We determine the eff ective electric properties of a composite with high contrast. The energy density is given locally in terms of a convex function of the gradient of the potential. The permittivity may take very large values in a fairly general distribution of parallel fibers of tiny cross sections. For a critical size of the cross sections, we show that a concentration of electric energy may arise in a small region of space surrounding the fi bers. This extra contribution is caused by the discrepancy between the behaviors of the potential in the matrix and in the fibers and is characterized by the density of the cross sections of the bers with respect to the cross section of the body in terms of some suitable notion of capacity. Our results extend those established in [7] in the periodic case for the p-Laplacian to a general nonlinear framework and a non-periodic distribution of fi bers.