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Constitutive Equations, Rheological Behavior, and Viscosity of Rocks

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Constitutive Equations, Rheological Behavior, and Viscosity of Rocks 2.18 Constitutive Equations, Rheological Behavior, and Viscosity of Rocks DL Kohlstedt, University of Minnesota, Minneapolis, MN, USA LN Hansen, University of Oxford, Oxford, UK ã 2015 Elsevier B.V. All rights reserved. This chapter is a revision of the previous edition chapter by D L Kohlstedt, Volume 2, pp. 389–427, © 2007, Elsevier B.V. 2.18.1 Introduction 441 2.18.2 Role of Lattice Defects in Deformation 442 2.18.2.1 Point Defects: Thermodynamics and Kinetics 442 2.18.2.1.1 Thermodynamics 442 2.18.2.1.2 Kinetics 444 2.18.2.2 Line Defects: Structure and Dynamics 445 2.18.2.3 Planar Defects: Structure and Energy 447 2.18.3 Mechanisms of Deformation and Constitutive Equations 449 2.18.3.1 Constitutive Equations 449 2.18.3.1.1 Diffusion creep 449 2.18.3.1.2 Deformation involving dislocations 450 2.18.3.1.3 Deformation mechanism maps 453 2.18.3.2 Role of Fluids in Rock Deformation 454 2.18.3.2.1 Role of melt in rock deformation 454 2.18.3.2.2 Role of water in rock deformation 457 2.18.3.3 Role of Texture in Rock Deformation 460 2.18.3.3.1 Texture development during deformation 460 2.18.3.3.2 Texture as a constraint on deformation mechanism 461 2.18.3.3.3 Effect of texture on rock viscosity 461 2.18.4 Upper Mantle Viscosity 463 2.18.4.1 Effect of Temperature and Pressure 464 2.18.4.2 Effect of Water 465 2.18.4.3 Effect of Melt and Melt Segregation 465 2.18.4.4 Comparison to Geophysical Observations 466 2.18.5 Concluding Remarks 466 Acknowledgments 467 References 467 2.18.1 Introduction Confident extrapolation of flow laws determined under lab- oratory conditions requires an understanding of the physical The rheological behavior of mantle rocks determines the processes involved in deformation. Minerals and rocks deform mechanical response of lithospheric plates and the nature of plastically by a number of different mechanisms, each of which convective flow in the deeper mantle. Geodynamic models of requires defects in the crystalline structure. Diffusion creep these large-scale processes require constitutive equations that involves the movement of atoms or ions and thus of point describe rheological properties of the appropriate mantle defects; dislocation creep entails the glide and climb of line rocks. The forms of such constitutive equations (flow laws defects; grain-boundary sliding necessitates motion along planar or combinations of flow laws) are motivated by micromecha- defects. Within any deformation regime, more than one defor- nical analyses of plastic deformation phenomena based on mation process may be important. For example, diffusion creep physical processes such as ionic diffusion, dislocation migra- can be divided into one regime dominated by diffusion along tion, and grain-boundary sliding. Critical parameters in flow grain boundaries and another dominated by diffusion through laws appropriate for rock deformation such as the depen- grain interiors (the so-called grain matrix). Grain size, tempera- dence of strain rate on stress, grain size, temperature (activa- ture, and pressure dictate which of these two processes is more tion energy), and pressure (activation volume) must be important. Likewise, in dislocation creep, deformation can be determined from well-designed, carefully executed laboratory divided into low-temperature and high-temperature regimes, experiments. Geophysical observations of the mechanical with glide of dislocations controlling the rate of deformation in behavior of Earth’s mantle in response to changes in the stress the former and climb of dislocations in the latter. As temperature environment due, for example, to an earthquake or the retreat decreases, differential stress necessarily increases if deformation of a glacier provide tests of the applicability of the resulting is to continue at a given rate. constitutive equations to deformation occurring under Plastic deformation is a thermally activated, kinetic, and irre- geologic conditions. versible process. Experimentalists frame laboratory investigations Treatise on Geophysics, Second Edition http://dx.doi.org/10.1016/B978-0-444-53802-4.00042-7 441 442 Constitutive Equations, Rheological Behavior, and Viscosity of Rocks of plastic deformation in terms of strain rate as a function of increases linearly with the addition of vacancies or self- differential stress, temperature, pressure, and grain size and interstitials, the entropy decreases nonlinearly. Hence, the other microstructural and chemical parameters. In contrast, geo- Gibbs free energy of a crystal is minimized not when the crystal dynamicists cast plasticity in terms of viscosity, the ratio of stress is perfect (i.e., defect-free) but rather when the crystal contains