In the paper the problem of homogenization of some integral functionals defined on functions subject to oscillating constraints on their gradients is considered. The motivation comes from the theory of elastic-plastic torsion. In abstract setting the problem concerns the asymptotic behaviour (for every open and bounded \(\Omega \subset {\mathbb R}^n\) and \(\beta \in L^1(\Omega)\), as \(h \rightarrow + \infty\)) of solutions to variational problems: \[ m_h(\Omega, \beta) = \min \biggl\{\int_{\Omega}f(hx,u,Du) dx + \int_{\Omega}\beta u dx \biggr\}, \] where \(\min\) is taken over all Lipschitz continuous functions \(u\) such that \(u = 0\) on \(\partial \Omega\), \(|Du(x)|\leq \varphi (x)\) for a.e. \(x \in \Omega\). Above, \(\varphi (\cdot)\) and \(f(\cdot,s,z)\) are measurable and \({]0,1[}^n\)-periodic functions satisfying some additional regularity assumptions. Using \(\Gamma\) convergence theory the authors find the integral representation of the \(\Gamma\) limit (in \(C_0^0(\Omega)\)) and prove also the convergence of minimal values of the problems to the minimal value of the limit problem.