Givenμ, κ, c>0, we consider the functional $$F(u) = \int_{\Omega \backslash S_u } {\left( {\mu |E^D u|^2 + \frac{\kappa }{2}(div u)^2 } \right)} dx + c\int_{S_u } {|u^ + - u^ - |} d\mathcal{H}^{n - 1} ,$$ defined on allR n -valued functionsu on the open subset Ω ofR n which are smooth outside a free discontinuity setS u, on which the tracesu +,u − on both sides have equal normal component (i.e.,u has a tangential jump alongS u).E Du=Eu − 1/3 (divu)I, withEu denoting the linearized strain tensor. The functionalF is obtained from the usual strain energy of linearized elasticity by addition of a term (the second integral) which penalizes the jump discontin uities of the displacement. The lower semicontinuous envelope $$\bar F$$ is studied, with respect to theL 1 (Ω;R n )-topology, on the spaceP(Ω) of the functions of bounded deformation with distributional divergence inL 2(Ω) (F is extended with value +∞ on the wholeP(Ω)). The following integral representation is proved: $$\bar F(u) = \int_\Omega {\left( {\varphi (\varepsilon ^D u) + \frac{\kappa }{2}(div u)^2 } \right)} dx + \int_\Omega {\varphi ^\infty } \left( {\frac{{E_s^D u}}{{|E_s^D u|}}} \right)|E_s^D u|, u \in P(\Omega ),$$ whereϕ is a convex function with linear growth at infinity. NowEu is a measure,ɛ Du represents the density of the absolutely continuous part of the absolutely continuous part ofE Du, whileE s D u denotes the singular part and ϕ∞ the recession function ofϕ. Finally, we show that $$\bar F$$ coincides with the functional which intervenes in the minimum problem for the displacement in the theory of Hencky’s plasticity with Tresca’s yield conditions.