Closed-form expressions for the effective coefficients of a fiber-reinforced composite with transversely isotropic constituents – I. Elastic and square symmetry

Abstract A two-phase parallel fiber-reinforced periodic elastic composite is considered wherein the constituents exhibit transverse isotropy. The fiber cross-section is circular and the periodicity is the same in two orthogonal directions. Simple closed-form formulae are obtained for the effective properties of this composite by means of the asymptotic homogenization method. Numerical computation of these is easy. The analytical solution of the required resulting plane- and antiplane-strain local problems, which turns out to be only three, makes use of potential methods of a complex variable and properties of Weierstrass elliptic and related functions with periods (1,0) and (0,1). Dvorak's universal type of relations for this composite are easily derived in an elementary new way without solving any local problem. This result also applies when the interface may be arbitrarily shaped, but compatible with the square symmetry. Comparison with experimental data is shown. The above results include the situation when one or both phases are isotropic.

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