Effective elasticity tensors of fiber-reinforced composite materials with 2D or 3D fiber distribution coefficients

A fiber-reinforced composite material \(\mathcal {N}\) consists of a matrix and numerous fibers. Besides their intrinsic properties and the volume fractions of matrix and fibers, the effective elasticity tensors of \(\mathcal {N}\) are also related to the two-dimensional (2D) or the three-dimensional (3D) fiber direction distributions. Herein the Fourier series and the Wigner D-functions are introduced as the 2D and the 3D fiber direction distribution functions (FDF), respectively. The expanded coefficients of the FDF are called the fiber distribution coefficients (FDC). When \(\mathcal {N}\) consists of an anisotropic elasticity matrix and numerous transversely isotropic fibers, we derive the effective elasticity tensor \(\widehat{\mathbf {C}}\) of \(\mathcal {N}\) by the self-consistent method with the 2D FDC or the 3D FDC. The FDC can be easily obtained via the fiber direction arrangements of \(\mathcal {N}\) for the fiber arbitrary or orthorhombic distributions of \(\mathcal {N}\). The procedure of deriving \(\widehat{\mathbf {C}}\) is simple because the Kelvin notation is used to compute tensor rotations. When both the matrix and the fibers are isotropic, for the 2D fiber distributions at least three direction arrangements of fibers are needed to build the fiber-reinforced transversely isotropic composite materials, and for the 3D fiber distributions at least six direction arrangements are needed to build the fiber-reinforced isotropic composite materials. The results of the FEM simulations are consistent with those of our expressions \(\widehat{\mathbf {C}}\).