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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