Tensor Ellipsoids

Tensor Ellipsoids
Hyper Streamline
Copyright (c) 1996 Prentice Hall PTR, Will Schroeder, Ken Martin, Bill Lorensen - The Visualization Toolkit

A symmetric second-order tensor can be visualized by defining an ellipsoid from its eigenvectors. The length of the eigenvectors is determined by the corresponding eigenvalues.
In above pictures a single point load is applied to an elastic material, resulting in singular stress and strain values. The top image visualizes the stress field with tensor ellipsoids. In stress analysis the axes of the ellipsoids are the principle direction of the stress. At the surface of the material the ellipsoids flatten because there is no stress perpendicular to the surface. Near the singularity the ellipsoids elongate and point in the direction of the load point.
The bottom picture shows the same tensor field visualized with hyperstreamlines. Note how the hypersteamlines point in direction of the point load and flare as they approach the stress singularity. Hyperstreamlines are explained in image 1