T12B-01 INVITED
Anisotropic Visco-Plastic Deformations in Geodynamics: Folding, Shear Banding and Mantle Convection
We give an outline of a constitutive relationship for transversely isotropic elastic-viscous materials. The formulation intended for materials with fine internal layering, which can be described by a single director orientation. This constitutive model is specifically designed for geological deformation problems involving very large deformations. Although there are more general descriptions possible, this formulation is, in fact, very broadly applicable to crustal rocks, where the preponderance of layering arises from deposition of one rock type onto another under gravity. We revisit the basic finite element formulation for viscous materials and demonstrate how the standard element vectors and matrices can be extended to include anisotropy. We explore scenarios from global to internal buckling in nonlinear finite element studies. These show that buckling can be induced at much lower viscosity contrasts between the matrix and the embedded beam or plate than would be the case for isotropic materials. Numerical solutions based on the standard continuum formulation assumed initially may become unstable if the contrast between the normal - and the shear viscosity becomes very severe. The director formulation can also be used to define a prefered plane for slip to occur given the local stress field. The simple-shear viscosity and the deformation can then be iterated to ensure that the yield criterion is always satisfied. We assume the Boussinesq approximation, neglecting any effect of dilatancy on the stress field. An additional criterion is required to ensure that deformation occurs along the plane aligned with maximum shear strain-rate rather than the perpendicular plane, which is formally equivalent in any symmetric formulation. Here we also allow for strain-weakening of the material. The material can remember both the accumulated failure history and the direction of failure. We have included this capacity in a Lagrangian-integration-point finite element code and show a number of examples of extension and compression of a crustal block with a Mohr–Coulomb failure criterion. The formulation itself is general and applies to 2-and 3-dimensional problems.
T12B-02 INVITED
Anisotropic ice in Glaciers and Polar ice Sheets: Experimental Results and Modelling
The slow motion of polar ice sheets is governed by the viscous deformation of anisotropic ices. Strain rates are typically between 10-9 and 10-13 s-1 and temperature above -57°C. At such low strain rates, the stress exponent can be lower than 2. But, basal slip is still the dominant deformation mode. At the both scales of the ice crystal and the polycrystal, the deformation is highly heterogeneous. This behavior is associated with the anisotropy of the ice crystal and the mismatch of slip at grain boundaries for the polycrystal. Due to the low lattice friction, dislocations glide cooperatively and long-range internal stresses develop, which are accommodated by excess dislocations, in contrast to statistical dislocations associated with short-range interactions between dislocations. The viscoplastic deformation induces the development of lattice preferred orientation (textures) giving a non- random orientation of the c-axes in the largest part of ice sheets. Initially isotropic ice formed after the transformation of snow into ice becomes anisotropic as textures develop. Depending on textures, the directional viscosity can be 10 times lower or 10 times higher than that for isotropic ice. Then, both the ice crystal and textured ices are characterized by an exceptional plastic anisotropy. Dynamic recrystallization can produce stress- controlled textures. In this case, ice must be considered as isotropic. The computation of the mechanical behaviour of anisotropic polycrystalline ice using self-consistent approaches is nowadays a standard approach. These models consist in regarding each crystal of the polycrystal as an inclusion embedded in an infinite homogeneous equivalent medium whose the behavior represents that of the polycrystal. Predictions of the mechanical response obtained with the second-order self-consistent approach are compared with full-field solutions. A good prediction of the mechanical behaviour of anisotropic ices is obtained on condition that stress and strain rates fluctuations within grains are introduced.
T12B-03
A Finite-Element Ice Flow Model for the Vicinity of Dome Fuji With Induced Anisotropy and Fabric Evolution
A three-dimensional, thermo-mechanically coupled flow model with induced anisotropy has been developed and applied to the vicinity of Dome Fuji, Antarctica. The model implements the full Stokes equations for the ice dynamics, and the system is solved with the finite-element method (FEM) using the open source multi-physics package Elmer (http://www.csc.fi/elmer/). The finite-element mesh for the computational domain has been created with two data sets, the fine resolution data obtained at Dome Fuji and the coarse resolution data sets RAMPDEM V2 and BEDMAP which cover the entire Antarctic ice sheet. The fine data which represent a 60 x 60 km area around the Dome Fuji station have been merged to the coarse data sets to create a single domain of about 200 x 200 km size. The mesh consists of a coarse resolution near the boundaries (20 km) and a mesh resolution refinement (up to 500 m) towards the position of the borehole located at the center of the domain. This procedure has been carried out in order to keep the lateral boundaries sufficiently far away from the dome, so that shallow- ice stresses can be prescribed there. At the base, no-slip conditions are assumed, and on the surface, the temperature is prescribed to be constant everywhere on the domain. A Continuum-mechanical, Anisotropic Flow model, based on an anisotropic Flow Enhancement factor (CAFFE model) is used for taking into account the flow-induced anisotropy in ice. The flow law is implemented in Elmer by means of second and fourth order orientation tensors that describe the c-axis orientation of the fabric. Similarly, the fabric evolution equation is written in terms of the evolution of the second order tensor, and it is solved with a Discontinuous Galerkin method using Picard type iterations for the non-linearity. Since the fabric evolution equation also depends on the fourth order orientation tensor, the IBOF (Invariant-Based Optimal Fitting) closure function is used for the computation of its components from the solution of the second orientation tensor. The questions to be investigated are (i) what is the evolution of the fabric over the last glacial cycle, (ii) how does the flow field change with the fabric, (iii) what is the effect of ice anisotropy at the site of the Dome Fuji ice core.
