2.Flow and Deformation

2 Flow and Deformation 2.1 Introduction 2.2 Terminology 2.3 Description and Reconstruction of Deformation 2.4 Reference Frames 2.5 Homogeneous and Inhomogeneous Flow and Deformation 2.6 Deformation and Strain 2.7 Progressive and Finite Deformation 2.8 Flow and Deformation in Three Dimensions 2.9 Fabric Attractor 2.10 Application to Rocks 2.11 Stress and Deformation 2.12 Rheology simple concepts and illustrations. Continuum mechanics is a subject that is con- sidered to be difficult by many students, and seen as too theoretical to be of practical use in the interpretation of geological structures. It is true that it is rarely possible to make detailed reconstructions of flow and flow history for a rock sam- ple, but it is crucial to have a basic understanding of the mathematical tools to describe the motion of particles in a continuum, and the interaction of forces and motion in a volume of rock. In this book, there is no space to give a detailed treatment of the subject, but we aim to treat at least the basic terminology so that the reader can work through the literature on microstructures unaided. In the first part of the chapter, reference frames are explained as a necessary tool to describe flow and deformation, and it is important to realize how a choice of reference frame can influence the description of deformation patterns. Then, flow, and deformation are treated including the important concepts of instanta- neous stretching axes, vorticity, and the kinematic vorticity number. The central part of this chapter explains kinematics and how to understand the motion of particles in a rock in two and three dimensions. Finally, the concepts of stress and rheology are briefly explained, and basic terminology of these subjects given. In this chapter, basic principles of continuum mechanics are explained in a non-mathematical way, assuming no previous knowledge of the subject and using 10 2 · Flow and Deformation 2.1 Introduction A hunter who investigates tracks in muddy ground near a waterhole may be able to reconstruct which animals arrived last, but older tracks will be partly erased or modified. A geologist faces similar problems to recon- struct the changes in shape that a volume of rock under- went in the course of geological time, since the end prod- ucts, the rocks that are visible in outcrop, are the only direct data source. In many cases it is nevertheless pos- sible to reconstruct at least part of the tectonic history of a rock from this final fabric. This chapter treats the change in shape of rocks and the methods that can be used to investigate and describe this change in shape. This is the field of kinematics, the study of the motion of particles in a material without regard to forces causing the motion. This approach is useful in geology, where usually very little information can be obtained concern- ing forces responsible for deformation. In order to keep the discussion simple, the treatment is centred on flow and deformation in two dimensions. 2.2 Terminology Consider an experiment to simulate folding using viscous fluids in a shear rig. A layer of dark-coloured material is inserted in a matrix of light-coloured material with an- other viscosity and both are deformed together (Fig. 2.1). The experiment runs from 10.00 to 11.00 h, after which the dark layer has developed a folded shape. During the experiment, a particle P in one of the fluids is displaced with respect to the shear rig bottom and with respect to other particles. At any time, e.g. at 10.10 h, we can attribute to P a velocity and movement direction, visualised by an arrow or velocity vector (Fig. 2.1). If we follow P for a short time, e.g. for 5 s from 10.10 h, it traces a straight (albeit very short) line parallel to the velocity vector. This line is the incremental displacement vector. At another time, e.g. 10.40 h, the velocity vector and associated incremental displacement vector of P can be entirely different (e.g. related to the folding of the dark layer). This means that the displacement path followed by P to its final position at 11.00 h is traced by a large number of incremental dis- placement vectors, each corresponding to a particular Fig. 2.1. Schematic presentation of the velocity, incremental displace- ment and finite displacement of a particle P in a deformation ex- periment in a shear box. Velocity of P at 10.10 h and 10.40 h can be illustrated as a velocity vector. If deformation proceeds over 5 sec- onds, the incremental displacement vector will be parallel to the ve- locity vector. The sequence of incremental displacement vectors gives the finite displacement path. The finite displacement vector is dif- ferent and connects initial (10.00 h) and final (11.00 h) positions of P 2.1 2.2 11 velocity vector. The displacement path is also referred to as the particle path. We can also compare the positions of the particle P at 10.00 and 11.00 h, and join them by a vec- tor, the