more on the stress vector, and the stress tensor

Here is the imaginary cube inside the rubber band, oriented so that the vertical faces are perpendicular to the pulling direction. The stress is perpendicular to the two vertical faces of the cube on the left and right sides, and parallel to all the other faces.

(I keep forgetting to mention, you can enlarge the drawing by pressing the ctrl and + keys, and then shrink back with crtl and -).

Imagine that the cube shrinks to an infinitesimal size. The normal and shear stresses will be associated with almost the same point in the material - and yet, they will be different because they represent the force on different surfaces, with different normal vectors n.

So we say that t is a function of both the position in the body (x), and the unit vector normal to the surface (n) under consideration. Thus the stress vector t is very different from most vectors in continuum mechanics, which are functions only of their position in space (and of time, in the case of time-dependent phenomena).

Putting it another way, it takes 6 numbers to specify the stress vector: 3 for the position x, and 3 for the surface with unit normal n, for which we want to calculate the force.

Thus each component of the stress vector (each of the x, y and z components) is a mathematical function of 6 variables. In order to develop a theory of stress, a new concept was invented which allowed an important simplification: a way of representing the stress vector with a new function, which depends only on the position x. This new idea has been called Cauchy's theorem,* and is stated in the beginning of the next post.

*According to Truesdell (The Classical Field Theories of Mechanics, by Clifford Truesdell and Richard Toupin, published in 1960 in Volume III of the Handbuch der Physik), it was the French mathematician Augustin-Louis Cauchy (1789–1857) who invented the concept of the stress tensor. As we get further into the theory, I think you will begin to appreciate the importance of this fundamental theorem, and the genius of the man who came up with it.