At 10:04 PM 2001/06/24 +0200, Falk Koenemann wrote:
>...
>The reason why Cauchy and I come to different conclusions regarding the
>components of stress, has nothing to do with homogeneity, but with the
>mathematical properties of planes and surfaces. Cauchy's tetrahedron is not
>a volume element in the sense that it is surrounded by a closed surface
>(with facets), but his planes are free surfaces that divide the universe
>into left and right etc.
I don't perceive any important physical difference between a planar element
that is part of the boundary of a system and a planar element that is
inside the same system. In the isotropic case at equilibrium (constant and
uniform P), counterbalanced forces act normal to each side of either type
of planar element. In the anisotropic case at equilibrium (constant and
uniform stress), counterbalanced forces act oblique (generally) to each
side of either type of planar element (and can be resolved into normal and
shearing components).
>Cauchy did not consider the relation of surface per mass, and I do.
Falk, may I remind you that your consideration of the relation of surface
per mass in a spherical thermodynamic system under isotropic stress (aka
hydrostatic stress, aka pressure) is still in contention. You posted a
table of calculations ostensibly based on the divergence theorem,
purporting to show that f/A across the system's boundary increases with
decreasing V at constant P. Several people have taken the trouble to argue
cogently that there must be some mistake in this result. If a new approach
cannot be shown to deal correctly with the isotropic case, a claim that it
deals correctly with the general case is likely to be less than credible.
>> Stress is not PdV work. Seems to me stress is simply a generalization of P
>> for solid systems in a state of homogeneous elastic strain.
>
>Consider isotropic compression only, for simplicity. Then you do work to get
>the system from the unloaded into the loaded state, right?
Agreed.
>P is a state function. It does not have spatial properties, ditto for work,
>temperature, volume, density.
If "spatial properties" means anisotropy, agreed.
>If you wish to describe the anisotropic shape of a body, you do it in terms
>other than volume. Both in isotropic and anisotropic loading an elastic
>potential develops. Isn't that what we call stress? If so, it is akin to
>PdV-work.
Disagree. Stress is an intensive variable akin to P. An elastic potential,
in common with any other kind of energy, must be the product of an
intensive and an extensive variable (e.g., PdV, TdS, fdl, ...
>The thermodynamic term for deformation work is sigma d epsilon,
>s_ij d e_ij.
If sigma d-epsilon is the product of a component of stress and a component
of conjugate infinitesimal strain, it looks like a correct expression for
elastic work and/or elastic potential energy.
>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. The point is, a term of the form T_ii
>includes radius-parallel components only. If radius-normal components are
>to be included it can only be done by integration over the surface of the
>system, the simple tensor invariants do not help you in this case.
There seems to be no dispute that a physically correct theory must take
account of the work done by shearing components of strain as well as normal
components, whether or not the deformation is volume-neutral. Rob Twiss
says the orthodox theory takes account of shearing work; Falk says it
doesn't. Not being conversant with tensor algebra, I can't make out whether
this disagreement involves physics or merely mathematical grammar/syntax.
Dugald M Carmichael Phone/V-mail: 613-533-6182
Dept of Geological Sciences and Geological Engineering
Queen's University FAX: 613-533-6592
Kingston ON K7L3N6 E-mail: [log in to unmask]
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