Page 3: Continuum mechanics is a perpetuum mobile theory (1)

In order to assess Cauchy's theory properly it helps to realize that in all the 16 papers by Cauchy which I have studied I have not found one single mention of physical work. That's too bad; he would have found out all by himself that he wrote a perpetuum mobile theory.

I have published three demonstrations to show that current texbook theory always leads to the conclusion that the work done in a volume-neutral deformation is zero <www.elastic-plastic.de/koenemann2008-2.pdf>. (Today I could quote a fourth source.) Why is this so?

All of classical physics (this side of Einstein & Planck) can be grouped into two very fundamental categories. Every system – a volume in space containing mass, e.g. a kinetic system with n bodies – contains a certain energy U. A process that does not change U is called conservative since U is "conserved", i.e. invariant. Commonly the system is isolated, which means that there is no exchange of mass or energy between system and surrounding across the system boundary.

The energy conservation law E_kin + E_pot = U = const defines a conservative process. Any process that observes this law will turn E_kin into E_pot and vice versa; the change is the work w. Hence all the work is done within the system. There is no exchange with a surrounding. Classical examples of a conservative process are the revolution of planets about the sun, or diffusion in water at rest at constant T.

If a process changes the system energy U, it is called non-conservative. It requires that energy fluxes take place between the system and its surrounding across the system boundary. Thus we need a new energy conservation law that takes account of the fluxes; this is the First Law of thermodynamics, dU = dw + dq. Non-conservative processes may be reversible or irreversible. In this case the work is done upon the system. It is commonly known as PdV-work.

The difference between a conservative and non-conservative process can be given by a simple mathematical condition.

  -  If there are no fluxes f, the divergence div f = 0. This is called the Laplace condition.

  -  If there are net fluxes f, the divergence div f = phi =/= 0. This is called the Poisson condition.

phi is the charge, which is a measure of the work done upon the system. Now, is elastic deformation a conservative or a non-conservative process? Clearly work is done upon the system such that it deforms, and energy is stored in the system, the elastic potential.

But it is well known, taught in every intro class, and found in every textbook, that the trace of the stress tensor tr s = s_11 + s_22 + s_33 = 0 for a volume-constant deformation. This is the Laplace condition. That is, the no-work condition is solidly built into Cauchy's stress theory.

How could this happen? Very simply: in the mid-18^th C when Euler thought about all this, Poisson's condition and the First Law of thermodynamics were still many decades in the future. How could he know that the energy of a system can be a variable? That was understood only after 1845. The only theoretical template Euler knew was Newton's mechanics which is rightfully conservative, but unsuited for any understanding of elasticity. 

Today we would start by asking: is elasticity conservative or non-conservative? Of course the latter. Thus we would turn to thermodynamics, assume a system with a given amount of mass, e.g. one mol, and then study the energetic fluxes between system and surrounding. And we would use div f = phi =/= 0 as a test to see whether we made a mistake – because it must be non-zero.

If the Gauss divergence theorem is the entrance gate into potential theory, the Laplace and Poisson conditions are its door wings. Potential theory is a wonderful and incredibly powerful theory, the backbone of all of classical physics, and the core of a myriad of methods in applied mathematics. Continuum mechanics has managed to ignore it completely and entirely.