GEOL 3700 Structure and Tectonics Department of Earth Science Utah Valley State College Spring 2003 M. Bunds

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1 GEOL 3700 Structure and Tectonics Department of Earth Science Utah Valley State College Spring 2003 M. Bunds 1. Introductory Lecture a. Format of Class i. Set of concepts and skills that are widely applicable in Earth science ii. Lecture and lab conceptually complement and supplement each other iii. Grade 1) based on both lecture and lab 2) details on the syllabus iv. Field trips - 1) tentative 2) if we go, they are required a) assignments will count b) testable material 3) Possible formats: a) Single day(s) b) Back-to-back Saturday-Sunday, but return to UVSC Saturday night c) 2+ day trip to somewhere like Moab 4) Will give notice of dates asap. v. Readings 1) Both lecture and laboratory readings a) Lecture readings are important supplements and are testable b) Laboratory readings are critical and short. This class will be easier and less time consuming if you do them before the laboratories vi. Keeping Up - do it! Life will be easier vii. Paper 1) choose a topic - from list or you own idea, but talk to me about it. 2) both draft and final versions will be graded b. Introduction to course topics i. So what is Structural Geology? 1) Study of geologic structures 2) structures are geometric features and/or patterns in rocks. 3) Look at some - bedding, folds, faults etc. 4) Define Primary and Secondary structures a) Primary structures i) formed when rock was formed; i.e., not from deformation b) Secondary i) formed subsequent to formation of the rock. ii) Result from deformation. iii) In this class we are primarily interested in secondary structures - although primary ones are at times crucial to understanding deformation because they can tell us about the shape, location and orientation of the rock before deformation ii. Tectonics? 1) Technically, its Earth (rock) movements 2) Today, deals with what its name says, which of course we look at through the plate tectonics paradigm. Deals with plate movements and interactions (i.e., at plate boundaries), mountain building, etc. 3) Rock (and plate) movements are often recorded in geologic structures, so the two tend to go together. However, virtually all fields of geology are critical to tectonic studies - which is part of the fun of them! iii. What use is structural geology GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 1 of 14

2 1) Earth History 2) How the Earth works 3) Rock strength 4) Earthquakes 5) Plate tectonics 6) Origin of mountains iv. What use is this class to you? We will deal with many basic geology skills 1) Map reading - geologic and topographic 2) Compass use 3) Map making 4) Cross-sections 5) Acquisition of basic geologic information - e.g., strike and dip 6) Stereonets - a tool for portraying and analyzing orientation data 7) And more 8) Rock mechanics v. Scale 1) In geology we deal with a wide range of scales - both in time and space. Lets talk spatial scales right now. 2) Some terminology that you need to know: a) Global (10,000 km; 10 7 m) b) Regional; state-sized areas (perhaps 100 to 1000 s of km; 10 5 to 10 6 m) c) macroscopic or map scale; size of a quadrangle; 1 to 100 km; 10 3 to 10 5 m) d) Mesoscopic or handsample size (perhaps 1 to 100 cm; 10-1 to 10 2 m) e) Microscopic; need a handlense or microscope (perhaps 0.01 to 10 mm) f) submicroscopic; need SEM, TEM (<0.01 mm) 3) In many cases, vastly different scale structures will relate and even mimic each other a) e.g., microscopic structures can provide critical information on regional scale structures and events b) Vastly different scale structures can even have very similar appearances - this is one of the main uses of 'fractals,' which is a mathematical method for describing how features appear or are shaped at different scales. This can be especially true of folds and faults. 4) Whether something was deformed as a continuum can depend on the scale of observation. vi. A simple structural geology problem - a homocline 1) How to record the structure a) strike and dip b) which way is up 2) Strike and dip - define [pot of water and a plank of wood) 3) Now the homocline has a linear feature in it - say a current-direction indicator a) Plunge and trend 2. Kinematic vs. Mechanical Approaches to Structural Geology a. 2 main approaches i. Mechanical 1) try to understand the forces (stresses) behind the deformation; 2) sort of an engineering approach draw sketch 3) this can be very difficult with complexly deformed rocks 4) we will look at this approach later 3. Kinematic a. try to understand the movements of particles of rock just look at the deformation, relate it to other aspects of geology (e.g., plate tectonics) b. Do not necessarily try to understand the stresses that caused the deformation c. Deformation Two main types i. rigid body deformation 1) translation, rotation with no change in the shape of the rock body 2) for example tilted bedding 3) movement of lithospheric plates GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 2 of 14

