Injectivity of the Cauchy-stress tensor along rank-one connected lines under strict rank-one convexity condition

TL;DR

This paper proves that in nonlinear elasticity, the Cauchy stress tensor is injective along rank-one connected lines if the material law is strictly rank-one convex, ensuring uniqueness of stress states under certain conditions.

Contribution

It establishes the injectivity of the Cauchy stress tensor along rank-one lines under strict rank-one convexity, a novel result in nonlinear elasticity theory.

Findings

Injectivity of the Cauchy stress tensor along rank-one lines.

Strict rank-one convexity implies uniqueness of stress states.

Provides a mathematical condition ensuring stress tensor injectivity.

Abstract

In this note, we show that the Cauchy stress tensor $\sigma$ in nonlinear elasticity is injective along rank-one connected lines provided that the constitutive law is strictly rank-one convex. This means that $\sigma(F+\xi\otimes \eta)=\sigma(F)$ implies $\xi\otimes \eta=0$ under strict rank-one convexity.