss-made unit cells for achieving optimum thermo-mechanical periodic truss metamaterials. Guth et al (2012Guth et al ( , 2015 optimized the layout of PTMMs including mechanical and thermal isotropic behavior as constraints in the optimization problem. All these works aimed at the optimization of material properties for a prescribed condition (shear stiffness, for instance) but they did not consider the pointwise response of a component made up of such material.…”

“…We organize this article in the following way: Sections 2 and 3 briefly review the theoretical bases of AH and EMsFEM for PTMs; Section 4 presents the unit cells previously obtained by Guth et al (2012) for extreme Poisson ratios; in Section 5 we show the EMsFEM numerical interpolation functions obtained for each of the unit cells presented in Section 4; in Section 6 we discuss numerical results comparing the AH and EMsFEM procedures. Finally, in Section 7 we close the article with concluding remarks.…”

Optimization of three-dimensional truss-like periodic materials considering isotropy constraints

“…In particular, thermal isotropy ensures diffusion isotropy, and this can be important while a preferential stiffness direction is prescribed. It is worth of note that in 2012, Guth et al (2012) had already presented results for the optimization of elastic and thermal homogenized tensors of PTMs using the same general guidelines adopted in this work. However, in that case the study was restricted to 2D materials subjected to plane stress and did not focus on the uncoupling of elastic and thermal symmetries.…”