Abstract
Asymptotic Homogenization (AH) and the Extended Multiscale Finite Element Method (EMsFEM) are both procedures that allow working on a structural macroscale that incorporates the effect of averaged microscopic heterogeneities, thus resulting in computationally efficient strategies. EMsFEM works directly on coupled finite micro and macroscales using numerically built discrete interpolation functions. Periodic Truss Metamaterials (PTMMs) are cellular materials formed by the periodic repetition of a truss-like unit cell and engineeringly tailored to show a given macroscopic response. In this work we analyze the numerical behavior of selected PTMMs that were designed for extreme Poisson ratios using AH theory. As a first issue, we study macroscopic structures made of finite unit cells and verify how close their average behavior coincides with the material properties predicted by AH. For comparison, we solve the macroscopic plane stress associate problems that employ the elastic constitutive tensor obtained by AH. The second issue is concerned with the ability of EMsFEM to reproduce the structural behavior of the full macro-micro model. We employ two versions of the EMsFEM, adopting linear (LBC) and periodic (PBC) boundary conditions to build the numerical interpolation functions. The third and most important aspect discussed in this research concerns evaluation of the EMsFEM downscaled displacement fields. We observe that according to the layout of the AH designed unit cell, to the use of LBC or PBC and, depending on the boundary conditions present in the macroscopic problem, spurious downscaled displacements might occur. Such spurious displacements are due to excessive compliance of the corresponding unit cell and can be detected when building the numerical interpolation functions. We conclude that the layout optimization of PTMM using AH must be carefully interpreted and that EMsFEM is a good tool to detect a macroscopic excessively compliant response at an early design stage.
Highlights
In nature, it is frequent to find organic materials that evolved along time adapting to behave efficiently in multipurpose tasks
We focus in the problems caused by excessive compliance shown by some Periodic Truss Metamaterials (PTMMs), which can be detected in the Extended Multiscale Finite Element Method (EMsFEM) by the presence of unexpected large downscaled displacements within unit cells when the material is subjected to external loading
We organize this article in the following way: Sections 2 and 3 briefly review the theoretical bases of asymptotic homogenization (AH) and EMsFEM for periodic truss materials (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
Summary
It is frequent to find organic materials that evolved along time adapting to behave efficiently in multipurpose tasks. In the last decades, such materials have been studied with an engineering point of view, inspiring researchers to develop metamaterials for multifunctional technological applications (Ashby, 1983; Gibson and Ashby, 1997). These bio-inspired materials include the so-called periodic truss materials (PTMs), a special class of cellular materials made by the periodic repetition of unit cells composed of bar elements. Lightness coupled to high capability to store strain energy are important features for automotive and aerospace industries, for instance (Yan et al, 2006) Owing to their spatial regularity, periodic cellular materials show little dispertion in their macroscopic properties, which can be an important engineering requirement.
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