The divergence of the stress tensor

The law of conservation of momentum usually says that the net force on an object is equal to its rate of change of momentum.  But for a region inside the object, there has to be an additional term, to account for the possible flux of momentum through the enclosing surface.  So the correction to the last equation from the previous blog, for conservation of momentum, must be as shown below.  The arrows over P and n are to remind us that the momentum and unit normal are vectors, and similarly for the tilde over the stress tensor.  This equation states that the rate of change of momentum of material within the volume V is equal to the combined effects of 1) a source term, given by the force density f, and 2) a flux term, given by the stress tensor dotted into the unit vector normal to the surface.  

 This equation fits the general form a conservation law in continuum mechanics, with the net rate of change of a quantity  (mass, momentum, energy) being the sum of a source within the region being considered, and the flux of the quantity in through the surface.  We have a minus sign for the flux term, because we take n to be the outer normal vector.  I should add that "continuum mechanics" is the study of materials whose properties, like mass, momentum and energy, are continuous functions of their spatial coordinates, i.e., x, y and z.  One more comment:  The volume V and its enclosing surface S are fixed in space.  It's hard to imagine much of a change in momentum in a solid object being slowly deformed - by stretching, compressing, bending, shearing, etc.  Yet the force required to cause the deformation can be very large, so this equation must represent the small difference between two very large numbers.

In calculating the work done in deforming a body, we assume that the rate of change of momentum, dP/dt, is so small it can be neglected, and therefore we can take the terms on the right hand side of the above equation as being equal.  

We now continue the discussion of work.  To get started, we make use of a formula from vector calculus, known as the divergence theorem (you can read about it in Wikipedia).  It states that the surface integral above can be transformed into a volume integral, by taking the divergence of the stress tensor:

The integrands on both sides are vectors, because the stress is a tensor.  By combining this this result with the previous equation (and with the rate of change of momentum equal to zero), we find:

Since the volume of integration is arbitrary, the integrands must be equal everywhere, and this leads to the second and third equations above (this time omitting the arrow and tilde).  The vector force acting on an infinitesimal volume of material is equal to the gradient of the stress tensor within that same small volume.  For a material in equilibrium, whose properties are the same in all directions (i.e., is isotropic), the divergence of the stress tensor is zero.  

We know that an elastic material, according to Cauchy's definition (see the blog about stretching a piece of rubber tubing), is one in which the stress depends only on the strain.  So this means we should be able to find an expression for the strain, set equal to zero.  

Since the strain depends on the gradient of the deformation (see the blog just referred to), we expect to find a set of three second order differential equations (for i = 1,2,3) for the deformation vector u of an isotropic material.  These equations, plus boundary conditions, should have a unique solution.

Unfortunately, nearly all boundary value problems in elasticity are very difficult to solve analytically, so there aren't very many examples that can be used just to clarify the concepts.  

Next time, we'll talk about Hooke's Law, the one we used for the rubber tubing example, generalized to three dimensions.