4

In Landau and Lifshitz's book on the theory of elasticity (vol 7, theoretical physics series), specifically section 2 of the first chapter, the authors present an argument for justifying the symmetry of the stress tensor. On page 6, they derive an expression for the net moment of the force acting on a portion of a body due to the other parts of the body.

$$M_{ik}=\int\left(F_ix_k-F_kx_i \right) dV$$

They write, "Like the total force on any volume, this moment can be expressed as an integral over the surface bounding the volume."

What is the logic behind this statement? I understand the net force being solely a surface integral since the molecular forces resulting from a deformation are essentially short-range. They go on to use this to prove that the stress tensor is symmetric.

Note: The only assumption declared prior to this argument is that the aforementioned internal forces are short-range. No other assumptions seem to have been declared. (Please correct me if I'm wrong)

Qmechanic
  • 220,844
Arkya
  • 703

2 Answers2

3

One way to interpret it is that they're assuming you can apply the divergence theorem backwards; that is, that the integrand $\left(F_ix_k-F_kx_i \right)$ is itself the divergence of some well-defined vector field $\mathbf{\tau}$, which is connected to the traction vector field $\mathbf{T}$.

(Curiously, most continuum mechanics books I've seen do it the other way around; they derive integrals of moment densities over a surface and then use the divergence theorem.)

aghostinthefigures
  • 3,287
1

When you add up all the forces acting on an assembly of particles (in continuum mechanics this comes down to the net force acting on a region or a volume) the net force should be due only to the interaction with external particles, because all the internal forces should cancel out due to Newton's 3rd Law. For the forces due to molecular interactions (described by the stress tensor) the net force coming from external particles should be coming from very close to the surface (because of their short range, with long-distance forces, as when the system is electrically charged, the argument is different). Since the stresses should only count at the surface, this means they should enter the problem as a divergence. This, in turn, allows us to write a conservation law for momentum in the form:

partial (momentum density) /partial t + div(some quantity) = 0

This "some quantity" turns out to be the explicit momentum flux (a tensor calculated from the density and two factors of velocity, something like rho v_i v_j) and the stress tensor. When we write a conservation law in this form, the conclusion is that the total value of the conserved quantity inside the volume can change only if something crosses the surface of that volume.

With momentum we are OK, because precisely for these reasons we have already decided that there must exist a tensor (the stress tensor) such that the net force (from close-range molecular forces) acting on a volume must be the divergence of that tensor.

Something similar should happen with the total moment (or torque) acting on a group of particles. The total torque on a volume due to molecular stresses should only come from the particles outside (because of the strong form of Newton's 3rd Law, that action-reaction force pairs are not only equal and opposite, but also act along the same line). This should lead to a similar conservation law for angular momentum, except that it is not so obvious that the cross product of r and the force (i.e. the cross product of r and the divergence of the stress tensor) is itself a divergence. In other words, we need:

vec(r) cross DIV(tensor(sigma)) = DIV(something)

or

r_i (div(sigma))_j - r_j (div(sigma))_i = div(something)_k

The obvious candidate for "something" in this case is r times sigma, but for this to work we need the antisymmetric part of sigma to be zero, although not quite.

This is where L.L. put in a clarification (I think this was introduced already without Landau). The point is that the antisymmetric part of sigma does not need to be necessarily zero; it can also be, itself, the divergence of some other (3rd rank) tensor. This gets a bit tricky, because it gets into arguments that the stress tensor is not uniquely determined (like there is some kind of gauge freedom) that I feel the authors could have explained better (one could counter-argue, for example, that what has real physical meaning is not just the divergence of the stress tensor, which is the net force on a volume, but also the stress tensor times the normal at any point of any, possibly imaginary, surface, which is often a boundary condition for the problem on that surface). As presented in L.L. it would seem that one could add, among other things, an arbitrary constant hydrostatic stress to any stress state without changing anything. I am not saying their argument is wrong, but it certainly deserves more discussion. The text is rather terse on the issue.

The authors cite this article (the argument is in Appendix A, the rest of the article is interesting, but not directly relevant):

Martin, P. C., O. Parodi, and Peter S. Pershan. 1972. Unified hydrodynamic theory for crystals, liquid crystals, and normal fluids. Physical Review A 6(6): 2401-2420. doi:10.1103/PhysRevA.6.2401 Permanent link: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10361959

I am actually interested in further clarification of the non-uniqueness of the stress tensor.

Otto Linsuain
  • 11