Index
Effective Rheology
Localization Instability
Indian Ocean
Wrinkle Ridges
Finite Element Modelling
Projects summaries
Effective
Rheology
The first paper out of my thesis... A pretty heavy thing, but that's with content! Have you ever wondered how to model a material that is not happy deforming in a distributed fashion but instead wants to localize deformation on a narrow shear zone, or a fault? I define a simple measure of material response to a local perturbation. That's the effective stress exponent. If it is negative, the material want to locale. The more negative it is, the stronger the tendency to localization. Of course, there are many processes that have been observed in nature and in lab experiments to produce localized deformation, so, I had to show that the effective stress exponent is indeed negative for these processes. By now, I've got quite a few mechanisms worked out (cohesion loss, non-associated elasto-plasticity, rate- and state- dependent friction, gouge effects, shear-heating in the ductile regime, and transition from dislocation- to diffusion-dominated creep). I give mathematical expressions (I know, Yack!) for the corresponding effective stress exponents and discuss the numerical values, when possible.
Montési,
L. G. J., and M. T. Zuber, A unified description of localization for
application to large-scale tectonics, Submitted to J. Geophys. Res, 2001
Conference Abstracts: fall AGU 1998Abstract
Full text (PDF version)
Localization Instability
The centerpiece of my work. We see in a few natural examples and in numerical models patterns of localized shear zones with characteristic fault spacing. In this paper, I present a theoretical treatment of a lithosphere models that shows that behavior. The model is originally unform, and is made of several layers with different material properties. In particular, the upper layer has a tendency to form faults. Then, I determine how fast tiny perturbations of that model grow, to see if these faults have characteristic spacing. Well, that works! I show a few simple expressions, or scaling relations, that allow a quick estimate of predicted fault spacing for application to the real world. The paper is just about to be sent. Just wait until I finish my thesis!
Montési,
L. G. J., and M. T. Zuber, Spacing of Faults at the Scale of the Lithosphere and Localization Instability 1: Theory, in preparation
Conference Abstracts:
Spring AGU 1998 ASME 1999
Indian Ocean
The central basin of the Indian Ocean is an interesting region in that
it is undergoing ditributed N-S shortening (Diffuse plate boundary between
Indian and Australia).Geoid and basements undulations can be explained
by classical buckling instability, but not the short wavelength faults. For that, we must use the localization instability theory that I developped. In the paper, we also explore the effect of different assumption about the structure of the lithosphere and the penetration of faults.
Montési,
L. G. J., and Zuber, M.T., Spacing of Faults at the Scale of the Lithosphere and Localization Instability 2: Application to the Central Indian basin, in preparation
Conference Abstracts:
fall AGU 2000
Wrinkle Ridges
Regularly-spaced faults are not limited to the Earth. On Mars, wrinkle ridges, which are underlain by faults, have a characteristic spacing. MOLA allowed us to identified new ridges in the Martian northern lowlands, that have a spacing roughly twice that of highland plains. The localization instability theory allow us to related the difference in fault spacing to the thickness of the crust, which is well constrained with MOLA and MGS data. A bonus: We can constrain the geotherm of Mars at the time of ridge formation!
Montési,
L. G. J., and M. T. Zuber, Clues to the lithospheric structure of Mars from wrinkle ridge spacing, in preparation
Conference Abstracts:
Tharsis Workshop 2000; LPSC 2001
Finite
Element Modeling
Of course, the analytical theory is quite simplified. Numerical methods, and in particular Finite Elements give an indenpend check on the theory, and permit to explore finite strains. In addition, we can use more complex configuration, for instance including lateral variations of geotherms or crustal thickness. That's quite important for the Martian wrinkle ridges!
