Abstract

A rigid inclusion perfectly embedded in a thin plate is considered as a two-dimensional elastostatic composite structure to solve the inverse problem of finding the inclusion shape around which the local maximum of the von Mises equivalent stresses attain the global minimum under shear loading at infinity. Absent optimality preconditions such as the equistress principle for bulk-type loading, a fast and accurate assessment of a given shape is developed by combining complex-valued series expansions with a new infinite summation scheme. This approach to solving the direct problem is then included into a genetic algorithm optimization over the set of shapes obtained from a circle by a finite-term conformal mapping with square symmetry. Compared to a circular inclusion, the stresses may thus be lowered by 15–25%, depending on the Poisson’s ratio of the plate. The numerical results presented allow us to conjecture that the von Mises stresses around the optimal shape are uniform. The inclusion that minimizes the induced energy increment is also identified by the same approach. Both shapes appear to be very similar, though not identical.

Keywords

plane elasticity problem, Kolosov–Muskhelishvili potentials, shape optimization, effective energy, extremal elastic structures, genetic algorithm

Authors