---
title: "Faults, Folds and Fabrics"
book: "University Earth and Space Sciences — Year 1"
subject: earth
language: en
chapter: 17
exercises: 12
source: https://one-course.com/books/earth/3/en/chapter/17-faults-folds-and-fabrics
license: CC-BY-NC-SA-4.0
credit: "One Earth and Space Sciences Book, One Course (one-course.com)"
---

# Chapter 17 — Faults, Folds and Fabrics

A road cutting shows a rock face polished like glass and scored with fine parallel grooves. Across the valley, a sea cliff shows beds of sandstone and shale bent into zigzags with straight sides and sharp corners. On a roof nearby, slates split into sheets a few millimetres thick, along planes that cross the layers of the mud they were made from. Each is a record of the deformation of rock, read at a different scale: the grooves give the direction in which two blocks slid past each other along a fault; the zigzags measure how much the beds were shortened; the slaty planes grew across the grain of the rock as it was squeezed. This chapter gives the geometry needed to measure the three records, and the mechanics of the previous chapter explains why faults take the [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) they do.

**You already know.**

The [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of a plane is its [dip direction](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) and [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude), written for instance $080/12$; its [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) follows by the right-hand rule, the plane dipping to the right of an observer looking along the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) ([Chapter 4](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#ch-b1-geological-maps)). A [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is a bend of once-planar layers; an [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is convex upward with the oldest rocks in its core, a [syncline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) concave upward; the [fold axis](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is the line along which the layers bend most ([Chapter 4](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#ch-b1-geological-maps)). The [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) and [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) of a fault are the vertical and horizontal components of its displacement ([Chapter 5](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#ch-b1-cross-sections)). A line has a [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) and a [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge), and the downward unit vector $(\cos p\sin t, \cos p\cos t, -\sin p)$ ([Chapter 6](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#ch-b1-field-lab-methods-1)). A rock obeying the Coulomb criterion breaks on planes at $45{}^{\circ} - \phi/2$ from $\sigma_1$, and existing faults slip according to [Byerlee’s law](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#prop-b1-stress-strain-rheology-byerlee) ([Chapter 16](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#ch-b1-stress-strain-rheology)). A [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) is in extinction between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) when its vibration directions are parallel to those of the polars ([Chapter 8](https://one-course.com/books/earth/3/en/chapter/8-optical-mineralogy#ch-b1-optical-mineralogy)). Faults were met in Book 1, grade 7; [reverse faults](#def-b1-faults-folds-fabrics-fault-types), thrusts and [foliation](#def-b1-faults-folds-fabrics-foliation) in Book 2, grade 12. The cross product of two vectors is perpendicular to both (mathematics, Year 1).

## 17.1 Faults: geometry and kinematics

A fault is a surface across which two blocks of rock have moved past each other. To describe the movement we need names for the two blocks and for the displacement, and a way to measure its direction within the plane of the fault.

**Definition 17.1 (Hanging wall, footwall, slip vector, net slip, rake).**

Let a fault be locally a plane that is not vertical. The *hanging wall* is the block above it, the *footwall* the block below it. Let $A$ be a point of the footwall and $A'$ the point of the hanging wall that was in contact with it before the movement. The *slip vector* $\overrightarrow{AA'}$ is the displacement of the hanging wall relative to the footwall; its length is the *net slip*. The *rake* $r$ (also called the pitch) of the slip vector is the angle, measured in the plane of the fault, from the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) direction given by the right-hand rule to the slip vector, positive when the slip vector points up the plane and negative when it points down it: $-180{}^{\circ} < r \le 180{}^{\circ}$. This is the seismologists’ convention, used again for earthquakes in [Chapter 18](https://one-course.com/books/earth/3/en/chapter/18-the-earthquake-source#ch-b1-earthquake-source); many field geologists record instead an angle between 0 and $180{}^{\circ}$, measured clockwise from the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) seen from above, and add the sense of movement in words.