Conference Abstracts:
Chapman conference 1997; Spring AGU 1999; LPSC 2000;
Conference
Abstracts
Montési,
L. G. J. and Zuber, M.T, Crustal Thickness Control on Martian Wrinkle Ridge Spacing, Lunar Planet. Sci. XXXII, abstr. 1879 , 2001
PDF version
Montési, L. G. J., and Zuber, M.T., Can the ~7-km spacing of faults in the central basin of the Indian Ocean have developed during the current N-S shortening? EOS Trans. Am. Geophys,. Un., 81, Fall Meet. Suppl., F1097, 2000
The central basin of the Indian Ocean is part of the diffuse plate boundary between the Indian and Australian plates. It has been undergoing roughly N-S compression expressed in particular by ~200km wavelength basement and geoid undulations. We focus on the set of the superimposed reverse faults, in particular on the origin of their ~5 km spacing.
It has been argued that the faults are normal faults formed at the mid-ocean ridge later reactivated with reverse motion [e.g. Bull and Scrutton, 1990]. However, no argument in favor of reactivation is indisputable, and we question whether these faults coul have formed during the early stages of the shortening. If reactivation was important, it occurred on only one out of four or five [Bull, 1990], mostly outward facing [Bull and Scruton, 1992], faults. In either case, the ~5 km wavelength must have formed early in the compressive stage.
The layer associated with faulting is at least 5 to 6 km thick, the depth at which faults are imaged. Possible upper limits are 25-30 km thick (depth of brittle-ductile transition from thermal modeling) or 40 km (maximum depth of earthquakes). Hence, the wavelength/thickness ratios is between 0.1 and 1. In contrast, the classical buckling instability of a layered visco-plastic lithospheric model, such as been used to explain the 200 km wavelength, predicts lambda/H>4 [e.g. Zuber, 1987; Johnson and Fletcher, 1994]. However, these models do not include the possibility of localization. Using an effective rheology for a medium undergoing localization (Montési and Zuber, submitted to JGR), we derive new growth spectra showing preferred growth at wavelength/thickness ratios between 1 and 0.1 if the effective stress exponent is of order -4 to -400 (Montési and Zuber, submitted to J. Fluid Mech.), consistent with laboratory experiments. Although the fault spacing may be modulated by pre-existing structures, explaining the variability in the data, the fault spacing is consistent with a dynamic origin during the shortening, once localization is taken into account.
Montési, L. G. J., and Zuber, M.T., The lithospheric structure of the margin of Tharsis as indicated by wrinkle ridges, Tharsis - Evolution in light of MGS data, USGS workshop, 2000
Montési,
L. G. J., Zuber, M.T, and Aharonson, O., Geometry of faults underlying
wrinkle ridges on Mars: Dynamic modeling and MOLA topography, Lunar
Planet. Sci. XXXI, abstr. 1927 , 2000
PDF
version
Montési, L. G. J., and Zuber, M.T, The importance of localization for the development of large-scale structures in the Earth's crust, ASME Mech. & Mat. Conf., p307, 1999
Montési, L. G. J., and Zuber, M.T., The evolution of fault patterns during orogeny, EOS Trans. Am. Geophys,. Un., 80, Spring Meet. Suppl., S345, 1999
The distribution of faults has an important control on orogeny as mountains are built by repeated slip on faults. As faults are readily observable from field studies as well as seismic investigations or remote sensing imagery, they can provide key constraints on the mechanical processes involved in the formation of a fold-and-thrust belt. However, they have been incorporated in most models at best as {\a priori} zones of weakness. We instead investigate their dynamic interaction with the deformation field and the evolution of predicted fault patterns with time.
We previously showed using an effective rheology for the localization of deformation on faults that two wavelengths of instability develop in a shortening crustal modal, one corresponding to single layer folding, the other to major fault spacing [Montési and Zuber (1998), EOS Trans. AGU 79, Spring Meet. Suppl., S346]. However, as finite deformation accumulates, faults rotate or flex to a point where the creation of a new fault is favored over the continued activation of the initial ones. We illustrate this process using a Finite Elements Model of an initially flat and uniform crust, where each element is free to deform either ductily or to fail at a stress level determined by the frictional strength of rocks. The brittle strength decreases with strain rate in a way inspired by the experimentally determined velocity weakening of the coefficient of friction, leading to localization [Neumann and Zuber (1995), Proc. 25th US. Symp. Rock. Mech., 191-198].
