Simple shear is not so simple

TL;DR

This paper derives the deformation form of homogeneous, isotropic, nonlinearly elastic materials under shear stress, revealing it is a combination of triaxial stretch and simple shear, not simple shear alone as in linear elasticity.

Contribution

It demonstrates that the deformation under shear stress involves a superposition of stretch and shear, challenging the traditional linear elasticity assumption.

Findings

Deformation includes a triaxial stretch superposed on simple shear.

Maximum shear angle achievable by pure shear stress is 45 degrees.

Tractions on inclined faces are calculated for parallelepiped deformation.

Abstract

For homogeneous, isotropic, nonlinearly elastic materials, the form of the homogeneous deformation consistent with the application of a Cauchy shear stress is derived here for both compressible and incompressible materials. It is shown that this deformation is not simple shear, in contrast to the situation in linear elasticity. Instead, it consists of a triaxial stretch superposed on a classical simple shear deformation, for which the amount of shear cannot be greater than 1. In other words, the faces of a cubic block cannot be slanted by an angle greater than 45deg by the application of a pure shear stress alone. The results are illustrated for those materials for which the strain energy function does not depend on the principal second invariant of strain. For the case of a block deformed into a parallelepiped, the tractions on the inclined faces necessary to maintain the derived deformation are calculated.