T12B-04 INVITED
Viscous and elastic anisotropy in partially molten rocks I: Experimental, field, and seismic observations
We present two papers on the causes and consequences of structural anisotropy in partially molten rocks. We show that viscous anisotropy can have significant effects on melt migration dynamics, with predictions that may be tested seismically by mapping the relationship between viscous and elastic anisotropy. This paper focuses on the observational motivations for the theory presented in the companion paper. End-member rheological behaviors of rocks in the Earth, elasticity and viscosity, are often approximated as isotropic. However, observations in the field, laboratory experiments, and seismic experiments all implicate significant anisotropy in rock properties. Nonethless, in geodynamics, definitive observational tests for viscous anisotropy have been difficult, so the approximation of isotropic viscosity is the norm. Structural anisotropy in mantle rocks comes from two main causes, 1) Lattice preferred orientation (LPO), or 2) Distribution of solid and/or fluid phases. 1) LPO is associated with dislocation creep. At high P-T, olivine a-axes generally align in the shear direction, but experiments at elevated pressure, stress and water fugacity show that each may affect LPO, presumably by affecting dislocation dynamics. Strain partitioning may also affect LPO. 2) Anisotropy due to phase distribution depends on the contrast in material properties between the two (or more) phases and their spatial distribution. Field observations of melt traces and dunites in ophiolites imply that melt is often aligned and segregated. In deformation experiments, spontaneous melt alignment and segregation during deformation is observed in a range of systems and deformation geometries. Basic theoretical aspects of this process are becoming understood. In experiments, a key observation is that melt appears to align before it segregates, which will cause a viscous anisotropy at the grain scale. In our companion paper, we show that this grain-scale anisotropy may be the cause of the instability that leads to longer-wavelength melt segregation. If such segregation and organization occur in the Earth, the viscous anisotropy (and reduction!) over a wide range of length scales should have macroscopic consequences for plate boundary dynamics. This hypothesis is becoming testable, using the granular model discussed in our companion paper to map between elastic and viscous properties using the same geometric description of melt. These models show that viscous properties are much more sensitive to anisotropy in melt than are seismic properties. This approach, combined with new seismic inversion methods and increased resolution, will enable new tests for the presence or absence of viscous anisotropy in the Earth.
T12B-05
Viscous and elastic anisotropy in partially molten rocks II: Significant role of viscous anisotropy in melt migration dynamics.
Observational motivations of our study on the causes and consequences of structural anisotropy in partially molten rocks are stated in the first of two companion papers. We present here a modeling approach to quantify the effects of grain scale microstructural anisotropy on the macroscopic elastic and viscous properties. Based on the results, we provide a quantitative method for mapping between elastic and viscous anisotropies and also demonstrate the importance of grain scale microstructural anisotropy as a cause of larger scale melt redistribution. Mechanical constitutive relations of partially molten rocks are derived based on a microstructural model with granular configuration. It is shown that both elasticity and viscosity depend on the grain-to-grain contact geometry, and that anisotropy of the contact geometry results in the anisotropy of these properties. By constraining the model with observations of contact geometry in experimentally deformed partially molten rocks (or rock analogue), direction and amplitude of viscous anisotropy are calculated under a given stress direction. The obtained viscosity tensor has off-diagonal components that define a coupling between shear and isotropic components, which do not exist in an isotropic viscosity tensor. One of the most remarkable consequences of this coupling is the enhancement of interaction between shear deformation and melt migration, which is demonstrated by solving the governing equations of solid-liquid two-phase system under some simple boundary conditions. In an example of a simple shear zone under constant stress conditions, perturbation in melt fraction parallel to the shear plane can grow due to the effect of viscous anisotropy on melt pressure. This process can produce the segregation observed in experiments and suggested by field observations. In a second example of rotary shear deformation, melt migrates up stress gradients in the solid framework: in other words, shear stress gradients can drive compaction. Such melt redistribution over distances greater than grain scale would significantly increase effective permeability and matrix deformability (lubrication), causing interaction between melt migration and deformation relevant to the scale of plate boundaries. Mapping between elastic and viscous properties enables us to test the possible occurrence of such interactions in the Earth.
