Material Symmetries in Homogenized Hexagonal-Shaped Composites as Cosserat Continua

Nicholas Fantuzzi,Patrizia Trovalusci,Raimondo Luciano

doi:10.3390/sym12030441

Nicholas Fantuzzi, Patrizia Trovalusci + Show 1 more

Open Access

https://doi.org/10.3390/sym12030441

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Abstract

In this work, material symmetries in homogenized composites are analyzed. Composite materials are described as materials made of rigid particles and elastic interfaces. Rigid particles of arbitrary hexagonal shape are considered and their geometry described by a limited set of parameters. The purpose of this study is to analyze different geometrical configurations of the assemblies corresponding to various material symmetries such as orthotetragonal, auxetic and chiral. The problem is investigated through a homogenization technique which is able to carry out constitutive parameters using a principle of energetic equivalence. The constitutive law of the homogenized continuum has been derived within the framework of Cosserat elasticity, wherein the continuum has additional degrees of freedom with respect to classical elasticity. A panel composed of material with various symmetries, corresponding to some particular hexagonal geometries defined, is analyzed under the effect of localized loads. The results obtained show the difference of the micropolar response for the considered material symmetries, which depends on the non-symmetries of the strain and stress tensor as well as on the additional kinematical and work-conjugated statical descriptors. This work underlines the importance of resorting to the Cosserat theory when analyzing anisotropic materials.

Highlights

In the study of composite materials, symmetries play a very important role in identify symmetry planes and peculiar material behaviors. It is well-known that composite materials can be investigated by directly analyzing their constituents in a micromechanical model or by homogenizing them as an equivalent continuum
The homogenization of complex interaction effects in composite materials needs internal scale parameters which are not negligible compared to the structural length scale
The paper [7] describes a historical overview of different homogenization techniques which have been extended to non-classical continua. As it is well known in the literature, the classical continuum, by lacking in internal lengths, leads to ill-posed problems

Summary

Introduction

In the study of composite materials, symmetries play a very important role in identify symmetry planes and peculiar material behaviors. The paper [7] describes a historical overview of different homogenization techniques which have been extended to non-classical continua As it is well known in the literature, the classical continuum, by lacking in internal lengths, leads to ill-posed problems. The couple-stress theory has been widely used for several applications [35,51,52], in this theory, micro and macro rotation coincide and when couple stresses are negligible, classical elasticity theory is derived (see appendix in [31]) As it has been recently analyzed by [53], micro-polar effects become prominent when geometrical or load singularities are present in the reference problem, such as concentrated loads, voids or material inclusions. Paper conclusions and remarks are given in the conclusion section

Framework of Cosserat Theory

Finite Element Implementation

Numerical Applications

Conclusions