The evolution of the fault pattern depends on the competition between the advection of weak zones, the inertia of the model strength adjustments, and the local stress state. As a result, the details of the fault pattern at any time are extremely model-dependent. However, the dominant wavelength is always that of the faulting instability. The separation between faults generated at different times still follows the faulting wavelength. It is therefore possible to compare the results of this model with actual examples of mountain belts, in spite of the integrated time history displayed in the geological record.
Montési, L. G. J., and Zuber, M.T, Modeling the development of faults: definition of the effective rheology of a continuum undergoing localization. EOS Trans. Am. Geophys. Un., 79, Fall Meet. Suppl., F846, 1998
Although faults and localized deformation are ubiquitous in at least the upper levels of the Earth, they have been incorporated in macroscale tectonic modeling in only a preliminary sense. A major difficulty arises because, while the coupling between different zones of deformation requires a continuum approach, strain is localized on discrete areas when faulting occurs. Consequently, two major classes of models have been used in modeling regional-scale deformation. Either faults are defined as {\it a priori} loci of deformation, or only the continuum flow aspect of the problem is solved. While these approaches have their range of validity, neither is entirely satisfactory if both the development and evolution of faults in a given tectonic setting are sought. Recent numerical studies using either elasto-plastic or self-lubricating rheologies have shown the limitation of the more classical approaches. Buckling wavelengths and growth spectra are also modified as faulting is incorporated.
The generation or activation of a fault localizes the deformation around it, where the material becomes effectively weaker. To reach the almost discontinuous flow field observed in nature, it is not enough to decrease the effective viscosity of the fault; the stress level itself must also decrease. This can be achieved by a variety of feedback mechanisms. In the most general case, we characterize the system composed of a rheology and a feedback scenario by an effective stress exponent. We show that this exponent must be negative for localization to occur.
For illustration, we construct the stability field of the following examples, based on experimental results on rocks: 1) velocity weakening of the coefficient of friction due to a change in surface or gouge characteristics of a fault; and 2) thermally activated creep laws, with softening by viscous heating. This latter example may justify the self-lubricating rheology used for modeling the formation of tectonic plates. Interestingly, we predict that localization is almost impossible for dry rock flow laws as may be relevant for Venus, in contrast to rock flow under terrestrial conditions, making plate boundary deformation unlikely to develop on Venus.
Montési, L. G. J., and Zuber, M.T., The influence of localization of tectonic strain on lithospheric buckling, EOS Trans. Am. Geophys,. Un., 79, Spring Meet. Suppl., S346, 1998
Modeling buckling at lithospheric scale can allow the retrieve of information on the crustal structure from geological observations or remote sensing data. For instance, depth of the brittle-ductile transition and surface heat flow can be estimated using wavelengths of lithospheric surface deformation. However, little attention has been directed towards the effects of faulting and tectonic strain localization on buckling. Preliminary numerical studies using finite-element methods [Montesi and Zuber, 1997] have shown that strain localization may decrease the dominant wavelength of deformation.
We present here a semi-analytical treatment of the amplification and evolution of infinitesimal perturbations at the interfaces of a rheologically stratified lithosphere that consists of a brittle layer that overlays an infinite half space with power law behavior. This model lithosphere is subjected to horizontal shortening. We investigate what differences from previous visco-plastic solutions arise when the localization of deformation is taken into account.
