Abstract
The Earth shows different modes of deformation in response to thermal or gravitational driving forces. The bulk mantle convects like a viscous fluid on the global scale, while the lithosphere is broken into several plates. They show little internal deformation, but change their shapes and relative positions. Oceanic plate material is generated at divergent margins and recycled into the mantle at subduction zones, on a regional scale. The buoyant continental crust resists subduction and develops meter-scale shear bands during deformation.
In this article we review Eulerian finite element (FE) schemes and a particle-in-cell (PIC) FE scheme [15]. Focussing initially on models of crustal deformation at a scale of a few tens of km, we choose a Mohr-Coulomb yield criterion based upon the idea that frictional slip occurs on whichever one of many randomly oriented planes happens to be favourably oriented with respect to the stress field. As coupled crust/mantle models become more sophisticated it is important to be able to use whichever failure model is appropriate to a given part of the system. We have therefore developed a way to represent Mohr-Coulomb failure within a mantle-convection fluid dynamics code.
With the modelling of lithosphere deformation we use an orthotropic viscous rheology (a different viscosity for pure shear to that for simple shear) to define a preferred 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 again assume the Boussinesq approximation - neglecting any effect of dilatancy on the stress field.
Subduction is modelled as a Rayleigh-Taylor instability with dense oceanic lithosphere sinking into less dense sublithospheric mantle. We use a linear viscous rheology for the mantle in this case. Parts of the lithosphere are viscous, others brittle. The values of the dynamic viscosity are different for lithosphere and mantle. The brittle behaviour of parts of the lithosphere can be modelled in the continuum limit by using a viscoplastic rheology.
Turning to the largest planetary scale, we present an outline of the mechanics of unified models plate-mantle models and then show how computational solutions can be obtained for such models using Escript. The consequent results for different types of convection are presented and the stability of the observed flow patterns with respect to different initial conditions and computational resolutions is discussed.
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References
Barr, A.H.: Superquadrics and angle-preserving transformations. IEEE Computer Graphics and Applications 1(1), 11–23 (1981)
Bird, P., Kong, X.: Computer simulations of california tectonics confirm very low strength of major faults. Geol. Soc. Am. Bull. 106, 159–174 (1994)
Brady, B.H.G., Brown, E.T.: Rock Mechanics. George Allen and Unwin, London (1985)
Bronkhorst, C.A., Kalidindi, S.R., Anand, L.: Polycrystalline plasticity and the evolution of crystallographic texture in fcc metals. Phil. Trans. Royal Soc. A 341, 443–477 (1992)
de Josselin de Jong, G.: The double sliding free rotating model for granular assemblies. Geotechnique 21, 155–163 (1971)
England, P., Houseman, G.: Role of lithospheric strength heterogeneities in the tectonics of tibet and neighbouring regions. Nature 315, 297–301 (1985)
England, P., Houseman, G., Sonder, L.: Length scales for continental deformation in convergent, divergent and strike-slip environments: Analytical and approximate solutions for a thin viscous sheet model. J. Geophys. Res. 90, 3551–3557 (1985)
England, P.C., Houseman, G.A.: The mechanics of the tibetan plateau. Philos. Trans. R. Soc. Lond. Ser. A-Math. Phys. Eng. Sci. 326, 301–320 (1988)
Gross, L., Bourgouin, L., Hale, A.J., Muhlhaus, H.-B.: Interface modeling in incompressible media using level sets in escript. Physics of the Earth and Planetary Interiors 163, 23–34 (2007)
