Irreducible decomposition of strain gradient tensor in isotropic strain gradient elasticity

TL;DR

This paper presents a method to decompose the strain gradient tensor into irreducible components under the orthogonal group in isotropic strain gradient elasticity, aiding in constitutive modeling.

Contribution

It introduces a systematic decomposition of the strain gradient tensor into irreducible parts using Young tableau methods, applicable in any dimension.

Findings

Decomposition into four irreducible pieces in 3D with 7+5+3+3 components.

Decomposition into three irreducible pieces in 2D with 2+2+2 components.

Provides a framework for improved constitutive relation formulation.

Abstract

In isotropic strain gradient elasticity, we decompose the strain gradient tensor into its irreducible pieces under the n-dimensional orthogonal group O(n). Using the Young tableau method for traceless tensors, four irreducible pieces (n>2), which are canonical, are obtained. In three dimensions, the strain gradient tensor can be decomposed into four irreducible pieces with 7+5+3+3 independent components whereas in two dimensions, the strain gradient tensor can be decomposed into three irreducible pieces with 2+2+2 independent components. The knowledge of these irreducible pieces is extremely useful when setting up constitutive relations and strain energy.