Injectivity in second-gradient nonlinear elasticity

Abstract 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.