A linearized model for the three-dimensional coupled kinematics of elastomeric sheets reinforced with fiber mesh: Complete analytical solutions

초록

A comprehensive mathematical framework for the three-dimensional mechanics of fiber-reinforced composite (FRC) sheet is presented within the context of superposed incremental deformations. Via the systematic linearization of the nonlinear governing shape equations, a decoupled equation system is derived, consisting of a normal and two tangential equations to respectively characterize the out-of-plane and in-plane responses. Complete closed-form analytical solutions are obtained by employing Ferrari's method, providing an explicit description of the concurrent deformations of the elastomeric matrix and the reinforcing meshwork. Particularly, the concurrent deformations are coupled via the infinitesimal deformation parameter in the spirit of infinitesimal loading leading to infinitesimal deformation. The resulting formulation and analysis establish a direct mechanistic link between microstructural fiber behavior-including stretching, bending, and twisting-and the global response of the composite. The comparative analysis indicates that the linear model qualitatively aligns with nonlinear formulations under infinitesimal loading, providing a computationally efficient and analytically tractable alternative for the infinitesimal deformation analysis of complex systems. To this end, three-point bending experiments were conducted on silicone-based FRCs, where theoretical predictions accurately captured experimental deflection patterns and changes in interaction angles of the network. The resulting framework provides a robust analytical tool for predicting coupled 3D matrix-fiber deformation in soft composite systems.

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