Effective Elasticity Tensor of Fiber-Reinforced Orthorhombic Composite Materials with Fiber Distribution Parallel to Plane

Abstract

An orthogonal composite material Ω with fibers consists of a matrix and orthothombic distribution fibers. In addition to the matrix properties, the fiber properties and the fiber volume fraction, the effective (macroscopic) elastic stress–strain constitutive relation of Ω is related to the fiber direction distribution. Until now, there have been few papers that give an explicit formula of the macroscopic elastic stress–strain constitutive relation of Ω with the effect of the fiber direction distribution. Taking the expanded coefficients of the Fourier series as the fiber direction distribution coefficients, we give a formula of the fiber direction distribution parallel to a plane computed through the fiber directions. By the self-consistent estimates, we derive an explicit formula of the macroscopic elastic stress–strain constitutive relation of Ω with the fiber direction distribution coefficients. Since all tensors are represented in Kelvin notation, the macroscopic elastic stress–strain constitutive relation of Ω can be derived and computed only by matrix manipulations. To check the explicit formula, we use the FEM computation to obtain the macroscopic elastic stress–strain relation of Ω for three examples. The computational results of the explicit formula for the three examples are consistent with those of the FEM simulations.