  We show that a sufficient condition for the weak limit of a sequence of
$W^1_q$-homeomorphisms with finite distortion to be almost everywhere injective
for $q \geq n-1$, can be stated by means of composition operators. Applying
this result, we study nonlinear elasticity problems with respect to these new
classes of mappings. Furthermore, we impose loose growth conditions on the
stored-energy function for the class of $W^1_n$-homeomorphisms with finite
distortion and integrable inner as well as outer distortion coefficients.
