Unified series solution for the anti-plane effective magnetoelectroelastic moduli of three-phase fiber composites

The anti-plane magnetoelectroelastic behavior of three-phase magnetoelectroelastic composites (fiber/interphase/matrix) with doubly periodic microstructures is dealt with. With the aid of the matrix notation, the anti-plane magnetoelectroelastic coupling problem is formulated as same as the anti-plane piezoelectric coupling problem. And then the eigenfunction expansion-variational method (EEVM) is extended to solve such a problem. Series solutions for the effective magnetoelectroelastic moduli are presented, which are in a unified form for generally periodic fiber arrays, different unit cell shapes as well as different constituent properties, and are applicable for high volume fraction of fibers. With the present solution, it is found that the effective magnetoelectric coefficient of a two-phase composite may have two local extrema rather than only one extremum predicted by the Mori-Tanaka method. By optimizing the volume fraction, permutation and the choice of the constituent phases, the maximum magnitude of the effective magnetoelectric coefficient of a three-phase composite can be much larger than that of any of the two-phase composites, and the sign of the magnetoelectric coefficient can be changed, which is not observed in a two-phase composite. For composites with a generally periodic array of fibers, the effective magnetoelectric moduli can be anisotropic.