Homogenization and boundary estimates of shear stiffness for the composites of the tetragonal structure

New formulas of longitudinal and transverse shear moduli are obtained for tetragonal composites with rectangular cross-section fibres. With the use of Lagrange's and Castigliano's variational principles in generalized form, it is proven that obtained approximate shear moduli give the lower and upper bounds of the exact values and lie within the Reuss–Voigt interval. Both analysis and computations show that by averaging these bounds the effective shear moduli can be established with a practically acceptable error. General technique of developing mathematical models, as formulated in the paper, can be also used in other homogenization problems.