The two names come from mining: a miner working a sloping vein stood on the [footwall](#def-b1-faults-folds-fabrics-fault-elements) and had the [hanging wall](#def-b1-faults-folds-fabrics-fault-elements) over his head. A displacement seen in a single cut is not, in general, the slip: a bed offset on a map or in a section shows a separation, which depends on the [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of the bed. The slip needs two points that were once one, such as the two halves of a line cut by the fault, for instance the edge of a dyke against a bed.

![The elements of a fault, drawn to scale for a fault dipping = 50 and a slip vector of rake r = -35. (a) The fault plane seen face-on: the slip vector AA' splits into a component along the strike and one down the dip. (b) The vertical section perpendicular to the strike: the dip-slip component splits into the throw and the heave.](https://one-course.com/images/onecourse/chapters/earth-3/b1-faults-folds-fabrics/fig-6293eaf0f3b8.svg)

*The elements of a fault, drawn to scale for a fault dipping $\delta = 50{}^{\circ}$ and a [slip vector](#def-b1-faults-folds-fabrics-fault-elements) of [rake](#def-b1-faults-folds-fabrics-fault-elements) $r = -35{}^{\circ}$. (a) The fault plane seen face-on: the [slip vector](#def-b1-faults-folds-fabrics-fault-elements) $\overrightarrow{AA'}$ splits into a component along the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) and one down the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude). (b) The vertical section perpendicular to the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude): the dip-slip component splits into the [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) and the [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw).*

**Proposition 17.2 (Components of the slip).**

Let a fault have [dip direction](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $\alpha$ and [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $\delta$, and a [slip vector](#def-b1-faults-folds-fabrics-fault-elements) of length $L$ and [rake](#def-b1-faults-folds-fabrics-fault-elements) $r$. With east, north and up components, let $\mathbf s = (\sin(\alpha-90{}^{\circ}), \cos(\alpha-90{}^{\circ}), 0)$ be the unit vector along the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) and $\mathbf u = (-\cos\delta\sin\alpha,
-\cos\delta\cos\alpha, \sin\delta)$ the unit vector up the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude). Then

$$
\overrightarrow{AA'} = L\,(\cos r\,\mathbf s + \sin r\,\mathbf u),
$$

its strike-slip component is $L\cos r$ and its dip-slip component $L\sin r$; the [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) is $L\,|\sin r|\sin\delta$, the [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) $L\,|\sin r|\cos\delta$, and the [slip vector](#def-b1-faults-folds-fabrics-fault-elements) [plunges](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) by $p$ with $\sin p = |\sin r|\sin\delta$.

**Proof.** The vector $\mathbf s$ is horizontal and points along the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude); $\mathbf u$ is the opposite of the downward unit vector of the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) line, which has [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $\alpha$ and [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $\delta$ ([Chapter 6](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#ch-b1-field-lab-methods-1)). Both lie in the fault plane, and $\mathbf s\cdot\mathbf u = -\cos\delta(\sin(\alpha -
90{}^{\circ})\sin\alpha + \cos(\alpha - 90{}^{\circ})\cos\alpha) = -\cos\delta\cos90{}^{\circ} = 0$, so they form an orthonormal basis of the plane. A vector of the plane of length $L$ at the angle $r$ from $\mathbf s$, counted toward $\mathbf u$, is $L(\cos r\,\mathbf s + \sin r\,\mathbf u)$. Its vertical component is $L\sin r\sin\delta$, whose size is the [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw); its horizontal part across the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) is $L\sin r$ times the horizontal part of $\mathbf u$, of length $\cos\delta$, which is the [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw); and the sine of its [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) is its vertical component divided by $L$. ∎

For a dip-slip fault, $|\sin r| = 1$ and the [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) is the [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) divided by $\tan\delta$, as in [Chapter 5](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#ch-b1-cross-sections). The [rake](#def-b1-faults-folds-fabrics-fault-elements) sorts faults into families.

