Dugald Carmichael schrieb:
Dugald, I must have left space for misunderstanding - my calculation was a
consideration of the internal energy at constant state as the scale (ie.
V) varies; no PdV-work is done. Mass is a variable in this case. If on the other
hand we consider PdV-work we assume that the mass is invariant so we can
calculate work per unit mass.
> Certainly "stress at a point" has no meaning in a real solid. A continuum
> differs from a real solid in being strictly homogeneous even at the
> subatomic scale. By the same token, "pressure at a point" and "temperature
> at a point" would be meaningless; P and T are properties of macroscopic
> systems. P or stress = f/A where A refers to the external surface of a
> system containing a finite quantity of "statistically homogeneous" matter.
Not quite. T, P, mass density or chemical composition at a point X all work by
the same principle: we choose a point X in space which indicates the place of a
thermodynamic system of arbitrary size. This system is finite and may enclose
heterogeneities. But if we now let V approach zero (such that X does not leave
V) we assume that the quantity under discussion approaches a finite value. We
then consider that value the density or chemical composition at X.
"Stress at a point" hits very much at the core of the problem. Euler had a
concept which does _not_ start with a volume element; Euler started with all
free planes containing X which are oriented in all possible directions, his
"group of planes". Cauchy then started with a volume element, but he sought to
transform it into Euler's group of planes. I cannot accept this because (a) in
doing so, Cauchy ignored the divergence theorem of Gauss (which was published 10
years earlier, and which was THE hot topic of the era), and (b) Cauchy started
with a closed surface and ended up with a group of free planes; this is not
possible, and besides, the magnitude of V _must_ reach zero in the process.
If stress is PdV-work the stress theory must be compatible with thermodynamic
principles. Currently this is not the case.
> >2. Measurability. Some standard unit volume must be assumed to which the
> >measurements refer. This may be the volume of some unit mass, or a unit
> volume
> >in the standard state. Then V is no longer a variable, and then f/A works.
>
> I would contend that P = f/A works for any thermodynamic system at
> equilibrium or during any reversible process, but -P = dU/dV works only for
> systems at constant mass and entropy.
Can we directly measure U at all?
> >The shape of the system is not arbitrary if rotational/shear forces can do
> >mechanical work. ...
>
> Agreed. The forces applied to the outer surface of the system, whatever its
> shape, must be mechanically balanced so as not to cause either linear or
> angular acceleration of the system. This is the condition that restricts
> the symmetry of stress to be not less than orthorhombic.
Not quite. You recall the monoclinic symmetry of my predictions for simple shear
fabrics (you commented on my home page a couple months ago, see the figure in
http://home.t-online.de/home/peregrine/hp-post.htm). The low symmetry is
possible only because I include the bonds between system and surrounding into my
calculations. The bonds certify that disequilibrium cannot exist in the elastic
case (as long as you don't break any bonds). This gives you one more degree of
freedom, and monoclinic symmetry for stress is possible (and, in my view, also
observed).
> >If we consider a thermodynamic system within a solid, ie. within a larger
> >volume that is continuously filled with mass, we get one more problem: ...
> >system and surrounding are permanently bonded to one another. This
> >point is of utmost importance for the equilibrium conditions; however it can
> >only be considered if the existence of bonds is indeed taken into account,
> >which is not done in continuum mechanics.
>
> Thermodynamics deals routinely with this type of system, taking account
> neither of bonds nor of strength of materials. The prevailing theory of
> metasomatic zoning assumes constant P during reactions that predicate
> finite changes of volume; hence it treats rocks as if they have zero
> strength. But a continuum is resistant to deformation, and hence continuum
> mechanics generalizes P (an isotropic scalar) into stress (a second-rank
> tensor with orthorhombic symmetry). Seems to me this is an important step
> towards taking account of the existence of bonds. Further steps are clearly
> desireable.
You can ignore strength and bonds if you consider chemical processes only (eg.
metasomatism). Furthermore, you can ignore bonds in the case of a compression of
a solid, but not in case of an expansion or shear. This point is very important.
Consider a solid, within the solid a subvolume which is our system; let it have
spherical shape in the XY-plane. Compress it isotropically. Whether the system-
surrounding- interface (SSI) is bonded or not, you will observe a negative
volume change in both cases. Now expand it isotropically; you get a system
expansion only if the SSI is bonded, but if it is not the surrounding expands,
but the system does not; we get a vacuum cavity with the unloaded system as a
discrete body inside. Reason is: contrary to gases, solids have a natural
finite zero pressure volume.
Now consider pure shear deformation, with contraction in Y and extension in X.
Along Y we observe shortening whether or not the SSI is bonded; but along X we
observe something very different: if the SSI is bonded we get a stretch in X to
the effect that the system expands in X in the same way as the surrounding; if
the SSI is not bonded, the system still expands, but not as much as the
surrounding, and there will be a cavity.
What does this tell us? First of all, we cannot assume by implication that the
bonds along the SSI are counted in, we have to put them in consciously. Second,
some part of the extension in X exists whether or not there is a surrounding or
the SSI is bonded, ie. the system squeezes out in X by itself; another part of
the extension is dependent on the existence of the bonds, ie. the surrounding
actively pulls the system out in X. That is, Poisson's ratio has two components
one of which is always there, the other component is observed only if the SSI is
bonded. Hence Poisson's ratio varies from within a solid towards the unloaded
faces. This thought experiment is so simple, but has anybody ever seen mention
of two independent components of Poisson's ratio? I haven't, and my theory takes
account of this.
Finally, fix a handle on the system perpendicular to the XY-plane and turn. No
other forces are applied. If the SSI is not bonded you can perform a free
rotation of the system within the surrounding; if the SSI is bonded you cannot.
Consequence: whether bonds exist or not, strongly influences the result and the
rotational equilibrium condition. Furthermore, what you will observe in the
second case is an expansion of the system (dilatancy). I can predict it, the
Euler-Cauchy theory cannot.
Falk Koenemann
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| Dr. Falk H. Koenemann Aachen, Germany |
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| stress elasticity deformation of solids plasticity strain |
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