  In this paper we study geometric aspects of Ball's classes in the context of
nonlinear elasticity problems. The suggested approach is based on the
characterization of Ball's classes $A_{q,q'}(\Omega)$ in terms of composition
operators on Sobolev spaces. This characterization allows us to obtain the
volume compression (distortion) estimates of topological mappings of Ball's
classes. We prove also that Sobolev homeomorphisms of Ball's classes which
possess the Luzin $N$-property are absolutely continuous with respect to
capacity.
