Exact relations for the anti-plane effective magneto-electro-elastic coefficients of two-phase fibrous composites

Two-phase fiber-reinforced magneto-electro-elastic composites are considered. The constituents exhibit transverse isotropy and the composite is assumed to have global monoclinic symmetry. The Milgrom–Shtrikman compatibility conditions are applied to obtain explicitly exact relations for the eighteen anti-plane effective coefficients. Such relations are written in terms of nine equalities of fourth-order determinants. These fourth-order determinants exhibit the regularity of a third-order minor formed by the response matrix of the matrix material and are completed by a row and/or column of the response matrices of the fibers material and the composite, respectively. Other two less explored alternative theories, namely, a second type of the Milgrom–Shtrikman conditions, which involve only effective coefficients, and Milgrom's version of the original Milgrom–Shtrikman conditions, are followed in order to derive twenty and forty exact relations, respectively. Particular and limit cases are recovered from the obtained relations.