to strain rate. In both cases, flow laws or constitutive equations a finite concentration of vacancies and self-interstitials (i.e., describe the relationships between these parameters, thus provid- when the crystal is imperfect) (see, e.g., Devereux, 1983, pp. ing the basis for extrapolating from laboratory to Earth condi- 296–300). In metals, the concentrations of these point defects tions. Tests of the appropriateness of laboratory-derived flow laws in thermodynamic equilibrium are determined by temperature to processes occurring in Earth at much slower strain rates (higher and pressure through their Gibbs free energies of formation viscosities) – and thus necessarily much lower stresses and/or (Schmalzried, 1981, pp. 37–38). In ionic materials, the situa- much lower temperatures and/or much coarser grain sizes – tion is more complicated because the concentrations of the include comparisons of microstructures observed in experimen- different types of point defects are coupled through the neces- tally deformed samples with those in naturally deformed rocks sity for electroneutrality, which requires that the sum of the and of viscosities derived from laboratory experiments with those concentrations of the various types of positively charged point determined from geophysical observations. defects equals the sum of the concentrations of all of the This chapter, therefore, starts with an examination of the role negatively charged point defects (Schmalzried, 1981, pp. 38– of defects in plastic deformation, including a discussion of the 42). The concentration of one type of positively charged point influence of water-derived point defects on kinetic processes. defect often greatly exceeds the concentrations of the other Next, a section on some of the important mechanisms of defor- positively charged point defects; a similar situation frequently mation introduces constitutive equations that describe plastic exists for negatively charged defects. This combination of pos- flow by linking thermomechanical parameters such as strain itive and negative defects defines the majority point defects or rate, differential stress, temperature, pressure, and water fugacity the defect type. The charge neutrality condition is then approx- with structural parameters such as dislocation density, grain size, imated by equating the concentration of the positively charged and melt fraction. Finally, profiles of viscosity versus depth in the to the concentration of the negatively charged majority point upper mantle are used to test the applicability of flow laws defects. An example serves to illustrate the salient elements of determined from micromechanical models and laboratory exper- point defect thermodynamics. iments to plastic flow taking place deep beneath Earth’s surface. Consider the case of a simple metal oxide, MeO, where Me is an ion such as Mg, Ni, Co, Mn, or Fe. Structural elements for this   • •• •• // // system include MeMe,OO,MeMe,Mei ,VO,VMe,andOi . The first 2.18.2 Role of Lattice Defects in Deformation two entries are termed ‘regular structural elements’ and the last five are called ‘irregular structural elements’ or point defects in Defects in rocks are essential for plastic deformation to proceed. the ideal crystal lattice. The Kro¨ger–Vink (1956) notation is used Zero-dimensional or point defects (specifically vacancies and to indicate the atomic species, A, occupying a specific crystallo- self-interstitials) allow flow by diffusive transport of ions, one- graphic site, S, and having an effective charge, C, relative to the C dimensional or line defects (dislocations) permit deformation ideal crystal lattice: AS . In this nomenclature, the superscripted by glide and climb, and two-dimensional or planar defects symbols Â, •, and / indicate neutral, one positive, and one (grain–grain interfaces) facilitate deformation involving grain- negative effective charges. Vacant sites are denoted by a V, and boundary sliding and migration. More than one type of defect ions located at interstitial sites are indicated by a subscripted i. • may be simultaneously involved in deformation. For example, The defect MeMe corresponds to a 3þ ion sitting at a site nor- diffusion creep involves not only diffusion by a point defect mally occupied by a 2þ ion, such as a ferric iron in a ferrous iron mechanism but also sliding on grain boundaries (Raj and site. In a strict sense, charge neutrality is given by Ashby, 1971), dislocation processes also frequently couple with h i h i h i hihi == == grain-boundary sliding (Langdon, 1994), and dislocation creep MeMe þ 2 VO þ 2 Mei ¼ 2VMe þ 2Oi [1a] necessitates glide on several slip systems often combined with where the square brackets, [ ], indicate molar concentration. In climb (Groves and Kelly, 1969; Paterson, 1969; von Mises, a Mg-rich system, one possible disorder type is the combina-
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