3 ii. non-rigid body deformation - strain 1) distortion and/or change in size (dilation) 2) during strain, in general the angles between material lines (lines connecting points in the rock body) will change. 3) examples -folded bedding. iii. continuum vs. non-continuum deformation 1) block deformed by faults like a deck of cards vs single fault 2) scale of observation affects whether we say this is a continuum type deformation on discontinuous (and possibly a translation if we study a single block) 3) terms penetrative and non-penetrative are similar d. Describing rigid-body deformation we use vectors i. Translations 1) can also be described with a vector and a sense 2) vector orientation gives the direction of the movement a) need both a trend and a plunge direction relative to north and relative to horizontal 3) vector length gives the distance (magnitude) of the displacement 4) sense of movement given by an arrow on the end of a vector indicates movement sense, e.g., from north to south (rather than vice versa). 5) use a photo from the racetrack for this? 6) Note that we may not know the true movement path of a rock, and certainly a linear vector does not describe a complex (curved) path from point A to B. a) incremental versus finite deformation (and strain) in the case above each increment (ideally infinitesimal) is quite different from the final, finite deformation of the object. 7) Slip on a fault can be described essentially as a translation a) Nee to identify two points on either side of a fault that were directly adjacent to each other prior to slip b) Then the line connecting them gives the true relative displacement (slip) on the fault. c) Note that this is a relative displacement we don t know if perhaps one point remained fixed in position relative to the center of the Earth while the other moved ii. Rotations 1) described with a rotation axis and the angle (amount) of rotation 2) so in principle can be described using a vector a) the vector position and orientation gives the rotation axis b) vector length gives the magnitude (angular amount) of the rotation 3) actually we will deal more with this later, at the end of the course when we look at lithospheric plate movements e. Strain i. deck of cards. ii. Homogeneous vs. heterogeneous strain 1) in homogeneous strain, all parts of the strained rock body underwent the same changes in shape. a) flatten a square with 4 circles in it into a rectangle. All 4 circles form the same shape ellipse after the strain. 2) in heterogeneous strain. different parts of the rock body undergo different strains. a) strain a rectangle with circles in it into a trapezoid. Circles will be transformed into different ellipses. 3) Rules of homogeneous (vs. heterogeneous) strain a) Lines that were straight before the deformation remain straight after the deformation ( straight lines remain straight ) b) Lines that were parallel before the deformation remain parallel after the deformation (parallel lines remain parallel ) 4) Homogeneous strain deforms perfect circles into perfect ellipses (and spheres into ellipsoids). In general, the lengths of lines and the angles between lines are changed when a rock body is strained. GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 3 of 14

4 5) Naturally deformed rocks commonly depart from ideal homogeneous strain, but we deal with this be dividing the rock volumes into domains that essentially deformed homogeneously. iii. Changes in the lengths of lines from strain 1) Extension: e = change in length divided by original length a percentage type thing. 2) Stretch: S = final length divided by original length 3) example. a) original length = 8 cm b) final length = 12 cm c) e = 4/8 = 0.25 (i.e., 50%) d) S = 12/8 = 1.5 4) S = 1+e [easy to derive] 5) note that e is negative for shortening 6) Apply to Basin & Range extension iv. Changes in angles during strain: Shear strain 1) First, a trigonometry primer a) sine b) cosine c) tangent 2) Shear strain is distortions that cause the angles between lines to change 3) Angular shear strain a) the change in the angle between two originally perpendicular lines b) to determine the angular shear strain along a line (i.e., direction), look at the poststrain angle between the line in question and a line that was perpendicular to it before being strained. c) angular shear strain is called psi ψ d) clockwise angle changes always are positive e) helpful to think of this by considering shearing in the sense of fault movement. f) note that the shear strain along the perpendicular direction (B in the figure) is always equal in magnitude and opposite in sign/sense. g) note how angles change during pure shear / simple flattening h) ranges from 0 to 90 o 4) Shear strain a) called gamma (γ); γ = tanψ b) recall what the tangent gives us. It tells us how far specific points are displaced by shearing. c) See the figure to the side; tanψ= x/y, so that x=ytanψ d) ranges from 0 to v. The Strain Ellipse: Summarizing strain in a rock body 1) Homogeneous strain in two dimensions can be described using the strain ellipse a) recall that homogeneous strain always distorts perfect circles into perfect ellipses. GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 4 of 14

