Explicit Harmonic Structure Of Bidimensional Linear Strain-Gradient Elasticity

TL;DR

This paper introduces a novel algorithm for explicit harmonic decomposition of higher-order tensors in 2D strain-gradient elasticity, enhancing understanding of anisotropic properties and physical interpretation of complex material behaviors.

Contribution

The paper presents the Clebsch-Gordan Harmonic Algorithm, enabling explicit, orthogonal, and unique harmonic decompositions of fifth- and sixth-order elasticity tensors in 2D.

Findings

Explicit harmonic decompositions of 5th- and 6th-order tensors are obtained.

The algorithm provides geometrically meaningful components simplifying physical interpretation.

First-time derivation of these decompositions in the context of Mindlin's strain-gradient elasticity.

Abstract

In the perspective of homogenization theory, strain-gradient elasticity is a strategy to describe the overall behaviour of materials with coarse mesostructure. In this approach, the effect of the mesostructure is described by the use of three elasticity tensors whose orders vary from 4 to 6. Higher-order constitutive tensors make it possible to describe rich physical phenomena. However, these objects have intricate algebraic structures that prevent us from having a clear picture of their modeling capabilities. The harmonic decomposition is a fundamental tool to investigate the anisotropic properties of constitutive tensor spaces. For higher-order tensors (i.e. tensors of order $n\geq$3), its establishment is generally a difficult task. In this paper a novel procedure to obtain this decomposition is introduced. This method, that we have called the \textit{Clebsch-Gordan Harmonic Algorithm}, allows to obtain \emph{explicit} harmonic decompositions satisfying good properties such as orthogonality and unicity. The elements of the decomposition also have a precise geometrical meaning simplifying their physical interpretation. This new algorithm is here developed in the specific case of 2D space and applied to Mindlin's Strain-Gradient Elasticity. We provide, for the first time, the harmonic decompositions of the fifth- and sixth-order elasticity tensors involved in this constitutive law.