The mechanics of thin elastic sheet reinforced with fiber mesh subjected to combined flexure and extension

Abstract

We present a three-dimensional continuum model aimed at investigating the mechanical performance of elastomeric materials reinforced with bidirectional fibers undergoing the combined out-of-plane bending moment and bilateral extension. To achieve this, we consider the Neo-Hookean strain energy model for the matrix material of the fiber composite and incorporate the strain energy of bidirectional fibers into the strain energy potential of the fiber composite. The bidirectional fibers’ strain energy is rigorously modeled by configuring the fiber kinematics on the fiber composite surface using differential geometry and accounting for stretching, bending, and twisting responses of the fibers via the computation of the first-order and second-order gradient of deformation. To establish equilibrium equations, we apply the variational principle to derive the constitutive relations, resulting in the normal and tangential shape equations of the fiber composite, and boundary conditions of bending and stretching. The model implementation emphasizes analyzing the combined out-of-plane bending and bilateral stretching effects on the fiber composite material, and the numerical results include the out-of-plane and in-plane deformation, strain-loading responses, bending, twisting, and stretching of fibers within the matrix material, as well as the deformation of the fiber network. Specifically, it is found that a larger bending moment results in intensified out-of-plane deformation while larger bilateral stretch decreases the out-of-plane deformation of fiber composite. In particular, bilateral stretches reduce the Lagrange strain (ɛ1) near the bilateral boundaries, unlike the intensity distribution of Lagrange strain (ɛ2) maintains almost unchanged due to no external tension applied on the upper and lower boundaries of the fiber while bending moments persist. These findings highlight the role of bilateral stretching acting as the prestress in reducing deflection, enhancing tensile strength and efficient use, and controlling crack generation of fiber-reinforced polymer (FRP) composites. More importantly, despite the scarcity of experimental data on fiber composite deformation at the microscopic, the theoretical demonstration of the extension and flexure of fiber units provide reasonable explanations for the resulting deformation of the overall fiber meshwork, validating the mechanism that the fiber microstructure deformation determines the overall deformation of fiber composites when subjected to combined bending moment and tension. Special attention should be given to the potential dislocation between the embedded meshwork and matrix material located in the center of the domain. Furthermore, the proposed model provides qualitatively reasonable explanations on the shaping of Bamboo Polylactic acid (PLA) composites, out-of-plane deformation of woven fabric, Sikken-type stiffener deformation, and fiber-reinforced thermoplastics via a series of numerical results.