5 2) The strain ellipse relates how a circle in the rock changed shape during deformation 3) assume no dilation, or if there is dilation, we remove its effects from the strain ellipse by re-scaling it so its area equals the area of the original, pre-deformation circle. 4) Several key elements a) principle strain axes i) e 1 and e 3 or S 1 and S 3 ii) directions of maximum and minimum extension or stretch iii) no shear strain along these axes they are perpendicular before and after strain b) lines of zero extension i) two directions of zero extension always exist in a strain ellipse (1) easy to see why they must exist. The radius of the ellipse in the e 3 direction is less than the radius of the pre-deformation circle, the ellipse's radius in the e 1 direction is greater, and the radius of the ellipse varies smoothly between e 3 and e 1. ii) they divide the ellipse into two fields: one of extension and one of shortening (1) all directions in the field that includes the e3 axis were shortened (2) all directions in the field that includes the e1 axis were extended c) shear strain is given by the angle between a radius of the ellipse and the tangent to the ellipse where the radius intersects the circumference i) recall that in a circle, a radius is always perpendicular to the circumference where it intersects it ii) thus a radius to the ellipse and the tangent are two lines that were perpendicular before deformation and we can obtain shear strain from them vi. Some types of strain 1) Plane strain a) plane strain is strain in which all movement - changes in length and angle - occur within a plane (the infinite set of directions that compose a plane). b) Essentially, there is no movement in or out of the plane c) Its 2-dimensional strain. Its much easier for us to wrap our brains around 2) Incremental vs Finite strain a) Incremental strain i) refers to the changes during a tiny increment of strain. ii) I.e., the orientations of the principal axes during a tiny bit of strain => incremental strain axes. These often are marked by a carrot-type hat. Like a derivative. iii) Also sometimes called 'infinitesimal strain.' b) Finite strain i) refers to the total strain imposed on a rock body by a strain event, or possibly multiple strain events throughout the life of a body. ii) 'Finite strain axes.' iii) finite strain can be very different from incremental strain. 3) Coaxial vs Non-coaxial strain a) Refers to two sets of axes: the incremental and finite strain axes. b) Coaxial strain is strain in which the incremental and finite strain axes remain parallel throughout the strain event c) Non-coaxial strain is strain in which the incremental and finite strain axes do not remain parallel 4) Pure shear vs. Simple shear; General shear a) These are two end-member types of plane strain b) Pure shear is co-axial plane strain. i) Just squishing of a cube into a rectangle ii) There is shortening in one direction and extension in the other throughout the strain event. iii) Particles of rock move in both the x and y directions iv) There is shearing, but equal amounts in both directions (symmetrical) v) The 3-d equivalent is pure flattening. There also is pure stretching. c) Simple shear is non-coaxial plane strain (about as non-coaxial as you can get) GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 5 of 14

6 i) Its deck-of cards shearing ii) With progressive strain, the finite strain axes rotate with the shearing iii) The incremental strain axes, however, are always at 45o to the maximum shearing direction (i.e., to the cards). iv) Shearing is not symmetrical. d) Note that you can't tell simple shear from pure shear by the finite strain ellipse alone - need a marker that shows either the rotation (or lack of), or the asymmetry of the shearing. e) General Shear i) Both pure and simple shear. In ductile faults, this probably is what usually happens ii) Simple or pure shear may dominate, but there probably is at least some of both. 5) Dilation - volume changes a) This does happen b) Compaction of sediment during burial c) dissolution or pressure solution, which can make things called stylolites or solution seams. 