Lamination exact relations and their stability under homogenization

TL;DR

This paper investigates the stability of lamination exact relations under homogenization in composite materials, providing examples in 2D and analyzing algebraic structures related to Jordan algebras.

Contribution

It introduces an example of an exact relation stable under lamination but not homogenization in 2D materials and explores algebraic representations of Jordan algebras in this context.

Findings

Example of a 2D exact relation stable under lamination but not homogenization.

Explicit algebraic description of Jordan algebra representations as symmetric matrices.

Analysis of the 4-chain relation in the context of Jordan algebras.

Abstract

Relations between components of the effective tensors of composites that hold regardless of composite's microstructure are called exact relations. Relations between components of the effective tensors of all laminates are called lamination exact relations. The question of existence of sets of effective tensors of composites that are stable under lamination, but not homogenization was settled by Milton with an example in 3D elasticity. In this paper we discuss an analogous question for exact relations, where in a wide variety of physical contexts it is known (a posteriori) that all lamination exact relations are stable under homogenization. In this paper we consider 2D polycrystalline multi-field response materials and give an example of an exact relation that is stable under lamination, but not homogenization. We also shed some light on the surprising absence of such examples in most other physical contexts (including 3D polycrystalline multi-field response materials). The methods of our analysis are algebraic and lead to an explicit description (up to orthogonal conjugation equivalence) of all representations of formally real Jordan algebras as symmetric $n\times n$ matrices. For each representation we examine the validity of the 4-chain relation|a 4th degree polynomial identity, playing an important role in the theory of special Jordan algebras.