Injectivity in second-gradient Nonlinear Elasticity

We study injectivity for models of Nonlinear Elasticity that involve the second gradient. We assume that $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a domain, $f\in W^{2,q}(\Omega,$\mathbb{R}$^n)$satisfies$|J_f|^{-a}\in L^1$andthat$f$ equals a given homeomorphism on $\mathrm{\partial }\mathrm{\Omega }$. Under suitable conditions on $q$ and $a$ we show that $f$ must be a homeomorphism. As a main new tool we find an optimal condition for $a$ and $q$ that imply that ${\mathcal{H}}^{n-1}(\{J_f=0\})=0$ and hence $J_f$ cannot change sign. We further specify in dependence of $q$ and $a$ the maximal Hausdorff dimension $d$ of the critical set $\{J_f=0\}$. The sharpness of our conditions for $d$ is demonstrated by constructing respective counterexamples.