#### Vol. 16, No. 4, 2021

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Stress-smoothing holes in a regularly perforated elastic plate with a given effective bulk modulus

### Shmuel Vigdergauz and Isaac Elishakoff

Vol. 16 (2021), No. 4, 511–525
##### Abstract

A thin elastic plate with a square lattice of traction-free holes is considered as a setup for the optimization problem of finding the hole shapes which minimize the (nonnegative) variation of the hoop stresses (MSV) induced by unit bulk load at given perforation rate and effective bulk modulus $\left({K}_{e}\right)$ of the structure. In this context, there is the well-known correspondence between the local and the averaged stress-strain field. The starting point of this study is the analytically proven existence of the specially shaped (equistress) holes which together provide the global maximum of the structure’s bulk modulus and a zero-variation (constant) stress distribution around the hole’s face. Here, using an effective optimization scheme we numerically analyze the ${K}_{e}$-to-MSV relation for nonoptimal ${K}_{e}$ and hence nonzero MSV across certain representative intervals of their values depending on the structure’s porosity. This is performed by explicitly finding the optimal hole shapes and the attendant stress distributions. The results obtained are detailed in tables and graphs.

##### Keywords
2D elastostatic problem, Kolosov–Muskhelishvili potentials, hoop stress variations, shape optimization, bulk effective modulus, genetic algorithm