A three-dimensional model for the mechanics of elastic/hyperelastic materials reinforced with bidirectional fibers is presented in finite elastostatics. This includes the constitutive formulation of matrix–fiber composite system and the derivation of the corresponding Euler equilibrium equation. The responses of the matrix material and reinforcing fibers are characterized, respectively, via the Neo-Hookean model and quadratic strain energy potential of the Green–Lagrange type. These are further refined by the Mooney–Rivlin strain energy model and the high-order polynomial energy potential of fibers to incorporate the nonlinear behaviors of the matrix material and fibers. Within the framework of differential geometry and strain-gradient elasticity, the general kinematics of bidirectional fibers, including the three-dimensional bending of a fiber and twist between the two adjoining fibers, are formulated, and subsequently integrated into the model of continuum deformation. The admissible boundary conditions are also derived by virtue of variational principles and virtual work statement. In particular, a dimension reduction process is applied to the resulting three-dimensional model through which a compatible two-dimensional model describing both the in-plane and out-of-plane deformations of thin elastic films reinforced with fiber mesh is obtained. To this end, model implementation and comparison with the experimental results are performed, indicating that the proposed model successfully predicts key design considerations of fiber mesh reinforced composite films including stress–strain responses, deformation profiles, shear strain distributions and local structure (a unit fiber mesh) deformations. The proposed model is unique in that it is formulated within the framework of differential geometry of surface to accommodate the three-dimensional kinematics of the composite, yet the resulting equations are reframed in the orthonormal basis for enhanced practical unitality and mathematical tractability. Hence, the resulting model may also serve as an alternative Cosserat theory of plates and shells arising in two-dimensional nonlinear elasticity.

有限弹性​​静力学中提出了双向纤维增强弹性/超弹性材料力学的三维模型。这包括基体-纤维复合材料系统的本构公式和相应的欧拉平衡方程的推导。基体材料和增强纤维的响应分别通过 Neo-Hookean 模型和 Green-Lagrange 类型的二次应变能势来表征。这些通过 Mooney-Rivlin 应变能模型和纤维的高阶多项式能量势进一步细化，以纳入基体材料和纤维的非线性行为。在微分几何和应变梯度弹性的框架内，制定了双向纤维的一般运动学，包括纤维的三维弯曲和两个相邻纤维之间的扭曲，并随后集成到连续变形模型中。允许的边界条件也是通过变分原理和虚功声明导出的。特别是，对所得的三维模型应用降维过程，通过该过程获得了描述用纤维网增强的弹性薄膜的面内和面外变形的兼容二维模型。为此，进行了模型实现并与实验结果进行了比较，表明所提出的模型成功预测了纤维网增强复合材料薄膜的关键设计考虑因素，包括应力应变响应、变形曲线、剪切应变分布和局部结构（单位纤维网格）变形。所提出的模型的独特之处在于，它是在表面微分几何框架内制定的，以适应复合材料的三维运动学，但所得方程在正交基础上重新构建，以增强实际的单一性和数学可处理性。因此，所得模型也可以作为二维非线性弹性中板壳的替代 Cosserat 理论。