#### Vol. 1, No. 2, 2006

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The stress-minimizing hole in an elastic plate under remote shear

### Shmuel Vigdergauz

Vol. 1 (2006), No. 2, 387–406
##### Abstract

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 $30%$ 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).

##### Keywords
plane elasticity problem, Kolosov–Muskhelishvili potentials, shape optimization, effective energy, extremal elastic structures, genetic algorithm
##### Milestones
Received: 29 September 2005
Revised: 16 December 2005
Accepted: 23 January 2006
Published: 1 June 2006
##### Authors
 Shmuel Vigdergauz R&D Division The Israel Electric Corporation, Ltd. P.O.Box 10 Haifa 31000 Israel