Dear colleagues,
Is it mathematically comprehensible to replace the cube with sphere ?
-----Original Message-----
From: Falk H. Koenemann <[log in to unmask]>
To: [log in to unmask] <[log in to unmask]>
Date: Saturday, June 16, 2001 12:26 PM
Subject: Re: stress/Koehn #3
Daniel Koehn schrieb:
> Sorry, this is not getting anywhere. I was refering to linear
> elasticity, using a small volume element which is presumably supposed
> to be cubic. Shurely if you go to large strain it becomes nonlinear
> and complicated. At lot of people have worked on that problem. I
> think I will pass it on.
I have explained in another posting why linear elasticity is not a helpful
concept.
There is a reason for using a cube as unit body. As long as you understand
pressure and stress as force per unit area you need that unit plane; and
thinking of a cube is then nearly inevitable. But this is an artifact of the
concept.
We have two definitions of pressure, P = f/A and P = dU/dV. The latter is by
far
the more fundamental one because pressure is a state function; and since it
is
one form of the most fundamental relation in physics - the ratio
energy/mass -
it is also the basis of all of potential theory. The two definitions are not
identical; the mathematical relation between the two is the subject of my
published paper. But if you manage to think of pressure only and exclusively
in
terms of energy density P = dU/dV, the shape of V is no longer
predispositioned
by other concepts.
Here just some food for thought. In discrete mechanics we have the concept
of
the lever. The longer length r, the less force is required for some unit
work
f x r. Vice versa, if the available lever is short, it takes a large f to
achieve something. It is one of the idiosyncrasies of continuum mechanics
that
the lever theory is never mentioned or even considered. If continuum
mechanics
were entirely compatible with Newtonian mechanics, this should not be so.
By assuming a priori that your unit volume is a cube, you thereby constrain
the
shape so that you have identical r in the principal directions. In effect
you
give away one degree of freedom in continuum mechanics - which you do have
in
discrete mechanics; but why is this done? What is the theoretical
justification?
In my 8 questions I pointed out that Newton's r is replaced by Euler's
surface-normal unit vector n without much explanation. I believe that here
is
the mistake.
What is a lever? It is a distance within a solid. So why is this ignored in
CM?
Because Cauchy's continuity approach in effect abandoned the concept of the
radius.
See my homepage for a new approach that avoids all these peculiarities.
Falk Koenemann
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| Dr. Falk H. Koenemann Aachen, Germany |
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