  We present a classification of strict limits of planar BV homeomorphisms. The
authors and S. Hencl showed in a previous work \cite{CHKR} that such mappings
allow for cavitations and fractures singularities but fulfill a suitable
generalization of the INV condition. As pointed out by J. Ball \cite{B}, these
features are physically expected by limit configurations of elastic
deformations. In the present work we develop a suitable generalization of the
\emph{no-crossing} condition introduced by De Philippis and Pratelli in
\cite{PP} to describe weak limits of planar Sobolev homeomorphisms that we call
\emph{BV no-crossing} condition, and we show that a planar mapping satisfies
this property if and only if it can be approximated strictly by homeomorphisms
of bounded variations.
