Chapter Sixteen - Fibrous composite with anisotropic constituents

Abstract

In this chapter, the local stress filed and effective elastic stiffness of a unidirectional fiber reinforced composite with anisotropic constituents is studied by the multipole expansion method. The geometry of the composite is modeled by a periodic structure with a representative unit cell containing multiple circular fibers. To obtain the complete displacement solution for the multiparticle model of the composite, the superposition principle is combined with the multipole expansion of perturbation fields of inhomogeneities in terms of elliptic harmonics. Following Lekhnitsky (1963), a general solution to the plane strain problem for an anisotropic solid is written in terms of complex potentials. An appropriate choice of these potentials and their multipole series expansion reduces the boundary-value problem for the multiple-connected domain to an ordinary, well-posed set of linear algebraic equations for the induced multipole moments of the inhomogeneities. The developed theory is valid for the general anisotropy of constituents and arbitrary orientation of the orthotropy axes. Exact expressions for the in-plane and out-of-plane components of the effective stiffness tensor of a fibrous composite with anisotropic constituents are obtained by analytical averaging of the strain and stress fields. An alternative, dipole moment based derivation of the effective elastic moduli in the framework of Rayleigh homogenization scheme is outlined.