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

$$\begin{aligned} &\bigl\langle S_{1}(F+\xi\otimes\eta) - S_{1}(F),\, \xi \otimes \eta\bigr\rangle = \bigl\langle \sigma(F+\xi\otimes\eta) \cdot \mathrm {Cof}(F+\xi \otimes\eta) - \sigma(F) \cdot \mathrm {Cof}F,\, \xi\otimes\eta\bigr\rangle \\ &\quad = \bigl\langle \sigma(F+\xi\otimes\eta),\, (\xi\otimes\eta) \bigl(\mathrm {Cof}(F+ \xi\otimes \eta) \bigr)^{T}\bigr\rangle - \bigl\langle \sigma(F),\, (\xi \otimes \eta) (\mathrm {Cof}F)^{T}\bigr\rangle \\ &\quad = \bigl\langle \sigma(F+\xi\otimes\eta),\; \xi \otimes \bigl(\mathrm {Cof}(F+\xi \otimes\eta). \eta \bigr)\bigr\rangle - \bigl\langle \sigma(F),\, \xi \otimes(\mathrm {Cof}F.\eta)\bigr\rangle \\ &\quad = \bigl\langle \sigma(F+\xi\otimes\eta),\; \xi\otimes \bigl(\mathrm {Cof}(F).\eta \bigr)\bigr\rangle - \bigl\langle \sigma(F),\, \xi\otimes(\mathrm {Cof}F.\eta)\bigr\rangle \\ &\quad = \bigl\langle \sigma(F+\xi\otimes\eta) - \sigma(F),\, \xi \otimes \bigl(( \mathrm {Cof}F).\eta \bigr)\bigr\rangle . \end{aligned}$$

$$\bigl(F\eta+ \lVert \eta \rVert ^{2}\xi \bigr)\otimes\xi= - \xi\otimes F\eta $$

$$F\eta+ \lVert \eta \rVert ^{2}\xi= \lambda\,\xi\,,\quad F\eta= -\lambda\, \xi \,, $$

$$\det(F+\xi\otimes\eta) = \det F + \bigl\langle \mathrm {Cof}(F)\eta,\,\xi\bigr\rangle = \bigl(1+ \bigl\langle \eta,\, F^{-1}\xi\bigr\rangle \bigr)\,\det F = \Bigl(1-\Bigl\langle \eta,\, \frac{2}{\lVert \eta \rVert ^{2}}\,\eta\Bigr\rangle \Bigr)\,\det F = -\det F\,, $$