A three-dimensional continuum model for the mechanics of an elastic medium reinforced with fibrous materials in finite elastostatics

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.