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

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.