6) Structural Compatibility a) Something to always keep in mind - does the strain/deformation that you are conceiving make sense? Does it all fit back together like a puzzle? This is something we should think about whenever we look at deformed rocks. 4. Mechanics a. Mechanics involves the study of the relationship between the forces that cause rocks to deform and the deformation that results, including the structures that are formed. b. Your book calls it dynamics. Mechanics probably is a better term. c. Considering the 'forces' behind the deformation, and trying to relate applied forces to changes in shape, orientation and position of rocks. d. Plan: force => pressure => stress => rock strength properties e. Force i. F = m*a ii. so force is the physical phenomenon that can accelerate a body. iii. W = f*d so a force applied through a distance does work, which is the transfer of energy. iv. Force is a vector quantity - it has magnitude and direction. v. Weight is a force - the mass of something times the acceleration due to gravity. vi. Units for the magnitude of force are Newtons (N), kg*m/s 2 (slugs are a unit of force also, but don t use them if you don t have to!). vii. If sufficient force is applied to a rock body, it can deform it. But we need more. book on a hand vs book on a needle on a hand example. f. Pressure - one dimensional stress i. F/A - force per unit area. ii. The area over which a force is applied makes a big difference in the deformation that happens. That s why we talk about pressure. iii. Twiss' dinosaur on a pedestal example iv. Units of pressure 1) Pascals = Pa = N/m 2 = kg/ms 2 = ma/m 2 = kg*m/s 2 /m 2 2) Megapascals = MPa = Pa * 10 6 = millions of pascals. A Pascal isn't a lot. 3) bars = b ~= one atmosphere of pressure at sealevel = 14.4 lbs/in 2 4) kilobars = kb = b * 10 6 = thousands of bars 5) 100 MPa = 1 kb v. Pressure is a one-dimensional quantity. Only has magnitude. vi. Lets do a calculation. 1) Lets assume that the pressure is equal in all directions within the Earth, like in water. What is the pressure at depth in the Earth? 2) Draw a column. What is the pressure at its base? a) What do we need to know?: depth, area, mass, g b) s = F/A = m*a/a = ρ*v*g/a = ρ*h*a*g/a = ρgh GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 6 of 14

7 c) at 10 km: ρ = 2600 kg/m3; h = m; g = 9.8 m/s 2 d) roughly, 2.6 * 10^3 * 10^4 * 10^1 = 2.6 * 10^8 or 100 MPa 3) Lets generalize to P(h) = ρgh a) Plot this. Its of the form y = ax. It s a line with slope ρg. b) Could be clever, help ourselves out, by making h positive down, then maybe we can more easily see how pressure increases with depth in the Earth. c) To do a better job, we would want to incorporate how ρ changes with depth. We could make blocky estimates, or better yet integrate. g. Stress: Pressure in three dimensions; direction and magnitude. i. The pressure acting on/in an object can vary with direction. There are several ways to go about dealing with this. We will mostly work in 2-dimensions because it is less messy. ii. To help us understand what stress is and some ways to represent it, lets consider the stress on a plane. The plane might represent a fault or bedding planes along which you are concerned there could be movement - a landslide. So you want to see what sorts of pressures are acting on it. iii. Lets say that we have a block of rock on which a vertical and a horizontal force are acting. 1) For now this is a simplification 2) Lets say that the vertical force is creating a horizontal stress of 100 MPa that we will call σ x and a horizontal stress of 50 MPa that we will call σ z 3) What is the pressure (stress) acting on a horizontal plane?: 50 MPa 4) What is the pressure (stress) acting on a vertical plane?: 100 MPa 5) NOW, what about an inclined plane? [see hand-written notes). a) first we find the vertical and horizontal stress acting on this plane b) then we find the stress vector acting on this plane c) then we can find the normal and shear stresses acting on this plane (tractions actually, at this point). iv. What have we learned from this exercise? 