I want to thank Dazhi Jiang, Scott Marshall and Robert Twiss for their thoughtful responses to my question. These certainly help.
In response to Scott and Dhazi - yes, point taken, making kinematic quantities the independent variables and dynamic quantities the dependent variables certainly avoids the logical problem, and is a better way of thinking about deformation anyway. Pages 544 - 546 in Twiss & Moores are an elegant statement of this. To encourage this type of thinking, and to discourage the tendency of students (encouraged by some first-year textbooks) to jump to dynamic conclusions, I always teach strain first and stress second.
Nonetheless, this does not altogether get round my problem. First, however much we think about strain as the cause and stress the effect, when dealing with brittle fracture it seems that we inevitably have to think about stress as a cause and failure as its consequence at some point. Yes, I know that once fracture occurs we are not dealing with a continuum any more, but the fact of the matter is that all our discussion of fracture criteria is bound up with stress concepts from continuum mechanics so I don't think it can be avoided entirely. Second, and more troubling to me, is that if a kinematic system (displacement or velocity gradient) described by 9 independent quantities is leading to a field of forces that can be described completely by a symmetric tensor of 6 independent components, is their any dynamic counterpart of the antisymmetric part of the tensor that describes the displacement field? I find it odd that there would be no equivalent at all. That would mean, Dazhi, that a small element in the middle of your layer, undergoing viscous simple shear between two plates, is subject to exactly the same force distribution as an element in the middle of a cylinder undergoing progressive coaxial deformation. The differences between the behaviour of the two elements would be determined only through the boundary conditions and the requirements of strain compatibility etc that you allude to. This is fine, but those requirements are much less elegantly stipulated than the constitutive laws that relate stress to distortion.
So, Robert Twiss's answer has set me thinking that there might be more to force distribution than the orthorhombic stress tensor. (However, I haven't read the references yet!) The argument that there is an antisymmetric component to stress is interesting, and new (to me). It set me thinking about the relationship of forces to planes on the one hand, and to lines on the other. In dealing with forces acting on the surfaces of our vanishingly small element, if we dealt with just force, we would end up arguing that the force on the surface vanished to zero as the size of the element under consideration was reduced to zero. To avoid this, we can define traction dF/dA which does not vanish to zero. The argument about moments seems to me to have some analogies. As we reduce the size of the cube to vanishingly small, the moment vanishes. Is there some non-vanishing quantity (dM/dr? where r is the length of the lever arm) which expresses the relationship between the forces acting (even within a continuum?) about a line, which could be related in a simple way to the kinematic vorticity? My thinking would be that these 'torque densities' (I have no idea if there is a word for this) would be described by an antisymmetric tensor. There then might be a pleasing symmetry between the strain and rotation tensors in the kinematic world, and the stress and torque-related tensors in the dynamic world. Perhaps I will find this in the papers on the micropolar model that you reference, Robert. I will take a look.
On 2011-Mar-27, at 11:31 AM, Robert J. Twiss wrote:
> For a discussion that amplifies Dazhi's points, see Twiss & Moores, Structural Geology, 2nd Ed., Section 18.1 (p.544-546). We can consider the boundary conditions as defining the 'cause' of the mechanical process because these are the conditions that are externally imposed on the deforming body. For a mechanically isotropic body, the symmetry principle [see Twiss & Moores, Section 17.8-ii, p.537] shows that if the stress is the cause of a deformation (stress boundary conditions), the resulting deformation can never have the low monoclinic symmetry of a simple shear. Only if velocity boundary conditions are applied can a simple shear result [Twiss & Moores, Section 17.8-iv (p.539)].
> I might add that it is important to understand the distinction between a real material and the continuum model of that material. The continuum formulation of deformation of a material is simply a mathematical idealization, a model, of the physical system, and should not be confused with the actual physical system itself. For the mathematical idealization, we can imagine taking a limit as we shrink a cube or a tetrahedron to an infinitesimal point. For a real material, however, such a process becomes meaningless as the size of the volume decreases because of the inherent discontinuities and heterogeneities in a real material. Thus we must always keep in mind what the correspondence is between an infinitesimal point in a mathematically idealized continuum and what that point represents in the real material.
> In particular, the value of a field quantity such as force at a point in a continuum is a mathematical idealization that is meant to represent an average of all real forces in the real material over a local volume around that point. This is the basic approach of statistical mechanics. When, in the mathematical idealization, we allow a volume to shrink to zero so that all moment arms vanish and torques become zero, we are using a mathematically convenient technique to express the physical situation that within a small local volume around a point in space, whatever torques there may be will average out to zero.
> Thus it is best to keep in mind the statistical mechanical basis for the relationship between a continuum model and what it is designed to represent in the real world. If the continuum model does not represent the real world adequately, we are free to use a different model. For example, a micropolar continuum model could provide a better representation of the behavior of a granular material than the classical continuum model.
> On Mar 27, 2011, at 6:19 AM, Dazhi Jiang wrote:
>> 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
>> 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.
>> -----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?
>> 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
>> Hope this helps.
>> 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
>> -----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
John Waldron, Department of Earth & Atmospheric Sciences, 1-26 Earth Sciences Building, University of Alberta, Edmonton, AB Canada T6G 2E3
Tel: 780-492-3892. Fax: 780-492-2030. [log in to unmask]
University spam filters may reject some sources of mail. If this happens, try me at [log in to unmask]