On the relaxation of integral functionals depending on the symmetrized gradient

We prove results on the relaxation and weak* lower semicontinuity of integral functionals of the form $${\cal F}[u]: = \int_\Omega f \left( {\displaystyle{1 \over 2}\left( {\nabla u(x) + \nabla u{(x)}^T} \right)} \right) \,{\rm d}x,\quad u:\Omega \subset {\mathbb R}^d\to {\mathbb R}^d,$$ over the space BD(Ω) of functions of bounded deformation or over the Temam–Strang space $${\rm U}(\Omega ): = \left\{ {u\in {\rm BD}(\Omega ):\;\,{\rm div}\,u\in {\rm L}^2(\Omega )} \right\},$$ depending on the growth and shape of the integrand f. Such functionals are interesting, for example, in the study of Hencky plasticity and related models.