1) stress involves trigonometry 2) Given an arbitrary state of stress (stress conditions, or stress field), we can determine the normal and shear stress acting on any arbitrary plane. It is useful to do this. It is the combination of shear and normal stress that causes slippage along a plane in rock. 3) We have seen that it takes 4 numbers to describe the state of stress in 2 dimensions. Actually, we can reduce this to 3 normally, but in general its 4. This can be seen throughout the example above. a) when sx and sz where given we had the magnitude and direction of sx, and then the magnitude of sz (direction is defined to be perpendicular to sx). b) sxp and szp we actually had an extra number because we specified the plane c) with sp we had the magnitude and direction of sp and the orientation of theplane d) with sn and ss, we had their magnitudes and the orientation of sp. e) Stress is a tensor quantity - one step above a vector. v. The Stress Tensor and Ellipse (oid) 1) Now lets imagine that we calculate s, sn and ss for planes at every 5 degrees to sz in the example above. a) We would find that sn is at a maximum and minimum for planes perpendicular to sz and sx, respectively, and that for these two conditions ss is zero. b) ss increases systematically as the angle between a plane and sz (or sx) increases, then reaches a maximum of 25 MPa at an angle of 45o. This is always the case: ss is maximum at 45o to a direction of no ss, and it equals the (max-min)/2 stress. c) Plotting all of the stresses (the single, total stress vector) for each plane so they pass through a single point produces an ellipse i) This ellipse is called the Stress Ellipse. It is an ellipsoid in 3-d ii) Stress is a tensor, not a vector. To describe stress at a point requires identifying the orientation and lengths of the major and minor axes of the stress ellipse (or ellipsoid). That s 3 numbers in 2-d and 6 numbers in 3-d. iii) The axes of the ellipse are called the principal stress axes GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 7 of 14

8 d) With this ellipse, we have fully given the state of stress at a point. This is a difference between forces and stress stress applies to a point, but a force applies to some area or volume. vi. Principal Stress Axes 1) These axes are all mutually perpendicular, like the x, y, and z axes in a 3-d cartesian coordinate system 2) There is no shear stress on planes oriented perpendicular (and parallel) to these directions. 3) Minimum principal stress = σ 3 this is the direction in which the pressure is the least 4) Maximum principal stress = σ 1 this is the direction in which the pressure is the greatest 5) Intermediate principal stress = σ 2 this is the direction that is perpendicular to s1 and s3 and is intermediate in magnitude between them. 6) When working in 2-dimensions - as we often do - we work in the plane that contains s1 and s3. 7) we cheated in the above example - there was no shear stress along the x and z directions - it was implicit in the way that we worked the problem, but we didn t state it up front. 8) However, along the direction of the plane, there was shear stress. In fact, the only directions in which there is no shear stress is the x and z directions - or more properly put, on planes in those directions. 9) Thus, sx and sz were principal axes of stress; they were perpendicular to planes on which there was no shear stress. These will always exist in a stress field. 10) This idea is very similar to the idea of principal strain directions. There is a slight difference in that we technically define shear stress along planes, so a principal stress axis as perpendicular to a plane of no shear stress. But because principal axes of stress are always perpendicular, it all comes out in the wash. vii. Hydrostatic Stress 1) Special stress condition in which stress is equal in all directions. This is what you may think of as pressure. It will exist in very weak materials that cannot support differential stress for example water or air. viii. Some scalar measures of stress 1) mean stress: the average of s1, s2 and s3 (in 3-d). This is the hydrostatic component of stress 2) deviatoric stress: (s1-s3)/2. This also is the maximum shear stress. The greater this is, the more impetus there is for a body to be strained. 3) Differential stress: s1-s3. The greater this is, the more impetus there is for a body to be strained. ix. The fundamental equations of stress and the Mohr Circle for stress 1) σ n = (σ 1 + σ3)/2 (σ 1 -σ 3 )/2*cos2θ 2) σ s = (σ 1 -σ 3 )/2*sin2θ 3) Note that these are the equations for a circle in radial coordinates 4) the Mohr Circle for stress a) Named for Christian Mohr ( ), a German Professor of mechanics and engineering b) Very useful for analyzing stress and rock strength c) Because the normal and shear stress on a plane can be written, in terms of s1 and s3 as above, on a plot of sn vs ss, (sn,ss) for planes of various orientations comprise a circle. d) draw the circle. i) s1, s3 (s1+s3)/2 as points on sn axis ii) a circle with center (s1+s3)/2 and radius (s1-s3)/2. iii) note that the center of the circle is the mean stress iv) the radius of the circle is the deviatoric stress v) the diameter of the circle is the differential stress e) Plot a plane i) determine theta, the angle (counter-clockwise positive but really just need to measure angle in the same sense on the Mohr circle as on a spatial diagram) between the normal to the plane and s1 GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 8 of 14

