I'd like to add a few more lines to what I sent around yesterday (below)
after reading R. J. Twiss's email.
In applying continuum mechanics, we assume that the continuum assumption is
valid for the problem. One may refer to many textbooks for this assumption.
Where this assumption is not valid, other formulations are necessary. But
John Waldron's question still must and can be answered in the context of
classic continuum mechanics.
Imagine a simple case where a Newtonian fluid is constrained between two
parallel rigid plates moving parallel to each other. The velocity field in
the fluid is a perfect progressive simple shear and is everywhere
monoclinic. But the stress tensor is everywhere orthorhombic. Where does
this unparallelism arise? I think the answer is that the velocity field is
not just driven by stress (the "deformation driven by stress" thinking). It
must satisfy the compatibility requirement and the boundary conditions as
well.
As we know, a complete set of equations for a continuum mechanics problem
includes: mechanic laws which ensure stress equilibrium and require that the
stress tensor be symmetric, constitutive equations (relating stress and
strain and strain rate etc.), kinematics (strain and compatibility etc.),
and the boundary conditions.
Cheers,
Dazhi
-----Original Message-----
From: Tectonics & structural geology discussion list
[mailto:[log in to unmask]] On Behalf Of Dazhi Jiang
Sent: Saturday, March 26, 2011 8:32 PM
To: [log in to unmask]
Subject: Re: a plea and a new question?
John,
Here is how I look at the stress and strain problem you have.
First, to say that deformation is driven by stress is incorrect, or at
least, incomplete. Let's limit ourselves to infinitesimal elastic
deformation first. One can say the strain is driven by the stress (through
the Hooke's law). Or equivalently, the other way around (left side equal to
right side of the Hooke's law). But deformation must be defined by the
complete displacement field, of which strain is only the symmetrical part.
The antisymmetric part of the displacement field is the rotation. Now to
answer your question, what determines the displacement field? It is the
combination of mechanical laws (balance of linear momentum, angular
momentum), stress-strain relation, compatibility, and the boundary
conditions. How does the vanishingly small cube 'know' about the boundary
conditions of the system in which it sits? It is through compatibility
requirement. The possible displacement field for a continuous body
deformation must make all parts compatible.
The about explanation applies to the deformation of any continuous body. For
a viscous body, just replace the displacement field by the velocity field.
When one moves from infinitesimal deformation to accumulative deformation,
one simply deals with the time integration of the displacement/velocity
field.
Hope this helps.
Dazhi
______________________________________________
Dr. Dazhi Jiang, Associate Professor
Department of Earth Sciences
The University of Western Ontario
London, Ontario
Canada N6A 5B7
Tel: (519) 661-3192
Fax: (519) 661-3198
www.uwo.ca/earth/people/faculty/jiang.html
___________________________________________________
-----Original Message-----
From: Tectonics & structural geology discussion list
[mailto:[log in to unmask]] On Behalf Of John Waldron
Sent: Saturday, March 26, 2011 5:51 PM
To: [log in to unmask]
Subject: a plea and a new question?
I have a plea and a question.
I have too found the discussion on this list very informative over time, and
I would like it to remain so. However, the last 18 posts have been about
the behaviour of people, not rocks. Public statements of opinion, or
announcements of intent to leave the list, however well-intentioned, may
contribute to the problem; more list members will be tempted to leave
because they don't want to read this stuff. So, I would make a plea based
on my experience on the Canadian list mentioned by Jürgen. If you intend to
leave the list, I would urge you to leave quietly, or to make your opinions
on individuals (on whichever side of the argument) known in private emails
or to the list owner (he may not thank me for this), rather than to the
whole list. I intend to stay on, and hope there will be enough expertise
left in the list to make it as informative in the future as it has been in
the past.
In that spirit (and lest I contribute to the same problem) I would like to
ask a question, that has been raised in my head by some of Dr. Koenemann's
comments. Like many members, I work in general field-based structural
geology, and am not an expert in continuum mechanics. However, I do teach
the basics of stress and strain in my undergraduate and graduate classes,
typically to students with even less background in physics and mathematics
than mine. Like most of us who teach this stuff, I take my students through
the hypothetical vanishingly small cubic element of a solid under stress,
and represent the three components of stress (or more properly traction) on
each surface so as to fill out the 9 components of the stress tensor.
Then comes the part that always leaves me with nagging doubts. There is an
argument in all the texts that the shear stresses sigma-x-y and sigma-y-x
are identical, based on the case that there is no net moment about the z
axis in this vanishingly small cube. When applied to all the off-diagonal
elements, this leads to a symmetrical stress tensor with 6 independent
terms, in contrast to the asymmetric deformation gradient tensor with 9
terms. I am uncomfortable with this contrast, which seems
counter-intuitive. If deformation is driven by stress, and the stress
tensor only controls the six terms that describe distortion (or distortion
rate) then how is the rotational part of deformation controlled? I realize
that rotation can be constrained by setting appropriate boundary conditions,
but my discomfort is that that vanishingly small cube doesn't 'know' about
the boundary conditions of the system in which it sits, so what controls its
rotation if not the state of stress? So I always end my lecture with the
feeling that the argument is sleight of hand - I have used phrases like
'arguments beyond the scope of this course lead to...', without feeling that
I actually have a proper grasp of those arguments.
This may be something that can be very simply answered, and that I simply
missed out on in my own education. However, Dr. Koenemann's discourses
raised the idea that we should be able to explain stress-strain
relationships in terms of forces that act along bonds between atoms, not
infinite imaginary surfaces within continua, so I am tempted to wonder
whether there are elements of his argument that might lead to a resolution
of my question, perhaps by including a rotational element into the
description of stress. If anyone has any suggestions or explanations that
help to make this make sense, and help me make sense of this to my students,
it would be most welcome.
John Waldron
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