  This work adresses the question of density of piecewise constant (resp.
rigid) functions in the space of vector valued functions with bounded variation
(resp. deformation) with respect to the strict convergence. Such an
approximation property cannot hold when considering the usual total variation
in the space of measures associated to the standard Frobenius norm in the space
of matrices. It turns out that oscillation and concentration phenomena interact
in such a way that the Frobenius norm has to be homogenized into a (resp.
symmetric) Schatten-1 norm which coincides with the Euclidean norm on rank-one
(resp. symmetric) matrices. By means of explicit constructions consisting of
the superposition of sequential laminates, the validity of an optimal
approximation property is established at the expense of endowing the space of
measures with a total variation associated with the homogenized norm in the
space of matrices.
