Mark (et al.)

        To my mind, the question of whether or not the principal axes of  
stress and instantaneous strain (or equivalently strain rate) are  
parallel during distributed brittle deformation of the crust is still  
an open question that has not satisfactorily been resolved.  It depends  
on the characteristics of the constitutive equations, which reflect the  
mechanical properties of the material.  There are arguments on both  
sides, and that has been the focus of much of this discussion: Is it  
justified to equate the orientations of principal axes and the ratio of  
principal value differences for the stress to those of the strain rate?  
  People who do 'stress inversions' in my opinion implicitly assume this  
equivalence, which means they implicitly assume the brittle crust  
behaves as an isotropic, rheologically linear material.  My only point  
is that this is a pretty big assumption and that we should try to test  
it rather than just implicitly assume it always holds.

        To test this equivalence is not straight forward, however.  It is not  
a question of doing a 'stress inversion' and then doing a 'strain rate  
inversion' and seeing whether they agree, or whether one gives a lower  
misfit to the data than the other.  The mathematics for calculating the  
orientation of the maximum resolved shear stress on a plane is the same  
as for calculating the maximum resolved shear strain rate.  Thus both  
assumptions give exactly the same formulation for the inversion, and  
one does not get different inversion solutions for the stress and the  
strain rate assumption.  The question then is whether the answer you  
get from the inversion is best interpreted as a stress tensor or a  
strain rate tensor.  I argue that the inversion solution is a solution  
for the strain rate (or the instantaneous strain), and the problem lies  
in the assumption that the solution can be interpreted as a solution  
for the stress.

        With regard to finite vs instantaneous strains:  The inversion of  
fault-slip data presumes the data reflect a small incremental strain,  
i.e. an 'instantaneous' strain (our theoretical formulation actually  
employs the strain rate).  The stress hypothesis implicitly does the  
same thing, because it assumes that slickenlines, once formed, remain  
in their original orientation and are not rotated into a different  
orientation by a finite strain.  In my experience, the best fault-slip  
data are generally found at the margins of shear zones where a large  
amount of shear has not destroyed the fabric.  Thus the strains are  
presumably small and the presumption of the inversion method is  
satisfactorily met.  Inversion of these data usually gives satisfactory  
results.

        I hope this has answered your question.

        cheers,

Rob Twiss


On Feb 10, 2005, at 6:16 PM, Mark Brandon wrote:

> Rob (and other brittle deformation aficionados),
>         There are two issues embedded in your discussion. The first is  
> if the constitutive relation for brittle deformation is associated
> or non-associated. For the record, associated flow means that the  
> principal stress and principal strain rates axes are coaxial.
> Non-associated flow means that these principals axes are not coaxial.   
> This issue is entirely at the level of stresses and strain rates.
> The second issue relates to finite brittle strains. In this case, we  
> know that a rotational deformation will cause the principal finite  
> strain axes
> are not equal to the principal stress axes. My objective in separating  
> these issues is that I want to understand your opinion on the first  
> issue.
> Do you think that brittle deformation is associated? If so, then for  
> small deformations, the stress methods and the strain rate methods  
> should agree.
> Cheers,
> Mark
>
>  At 08:01 PM 2/10/2005, you wrote:
>> Steve, Heather, Mark, and others interested,
>>
>>         First I'd like to make a point about inversions using the  
>> stress hypothesis vs the instantaneous strain (or strain rate)  
>> hypothesis, then I'll comment on Steve's questions.
>>
>>                 Both the stress and instantaneous strain tensors are  
>> second rank symmetric tensors, and the components of both tensors  
>> plot as Mohr circles.  The mathematics for calculating the  
>> orientation of the direction of maximum resolved shear stress on a  
>> plane is identical to that for calculating the orientation of the  
>> maximum resolved instantaneous shear on a plane.  Thus, an inversion  
>> process that minimizes the misfit between observed slip-directions  
>> and the theoretical orientations for either maximum resolved shear  
>> stress or maximum resolved instantaneous shear gives exactly the same  
>> answer. Thus all published 'paleostress' inversions can equally well  
>> be interpreted as 'instantaneous-strain' inversions.  The fundamental  
>> question, however, is, which tensor does the solution actually  
>> represent, because in nature, the principal axes of stress and strain  
>> are not necessarily parallel, nor is the ratio of differences for the  
>> two sets of principal values necessarily the same.
>>
>>         With regard to the paper Steve mentioned by Sibson and Xie:
>>                 They assume plane strain, and they assume a  
>> horizontal maximum compressive stress.  Neither condition necessarily  
>> represents the actual situation in the earth, so the histograms  
>> should be taken with a grain of salt.  Moreover, might their  
>> histograms possibly be interpreted as a near-normal distribution with  
>> a maximum at about 45°, consistent with maximum resolved shear, but  
>> not with the Coulomb fracture criterion?  Admittedly there is  
>> actually a 'hole' in the histogram at 45°, but then their sample size  
>> is only 31, so irregularities in the distribution are not unlikely.   
>> I also agree with Heather's comments.  So while the paper is an  
>> interesting investigation, in terms of settling the question at hand,  
>> it seems too limited by the assumptions to be very reliable.
>>
>>         With regard to Lisle and Srivastava, 2004:
>>                 Lisle and Srivastava have shown some consistency  
>> between their predictions based on friction and the observations of  
>> natural fault-slip data, from which they conclude that the natural  
>> fault-slip data reflect the orientation of the principal stresses.   
>> There is, however, no measure of how good the correspondence is.  It  
>> seems to me a better test would have been to take the observed  
>> shear-planes, use the friction model to calculate the theoretical  
>> direction of shear on those planes, and measure the misfit between  
>> the theoretical and the observed slip directions.  That would at  
>> least have given a quantitative measure of the correspondence between  
>> the theory and observation.  As it is, all we are given is contoured  
>> plots of theoretically chosen shear-planes and their slip-directions  
>> to compare with scatter plots of observed shear-plane/slickenline  
>> data.   Why not contour the scatter plots of the data?  It is  
>> difficult to see from the scatter plots where the maxima really are;  
>> contouring would have made this more obvious.  Is it possible that  
>> the correspondence is actually not as close as it seems?
>>
>>         Despite these criticisms, it may be that these studies,  
>> however imperfect, are pointing toward the validity of a stress  
>> interpretation of fault-slip data.  To that extent, they are  
>> interesting papers, but to me they are not really conclusive.   
>> Testing the difference between the hypotheses is difficult because  
>> any possible means of independently measuring the stress depends on  
>> knowing some constitutive relation between stress and the  
>> instantaneous strain or (equivalently) the strain rate.  There are  
>> some data that suggest the strain interpretation is better, but they  
>> are not conclusive either. Other tests should be made, including  
>> investigations of the rheology of cataclastic flow.  (I know the  
>> modelers would be delighted if we found that the brittle crust  
>> actually behaved as an isotropic linear newtonian viscous material!)   
>> But we don't really know at this point, and I think it is a  
>> significant question for research.
>>
>>         There is also the question of how good a fit is good enough.   
>> Does it matter all that much if the 'true' stress axes are, for  
>> example, 20° away from where the stress interpretation of the  
>> fault-slip inversion says they are?  The confidence limits for the  
>> inversions commonly allow at least that much latitude.  Perhaps for  
>> the uses these results are put to (e.g. the world-wide 'stress' map,  
>> which is really a strain map), it does not matter that much, but for  
>> me it is a question of good science, and really understanding the  
>> system one is trying to work with.  Wrong hypotheses that give a  
>> 'good enough' answer will certainly not lead to reliable progress in  
>> our understanding.
>>
>>         I don't have any vested interest in whether or not the stress  
>> axes are parallel to the instantaneous strain axes, but I do think we  
>> should know whether they are or not.  Until we have a good answer, my  
>> bias is that researchers should either use the strain axes as the  
>> most direct interpretation of the inversion solutions, or at least  
>> state up front that they assume an isotropic linear constitutive  
>> relation for cataclastic flow to interpret the inversion solutions as  
>> stress axes.
>>
>>         Apologies for the length of this reply.  For those  
>> interested, most of these points have been discussed at even greater  
>> length in Twiss and Unruh, 1998, Jour. Geophys. Res.
>>
>>         Cheers,
>>
>> Rob Twiss
>>
>>
>> _/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/ 
>> _/_/
>>
>>             Robert J. Twiss, Prof. Emeritus           email:    
>> [log in to unmask]
>>             Geology Department                          telephone:  
>> (530) 752-0179
>>             University of California at Davis          FAX:         
>> (530) 752-0951
>>             One Shields Ave.                               website:  
>> www.geology.ucdavis.edu/
>>             Davis, CA 95616-8605, USA                                  
>>  faculty/twiss.html
>>
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>> _/_/
>>
>>
>> </blockquote></x-html>
>
>
> _______________________________________________________________________ 
> _
> Mark Brandon, Professor, Dept. of Geology and Geophysics
> Yale University, P.O. Box 208109, 210 Whitney Avenue, New Haven, CT  
> 06520-8109
> e-mail: [log in to unmask]
> wk. phone: +203-432-3135, wk. fax: +203-432-3134
> Dept. Web site: http://www.geology.yale.edu
> Brandon's site: http://www.geology.yale.edu/~brandon
> _______________________________________________________________________ 
> _
>
>


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             Robert J. Twiss, Prof. Emeritus           email:     
[log in to unmask]
             Geology Department                          telephone:  
(530) 752-0179
             University of California at Davis          FAX:          
(530) 752-0951
             One Shields Ave.                               website:   
www.geology.ucdavis.edu/
             Davis, CA 95616-8605, USA                                    
faculty/twiss.html

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