9 ii) measure 2theta, counter-clockwise positive, from s1 on the circle. iii) where a radius at angle 2theta intersects the circle is (sn,ss) for the plane. iv) sinistral (counterclockwise shear) is positive. Note this is opposite to the book. f) Dissect the diagram i) mean stress is the center of the circle ii) deviatoric stress is the radius of the circle iii) planes at 90 and 0 degrees to s1 have ss=0 iv) maximum shear stress is at theta = 45. v) hydrostatic stress would appear as a dot. g) Some (more) specific states of stress and their names i) hydrostatic s1=s2-s3 ii) uniaxial compressive stress s1>s2=s3=0 iii) axial stress s1>s2=s3>0 or s1=s2>s3>0 iv) triaxial stress s1>s2>s3 (general case) v) effective stress = actual stress pore pressure. (1) this is critical to account for in the upper crust. (2) the pore spaces of rocks are filled with water below the water table i.e., most of the crust. (3) the fluid in the pores exerts outward pressure against the rock. This pressure counteracts the compressive stress present virtually everywhere in the crust. (4) Pore pressure is usually denoted P f. (5) Usually denoted with a star or asterisk, i.e., σ 1 * (6) at a minimum, and as a standard approximation, Pf is considered to equal the pressure from the weight of a water overburden equal in height to the depth under consideration i.e., ρgh. (7) Changes (reduces) mean stress w/o altering differential stress, so effect is to slide Mohr circle to the left, w/o changing its diameter. (8) Lets do an example for 4 km depth. (9) lets say s1 is vertical for simplicity. (a) s1 = qgh (b) s3 see spreadsheet (c) do calcs, plot mohr circle h. Rock Mechanics i. Goal here is to take a first look at how rocks respond to applied stress in other words their strength properties. 1) Information on the mechanical properties of rocks comes primarily from experimental and to a lesser extent theoretical studies. 2) Experiments reveal to us the response of rocks to applied stress states a) strain b) strain rate c) structures formed 3) The study of the relationship of strain to stress is called rheology 4) The mathematical equations that relate strain to stress are called constitutive equations. ii. Basic Rock Deformation Experiment Methodology 1) Utilizes a plug of rock, usually small, perhaps 2 cm by 5 cm 2) Experimental Conditions a) Temperature b) Stress i) uniaxial ii) triaxial confining pressure, but normally s2=s3 c) Sometimes strain rate is specified, machine applies necessary s1-s3 3) What s measured a) axial displacement, which can be converted to extension i) extension = l f /l o b) s1-s3 (differential stress) c) time (so a strain rate can usually be calculated) GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 9 of 14

10 4) Results a) Initially, i.e., relatively small shortening, displacement increases linearly with increasing applied stress, and if the applied stress is removed, the rock returns to its original shape i) This is called elastic behavior ii) Elastic strain is (virtually) instantaneous. I.e., if a stress is applied, the material instantly deforms and then instantly returns to its original shape when the stress is removed. iii) The relationship of the applied stress (either a normal or shear stress) to the strain is linear. iv) The complete relationship of applied stress to elastic strain is given by the elasticity tensor, which is fourth order and contains 81 components in its most general form (reduces to 36 parameters because the stress and strain tensors are symmetric)! However we usually can make some assumptions and simplify it to 4 important parameters: v) Young s Modulus, which relates the axial shortening to applied normal stress: σ = E e. E is called the Young's Modulus. (1) This relationship is analogous to Hooke's Law for a spring. vi) Poisson s ratio: (1) when a cube of rock is elastically compressed, it will extend outwards in the direction perpendicular to the compression direction (2) Poisson s ratio expresses the ratio of this lateral extension to the compression: ν = e t /e l (3) Shear modulus: σ s = Ge s (4) Bulk modulus, for dilation (or contraction): σ = Ke 5) Beyond Elastic Behavior a) After sufficient (differential) stress is applied, the rock can no longer behave elastically, and several things can happen. In all cases, greater strains can be achieved with less increase in stress than during elastic deformation. Most rocks can deform elastically only a very small amount b) Present the several standard resulting curves. i) Simple brittle ii) Brittle followed by stick-slip iii) Brittle followed by stable sliding iv) Perfectly plastic v) Generic ductile - strain hardening followed by strain weakening 6) Lets focus on brittle for a bit. a) Brittle refers to breaking, fracturing of the rock i) this may result in significant, sudden, decrease