We study injectivity for models of Nonlinear Elasticity that involve the second gradient. We assume that \(\Omega \subset {\mathbb{R}}^n\) is a domain, \(f\in W^{2,q}(\Omega ,{\mathbb{R}}^n)\) satisfies \(|J_f|^{-a}\in L^1\) and that f equals a given homeomorphism on \(\partial \Omega\) . 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.
