The Mechanics of Deforming Metamorphic Rocks
E-Book, Englisch, 680 Seiten, Format (B × H): 191 mm x 235 mm
ISBN: 978-0-12-407933-5
Verlag: Academic Press
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
This book stands alone in unifying deformation and metamorphism and the development of the mineralogical fabrics and the structures that we see in the field. This reflects the thermodynamics of systems not at equilibrium within the framework of modern nonlinear solid mechanics. The thermodynamic approach enables the various mechanical, thermal, hydrological and chemical processes to be rigorously coupled through the second law of thermodynamics, invariably leading to nonlinear behavior. The book also differs from others in emphasizing the implications of this nonlinear behavior with respect to the development of the diverse, complex, even fractal, range of structures in deformed metamorphic rocks.
Building on the fundamentals of structural geology by discussing the nonlinear processes that operate during the deformation and metamorphism of rocks in the Earth's crust, the book's concepts help geoscientists and graduate-level students understand how these processes control or influence the structures and metamorphic fabrics-providing applications in hydrocarbon exploration, ore mineral exploration, and architectural engineering.
- Authored by two of the world's foremost experts in structural geology, representing more than 70 years of experience in research and instruction
- Nearly 300 figures, illustrations, working examples, and photographs reinforce key concepts and underscore major advances in structural geology
Zielgruppe
<p>Primary audience includes geologists, economic and petroleum geologists, mineralogists and exploration geoscientists globally conducting research in hydrocarbon exploration and structural geology.</p>
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
Preface
Acknowledgements
1. Introduction
Part I: THE MECHANICS OF DEFORMING SOLIDS: OVERVIEW OF SECTION A
2. Geometry: The Concept of Deformation
3. Kinematics e Deformation Histories
4. The Balance Laws: Forces Involved in Deformation
5. Energy Flow e Thermodynamics
6. Constitutive Relations
7. Nonlinear Dynamics
Part II PROCESSES INVOLVED IN THE DEVELOPMENT OF GEOLOGICAL STRUCTURES: OVERVIEW OF SECTION B
8. Brittle Flow
9. Visco-Plastic Flow
10. Damage Evolution
11. Transport of Heat
12. Fluid Flow
13. Microstructural Rearrangements
14. Mineral Reactions: Equilibrium and Non-equilibrium Aspects
15. Models for Mineral Phase Nucleation and Growth
Epilogue
Appendix A: Commonly Used Symbols
Appendix B: Vectors, Tensors and Matrices
Appendix C: Some Useful Mathematical Concepts and Relations
References
Index
Chapter 2 Geometry
The Concept of Deformation
Abstract
This chapter introduces the geometrical concept of deformation and the associated concepts of strain and rotation. The geometrical description of deformation is independent of the applied forces, velocities and histories of these quantities. We consider the concept of the deformation gradient and how that quantity describes the changes in the positions of material points, the lengths and orientations of lines, the distortion of arbitrary surfaces and of volume elements. The concept of a phase portrait is introduced as a precursor to its use in dynamical systems. Also considered are special deformations associated with buckling, localisation and plastic slip within crystals. Conditions for compatibility of deformations across a surface are defined. Worked examples for simple shear and inhomogeneous deformations are presented. Keywords
Deformation; Deformation compatibility; Deformation gradient; Localisation; Phase portrait; Rotation; Simple shear; Strain Outline 2.1 Deformations 28 2.1.1 Deformation of a Surface Element 36 2.2 Distortion and Rotation 37 2.3 Deformation and Strain Tensors and Measures 39 2.3.1 Physical Significance of C 42 2.4 Distortion and Volume Change 44 2.5 Example 1: The Geometry of a Simple Shear deformation 46 2.5.1 Deformation of an Arbitrary Line Element in Simple Shear 49 2.5.2 Deformation of an Arbitrary Volume Element 49 2.6 Pseudo Phase Portraits for Affine Deformations 50 2.7 Example 2: Non-affine Deformations 52 2.7.1 Isochoric Non-affine Deformations 52 2.7.2 Non-affine Deformations with Volume Change 53 2.7.3 Type IC Folds 57 2.7.4 Localised Shearing 57 2.8 The Deformation Arising from Slip on a Single Plane 58 2.9 Incremental Strain Measures 59 2.10 Compatibility of Deformations 61 Recommended Additional Reading 65 2.1. Deformations
The deformation of rocks involves the relative motion of elements of the rock so that new arrangements of these elements develop. The motions comprise rigid rotations, rigid translations and distortions (changes in shape) of the elements. The motions are a function of time but it is convenient as a first step to neglect time and consider only the deformed configuration relative to what we imagine to be the undeformed or some reference configuration. The relation between these two configurations is called deformation. If the deformation is the same everywhere (that is, homogeneous) then it does not really matter if one considers the rotations and translations since they make little difference to what we observe within the rock in the deformed state. However, if the deformation varies from place to place (that is, the deformation is inhomogeneous) as in folded rocks or rocks with localised shear zones then the local rotations and translations need to be considered since these local adjustments enable variously distorted regions to