finite displacement vector (Fig. 2.1). This vector carries no information on the displacement path of P. If the behaviour of more than one particle is con- sidered, the pattern of velocity vectors at a particular time is known as the flow pattern (Fig. 2.2). The pattern of in- cremental displacement vectors is known as the incremen- tal deformation pattern. The pattern of displacement paths is loosely referred to as the deformation path and the pattern of finite displacement vectors is the finite de- formation pattern. The process of accumulation of defor- mation with time is known as progressive deformation, while finite deformation is the difference in geometry of the initial and final stages of a deformed aggregate. The difference between flow and deformation can be visualised by the example of cars in a town. If we compare the positions of all red cars in a town on aerial photographs at 8.30 and at 9.00 h, they will be vastly different; the difference in their initial and fi- nal positions can be described by finite displacement vectors (Fig. B.2.1a). These describe the finite deformation pattern of the distribution of cars in the town. The finite deformation pattern carries no information on the finite displacement paths, the way by which the cars reached their 9.00-h position (Fig. B.2.1b). The finite displacement paths depend on the velocity and movement direction of each individual car and its change with time. The ve- locity and movement direction of each car at 8.33 h, for example, can be described by a velocity vector (Fig. B.2.1c). The combined field of all the velocity vectors of all cars is known as the flow Fig. B.2.1a–d. Illustration of the concepts of flow and displacement or deformation using cars in a town Box 2.1 Terminology of deformation and flow; a traffic example pattern at 8.33 h (Fig. B.2.1c). Flow of the car population there- fore describes the pattern of their velocity vectors. At 8.52 h (Fig. B.2.1d) the flow pattern will be entirely different from that at 8.33 h and the flow pattern is therefore described only for a spe- cific moment, except if the cars always have the same direction and velocity. If we register the displacement of cars over 2 seconds, as a vector field starting at 8.33 h, this will be very similar to the ve- locity vector field at 8.33 h, but the vectors now illustrate displace- ment, not velocity. These vectors are incremental displacement vectors that describe the incremental deformation pattern of the distribution of cars in the town. The incremental deformation pattern is usually different from the finite deformation pattern. If we add all incremental displacement vectors from 10.00 to 11.00 h, the sum will be the finite displacement paths (Fig. B.2.1b). 2.2 · Terminology 12 2 · Flow and Deformation Fig. 2.2. Schematic presentation of the reconstruction of patterns of flow, incremental deformation and finite deformation based on a film of the experiment in Fig. 2.1. At the top is shown how incremental deformation patterns can be determined from adjacent images on the film: flow patterns and finite displacement paths can be constructed from these incremental deformation patterns. At the bottom is shown how finite deformation patterns can be constructed from images that are further separated in time. Black dots are marker particles in the material 13 2.3 Description and Reconstruction of Deformation It is interesting to consider how we could accurately de- scribe velocities and displacement of particles in the ex- periment of Fig. 2.1 using a film (Fig. 2.2). Intuitively, one would assume that the film gives a complete and accu- rate picture of the experiment, and that no further prob- lems occur in reconstruction of flow and deformation. However, such a reconstruction is more difficult than it would seem. If we compare stages of the experiment that are far apart in time, e.g. at 10.00, 10.30 and 11.00 h, we can connect positions of particles by vectors which de- fine the finite deformation pattern (Fig. 2.2 bottom). However, these finite deformation patterns carry no in- formation on the history of the deformation, i.e. on the displacement paths of individual particles. Finite displace- ment paths have to be reconstructed from incremental deformation patterns; if we take two stages of the ex- periment that are close together in time, e.g. two subse- quent images of the film (Fig. 2.2 top), these can be used to find the incremental deformation pattern. Finite dis- placement paths can be accurately reconstructed by add- ing all incremental deformation patterns. This is obvi- ously impossible in practice. An approximation can be obtained by adding a selection of incremental deforma- tion patterns, or a number of finite deformation patterns which represent short time periods. The flow pattern at particular stages of the deformation can be reconstructed from the incremental deformation patterns since these have the same shape. 