Robert Twiss schrieb:
> A basic introduction to continuum mechanics and continuum
> thermodynamics can be found in Eringen, A.C., 1967, 'Mechanics of
> Continuua', John Wiley & Sons. Ch.4 is titled 'Thermodynamics of
> continuous media'.
Chapter 4 starts with a rather idiosyncratic statement of the First Law:
dK/dt + dE/dt = w + q
(Eringen used Newton's dot notation) where K is the kinetic energy and E is the
internal energy.
This is not what I learned. The First Law is Delta E = w + q, or dE = dw + dq.
Nowhere does the First Law say anything about kinetic energy.
The difference is significant. By using Newton's notation Eringen makes it clear
that he considers the change of the respective energies as a function of time;
moreover, time is an inherent parameter which is indispensable for the
understanding of the problem. This is true for the kinetic energy of a discrete
body in empty space, it is certainly not true for the internal energy: in the
complete equation of state PV = nRT time is not a parameter. Of course
thermodynamic changes of state take place in time; but nevertheless, changes of
state are time-independent: whether a particular loading takes place in a short
or a very long time, it is the identical work.
In Newton's mechanics, the energy conservation law (Bernoulli's Law) is E_kin +
E_pot = const = H. H is the total energy of a kinetic system (containing n
discrete bodies in empty space). An indispensable parameter of such a system is
indeed the velocity; a kinetic system has thus a velocity potential. In the case
of a mono-atomic gas, if the electromagnetic long-range forces are ignored by
which atoms repel one another, H is equivalent to the internal energy of the
system. But this IF is not permmissible: electromagnetic forces are derived from
an electromagnetic potential F; they are just irrelevant in the mechanics of
larger discrete bodies (eg. planetary systems); in the case of a gas they are
not at all irrelevant, hence H < E even for a gas. The full internal energy of a
gas (let's call it U = F + H) is thus a thermodynamic potential involving
electro-magnetic potentials, not just a velocity potential. (In a mono-atomic
gas H may be the larger term; in solids it is without significance since bonds
in solids are electromagnetic forces, and immensely larger than the kinetic
energy of atoms bouncing around in their limits confined by the bonds.)
The above statement of the First Law is a case of apples & oranges, and a
dangerous one. K is a Newtonian term and conservative (i.e. it refers to
Bernoulli's Law), E (whether E is my H or my U) contains K and is a
thermodynamic term and non-conservative (it refers to the First Law). Through
statements such as this one the profound difference between Newtonian mechanics
of discrete bodies and thermodynamics is blurred, it just confuses the student:
either we discuss work done by displacing or accelerating a discrete body with
given mass; the body is then part of a kinetic system, and work is linear, it is
done against inertia and _in_ a system, and path-independent in Euclidean space.
Or else we consider work done _on_ a system, then we change the energy of the
entire system; work is done against an electromagnetic potential, it is
logarithmic, and path-independent in PV-space (in the reversible case).
I have elaborated on this in some detail because the mixup of the two entirely
different theories is characteristic for current continuum mechanics.
> We have seen it argued recently that according to classical
> theory no work is done in constant volume deformation. This is dead
> wrong. The work is simply t(i,j) e(i,j) (summed on indices i and j)
> where t(i,j) are the components of the stress tensor, and e(i,j) are
> the components of the infinitesimal strain tensor (the symmetric part
> of the displacement gradient tensor). This term appears in the
> continuum equation for the conservation of energy (Eringen, eq.
> 5.5.15) as the rate of work t(i,j) d(i,j) where d(i,j) are the
> components of the strain rate tensor (also called the deformation
> rate tensor; the symmetric part of the velocity gradient tensor).
>
>> Koenemann wrote:
>> 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.
>
>
> However, t(i,i) and e(i,i) (summed on i) are NOT the
> divergences of the stress and strain respectively. They are called
> the 'trace' or the first scalar invariant of each tensor. The
> divergences would be =8Ft(i,j)/=8Fx(j) and =8Fe(i,j)/=8Fx(j) (summed on j).
Whatever that is (this is how it arrived here), the trace of a tensor is its
divergence.
> It is not clear what quantity Koenemann refers to in his reference to
> the divergence being a measure of work done. t(i,j) e(i,j) is the
> work done. Expanding the implied summation, we have for the work:
> t(1,1)e(1,1) + t(1,2)e(1,2) + t(1,3)e(1,3)+
> t(2,1)e(2,1) + t(2,2)e(2,2) + t(2,3)e(2,3) +
> t(3,1)e(3,1) + t(3,2)e(3,2) + t(3,3)e(3,3)
> Clearly this work term is not zero even if
> e(i,i) = e(1,1) + e(2,2) + e(3,3) = 0
> and t(i,i) = t(1,1) + t(2,2) + t(3,3) = 0
Both t(i,j) and e(i,j) are "symmetric", i.e. orthogonal, i.e. the off-diagonal
terms can be taken to be zero, hence if e(i,i) = 0 and/or t(i,i) = 0, there is
no way out: t(i,j) e(i,j) = 0.
> The constitutive equation for elasticity is
> t(k,l) = L e(m,m) =8F(k,l) + 2=B5 e(k,l) (summed on m)
> where L and =B5 are the Lamé elastic constants for isotropic elastic
> materials, =8F(k,l) are the components of the Kronecker delta, which
> equals 1 if k=3Dl and otherwise is zero, and e(k,l) are the components
> of the infinitesimal strain tensor. Thus even if the first invariant
> of strain e(m,m) and of stress t(m,m) (summed on m) are 0, the work
> done would still not be zero.
There are a bunch of assumptions in this which are not obvious to the casual
reader. Both the Lamé constants and Poisson's ratio are observed terms, i.e.
they are phenomenological, and dependent on a specific experimental set of
boundary conditions (which are then implicitly believed to be ideal). Whether or
not these terms are necessary is largely a question of the particular
mathematical structure of the approach one has chosen to follow. I have always
felt that a properly derived theory should be without such phenomenological
terms, they are like fudge factors.
But the real problem is a different one. Twiss uses tensor terminology, and the
question is whether or not t_ij exists at all as a mathematical term. (By
existence of a term I mean: a term exists only if it can be shown that ist
derivation is in line with general mathematical logic.) I believe that it does
not; the question relates to the validity of Cauchy's continuity approach. If
t_ij does not exist, Twiss's argument is void.
> I have pointed out in recent postings, a number of
> misconceptions, logical errors, and incorrect conclusions that are in
> Koenemann's postings. I do not choose to spend my time analyzing in
> detail a theory that apparently includes conclusions that are
> incorrect. If there are demonstrable errors in the results, then I
> do not need to find out where the problem is in the derivations. I
> have only decided to make these comments to give others the
> opportunity to make this choice if they are so inclined, but for my
> part, it is not something I want to spend more time on.
>
> Rob Twiss
That attitude works in the office. Here the audience is too large.
Falk Koenemann
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| Dr. Falk H. Koenemann Aachen, Germany |
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