Relative volume change and the trace of the strain tensor

Suppose we have a cube subject only to normal stresses on its six faces, as shown below.  We expect the cube to expand, to form a slightly larger body with parallel sides (although not necessarily a cube).  

Let the cube have sides of dimension lx, ly and lz in the x, y, and z directions, and the displacements of the x, y and z planes (those with normals in the x, y, and z directions, respectively) be ux, uy, and uz.  Then the relative change in volume due to the normal stresses will be

Since these are the diagonal terms of the strain tensor, the right hand side is called the trace of the strain tensor.

From the drawing at the top, it might appear that Hooke's law would have a term in which the trace of the stress tensor was a linear function (i.e., a constant times) the trace of the strain tensor (and vice versa).  In fact, this is the case, and

In the next blog, we'll look at how to combine the shear and compression terms into a single expression for Hooke's law that covers all cases.