2.4 Reference Frames The flow, incremental and finite deformation patterns in Fig. 2.2 were produced with a camera fixed to an immo- bile part of the shear rig. The shear rig acts as a refer- ence frame. However, the patterns would have a different shape if another reference frame were chosen. Figure 2.3 shows three possible arrangements for reconstruction of finite deformation patterns from two photographs taken at 10.10 and 10.50 h (Fig. 2.3a). For most studies of flow and deformation it is advantageous to choose a refer- ence frame fixed to a particle in the centre of the do- main to be studied, since this produces symmetric pat- terns around the central particle. An example is shown in Fig. 2.3d, where one particle P is chosen to overlap in both photographs, and the edges of the photographs are parallel; we have now defined a reference frame with or- thogonal axes parallel to the sides of the photographs (and therefore to the side of the shear box), and with an origin on particle P. The patterns in Fig. 2.3b and c are not wrong, but less useful; they have additional transla- tion and rotation components that have no significance Fig. 2.3. Illustration of the influence of different reference frames on the finite deformation pattern for two stages in the experiment of Fig. 2.1. a Two photographic enlargements of the same segment of material at 10.10 h and 10.50 h. Arrows indicate the distance between two particles in the two deformation stages. b, c and d show three different ways of constructing finite deformation patterns from the two images. In b no particle in both photographs is overlapping and the finite deformation pattern has a large component of translation. In c and d one particle P is chosen to overlap in both photographs. In d, the sides of the photographs are chosen parallel as well. Since the photographs were taken with sides parallel to sides of the shear box, d is selected as the most useful presentation of the finite defor- mation pattern in this case. e Illustration of the concept of stretch. lo is original length; l1 is final length 2.3 2.4 2.4 · Reference Frames 14 2 · Flow and Deformation in the experimental setup described here, and therefore obscure the relative motion of the particles with respect to each other. Flow and deformation patterns have certain factors that are independent of the reference frame in which they are described. For example, the relative finite displacement of two particles in Fig. 2.3 can be found from the distance between pairs of particles in both photographs. The final distance divided by the initial distance is known as the stretch of the line connecting the two particles (Fig. 2.3e); this stretch value does not change if another reference frame is chosen (cf. Fig. 2.3b, c and d). In the case of flow, stretching rate (stretch per time unit) is equally independ- ent of reference frame. 2.5 Homogeneous and Inhomogeneous Flow and Deformation 2.5.1 Introduction Usually, flow in a material is inhomogeneous, i.e. the flow pattern varies from place to place in the experiment and the result after some time is inhomogeneous deforma- tion (e.g. Fig. 2.2). The development of folds and boudins in straight layering (Figs. 2.1, 2.2) and the displacement pattern of cars in a town (Fig. B.2.1) are expressions of inhomogeneous deformation. However, the situation is not Box 2.2 How to use reference frames The world in which we live can only be geometrically described if we use reference frames. A reference frame has an origin and a particular choice of reference axes. If a choice of reference frame is made, measurements are possible if we define a coordinate sys- tem within that reference frame such as scales on the axes and/or angles between lines and reference frame axes. Usually, we use a Cartesian coordinate system (named after René Descartes) with three orthogonal, straight axes and a metric scale. In daily life we intuitively work with a reference frame fixed to the Earth’s surface and only rarely become confused, such as when we are in a train on a railway station next to another train; it can then be difficult to decide whether our train, the other train, or both are moving with respect to the platform. As another example, im- agine three space shuttles moving with respect to each other (Fig. B.2.2, ×Video B.2.2). The crew in each of the shuttles can choose the centre of its machine as the origin of a reference frame, choose Cartesian reference axes parallel to the symmetry axes of the shuttle and a metric scale as a coordinate system. The three shut- tles use different reference frames and will therefore have different answers for velocity vectors of the other shuttles. Obviously none of them is wrong; each description is equally valid and no refer- ence frame can be favoured with respect to another. Note that the reference frames are shown to have a different orientation in each diagram of Fig. B.2.2 (×Video B.2.2), because we see them from outside in