Dear Steve and others,
In response to your recent comment about the interpretation of
fault-slip data in terms of strain or stress...
I am certainly not opposed to analyzing the mechanics of faulting or
to the use of stress for that purpose. But unless we are clear about
what we are actually measuring, and what the assumptions are that go
into an interpretation, then we are headed for some confusion (which I
believe already exists). I think we agree that measurement of
fault-slip data and of displacements from interferometric images both
give direct information about the STRAIN. There are two points to be
made here:
1) It seems to be a reasonable approximation for interferometric data,
to assume that we can treat the crust as a linear isotropic elastic
material, and therefore to infer that the stress axes are parallel to
the observed strain axes. But to say that we are measuring the stress
axes and to skip the essential assumption of a constitutive relation is
very misleading, as will be evident from point (2) below. It is a fact
that one can NEVER actually measure the stress. All methods of
purportedly measuring stress actually depend on measuring some type of
strain, which is then related to stress through an established
constitutive relation, either directly through calculation, or
implicitly through calibration (which ultimately is how the
constitutive relation is established in the first place).
2) The strain measured by fault-slip data and the strain measured from
interferometric images are actually not the same thing. Fault-slip
data give information about a permanent non-recoverable deformation
that has accumulated through cataclastic flow (distributed brittle
deformation). Interferometric data, on the other hand, record at least
in large part a recoverable, i.e. elastic, deformation. Thus the
constitutive equations used to extract information about stress from
the one set of data do not necessarily apply to another. If we simply
assume both data sets provide a measure of stress, that leads us not to
consider the constitutive relations that in fact relate the observed
strain to the stress, and thus to ignore the essential difference
between the two types of data. Thus the argument that an isotropic
linear elastic constitutive relation applies to the interpretation of
permanent non-recoverable deformation recorded by fault-slip data is
not in fact valid. By analogy, I suspect no one would ever try to use
linear isotropic elasticity to try to interpret the origin of the shape
of deformed ooids in a marble.
The situation of deformation by distributed brittle faulting in the
crust is analogous to the deformation of a crystalline material through
the propagation of dislocations. One can analyze the strain and stress
field around a dislocation as if it were in a linear isotropic elastic
material, and the results are very useful for understanding static
dislocations and their static interactions with one another. That
analysis, however, does not give you the dynamic rheological properties
of the material that result from the aggregate of motion,
multiplication, and annihilation of those dislocations within the
lattice. The propagation of slip events on a fault in the earth's
crust is like the expansion of dislocation loops on a glide plane in a
crystal lattice. One can use linear isotropic elasticity to understand
the stresses around an individual static fault and its static
interactions with other faults, but that is a static analysis. Once
slip starts accumulating on faults, and the faults start propagating
and multiplying and healing, one is no longer dealing with an elastic
material but with a quasi-ductile material, just as in the case with
dislocations. The constitutive relations that govern the ductile
behavior resulting from the aggregate dynamics of propagation of
fault-slip events, multiplication of fault planes, and healing of fault
planes have little to do with the constitutive relations that govern
the static stress fields around the faults, just as the linear elastic
relations that describe the stress field around dislocations have
little to do with the power-law relations that describe the ductile
flow of the crystalline material.
I would argue that we do not know very well what the constitutive
relations are for cataclastic flow (quasi-ductile deformation) in the
brittle crust. The fact that cataclastically deformed materials often
(always?) have a strong preferred orientation of fractures, however,
argues that it is unlikely to show isotropic mechanical properties for
cataclastic flow, and the complexities of the friction that presumably
govern slip on brittle shear fractures suggests that the rheology for
cataclastic flow may well not be linear.
So this highlights the confusion that has arisen by assuming
fault-slip data provide direct evidence for the stress. In fact, for
both fault-slip data and for interferometric data, we are measuring
strain, but in these two cases, the accumulation of strain has been by
two very different processes that are described by two very different
constitutive relations. One can only infer stress from the strain by
applying the appropriate constitutive relation, and in the case of
fault-slip data, I would argue that linear isotropic elasticity is not
the correct constitutive relation to apply.
Bottom line, just as in the case of dislocations in crystalline
lattices, the static case in which linear isotropic elasticity can be
used, provides little guidance for inferring the rheology of
cataclastic (or ductile) flow, and when we examine fault-slip data, we
are examining the results of cataclastic flow. I and Unruh discuss
these points in our JGR paper.
Rob Twiss
On Feb 8, 2005, at 9:30 PM, steven micklethwaite wrote:
> Dear Rob,
>
> You make a good point that fault-slip data speaks directly about
> incremental strain axes and not stress. On the other hand, with
> fault-slip inversion of data from fault systems in the brittle upper
> crust the validity of its assumptions is scale dependent.
> Interferometric images allow us to measure coseismic displacements of
> crust around earthquakes in geologically complex parts of the world.
> Despite geological complexity these displacements have been matched
> to a very accurate degree by simple models where the crust behaves as
> an isotropic elastic medium. Therefore, on the scale of a fault
> system in the brittle upper crust, it seems to be quite valid to
> assume a mechanically isotropic, rheologically linear material. From
> your work this would indicate that the principle axes from inversion
> are stress axes.
>
> Many of the applications I have seen of fault-slip inversion do not
> go beyond constructing a deformation history for an area. In these
> cases it is totally unnecessary to attempt to talk about stress axes.
> It would be foolish however to be dualistic in our interpretations of
> fault systems and the rock record i.e. thinking of fault systems only
> in terms of either strain or stress. Strain will not tell us anything
> about where there were greater accumulations of aftershock events
> around fault systems over time; a question which has important
> implications for fluid flow associated with those fault systems (Cox
> & Ruming JSG v26 p1109+, Micklethwaite & Cox Geology v32 p813+). If
> we want to research fault mechanics from field observations then
> stress is more useful. Likewise understanding fault systems in
> mineralised terranes has been held up for many years because
> geologists have tended to approach them with simple concepts of
> strain.
>
> Cheers,
>
> Steve
> --
> Steven Micklethwaite
> Postdoctoral Fellow
> Rock Physics,
> Research School of Earth Sciences,
> Mills Road, ANU
> Canberra, ACT 0200
>
> T: +61 2 61255169
> F: +61 2 61258253
>
> http://rses.anu.edu.au/petrophysics/Staff/StevenMHome.html
>
>
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Robert J. Twiss, Prof. Emeritus email:
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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
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