Cross Relations Between the Planar Elastic Moduli of Perforated Structures

The effective compliance moduli of a plate with a doubly periodic set of traction-free holes are considered. Attention is drawn to the perturbation form in which they are expressed by applying the complex variable methods in two-dimensional elasticity. This permits one to derive specific dimensionless combinations of the effective moduli, which are independent of the solid Poisson ratio. Using them saves computations of the structure moduli by FEM-like methods and helps one to evaluate their practical accuracy. Thus far, the only result of this kind has been observed numerically by Day, Snyder, Garboczi, and Thorpe (J. Mech. Phys. Solids. 40, pp. 1031–1051, 1992) and later proved by Cherkaev, Lurie, and Milton (Proc. R. Soc. London, Ser. A 458, pp. 519–529, 1992).