We consider vectorial variational problems of the form E[u]=∫W(Du)dx, typical for example of nonlinear elasticity and plasticity, which include constraints on the determinant. Specifically, the energy density W is assumed to diverge outside of the set of matrices with positive determinant, or, alternatively, outside of the set of matrices with determinant equal 1. If W is not quasiconvex then E is not lower semicontinuous and does not, in general, have minimizers. Low-energy states can be studied via the relaxation of E. We discuss how, in some situations of physical interest, the relaxation of E can be explicitly characterized in terms of the quasiconvex envelope of W. This talk is based on joint work with Georg Dolzmann (Regensburg).