  We prove the local H\"{o}lder continuity of strong local minimizers of the
stored energy functional \[E(u)=\int_{\om}\lambda |\nabla u|^{2}+h(\det \nabla
u) \,dx\] subject to a condition of `positive twist'. The latter turns out to
be equivalent to requiring that $u$ maps circles to suitably star-shaped sets.
The convex function $h(s)$ grows logarithmically as $s\to 0+$, linearly as $s
\to +\infty$, and satisfies $h(s)=+\infty$ if $s \leq 0$. These properties
encode a constitutive condition which ensures that material does not
interpenetrate during a deformation and is one of the principal obstacles to
proving the regularity of local or global minimizers. The main innovation is to
prove that if a local minimizer has positive twist a.e\frenchspacing. on a ball
then an Euler-Lagrange type inequality holds and a Caccioppoli inequality can
be derived from it. The claimed H\"{o}lder continuity then follows by adapting
some well-known elliptic regularity theory. We also demonstrate the
regularizing effect that the term $\int_{\om} h(\det \nabla u)\,dx$ can have by
analysing the regularity of local minimizers in the class of `shear maps'. In
this setting a more easily verifiable condition than that of positive twist is
imposed, with the result that local minimizers are H\"{o}lder continuous.
