Pure shear axes and elastic strain energy

TL;DR

This paper characterizes all pure shear bases, derives vector functions to generate them, and discusses their implications for strain energy optimization in elastic solids with cubic symmetry.

Contribution

It introduces new results that describe all possible pure shear bases and their generation, extending understanding of stress and strain energy in elastic materials.

Findings

Pure shear bases can be generated from any basis using derived vector functions.

Strain energy extrema occur when axes align with principal stress or pure shear bases.

Optimal orientation for stress minimization or maximization depends on material parameters.

Abstract

It is well known that a state of pure shear has distinct sets of basis vectors or coordinate systems: the principal axes, in which the stress is diagonal, and pure shear bases, in which diag(stress)=0. The latter is commonly taken as the definition of pure shear, although a state of pure shear is more generally defined by tr(stress)=0. New results are presented that characterize all possible pure shear bases. A pair of vector functions are derived which generate a set of pure shear basis vectors from any one member of the triad. The vector functions follow from compatibility condition for the pure shear basis vectors, and are independent of the principal stress values. The complementary types of vector basis have implications for the strain energy of linearly elastic solids with cubic material symmetry: for a given state of stress or strain, the strain energy achieves its extreme values when the material cube axes are aligned with principal axes of stress or with a pure shear basis. This implies that the optimal orientation for a given state of stress is with one or the other vector basis, depending as the stress is to be minimized or maximized, which involves the sign of one material parameter.