Daniel Koehn schrieb:
> >The theory as outlined by Landau & Lifschitz permits the conclusion that a
> >volume-neutral deformation does not cost work; and this cannot possibly be
> >correct.
>
> This is not true since the work is defined in terms of changes in the
> strain tensor, the whole tensor not only its diagonal components
> which define volume changes. Shape changes in the body also cost work
> according to elastic theory.
Because of the mathematical symbols I give the full quote of Landau and
Lifschitz in the attached WinWord-file. There I prove my claim above. The file
is so small that I may be forgiven for sending an attachment.
Daniel,
consider volume-neutral deformation only. Positive work is done in the
directions in which radius lengths are shortened. Negative work is done in the
directions in which radius lengths are stretched. For a volume-constant
deformation the effect must cancel exactly. For a volume-neutral deformation
both the trace of the stress tensor as well as the trace of the strain tensor
must be zero. Hence no work is done.
The fallacy in this logic is: the trace of a tensor only considers normal
components. The trace (the divergence) of a tensor is a measure of the total
work done only and only if radius-normal components can be excluded from
consideration (because from the nature of the physical problem they do not occur
[such as in gravity problems], or because the boundary conditions are such that
they do not occur [isotropic compressison]). In case of a deformation (V =
const) however, it must be considered that shear forces do work, ie. in such a
case the divergence is no longer a measure of the _total_ work done. Thus
considering the diagonal terms only is an incomplete answer to the question.
How do we include the work done by shear forces? This cannot be done by looking
at tensor invariants, this can only be done by integrating over the surface of
the volume element. But if we want to do this we must have information about the
shape of the volume element. This point is nowhere considered in the literature
on the Euler-Cauchy theory, in fact it was considered an advantage of the
EC-approach that due to Cauchy's continuum approach, the shape was supposed to
vanish. This is not the case; the shape can vanish only with the volume element,
ie. V must reach zero. This in turn is against the intent of a limit operation.
Falk Koenemann
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| Dr. Falk H. Koenemann Aachen, Germany |
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| URL: http://home.t-online.de/home/peregrine/hp-fkoe.htm |
| stress elasticity deformation of solids plasticity strain |
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