Explicit harmonic structure of bidimensional linear strain-gradient elasticity

Abstract

In the perspective of homogenization theory, strain-gradient elasticity is a strategy to describe the overall behavior 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≥3), their determination is generally a difficult task. In this paper, a novel procedure to obtain this decomposition is introduced. This method, which we have called the Clebsch–Gordan Harmonic Algorithm, allows one to obtain explicit harmonic decompositions satisfying good properties such as orthogonality and uniqueness. 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. The Clebsch–Gordan Harmonic Algorithm is not restricted to strain-gradient elasticity and may find interesting applications in different fields of mechanics which involve higher-order tensors.