On the parameterisation of a class of doubly periodic lattices of equally strong holes

Abstract

Summary We construct an exact, explicit parameterisation of a class of doubly periodic lattices of equally strong holes in an infinite elastic plate that is in a state of plane stress. This parameterisation assumes no symmetries of the lattices’ holes and allows for any finite number of holes per period cell. It is stated in terms of a conformal map from a circular domain. We construct this map in terms of the integrals of the first kind that are associated with a Schottky group that is generated from this circular domain. Key to our derivation of this parameterisation is the observation that a doubly periodic lattice of equally strong holes is characterised by the property that the Schwarz functions of all of its holes’ boundaries are identical up to additive constants. We also conjecture a condition that is necessary and sufficient for the existence of the class of lattices that are described by this parameterisation, although we are only able to verify this condition numerically here. We also present a selection of examples of such lattices, computed using this parameterisation.