An analytical method for the plane problem of doubly periodic circular cross-section fiber composite materials

Combining the theory of doubly periodic and doublyquasi-periodic Riemann boundary value problems and Eshelby's equivalentinclusion method, an analytical method for the plane problem of compositematerials with a doubly periodic array of circular cross-section fibers ispresented. The stresses expressions in series are obtained in the fibers andmatrix and a comparison with the finite element calculations is done. Thetransverse tensile and shear moduli are predicted for a unidirectionalfiber-reinforced composite with an doubly periodic array of circular fibers.It is found that for a composite with hard fibers and a soft matrix under asame fiber volume fraction, the effective moduli for a square array offibers are larger than those for a hexagonal array of fibers. The presentmethod provides an efficient tool for analyzing the mechanical properties ofinhomogeneous materials and designing microstructures of compositematerials, and can also be used to evaluate the precision of other numericaland approximate methods such as the finite element method.