in strength following elastic limit especially at lower confining pressure (s2 = s3) ii) involves the formation of fractures, faults at some scale. In experiments, almost always one or two primary faults are formed b) The fault(s) are formed at about 30o to s1. i) Why not at 45o? ii) show this on a Mohr circle. Angle between planes that experience max shear stress (45o to s1) and actual faults is called φ, the angle of internal friction. c) At higher confining pressure, more differential stress is required to break the rock. i) can generate a set of stress-strain curves at various differential stress by doing experiments, ii) plot differential stress at failure for each experiment on a mohr circle. iii) can also plot the failure planes formed in the rocks on the mohr diagram iv) the points connect into a line (1) this line is the Coulomb Navier failure criterion (2) τ = µσ n + C o (3) always, always, remember this. GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 10 of 14

11 (4) this relates the shear stress on a plane at failure to the frictional properties of the material involved (m, the coefficient of friction), the normal stress on the plane, and the cohesive strength of the material (5) for pre-existing faults/fractures, Co can go to zero. (6) Amontons first put forth some of these ideas, when he, to great controversy, proposed in 1699 that frictional resisting force is proportional to the normal force and frictional resisting force is independent of the area of surface contact. v) Byerlee s Law (1) In 1978, Jim Byerlee at the USGS compiled his own and other peoples work on the frictional properties of rock, making important adjustments for variations in the experimental equipment used. Experiments were performed on samples with pre-cut fractures/faults. (2) He showed that most all types of rocks have very similar coefficients of friction. (3) Below about 300 MPa, t = 0.85sn (4) above about 300 MPa, t = 0.6sn + 50 MPa (5) clay minerals are the major exception. 7) Ductile yielding a) Strain hardening and strain weakening evident in stress-strain curves b) But in general, ductile deformation is time dependent. Imagine smashing silly putty at three different normal stresses. Would get three different responses of accumulated strain over time: c) The strain rates would be different. Strain rate is just strain divided by time: e/t, so it has units of inverse time (strain is unitless). It is denoted by ė. In geology, ductile strain rate easily can vary from to s -1. d) Ductile deformation can result from a number of different internal, atomic-scale processes in the rock, termed deformation mechanisms. 5. Deformation mechanisms a. Note that we really already have examined 3 important brittle deformation mechanisms i. elastic stretching of atomic bonds ii. fracturing breaking of atomic bonds along continuous surfaces iii. frictional sliding at atomic scale involves breaking of atomic bonds a contact points between sliding surfaces b. Ductile deformation mechanisms i. Cataclastic flow 1) In a granular medium, frictional sliding, rolling, and fracturing of grains can result in a macroscopically ductile response. GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 11 of 14

12 2) Thus, at the atomic scale it is brittle, but its macroscopically ductile 3) actually involves brittle deformation mechanisms 4) rheology is complicated so lets leave it as a potentially ductile deformation process ii. Pressure Solution / Diffusive mass transfer 1) Three step sequence: Dissolution of material at contact points between grains, diffusion of material open areas, re-precipitation of material 2) Dissolution occurs because high normal stresses increase solubility. Tends to happen at contacts between grains. 3) Dissolved material diffuses away from dissolution sites because there is a higher concentration of dissolved ions at the dissolution site. a) Diffusion chemical type of diffusion acts to even out the distribution of dissolved material. b) In rocks, happens slowly, but less slowly through a fluid. So the presence of even a very thin film of fluid along grain boundaries greatly increases the rate of diffusion 4) Re-precipitation occurs at areas of low normal stress a) different minerals can be reprecipitated than were dissolved so this process can go hand-in-hand with metamorphism/alteration 5) In some cases the material is flushed away to other locations (advected away). This results in volume loss in the rock and is true pressure solution. 