Shape optimization in an elastic plate under remote shear: from single to interacting holes

Abstract

An elastic plate with two closely spaced identical holes of fixed area is taken as a two-dimensional sample geometry to find the interface shape which minimizes the energy increment in a homogeneous shear stress field given at infinity. This is a transient model between a single energy-minimizing hole and a regularly perforated plate, both numerically solved by a genetic optimization algorithm together with a fast and accurate fitness evaluation scheme using the complex-valued elastic potentials which are specifically arranged to incorporate a traction-free hole boundary. Here the scheme is further enhanced by a novel shape-encoding procedure through a conformal mapping of a single hole rather than both holes simultaneously as is done in standard practice. The optimized shapes appear to be slightly rounded elongated quadrangles aligned with the principal load axes. Compared to the single (square-like) optimal hole, they induce up to 12% less energy depending on the hole spacing. Qualitatively, it is also shown that the local stresses, computed along the optimal shapes as a less accurate by-product of the optimization, exhibit a tendency to be piecewise constant with no local concentration.