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 $$\mathcal{F}[u]:={\int }_{\mathrm{\Omega }}f\left(\frac{1}{2}\right(\mathrm{\nabla }u(x)+\mathrm{\nabla }u(x{)}^T))\mathrm{d}x,u:\mathrm{\Omega }\subset {\mathbb{R}}^d\to {\mathbb{R}}^d,$$ over the space $\mathrm{B}\mathrm{D}(\mathrm{\Omega })$ of functions of bounded deformation or over the Temam-Strang space $$\mathrm{U}(\mathrm{\Omega }):=\{u\in \mathrm{B}\mathrm{D}(\mathrm{\Omega }):\mathrm{d}\mathrm{i}\mathrm{v}u\in {\mathrm{L}}^2(\mathrm{\Omega })\},$$ 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.