Project_08
Elastoplastic Torsion
Twist and Stress


Project 8 makes a model of the behavior of a long thin rod that is twisted. Small amounts of twisting are reversible, and the rod will return to its original shape when the twisting force is released. This is called elastic behavior. However, large amounts of twisting can cause the rod to be permanently deformed. In such cases, the rod is said to be undergoing plastic deformation.

In the model to be constructed and examined here, the amount of twisting force will vary along the rod; over the rod's extent there may be both elastic and plastic deformation regions.

This case study is a chance to consider the engineering concepts of stress and shear for physical materials, and numerical methods for modeling the behavior of materials to which forces are applied.

Surprisingly, one part of the model requires us to compute the distance between an arbitrary point and an ellipse. This problem is considerably harder than what happens when the shape is a circle. In fact, this is an example of a nonlinear equation for which there is no formula for the solution. Instead, we need to apply some sort of approximate iterative method. Finding nearest points to geometric objects is a common problem in Computational Geometry.

Reference:

  1. Dianne O'Leary,
    Elastoplastic Torsion: Twist and Stress,
    Computing in Science and Engineering,
    Volume 6, Number 4, July/August 2004, pages 74-76.
  2. Dianne O'Leary,
    Scientific Computing with Case Studies,
    SIAM, 2008,
    ISBN13: 978-0-898716-66-5,
    LC: QA401.O44.


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Last revised on 10 February 2009.