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 $\mathcal{F}[u] := \int_Ω f \bigg( \frac{1}{2} \bigl( \nabla u(x) + \nabla u(x)^T \bigr) \bigg)\,\mathrm{d} x, \qquad u : Ω\subset \mathbb{R}^d \to \mathbb{R}^d,$ over the space $\mathrm{BD}(Ω)$ of functions of bounded deformation or over the Temam-Strang space $\mathrm{U}(Ω):=\bigl\{u\in \mathrm{BD}(Ω): \ \mathrm{div} \ u\in \mathrm{L}^2(Ω)\bigr\},$ 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.

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