Conformal mappings provide an elegant formulation for planar elastostatic problems.
Here, the mapping function coefficients are used in a new manner as design variables
in the genetic-algorithm (GA) approach to find a piecewise smooth optimal shape of
a single traction-free hole in an elastic plate that minimizes the local stresses under
remote shear. This scheme is sufficiently fast and accurate to numerically show that
the sought-for shape generates tangential stress of constant absolute value, equal to
less
than the stress concentration factor (SCF) for the commonly used circular hole. The
shape has four symmetrically located corners, and the stress changes sign while
remaining finite as it rounds each corner. This is the same shape as the
energy-minimizing contour identified in 1986 by the author and Cherkaev for the
same load. Other nontrivial examples are given to demonstrate the potential of the
approach. Methodologically, this article continues the optimization study first
conducted by the author and Cherkaev (J. Appl. Math. Mech. 50:3 (1986), 401–404)
and subsequently by Cherkaev et al. (Internat. J. Solids Structures (35):33,
4391–4410).
