“Van Fo Fy Method” in the Micromechanics of Fibrous Composites

A method for calculating the stress-strain state in “fiber-matrix” microstructures (“Van Fo Fy (VFF) method”) was developed by the famous scientist in mechanics G. A. Vanin. The method is based on the satisfaction of boundary conditions on the interfaces with application of series of derivatives of doubly-periodic elliptic Weierstrass functions. Problem solutions for various loading conditions of unidirectional fibrous composites have an analytical form expressed in functional series, but their calculation presents considerable difficulties, which, until recently, limited the application and development of the VFF-method. In this paper, in order to explain this method, the derivation of the basic micromechanics relations that make it possible to calculate the stress and strain fields in unidirectional composite structures and, on their basis, to estimate the effective elastic properties of composites with a double-periodic arrangement of fibers was briefly presented. The method also allows us to estimate other effective (averaged over the periodicity cell) physical properties: thermal conductivity, electrical conductivity, magnetic permeability, etc. VFF-method admits extension to multilevel hierarchically organized structures reflecting the structure of natural biocomposites. As illustrations, the paper presents solutions of the problems for unidirectional composites with circular fibers, although the method allows one to solve problems for elliptical, hollow, arbitrarily arranged fibers. For the numerical implementation of the method, which was not carried out before, the analytical relations obtained were brought to algorithms and programs by means of computer algebra. The results of calculations of microstress fields in composite structures with different variants of periodicity of arrangement of elementary cells were presented as the particular examples. Calculations by the VFF-method and the standard numerical finite element method were compared and a good agreement of the results was demonstrated.