This discussion about stress etc. has stimulated a lot of
postings, but it seems to me there is considerable lose logic
floating around. Following up on my previous posting, perhaps some
other comments on one of Koenemann's recent postings can shed some
light.
> > 3) A simplifying feature of using flat surfaces is that it's easy to
> > resolve force vectors into normal and parallel components. When you're
> > using the surface of a sphere, this would seem to make balance-of-force
> > computations much more difficult to do correctly.
>
>I think that this is due to the definition of a shear force as a
>force parallel to a surface, according to Euler. I cannot see why
>this should be correct, and I believe that this is simply a faulty
>definition, for the following reason:
>
>Newton defined the rotational momentum as f x r where r is the
>distance from the center of mass of a discrete body to the point of
>action of f on the surface of the body.
Actually, f x r is the torque, which is the rate of change of angular momentum.
>The rotational momentum is thus independent of the orientation of
>the surface, what matters is the shape of the body. Euler defined
>the shear force as f x n where n is the surface-normal vector. The
>stress tensor is assumed to be orthogonal; this assumption is based
>on the physical significance of f x n.
I don't know what Koenemann's reference is, but I doubt Euler used
this definition. f x n is a vector that is normal to both f and n.
Thus f has no component in the direction of f x n and thus f x n
cannot possibly be the shear component of the force acting on the
surface. Perhaps Koenemann means n x f x n
>I cannot see how Newton's r can be replaced by Euler's n, and the
>transformation must be made if f x n is to have a physical
>significance. If it is assumed that f and n are physically of
>similar relevance it is like saying that since n is by definition a
>unit vector, so is r, hence the shape of the body is a sphere! Then
>what do you do with an elliptical body? Say, its principal axes have
>length x = 5 and y = 2; then radius-normal forces with magnitude 2
>and -5, respectively, would balance the rotational momentum. But a
>vector field with maximum radius-normal components in the directions
>x and y with the magnitudes given above is not orthogonal. Hence it
>is possible to balance the momentum of a non-orthogonal force field
>with an elliptical body.
Koenemann is confusing the use and significance of the
position vector of a point in the continuum r and the unit normal to
a surface n which simply defines the orientation of a particular
surface. Taking moments about the vector origin requires the use of
the position vector, not the unit surface normal. n x f x n
provides the maximum component of f parallel to the surface whose
normal is n, a simple resolution of a vector onto a plane. This has
NOTHING to do with the moment of the force. The two vectors r and n
are not the same, and they cannot be interchanged. Thus the whole
discussion based on this misconception is moot.
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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