fit together with no gaps. In much of the geological literature to date the discussion has concerned homogeneous deformations and so the emphasis has been on just the distortional part of the deformation. This is commonly called strain. Thus the study of strain in deformed rocks has occupied much of the literature on geological deformations over the past 45 years or so since the publication of Ramsay's classical book in 1967 (Ramsay, 1967). In emphasising the mechanics of metamorphic rocks it will become apparent that the total deformation picture as described by local rotations, displacements and strains becomes important. For instance, in describing the motions that are developed in rocks by imposed forces the strain plays little role. To paraphrase Truesdell and Toupin (1960, p 233) regarding the behaviours involved in some familiar constitutive relations: • If the distances between particles do not change this is a rigid body. • If the stress is hydrostatic this is a perfect fluid. • If the stress may be obtained from the rate of stretch alone this is a viscous fluid or a perfectly plastic body. • If the stress may be obtained from the strain alone this is a perfectly elastic body. Thus, as far as the mechanics of deformation is concerned, it is only in elastic theory that the strain becomes important. For the constitutive relations that describe the behaviour of deforming rocks (combinations of elasticity, plasticity and viscosity) some measure of the rate of deformation (not the rate of strain) becomes important and much of the development in that area is based on what is defined below as the deformation gradient. The strain is a quantity that accumulates during the deformation history; it is not a quantity that controls the behaviour of materials (other than perfectly elastic materials). Hence in this chapter we assume that much of the development in the analysis of strain in deformed rocks is already covered by books such as Ramsay (1967), Means (1976) and Ramsay and Huber (1983) and we concentrate on the subject of deformation. It is also commonplace in the geological literature to begin the study of deformation with infinitesimal strain theory and work progressively towards finite strain theory (Hobbs et al., 1976; Jaeger, 1969; Means, 1976; Ramsay, 1967). There are two aspects to this approach: (1) The interest in infinitesimal strain theory is based on classical engineering approaches where small deformations of elastic materials were once the common-place application. (2) The approach attempts to progress from the particular to the general situation. First, the interesting deformations in geology are finite in scale and the relations that describe how a material behaves under the influence of imposed forces or displacements involve quantities related to the deformation rather than to the strain so we begin our approach with this in mind. Second, it sometimes is difficult to generalise from the particular to the general; this we will see is particularly relevant in thermodynamics (Chapter 5) where equilibrium thermodynamics appears as a special case of generalised thermodynamics. Interesting deformations in geology are commonly finite and it turns out that a logarithmic measure of deformation is more useful and rigorous than classical infinitesimal measures of strain, although more difficult to calculate. In this chapter infinitesimal strain theory appears as a final approximation only sometimes relevant to the behaviour of geological materials and in some cases (as in the theory of buckling) does not incorporate the geometry of the deforming process. We consider a body of rock , with a bounding surface , and seek to describe the deformation of this body as is loaded by forces or is displaced in some manner (Figure 2.1). The subject of deformation is commonly dealt with under the heading of kinematics in the mechanics literature even though the word kinematics carries with it the connotation of movement or motion. To a structural geologist the distinction between geometry and movement is paramount since the geometry of the deformed state is commonly the only feature that can be directly observed. Hence we prefer to divide the subject into two distinct areas of study, namely, deformation, the subject of this chapter and concerned with the geometrical features that can be observed in deformed rocks, and kinematics (the subject of Chapter 3), to do with the movements that a rock mass has experienced. As so defined we emphasise that the subject of deformation is concerned only with the geometrical relations between a deformed body and some reference state. It has nothing necessarily to do with the movements and stresses that the body has experienced and the histories of these quantities. In general much more than just the geometry is required in order to establish these relationships as discussed in Chapters 3 and 4. It is important to understand that simply because one can prescribe a particular deformation does not mean that this deformation can actually be achieved for a particular material and still satisfy the boundary conditions. In other words, the deformation may be geometrically admissible but not necessarily dynamically admissible (Tadmor et al., 2012, pp. 247–248).
Figure 2.1 Reference and deformed states. (a) Reference state for the body with bounding surface . The coordinate axes are X1, X2, X3 and we consider a layer . (b) The deformed state with coordinate axes, x1, x2, x3. The body has been inhomogeneously deformed to become d with bounding surface d. P is deformed to become p, dL is deformed to...
