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

Abstract

In this note, we show that the Cauchy stress tensor $\ddot{I}ƒ$ in nonlinear elasticity is injective along rank-one connected lines provided that the constitutive law is strictly rank-one convex. This means that $\ddot{I}ƒ(F+\hat{I}¾\otimes \hat{I}\cdot )=\ddot{I}ƒ(F)$ implies $\hat{I}¾\otimes \hat{I}\cdot =0$ under strict rank-one convexity.