Analysis of Boundary Layer Influence on Effective Shear Modulus of 3-1 Longitudinally Porous Elastic Solid

Adair Roberto Aguiar,Uziel Paulo Da Silva,Edmar Borges Theóphilo Prado

doi:10.1590/1679-78255960

Adair Roberto Aguiar, Uziel Paulo Da Silva + Show 1 more

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https://doi.org/10.1590/1679-78255960

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Abstract

The evaluation of the effective properties of nonhomogeneous solids using analytical methods is, in general, based on the assumption that these solids have infinite dimensions. Here, we investigate the influence of both the number of holes and the boundary layer of a solid with finite dimensions on the determination of these properties. We use the Asymptotic Homogenization Method (AHM) to determine the effective shear modulus of an elastic solid with infinite dimensions containing a uniform and periodic distribution of circular cylindrical holes arranged on a hexagonal lattice. We also use the Finite Element Method (FEM) to determine this modulus in the case of a solid with finite dimensions containing the same uniform distribution of cylindrical holes away from its boundary. Near the boundary, we consider a layer of material with no holes, which is usually left in the fabrication process of samples. Both solids have the same elastic properties and are subjected to similar anti-plane shear loadings. For the finite medium, we study two sequences of domains discretized by the FEM, which are called the Fixed Layer Sequence (FLS) and the Fixed Domain Sequence (FDS). For the FLS, the layer thickness is kept fixed and both the dimensions of the domain and the number of holes vary. For the FDS, the dimensions of the domain are kept fixed and both the number of holes and the layer thickness vary. Results obtained from numerical simulations are then used to generate graphs of the effective shear modulus versus void volume fraction. It is observed that, in the FLS case, the shear modulus obtained from the numerical simulations converges to the analytical solution obtained via AHM. It is also observed that, in the FDS case, the shear modulus obtained from the numerical simulations converges to a limit function, which is close to the analytical solution obtained via AHM. For comparison purposes, we have also calculated the effective shear modulus of porous elastic solids containing a square array of circular cylindrical holes. We then show graphs of this modulus versus void volume fraction for both hexagonal and square arrangements that are very close to each other up to void volume fraction of 0.5.

Highlights

The study of the behavior of nonhomogeneous solids, such as biological tissues, requires the determination of their effective properties via analytical methods, computational methods, or a combination of both
(1996), Ghosh et al (2001), Marino et al (2019), and Lo Cascio et al (2020), who have considered random distributions of heterogeneities. These authors study how spatial distributions of heterogeneities and volume fraction influence the effective elastic properties of nonhomogeneous solids by identifying a Representative Volume Element (RVE) and determining the proper boundary conditions that should be applied on the boundary of the RVE
We use the Fixed Domain Sequence (FDS) approach, with fixed dimensions of MC given by L × H = 0.1× 0.1m2 and fixed layer thickness given by a = 0.0001 m, and consider a sequence of decreasing values of the hexagon dimension h in the set

Summary

Introduction

The study of the behavior of nonhomogeneous solids, such as biological tissues, requires the determination of their effective properties via analytical methods, computational methods, or a combination of both. (1996), Ghosh et al (2001), Marino et al (2019), and Lo Cascio et al (2020), who have considered random distributions of heterogeneities These authors study how spatial distributions of heterogeneities and volume fraction influence the effective elastic properties of nonhomogeneous solids by identifying a Representative Volume Element (RVE) and determining the proper boundary conditions that should be applied on the boundary of the RVE. Different from these works, here, we apply conditions on the boundary of the whole domain

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