Musson, Roger MW schrieb:

> There exist, undeniably, a
> number of eccentrics who make it their business to come up with abstruse
> alternative science which they then attempt to promulgate. Sometimes these
> heresies are obviously unsound, sometimes they are so unintelligible that
> they cannot be easily refuted. Either way, taking the effort to try and
> refute them is inevitably time wasted.

I pose the following eight questions on the Euler-Cauchy theory. I hope that
they are sufficiently clear to be understood by everyone. A reasonable answer to
any of these questions would amount to proper defense of the Euler-Cauchy
theory, and hence a refutation of my views (which at this point I haven't even
offered). Go ahead.


1. Why is an equation of motion used in continuum mechanics and not an equation
of state, despite the fact that elastic deformation is by nature a change of
state in the sense of the First Law of Thermodynamics, meaning that work is done
upon a system such that its internal energy changes from the unloaded into the
loaded state? Which aspect of the continuum mechanics theory indicates that
e.g., elastic deformation is a non-conservative, yet reversible process, but not
a conservative process (like celestial mechanics) according to Hamilton and
Bernoulli?

      Short comment. I asked this question for the first time in class as a
      graduate student 20 years ago out of sheer innocence. In the morning I
      took a class in thermodynamics and was taught everything about the
      equation of state and how to handle it, in the afternoon I took a class in
      Stress and Deformation which started with an equation of motion; I was
      just puzzled. But in 20 years I have not found a single discussion partner
      who was willing to openly address the question. It is the instant death
      question. Whoever answers it cannot avoid to abandon Euler's approach.


2. How can the derivation of the Cauchy stress tensor and the entire theory
of continuum mechanics be reconciled with the Gauss Divergence Theorem and
potential theory in general? Special attention is to be paid to the criteria by
which a distributed source is identified as such.

      The paper which I recently published (see my homepage or previous post)
      contains the proof that the derivation of the stress tensor cannot be
      reconciled with the divergence theorem.


3. Which physical argument justifies Lagrange's conjecture that it is strain
that is causally related to stress despite the fact that all experimental and
natural evidence points towards a cause-effect relation of stress to
displacement? Strain is by definition a tensor; according to current
understanding stress is assumed to be a tensor; displacement is a vector field.


4. Newton defined a rotational force as a force acting upon the surface of a
body such that the force is oriented perpendicular to the radius vector
(location vector of the point of action of the force relative to the center of
mass of the body). Euler defined a shear force as a force parallel to a plane
and perpendicular to the plane orientation vector. Both are required to balance
if they are integrated over the surface of the body (Newton) or all planes
passing through the point of interest (Euler's group of planes). The two are
treated as physically equivalent. Thus it must be possible to transform one into
the other, i.e. it must be possible to replace Newton's radius vector by Euler's
plane orientation vector without causing inconsistencies. How is this
transformation done?

5. Euler's cut model considers "stress at a point" by applying the group of
planes concept: an infinite number of planes oriented in an infinite number of
orientations all share one commmon point. Cauchy considered a volume element
which he let approach zero; in the transition the existence of the stress tensor
is proved. In my discussions I have encountered two schools of thought: some
maintained strongly that the volume element approaches zero, but it must not
reach it; others maintained just as vehemently that it does reach zero,
and this is ok. I believe it must reach zero, but this is proof of an error. How
is it possible to perform a limit operation such that the two opposite faces of
e.g., a cube merge without the volume reaching zero? More generally, how can
Cauchy's volume element be transformed into Euler's group of planes without the
volume vanishing identically?


6. How can Newton's 3rd law be transformed into the thermodynamic equilibrium
condition P_system + P_surrounding = 0?


7. Navier-Stokes equations: by what argument is it justified to use conservative
terms from Newtonian mechanics and non-conservative thermodynamic terms side by
side in the same equation - despite the fact that they have different mass
terms, consider entirely different work terms, are subject to different
equilibrium conditions, and refer to different potentials?


8. Which detail in the continuum mechanics theory indicates that there are bonds
in solids? How are the externally controlled loading forces and the bonding
forces characterized as separate in nature, and independently existing? How is
the differing nature of the two types of forces taken account of in a
consideration of the equilibrium condition?


Note. The terms conservative and non-conservative have precise meaning in
theoretical physics.

Any process for which Bernoulli's law U_kin + U_pot = const holds, is said to be
conservative because the total energy of a system is invariant (ie. constant).
For such processes Bernoulli's law is the energy conservation law. The proper
physical theory is Newtonian mechanics. Work is done within a system (by one
particle within a kinetic system upon another, yet such that U_tot = const).

If the total energy of a system is a variable, the above principles cannot be
applied. In that case the proper energy conservation law is the First Law
dU = dw + dq. Such processes are said to be non-conservative - this does not
mean that energy is lost, just that it is not constant. A non-conservative
process may be reversible or irreversible. The theoretical framework is that of
thermodynamics. Work is done upon the system.


Shoot me down if you can.

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.                              |
|_____________________________________________________________________|