T12B-06 INVITED
Influence of anisotropic rheology on Rayleigh-Benard convection
Dislocation creep, which is the dominant deformation mechanism in the upper mantle, results in a non-Newtonian anisotropic rheology. The implication of non-Newtonian rheology has been quite extensively studied in geodynamic models but the anisotropic aspect remains poorly in- vestigated. We propose to fill this gap by (1) introducing a simple mathematical description of anisotropic viscosity and (2) illustrating the link between plastic crystal deformation and bulk material rheology. The study relies on the two systems having highest symmetry, i.e. cubic and hexagonal being characterised by one and three anisotropic perturbation parameters, respectively. First-order implications of anisotropy are quantitatively explored as functions of these parameters. The effective rheology of the material is described as a function of the orientation of the crystals and of the imposed stress and the validity of the isotropic approximation is discussed. The model, applied to ringwoodite, a cubic crystal with spinel-type structure, predicts that the dynamics of the transition zone in the Earths mantle is going to be strongly affected by mechanical anisotropy. Similarly, the dynamics of materials having an hexagonal symmetry like in icy satellites and layered media is affected by anisotropy. Using linear stability analysis on Rayleigh-B?©nard problem, we show how the critical Rayleigh number, the length scale of the first instability and the associated roll direction quantitatively depend on anisotropy.
T12B-07
Anisotropic viscosity in geodynamical flow models - Rayleigh-Taylor instabilities as a text example
Rocks often develop fabric when subject to deformation, and this fabric causes anisotropy of physical properties including effective viscosity and seismic velocities. While predicting anisotropy from mantle flow models is becoming increasingly popular, the effect of viscous anisotropy on the flow has yet to be investigated thoroughly. We employ analytical solutions and two-dimensional numerical flow models to determine the effects of anisotropic viscosity on the development of Rayleigh-Taylor instabilities, a process strongly connected to lithospheric instabilities. Anisotropic viscosity has significant effects on the development of instabilities --- their timing, location, and, most notably, their wavelength, are strongly affected by the initial fabric. We find a significant increase in the wavelength of instability in the presence of anisotropic viscosity orientated to favor horizontal shear. We also find that interplay between regions with different initial fabrics gives rise to striking irregularities in the downwellings. An inherent part of our investigation is the development and tracking of anisotropic fabric by the flow. We compare three methods for tracking fabric development - directors, finite strain ellipses and a kinematic crystallographic method. We find the directors method to provide a reasonably accurate and efficient prediction tool, appropriate for incorporation into self-consistent anisotropic mantle flow models. Our study shows that for investigations of lithospheric instabilities, and likely of other mantle processes, the approximation of isotropic viscosity may not be adequate. Thus anisotropic viscosity should, and can fairly easily, be included.
T12B-08
The anisotropic signature of western Canada: Probing the mechanics of lithospheric deformation
Spatial variations in the rigidity of continental plates (expressed in terms of an effective elastic thickness, Te ) can have a profound influence on the style of deformation of ocean- continent convergent margins. Depending on where the largest gradients in Te occur, strains related to plate boundary forces can become concentrated where Te is small. We estimate Te and Te anisotropy in western Canada by inverting the coherence between Bouguer gravity and topography. In addition to a nearly stepwise change in Te from the Cordillera to the craton, we show that weak axes of Te anisotropy are parallel with most compressive directions of horizontal stress components and fast axes of upper mantle seismic anisotropy. Maxima in the magnitude of Te anisotropy are spatially correlated with the locations of most earthquakes and the locations and directions of maxima in electrical anisotropy and conductivity. The pattern of brittle failure and seismicity is explained as a response to plate boundary forces acting on a plate of variable rigidity and is expected to induce the observed mechanical anisotropy and enhance the flow of crustal fluids, resulting in locally high electrical conductivities. Our combined results thus suggest for the first time that the elastic properties of continental lithosphere have a leading order influence on the deformation and evolution of convergent plate boundaries.