Effective transport properties for periodic multiphase fiber-reinforced composites with complex constituents and parallelogram unit cells

•Two-scale asymptotic homogenization method is used to find formulas for effective properties.•The constituents have complex-valued transport properties and parallelogram unit cells.•Antiplane linear elasticity problem is formulated in the frame of transport properties.•Shear effective coefficients are explicitly obtained for n-phase fiber-reinforced composites.•Also, the example of gain enhancement of inertial mass density is looked into. The two-scale asymptotic homogenization method is used to find closed-form formulas for effective properties of periodic multi-phase fiber-reinforced composites where constituents have complex-valued transport properties and parallelogram unit cells. An antiplane problem relevant to linear elasticity is formulated in the frame of transport properties. The application of the method leads to the need of solving some local problems whose solution is found using potential theory and shear effective coefficients are explicitly obtained for n-phase fiber-reinforced composites. Simple formulae are explicitly given for three- and four-phase fiber-reinforced composites. The broad applicability, accuracy and generality of this model is determined through comparison with other methods reported in the literature in relation to shear elastic moduli and several transport problems of multi-phase fiber-reinforced composites in their realm, such as conductivity in a biological context and permittivity leading to gain and loss enhancement of dielectrics. Also, the example of gain enhancement of inertial mass density is looked into. Good agreement with other theoretical approaches is obtained. The formulas may be useful as benchmarks for checking experimental and numerical results.