Static Model Describing the Inhomogeneous Deformation of Unidirectional Fibrous Composites under Plane Strain

Abstract

A mathematical model is proposed to describe the static deformation of a unidirectional fiber-reinforced composite under plane-strain conditions in a plane perpendicular to the fiber direction. The model is based on the stochastic static equations for an elastic homogeneous two-component medium with nonzero body forces. Applying the method of conditional moments and double Fourier transform yields a nonlocal model in the form of integro-differential equations. Expanding the integral kernels into series in expansion coefficients yields differential equations whose order depends on the number of terms in the series. The zero-order approximation leads to the theory of effective moduli, the first-order approximation to the refined theory of effective moduli, and the second-order approximation to fourth-order equilibrium equations for mean displacements (mathematical expectations) and formulas for displacements, strains, and stresses averaged over the composite and its components. All the coefficients in theses formulas are expressed in terms of the elastic constants of the components and the geometric parameters of the structure. The model is valid for heavy stress gradients. In a particular case, the static theory of two-component elastic mixtures follows from the model