In the paper the problem of finding variational models for low dimensional elastic structures (more precisely for the junctions of elastic materials with different dimensions) is considered. The authors describe the structure by means of a measure \(\mu\) on \(\mathbb R^n\), the energy functional being initially defined by \[ F(u) = \int f(x,Du) d\mu\quad (u \in C^1(\mathbb R^n;\mathbb R^m)) \] with \(n\)-dimensional energy density integrand \(f\). The low dimensional elastic model is obtained by a suitable relaxation \(\overline{F}\) of the functional \(F\); i.e., \[ \overline{F}(u) = \int f_{\mu}(x,D_{\mu}u) d\mu \quad (u \in W_{\mu}^{1,p} (\mathbb R^n;\mathbb R^m) \] with \(f_{\mu}\) being the relaxed integrand and \(D_{\mu}\) being the “tangential gradient” operator with respect to \(\mu\). Given a low dimensional manifold \(S\) of dimension \(k\) it will suffice to take \(\mu = H^k \angle S\) (the restriction of the \(k\)-dimensional Hausdorff measure to the manifold \(S\)) in order to obtain the desired elasticity model on \(S\).