our own, external reference frame, e.g. fixed to the earth. Similar problems are faced when deciding how to describe flow and deformation in rocks. In experiments, we usually take the shear box as part of our reference frame, or the centre of the deforming sample. In microtectonics we tend to take parts of our sample as a reference frame. In the study of large-scale thrusting, however, it may be more useful to take the autochthonous basement as a refer- ence frame, or, if no autochthonous basement can be found, a geo- graphical frame such as a town or geographical North. Fig. B.2.2. Illustration of the concept of reference frames. If three space shuttles move with respect to each other in space, observ- ers in each one can describe the velocities of the other two (black arrows) as observed through the windows; the reference frame is fixed to the observing shuttle in each case. The results are differ- ent but all correct. The circular arrow around the white shuttle at right indicates that it rotates around its axis in the reference frames for each of the other two shuttles. Grey arrows represent addition of velocity vectors in order to show how they relate Z Y X Y X Y 2.5 15 as complex as may be supposed from Fig. B.2.1 since, con- trary to cars, the velocities of neighbouring particles in an experiment or deforming rock are not independent. Flow in nature is generally inhomogeneous and diffi- cult to describe in numbers or simple phrases. However, if considered at specific scales (Fig. 2.4), flow may be ap- proximately homogeneous with an identical flow pattern throughout a volume of material, wherever we choose the origin of the reference frame (Fig. 2.4a). The result after some time is homogeneous deformation. Characteristic for homogeneous deformation is that straight and parallel marker grid lines remain straight and parallel, and that any circle is deformed into an ellipse. Ho- mogeneous flow or deformation can (in two dimensions) be completely defined by just four numbers; they are ten- Box 2.3 Tensors All physical properties can be expressed in numbers, but dif- ferent classes of such properties can be distinguished. Tem- perature and viscosity are independent of reference frame and can be described by a single number and a unit, e.g. 25 °C and 105 Pas. These are scalars. Stress and homogeneous finite strain, incremental strain, finite deformation, incremental de- formation and flow at a point need at least four mutually in- dependent numbers to be described completely in two dimen- sions (nine numbers in three dimensions). These are tensors. For example, the curves for the flow type illustrated in Fig. 2.6a need at least four numbers for a complete description, e.g. amplitude (the same in both curves), elevation of the Ö-curve, elevation of the ω -curve, and orientation of one of the maxima or minima of one of the curves in space. We might choose another reference frame to describe the flow, but in all cases four numbers will be needed for a full description. Homogeneous deformation can be expressed by two equa- tions: x' = ax + by y' = cx + dy where (x', y') is the position of a particle in the deformed state, (x, y) in the undeformed state and a, b, c, d are four parameters describing the deformation tensor. Homogeneous flow can be described by similar equations that give the velocity compo- nents vx an vy in x and y direction for a particle at point x, y: vx = px + qy vy = rx + ty p, q, r, t are four parameters describing the flow tensor. Both tensors can be abbreviated by describing just their param- eters in a matrix as follows: Multiplication of these matrices with the coordinates of a particle or a point in space gives the complete equations. Ma- trices are used instead of the full equations because they are easier to use in calculations. Fig. 2.4. Illustration of the concepts of homogeneous and inhomo- geneous deformation. a For homogeneous deformation, straight and parallel lines remain straight and parallel, and a circle deforms into an ellipse, the axes of which are finite strain axes. Inhomogeneous and homogenous deformation occur on different scales. b Five scales of observation in a rock. From top to bottom Layering and foliation on a km scale – approximately homogeneous deformation; layering and foliation on a metre scale – inhomogeneous deformation; folia- tion on a cm scale – approximately homogeneous deformation; thin section scale – inhomogeneous deformation; crystal scale – approxi- mately homogeneous deformation 2.5 · Homogeneous and Inhomogeneous Flow and Deformation 16 2 · Flow and Deformation sors (Box 2.3). It is therefore attractive to try and describe natural flow and deformation as tensors. This is possible in many cases, since deviation of flow from homogeneity is scale-dependent (Fig. 2.4b); in any rock there are usually parts and scales that can be considered to approach homo- geneous flow behaviour for practical purposes (Fig. 2.4b). 2.5.2 Numerica