Dugald Carmichael schrieb:
> Neither P nor U/V can vary with scale at constant state. I doubt you have
> disproven the first law of thermodynamics; there must be some mistake.
I don't want to disprove the First Law - quite the opposite, I want to apply it!
All I do is to reformulate it from scalar form (V, P, T) to vector form (f, r),
but we haven't discussed this here yet.
> By the same dubious token (i.e., assuming that matter is homogeneous at the
> atomic and subatomic scale), Cauchy contended that components of stress
> approach finite values as V approaches zero.
This is two problems in one sentence.
1. In order to apply calculus to thermodynamics we must be sure that the
properties of matter are differentiable at all scales. If not, the terms dV or
dm (mass differential) are meaningless. Therefore it is assumed that mass is
perfectly homogeneously distributed in space. The simplest and best-known
example of this concept is the ideal gas which does not have atoms: it is a
mathematical abstraction of reality that works at macroscopic scales, ie. larger
than a couple 1000 atoms. Necessarily, such a theory must break down at the
atomic scale where you need statistical mechanics; but as you increase the scale
of consideration, the two theories must merge.
Both Cauchy and I apply the same concept to solids, and I see nothing wrong with
it. In the literature one commonly finds the assumption that a distribution (of
mass, temperature, etc.) is "twice differentiable"; this statement has the
purpose to exclude the existence of discontinuities at all scales.
2. The reason why Cauchy and I come to different conclusions regarding the
components of stress, has nothing to do with homogeneity, but with the
mathematical properties of planes and surfaces. Cauchy's tetrahedron is not a
volume element in the sense that it is surrounded by a closed surface (with
facets), but his planes are free surfaces that divide the universe into left and
right etc. Cauchy did not consider the relation of surface per mass, and I do.
> >If stress is PdV-work the stress theory must be compatible with thermodynamic
> >principles. Currently this is not the case.
>
> Stress is not PdV work. Seems to me stress is simply a generalization of P
> for solid systems in a state of homogeneous elastic strain.
Consider isotropic compression only, for simplicity. Then you do work to get the
system from the unloaded into the loaded state, right? P is a state function. It
does not have spatial properties, ditto for work, temperature, volume, density.
If you wish to describe the anisotropic shape of a body, you do it in terms
other than volume. Both in isotropic and anisotropic loading an elastic
potential develops. Isn't that what we call stress? If so, it is akin to
PdV-work.
> To my knowledge, none of the many attempts to reconcile stress/strain
> theory with thermodynamic principles found it to be incompatible with them.
Dugald, there _are_ incompatibilities between thermodynamics and elasticity
theory. The thermodynamic term for deformation work is sigma d epsilon,
s_ij d e_ij. The problem with this term is: for a volume-neutral deformation
both s_ii (the divergence of stress) and e_ii (the divergence of strain) must be
zero. However, potential theory says that the divergence is a measure of the
work done. Hence no work is done in a volume-neutral deformation, according to
the Cauchy-theory. The point is, a term of the form T_ii includes
radius-parallel components only. If radius-normal components are to be included
it can only be done by integration over the surface of the system, the
simple tensor invariants do not help you in this case.
Falk Koenemann
_____________________________________________________________________
| Dr. Falk H. Koenemann Aachen, Germany |
| |
| Email: [log in to unmask] Phone: *49-241-75885 |
| |
| URL: http://home.t-online.de/home/peregrine/hp-fkoe.htm |
| stress elasticity deformation of solids plasticity strain |
|~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~|
| The rain, it raineth on the Just |
| And on the Unjust fella. |
| But chiefly on the Just because |
| The Unjust stole the Just's umbrella. |
|_____________________________________________________________________|
|