Mathematics, Vol. 13, Pages 2201: On the Mechanics of a Fiber Network-Reinforced Elastic Sheet Subjected to Uniaxial Extension and Bilateral Flexure

Mathematics doi: 10.3390/math13132201

Authors:
Wenhao Yao
Heung Soo Kim
Chun Il Kim

The mechanics of an elastic sheet reinforced with fiber mesh is investigated when undergoing bilateral in-plane bending and stretching. The strain energy of FRC is formulated by accounting for the matrix strain energy contribution and the fiber network deformations of extension, flexure, and torsion, where the strain energy potential of the matrix material is characterized via the Mooney–Rivlin strain energy model and the fiber kinematics is computed via the first and second gradient of deformations. By applying the variational principle on the strain energy of FRC, the Euler–Lagrange equilibrium equations are derived and then solved numerically. The theoretical results highlight the matrix and meshwork deformations of FRC subjected to bilateral bending and stretching simultaneously, and it is found that the interaction between bilateral extension and bending manipulates the matrix and network deformation. It is theoretically observed that the transverse Lagrange strain peaks near the bilateral boundary while the longitudinal strain is intensified inside the FRC domain. The continuum model further demonstrates the bidirectional mesh network deformations in the case of plain woven, from which the extension and flexure kinematics of fiber units are illustrated to examine the effects of fiber unit deformations on the overall deformations of the fiber network. To reduce the observed matrix-network dislocation in the case of plain network reinforcement, the pantographic network reinforcement is investigated, suggesting that the bilateral stretch results in the reduced intersection angle at the mesh joints in the FRC domain. For validation of the continuum model, the obtained results are cross-examined with the existing experimental results depicting the failure mode of conventional fiber-reinforced composites to demonstrate the practical utility of the proposed model.

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