ON SETS OF MULTIPLE EQUALLY STRONG HOLES IN AN INFINITE ELASTIC PLATE: PARAMETERIZATION AND EXISTENCE

This paper addresses the inverse problem of determining sets of any finite number of "equally strong" holes (with smooth boundaries) in an infinite, homogeneous, isotropic elastic plate that is subjected to uniform in-plane loadings along the holes' boundaries and at infinity. Our two main results are (i) we construct an exact, explicit parameterization that describes all such sets of holes in terms of a one-to-one conformal map from a circular domain; (ii) we derive a condition on the loading parameters that is both necessary and sufficient for such a set of holes to exist. We derive result (i) by considering the Schwarz functions of the holes' boundaries, introducing a Schottky group that is generated from our preimage circular domain, and making use of the theory of automorphic functions. We derive result (ii) by making use of our parameterization and exploiting existing results relating to univalent conformal maps. In addition to presenting results (i) and (ii), we discuss a connection to quadrature domain theory.