Principal Axes and Invariants

Abstract

We have seen (on p. 46) how the force transmitted in an arbitrary area element dA can be described in terms of the nine components σ^ij of the stress tensor. We may ask whether, among all the area elements passing through a given point, one can find a special one for which the force σ^ijdA _i happens to be normal to the area on which it is acting. In that case the stress would be a simple tension or compression, not accompanied by a shear stress. The normal to such an area element and, hence, the direction of the stress in it, is called a principal direction or a principal axis of the stress tensor σ^ij.