Tim Wynn wrote:
>I also suspect that the physics is too unfamiliar or advanced for many list
>members. To that end, maybe the main protagonists (Rob Twiss, Dugald
>Carmichael & Falk Koenemann) could recommend some good, basic text books
>and/or papers on thermodynamics, tensors and elasticity for the rest of us
>to chase down.
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'. The constraints that thermodynamic principles
place on constitutive equations are discussed in Ch. 5: Constitutive
equations. In particular, the relation is derived in Ch. 5 for
elastic materials that the stress is proportional to /C(K,L) where
is the free energy and C(K,L) are the components of the Green
deformation tensor (eq. 5.5.21).
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 t(i,j)/x(j) and e(i,j)/x(j) (summed on j).
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
The constitutive equation for elasticity is
t(k,l) = L e(m,m) (k,l) + 2΅ e(k,l) (summed on m)
where L and ΅ are the Lamι elastic constants for isotropic elastic
materials, (k,l) are the components of the Kronecker delta, which
equals 1 if k=l 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.
Thus it seems to me that Koenemann's conclusion is based on a
misconception of what the divergence is in this case.
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
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Robert J. Twiss email: [log in to unmask]
Geology Department telephone: (530) 752-1860
University of California at Davis FAX: (530) 752-0951
One Shields Ave.
Davis, CA 95616-8605, USA
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