An Irreducible Polynomial Functional Basis of Two-dimensional Eshelby Tensors

TL;DR

This paper establishes a minimal irreducible polynomial functional basis for two-dimensional Eshelby tensors, providing a fundamental tool for representation in mechanics and advanced material design.

Contribution

It introduces a minimal integrity basis of ten isotropic invariants for 2D Eshelby tensors, confirming it as an irreducible functional basis.

Findings

Identified a minimal integrity basis of ten isotropic invariants.

Confirmed the basis as an irreducible functional basis.

Applied the basis to representation problems in mechanics.

Abstract

Representation theorems for both isotropic and anisotropic functions are of prime importance in both theoretical and applied mechanics. The Eshelby inclusion problem is very fundamental, and is of particular importance in the design of advanced functional composite materials. In this paper, we discuss about two-dimensional Eshelby tensors (denoted as $M^{(2)}$). Eshelby tensors satisfy the minor index symmetry $M_{ijkl}^{(2)}=M_{jikl}^{(2)}=M_{ijlk}^{(2)}$ and have wide applications in many fields of mechanics. In view of the representation of two-dimensional irreducible tensors in complex field, we obtain a minimal integrity basis of ten isotropic invariants of $M^{(2)}$. Remarkably, note that an integrity basis is always a functional basis, we further confirm that the minimal integrity basis is also an irreducible function basis of isotropic invariants of $M^{(2)}$.