  This article is about hyperelastic deformations of plates (planar domains)
which minimize a neohookean type energy. Particularly, we investigate a stored
energy functional introduced by J.M. Ball in his seminal paper "Global
invertibility of Sobolev functions and the interpenetration of matter". The
mappings under consideration are Sobolev homeomorphisms and their weak limits.
They are monotone in the sense of C. B. Morrey. One major advantage of adopting
monotone Sobolev mappings lies in the existence of the energy-minimal
deformations. However, injectivity is inevitably lost, so an obvious question
to ask is: what are the largest subsets of the reference configuration on which
minimal deformations remain injective? The fact that such subsets have full
measure should be compared with the notion of global invertibility which deals
with subsets of the deformed configuration instead. In this connection we
present a Cantor type construction to show that both the branch set and its
image may have positive area. Another novelty of our approach lies in allowing
the elastic deformations be free along the boundary, known as frictionless
problems.