6) Constitutive Law is something of the sort a) ė = k(t)( σ 1 -σ 3 )/d n. Where k is constant that varies with temperature, and d is grain size raised to some power. k(t) and n, in principle, should be experimentally determined. b) In reality, we have little idea of k(t) and n because the experiments take too long (months or more). Details of the chemistry could be just as important and variable as k(t) and n. c) strain rate varies linearly with differential stress, which makes this a linear viscous rheology. Anything that is linear viscous is a Newtonian fluid. d) No yield stress in theory, if the temperature is high enough (and the grain size small enough), if a differential stress is applied, then the rock will begin to deform by this mechanism if a fluid is present e) Observations support this idea we see lots of evidence for this mechanism in rocks. i) stylolites ii) clay seams iii) microscopic textures iii. Crystal Plasticity / Dislocation creep 1) This is a class of deformation mechanisms that all involve the movement (diffusion) of crystal lattice defects through a crystal in response to applied stress, which causes the crystal to change shape. 2) There are three types of defects a) point defects i) impurities ii) missing atoms iii) for them to move through a crystal requires high temperatures (i.e., 0.5 to 0.85 of melting temperature). So this is important in the lower crust and especially the mantle. b) planar defects i) twins which can form from deformation, especially in carbonates ii) grain boundaries c) line defects, also called dislocations i) these are most important for deformation by crystal plasticity, especially in the crust. ii) Why? iii) In effect, only a little of the crystal has to move at once, which is easier than moving the whole thing at once. Another way to look at it is that only a few GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 12 of 14

13 bonds must be broken at once, rather than a whole bunch for fracturing and frictional sliding. 3) Defects migrate through crystal and cause it to deform. a) at low temperature, can only move by glide. i) this leads to pinning and tangling of dislocations and strain hardening ii) potentially, a bit of deformation of this type can happen, and then the rock breaks brittley. b) at higher temperature, dislocation tangles are eliminated/overcome by recovery mechanisms. c) Dislocations can climb over each other by movement (diffusion) of vacancies. i) this is called dislocation climb. ii) this does not result in strain hardening. iii) leads to formation of subgrains and rotation recrystallization. grain/crystal size of the rock is reduced d) Less strained grains can grow and consume more strained neighboring crystals i) This is called grain-boundary migration recrystallization ii) generally requires slightly higher temps than dislocation climb. iii) acts against reduction in grain size; if this process is going well in a rock, crystals wont become as small as with rotation recrystallization e) General flow law for dislocation creep is e(dot) = k(s1-s3) n exp(-ea/rt). i) note huge temperature dependence it resides as an exponent; increase the temperature by 10, increase strain rate by almost 8000x!!. ii) k, n must be experimentally determined, which has been done for a fair number of mineral/rocks. Can do these experiments because at elevated T, strain happens reasonably fast. But must assume that relations/processes are true for geologic strain rates and lower temperatures. This assumption is supported by similarity of textures in experimentally and naturally deformed rocks. iii) called power-law creep, because strain rate varies by differential stress raised to a power. 4) Resulting textures a) macroscopic/handlens scale i) mylonite ii) foliation b) microscopic i) undulatory extinction ii) deformation lamellae iii) subgrains iv) core and mantle structure v) crystallographic preferred orientations iv. Two other high temperature ductile/plastic deformation mechanisms 1) Coble Creep (grain-boundary diffusion creep in the text) a) Works by diffusion of atoms along grain boundaries, but not through a liquid phase, as with diffusive-mass-transfer. Does not involve dissolution and re-precipitation. Requires relatively high temperatures. 2) Nabarro Herring Creep (Volume - diffusion creep in the text) a) Works by diffusion of atoms through the crystal lattice. Requires very high temperatures. 3) 6. Folds a. Description i. Antiforms, anticlines, synforms, synclines ii. Elements of a fold 1) limbs 2) hinge 3) inflection points 4) hinge surface/plane (axial surface/plane) GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 13 of 14

14 5) fold axis iii. Orientation of a fold 1) hinge: plunge/trend 2) hinge plane: strike/dip a) upright b) overturned iv. Shape 1) tightness: interlimb angle a) open b) tight c) isoclinal 2) cylindrical vs. conical 3) Styles a) chevron b) box c) disharmonic 4) curvature of adjacent horizons a) concentric b) similar v. Stereographic analysis of folds 1) Pi - diagrams GEOL 3700 Structure and Tectonics - lecture notes - M. Bunds, Dept. of Earth Science, UVSC, 03/07/2003 page 14 of 14