Plastic Torsion and Related Problems

A simple algorithm to calculate the maximum torsional load for a cylindrical shaft is presented. The algorithm is based on the notion of viscosity solutions to the eikonal equation, and is not restricted to simply-connected cross-sections. Applications to other, related problems, such as ferromagnetic thin lms, and elastic buckling of thin lm blisters are also discussed. 1. Introduction A classical problem in Plasticity (more speciically, in Limit Analysis) is that of determining the maximum admissible torsional load for a cylindrical shaft. If the yield criterion is von Mises', and if the cross section is simply connected, the solution can be obtained through Nadai's \sand heap" analogy 8]. When the cross-section is not simply connected, the sand heap construction can still be used, provided that constant heights are judiciously assigned on the contours of the internal holes 1. Mathematically, constructing sand heaps amounts to solving the eikonal equation. There are many solutions to this equation, typically with discontinuous slopes. However, the tallest admissible sand heap, i.e., the maximal pointwise Lipschitz solution of the eikonal equation can be obtained as its viscosity solution (an introduction to the notion of viscosity solution can be found in 3]). Very eecient computational methods for nding viscosity solutions have become available in the last decade, and they are easy to implement 10]. Inspired by these recent advances, we present an algorithm to compute the limit plastic torque for a cylindrical shaft with multiply connected cross-section. We also discuss the application of the algorithm to a few more physical systems, whose behavior is also modelled by the eikonal equation. We hope that the few lines of Matlab code provided herein will entice many to experiment with plastic torsional loads, isotropic thin lm ferromagnets, and elastically buckled thin lm blisters.