How to handle onedimensional data in a threedimensional world.
By Yung S. Oh
Consultant
Grayslake, Ill.
Edited by Kenneth Korane

The fields of finiteelement analysis, failure analysis, and solid mechanics all express stresses in terms of tensors. Engineering literature usually treats a tensor as a symbol dressed with indices satisfying certain rules. It looks more complicated than a vector and its true nature seems evasive, raising the question as to why such a complicated concept is needed to describe stress.
After all, stress is basically defined as:
α= F⁄A
where F = force and A = area. This seems to be the same as pressure, but pressure is scalar, that is, a number. Actually, pressure is a special case of stress. To help explain why engineers need tensors, consider a fixed threedimensional object subject to external force F. Let's look at the internal forces induced by F.
If the object is cut by a plane passing through internal point P, a vector perpendicular to the plane, called a unit normal vector N, defines the plane. The internal stress is the induced force vector over the unit area at P, called a traction vector, T.
Stress can therefore be interpreted as internal tractions that act on a defined internal plane. One cannot measure stress without first specifying the datum plane.
Stating the internal traction stress in terms of the unit normal vector,
T = αN.
But N and T are threedimensional vectors. From linear algebra, any linear transformation of one vector to another can be described as a 3 X 3 matrix, or
Therefore, the matrix components i,j for i,j = 1,2,3 completely determine the transformation. These are in fact the stress tensors. Knowing all nine components _{i,j }(actually, only six are needed due to symmetry), one can find any internal stress. To be more precise, the internal stresses are the traction vectors and the stress tensors are the transformation of the unit normal vector to the internal traction stress. In most engineering literature, these concepts are often used interchangeably.
Going back to the case of pressure, stress tensors are not needed because the surface is already given — there is only one unique unit normal vector N to the surface and a unique traction vector T, the pressure. Because T is always parallel to N, the magnitude of T describes the pressure. In other words, there is no point of bringing in the concept of stress tensors to describe pressure.
The six stress tensors completely determine internal stresses. The next step is relating them to actual stress analysis of a specific material. Experimental tensile testing typically determines material properties, with results given in terms of yield, ultimate, and other stresses. But all these stresses are one dimensional, scalar, along the axial direction. Relating the six stress components to a single scalar quantity brings into play the von Mises stress.
The equivalent, or von Mises stress _{M }is based on stress invariant theory, and defined as:
The essential point is that the equation combines six stress components into a single number, which can be compared with a tensile specimen's yield stress. Think of the von Mises equation as a bridge between 3D objects and the 1D tensile specimen.
The equivalentstress equation works well as a failure criterion for ductile materials. However, it may not be appropriate for brittle materials. In this case tensionstress components are more important than the "average" value. Thus, the maximum tension/compression stresses are critical parameters for the stress analysis. These are the three principal stresses, which should be compared with the material yield stresses.
To determine principal stresses, the traction vector T must be exactly normal to the cutting plane, that is, parallel to N. (Note that in general, T and N are not parallel). In this special case the traction vectors are the principal stresses.
This is the real reason behind the concept of stress tensors. In 3D objects or space, engineers need a coordinate system to express the stresses. But the coordinate system is manmade. Stress expressed in one coordinate system may differ from the same stress in another coordinate system, even though they are identical. Therefore, one must impose covariant (or contravariant) transformation rules so one can recognize them as equal even though they look different.
The essence of the tensor is expressing 3D stresses independent of a particular coordinate system. The device is so useful that it is ubiquitous in engineering and physics literature. Onedimensional objects already have a natural coordinate system, namely the axis. And thus stress is expressed uniquely in terms of this coordinate system, and tensors are not needed.
There are other moresophisticated stress tensors in solid mechanics, such as PiolaKirchhoff stress tensors. These consider not only the externally applied forces but also the object's deformation induced by such forces. If applied forces are small and the deformation minimal, the above consideration is usually sufficient.