יום חמישי, 24 בדצמבר 2009

Integral Criteria for Diminishing the Boundary Stresses in Elastic Domains with Holes

S. Vigdergauz

Israel Electric Corp., Ltd.

The following linearly elastostatic optimization problem for a 2D region with holes is considered:

Given an external load, find the holes areas and their mutual arrangement and shapes in order to provide a favorable hoop stress distribution.

Here, the engineering aim is usually either (A) to minimize the maximum stress along the holes or (B) to smooth them to avoid both occurrence of local stress concentrations and underloading of certain parts of the boundary. The absolutely non-trivial and purely analytical example here are the equi-stress shapes (ESS) along which the stresses are simultaneously constant and globally minimal [1]. However, they exist only in an infinite plane and under a bulk-dominating far loading. In any other case a numerical analysis should be implemented when both criteria require the point wise stress estimation with substantial accuracy. This is a hard computational task especially within a repetitive optimization problem. Instead, we propose to use the integral counterparts of the above mentioned local estimations:

(A') - Ln norm of the hoop stresses which reaches the stress maximum (A) as n tends to infinity and (B')- the hoop stresses variation V fully corresponding to (B). In the ideal limit the variation goes to zero or, equivalently, to the equi-stress condition.
In contrast to the ESS, the (A') – and (B')-optimal holes exist under much weaker conditions. On the other hand, the integral criteria are computationally more stable than the point wise quantities and hence can be easily embedded into stochastic optimization. For a variety of geometries, this two-fold advantage permits us to numerically solve the shape optimization problem at hand by a standard genetic algorithm approach. As an example, a circular disk with two identical side holes under a uniform compression is considered. The results obtained show that the optimal holes are kidney-like in shape. Compared to the widely used circular holes they diminish the hoop stresses up to tens of percents depending on the holes mutual arrangement. It should be remarked that the newly defined criterion is taken over the total region boundary including the fixed outer contour of the disk.
The proposed criterion of the stress field optimality may find further use in arious continuum mechanics problems.

Literature:
1. Cherepanov, G.P, 1974, "Inverse problem of the plane theory of elasticity," J. Appl. Math. Mech. (38) (5), 913-931.

2. Vigdergauz, S.B, 1976, "Integral equation of the inverse problem of the plane theory of elasticity," J. Appl. Math. Mech. (40) (3), 566-569.

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