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. As a consequence of this seemingly unnoticed observation, it follows that rank-one convexity and a homogeneous Cauchy stress imply that the left Cauchy-Green strain is homogeneous, as is shown in Mihai and Neff (Int. J. Non-Linear Mech., 2016, to appear).