Stress and strain in symmetric and asymmetric elasticity

TL;DR

This paper develops a tensorial elasticity theory using arbitrary coordinates, accounting for asymmetric stress and Cosserat media, with a focus on measurable quantities and proper deformation and strain tensors.

Contribution

It introduces a tensorial elasticity framework that handles asymmetric stresses and Cosserat media without micro-rotations, emphasizing measurable quantities and proper deformation tensors.

Findings

The deformation tensor is related to the deformation velocity via the matricant.

Strain is defined as the logarithm of the deformation tensor.

The stiffness tensor components are time-independent in material coordinates.

Abstract

Usual introductions of the concept of motion are not well adapted to a subsequent, strictly tensorial, theory of elasticity. The consideration of arbitrary coordinate systems for the representation of both, the points in the laboratory, and the material points (comoving coordinates), allows to develop a simple, old fashioned theory, where only measurable quantities -like the Cauchy stress- need be introduced. The theory accounts for the possibility of asymmetric stress (Cosserat elastic media), but, contrary to usual developments of the theory, the basic variable is not a micro-rotation, but the more fundamental micro-rotation velocity. The deformation tensor here introduced is the proper tensorial equivalent of the poorly defined deformation "tensors" of the usual theory. It is related to the deformation velocity tensor via the matricant. The strain is the logarithm of the deformation tensor. As the theory accounts for general Cosserat media, the strain is not necessarily symmetric. Hooke's law can be properly introduced in the material coordinates (as the stiffness is a function of the material point). A particularity of the theory is that the components of the stiffness tensor in the material (comoving) coordinates are not time-dependent. The configuration space is identified to the part of the Lie group GL(3)+, that is geodesically connected to the origin of the group.