A continuum model for the mechanics of elastomeric sheets reinforced with extensible bidirectional fibers resistant to lateral pressure

We investigate the concurrent three-dimensional (in-plane and out-of-plane) deformations of fiber-reinforced composite (FRC) sheets undergoing lateral pressure. This involves the utilization of the Neo-Hookean strain energy model for the matrix material and computing the strain energy of bidirectional fibers by accounting for the stretching, bending, and twisting responses of the fibers. The kinematics of FRC are formulated within the framework of differential geometry on FRC surfaces, including the computations of the first and second gradient of deformation. By employing the variational principles, we derive the Euler equations describing the mechanics of the fiber–matrix composite system subjected to lateral pressure. The resulting three-dimensional continuum model theoretically predicts the deformation of the matrix material and it is found that the maximum deformation of matrix material occurs in the diagonal direction of the domain, yet, the center of the domain suffers weak in-plane deformation because of surface tension equilibrium. In addition, the stretching, bending, and twisting kinematics of fiber units are computed to investigate the effects of the individual fiber’s kinematics on the overall deformation of fiber meshwork. Lastly, it is found that the lateral pressure on the FRC surface induces fiber flexure in the vicinity of domain boundaries and fiber stretch inside the domain, corresponding to the intensified shrinking strain near the edges and stretching strain in the middle of the domain. The theoretical results provide phenomenologically meaningful insights into comprehending the damage patterns of the fiber-reinforced building material, the hemispherical dome shaping results of bamboo Poly (lactic) acid (PLA) composites and the out-of-plane deformation of woven fabric.