Stephen White schrieb:
> Koenemann asked:
> "3. Which physical argument justifies Lagrange's conjecture
> that it is strain that is causally related to stress despite
> the fact that all experimental and natural evidence points
> towards a cause-effect relation of stress to displacement?
> Strain is by definition a tensor; according to current
> understanding stress is assumed to be a tensor; displacement
> is a vector field."
I thank Steve for taking this into the public.
The disparity between handling infinitesimal and finite strain is well-known.
The problem is typical for situations involving simple shear because the strain
term does not account of rotations. Commonly it is held that a general
deformation can be decomposed into a strain, a rotation, and a translation.
Let's concentrate on the two first ones.
In the case of pure shear the eigendirections of stress, strain, and
displacement are all orthogonal and parallel to one another. Thus in this case
it cannot be decided which term is the relevant one. In simple shear the
eigendirections of strain and displacement differ; strain is orthogonal by
definition whereas the displacement field is surely non-orthogonal. The problem
is how to relate this to stress which is so far thought to be orthogonal by
definition; as long as this is unchallenged, a correlation of orthogonal stress
with orthogonal strain appears to be natural.
The critical question is work done. If the rotation in the departitioning above
is an external rotation it does not cost work because it is a free rotation (and
unbalanced). But in solids we cannot have a free rotation. The partitioning is
geometrically correct, but physically irrelevant.
If simple shear and pure shear differ in the work done per unit strain it is
proof that displacement is the physically relevant term, and not strain.
Experimental evidence indicates that it is so:
- in the elastic field, simple shear costs ca.10% more work per unit strain than
pure shear;
- in the plastic field, simple shear costs ca.30% less work per unit strain than
pure shear.
These patterns cannot be predicted by the Euler-Cauchy theory. I can predict
them.
Falk Koenemann
_____________________________________________________________________
| Dr. Falk H. Koenemann Aachen, Germany |
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