  In this work, we provide a characterization result for lower semicontinuity
of surface energies defined on piecewise rigid functions, i.e., functions which
are piecewise affine on a Caccioppoli partition where the derivative in each
component is a skew symmetric matrix. This characterization is achieved by
means of an integral condition, called $BD$-ellipticity, which is in the spirit
of $BV$-ellipticity defined by Ambrosio and Braides. By specific examples we
show that this novel concept is in fact stronger compared to its $BV$ analog.
We further provide a sufficient condition implying $BD$-ellipticity which we
call symmetric joint convexity. This notion can be checked explicitly for
certain classes of surface energies which are relevant for applications, e.g.,
for variational fracture models. Finally, we give a direct proof that surface
energies with symmetric jointly convex integrands are lower semicontinuous also
on the larger space of $GSBD^p$ functions.
