In this paper, we aim to explore the mechanical potentialities of a material made of an orthogonal net of fibers arranged in logarithmic spirals. Therefore, an annular plate described with a second-gradient model is envisaged to evaluate the behavior of such material in a nonlinear elastic regime when large displacements and deformations occur. Several mechanical tests are performed numerically under the finite element method approximation obtained directly with a weak formulation based on the elastic energy that it is assumed to be predictive for this kind of network system of fibers. Plots reporting the mechanical characteristics in all the considered tests are provided to illustrate the overall mechanical behavior of the evaluated system.