Post-failure brittle behavior is modeled by a continuum rheology derived from the experimentally determined velocity dependence of the friction coefficient: the decrease of stress as strain rate increases leads to localization of strain at lithospheric scale. We relate this new rheology to generic nonlinear laws through an effective stress exponent n, which is negative. The resulting flow solutions lack the characteristic decay depth associated with viscous or power law media (n>1). A resonance develops between the upper and lower boundaries of the brittle layer, resulting in short wavelength structures associated with faulting printed over viscoplastic (n=±\infty) buckling. If this effect were not taken into account, the use of fault spacing as an indication of buckling wavelength would underestimate the thickness of the deforming brittle layer. Applied to Venusian crust, this effect can eliminate the discrepancy between observed wavelengths of deformation and crustal structure as indicated by dry diabase rheology and an Earth-like geotherm.
Montési,
L. G. J., and Zuber, M.T, Short-wavelength compressional deformation in a strong Venusian lithosphere, Chapman Conf. Geodynamics of Venus: Evolution and Current State, p.14 1997
Mountain belts on Venus are characterized by complex deformation patterns. However, many fold and thrust structures exhibit parallel ridge structures with spacings typically between 3 and 10 km. Models of unstable buckling of a brittle layer above a viscous substrate predict a dominant wavelength of a least 3.5 times the thickness of that brittle layer, which requires a rther shallow brittle-ductile transition (BDT). Alternatively, the strength of the lithosphere as implied by experiments on the ductile strength of dry diabase and on Magellan gravity/topography relationships imply a significantly deeper BDT, at least for the present day. We are motivated to question whether the tectonics preserves the record of an earlier weak lithosphere or whether shorter wavelength deformation than predictied by simple flexure and infinitesimal amplitude instability models is possible given dry rheological data.
We consider a model for the development of compressional deformation inside a thick brittle layer undergoing finitie amplitude strain. In addition to the usual Byerlee's friction law and power law creep for rocks of reasonable composition, we include in our formulation a strain rate weakening factor in the brittle layer that results in strain localization in linear regions of finite extent, resembling fault zones at crustal scale. At small strains, localization develops at the dominant wavelength. However, with increasing strain, conjugate and secondary high strain zones are generated, so that the overall pattern shows both the buckling wavelength and a second deformational length scale, of the same order as the brittle layer thickness. Our non-linear finite element model in numerically stable up to 30% strain. as deformation develops, the localization pattern evolves, with some weak zones being deactivated, while others are generated. The general trend is to concentrate the strain in one or two major parallel faults. However, the topographic expression of previously active zones is usually preserved. Thus, in areas of lithospheric shortening that have experienced high strains, the depths of the BDT as inferred from models that utilize the flow laws of dry diabase are not necessarily incompatible with short tectonic wavelengths observed in Magellan images.
Paper Abstracts
Montési,
L. G. J., and Zuber, M.T, A unified description of localization for
application to large-scale tectonics J. Geophys. Res, Submitted, 2001
Localized regions of deformation such as faults and shear zones are ubiquitous in the lithosphere of the Earth. However, we lack a simple unified framework of localization that is independent of the mechanism or scale of localization. We address this issue by introducing the effective stress exponent, $\NE$, a parameter that describes how a material responds to a local perturbation of an internal variable being tested for localization. The value of $n_e$ is based on micromechanics. A localizing regime has a negative $\NE$, indicating a weakening behavior, and localization is stronger for more negative $\NEinv$. We present expressions for the effective stress exponent associated with several mechanisms that trigger localization at large scale: brittle failure with loss of cohesion, elasto-plasticity, rate- and state- dependent friction, shear heating, and grain-size feedback in ductile rocks. In most cases, localization does not arise solely from the relation between stress and deformation but instead requires a positive feedback between the rheology and internal variables. Brittle mechanisms (failure and friction) are generally described by $\NE$ of the order of -100. Shear heating requires an already localized forcing, which could be provided by a brittle fault at shallower levels of the lithosphere. Grain size reduction, combined with a transition from dislocation to diffusion creep, leads to localization only if the grain size departs significantly from its equilibrium value, either because large-scale flow moves rocks through different thermodynamic environments, or new grains are nucleated. When shear heating or grain-size feedback produce localization, $\NEinv$ can be extremely negative and control lithospheric-scale localization.
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