Hill, R.: The Mathematical Theory of Plasticity. Oxford University Press, Oxford (1998)
Humphreys, F.J., Hatherly, M.: Recrystallization and related annealing phenomena. Pergamon Press, U.K.(1995)
Karato, S., Wu, P.: Rheology of the upper mantle - a synthesis. Science 260, 771–778 (1993)
Kolymbas, D., Herle, I.: Shear and objective stress rates in hypoplasticity. Int. J. Numer. Anal. Methods Geomech. 27, 733–744 (2003)
Kong, X., Bird, P.: Thin-shell finite- element models with faults. In: Yin, A., Harrison, T.M. (eds.) The Tectonic Evolution of Asia, pp. 18–34. Cambridge University Press (1996)
Moresi, L., Dufour, F., Muhlhaus, H.B.: A lagrangian integration point finite element method for large deformation modeling of viscoelastic geomaterials. Journal of Computational Physics 184, 476–497 (2003)
Moresi, L.N., Solomatov, V.S.: Mantle convection with a brittle lithosphere: thoughts on the global tectonic styles of the earth and venus. Geophysicical Journal International 133, 669–682 (1998)
Muhlhaus, H.-B., Regenauer-Lieb, K.: Towards a self-consistent plate mantle model that includes elasticity: simple benchmarks and application to basic modes of convection. Geophys. J. Int. 163(2), 788–800 (2005)
Muhlhaus, H.-B., Dufour, F., Moresi, L., Hobbs, B.: A director theory for viscoelastic folding instabilities in multilayered rock. Int. J. Solids Structures 39, 3675–3691 (2002)
Muhlhaus, H.B., Moresi, L., Hobbs, B., Dufour, F.: Large amplitude folding in finely layered viscoelastic rock structures. Pure and Applied Geophysics 159, 2311–2333 (2002), doi:10.1007/s00024-002-8737-4
Muhlhaus, H.B., Moresi, L., Cada, M.: Emergent anisotropy and flow alignment in viscous rock. Pure Appl. Geophys. 161, 2451–2463 (2004)
Rudnicki, J.W., Rice, J.R.: Conditions for the localization of deformation in pressure sensitive dilatant materials. J. Mech. Phys. Solids 23, 371–394 (1975)
Solomatov, V.S.: Scaling of temperature- and stress- dependent viscosity convection. Phys. Fluids 7, 266–274 (1995)
Tackley, P.J.: Three-dimensional simulations of mantle convection with a thermo-chemical basal boundary layer: D”? Americal Geophysical Union (1998)
Tackley, P.J.: Self-consistent generation of tectonic plates in threedimensional mantle convection. Earth Planet. Sci. Lett. 157, 9–22 (1998)
Vardoulakis, I., Sulem, J.: Bifurcation Analysis in Geomechanics. Blackie Academics and Professional (1995)
Watts, A.B., Bodine, J.H., Ribe, N.M.: Observations of flexure and the geological evolution of the pacific basin. Nature 283, 532–537 (1980)
Watts, A.B., Bodine, J.H., Steckler, M.S.: Observation of flexure and the state of stress in the oceanic lithosphere. J. Geophys. Res. 85, 6369–6376 (1980a)
Zhong, S.J., Gurnis, M.: Controls on trench topography from dynamic-models of subducted slabs. J. Geophys. Res.-Solid Earth 99, 15683–15695 (1994)
Zhong, S.J., Gurnis, M.: Towards a realistic simulation of plate margins in mantle convection. Geophys. Res. Lett. 22, 981–984 (1995)
Zhong, S.J., Gurnis, M.: Mantle convection with plates and mobile, faulted plate margins. Science 267, 838–843 (1995)
Zhong, S.J., Gurnis, M.: Interaction of weak faults and non-newtonian rheology produces plate tectonics in a 3d model of mantle flow. Nature 383, 245–247 (1996)
Zhong, S.J., Gurnis, M., Moresi, L.: Role of faults, nonlinear rheology, and viscosity structure in generating plates from instantaneous mantle flow models. J. Geophys. Res.-Solid Earth 103, 15255–15268 (1998)
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Mühlhaus, HB., Moresi, L., Davies, M., Gottschald, K., Hale, A. (2013). Instabilities across the Scales: Simple Models for Shear Banding, Plate Subduction and Mantle Convection in Geodynamics. In: Denier, J., Finn, M. (eds) Mechanics Down Under. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5968-8_11
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