**Definition 17.3 (Normal, reverse, thrust, strike-slip and oblique-slip faults).**

A fault whose [slip vector](#def-b1-faults-folds-fabrics-fault-elements) lies along the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) line ($r = \pm90{}^{\circ}$) is a dip-slip fault: a *normal fault* when the [hanging wall](#def-b1-faults-folds-fabrics-fault-elements) has moved down ($r = -90{}^{\circ}$), a *reverse fault* when it has moved up ($r = 90{}^{\circ}$). A *thrust fault* is a reverse fault that [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $45{}^{\circ}$ or less. A fault whose slip is parallel to its [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) ($r = 0$ or $180{}^{\circ}$) is a *strike-slip fault*; such faults are usually near vertical. A strike-slip fault is *dextral* (right-lateral) when an observer standing on one side sees the other side moved to the right, *sinistral* (left-lateral) when to the left. A fault whose slip has both a strike-slip and a dip-slip component is an *oblique-slip fault*; in practice a fault is named after its larger component, so that a [rake](#def-b1-faults-folds-fabrics-fault-elements) near $-90{}^{\circ}$ is called normal.

**Proposition 17.4 (The sense of a strike-slip fault).**

The sense of a [strike-slip fault](#def-b1-faults-folds-fabrics-fault-types) does not depend on the side the observer stands on. A [rake](#def-b1-faults-folds-fabrics-fault-elements) near $0$ is [sinistral](#def-b1-faults-folds-fabrics-fault-types), a [rake](#def-b1-faults-folds-fabrics-fault-elements) near $180{}^{\circ}$ [dextral](#def-b1-faults-folds-fabrics-fault-types).

**Proof.** If an observer on block 1 sees block 2 move by $\mathbf v$, an observer on block 2 sees block 1 move by $-\mathbf v$. Turning round to face block 1, the second observer has exchanged left and right. The two changes cancel: both see the other side go the same way, to the right or to the left. For the second part, stand on the [footwall](#def-b1-faults-folds-fabrics-fault-elements) and face the [hanging wall](#def-b1-faults-folds-fabrics-fault-elements): one then looks down the [dip direction](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude), and since the plane [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) to the right of its [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude), the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) direction points to one’s left. A [rake](#def-b1-faults-folds-fabrics-fault-elements) of $0$ moves the [hanging wall](#def-b1-faults-folds-fabrics-fault-elements) along the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude), to the left; a [rake](#def-b1-faults-folds-fabrics-fault-elements) of $180{}^{\circ}$ moves it to the right. ∎

A fault surface is often itself a record of the movement: the scratches and fibres on it lie along the slip. Around the surface, a fault zone may be filled with rock broken by the movement.

**Definition 17.5 (Slickenside, slickenlines, fault rocks).**

*Slickenlines* are scratches or fibres on a fault surface, parallel to the slip; a *slickenside* is a fault surface that carries them, often smoothed by the slip. A *fault rock* is a rock made by the movement on a fault. A *fault breccia* is made of angular fragments of the walls of the fault, rotated and moved out of place, from huge blocks down to about $2\,\mathrm{mm}$; a *cataclasite* is made of fragments of sand size or smaller; neither has a fabric, the fragments lying at random. A *fault gouge* is a fault rock rich in clay, very soft in [outcrop](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-geological-map). A *mylonite* is a fine-grained, strongly foliated rock made in a shear zone by ductile shearing, the grains shrunk by recrystallisation rather than by milling.

The [fault rocks](#def-b1-faults-folds-fabrics-fault-rocks) are a depth gauge, and they follow the [strength envelope](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-strength-envelope) of [Chapter 16](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#ch-b1-stress-strain-rheology). Gouge, breccia and [cataclasite](#def-b1-faults-folds-fabrics-fault-rocks), with their random fabric, are made by frictional sliding and fracture in the brittle [crust](https://one-course.com/books/earth/3/en/chapter/9-the-layered-earth#def-b1-layered-earth-layers); [mylonites](#def-b1-faults-folds-fabrics-fault-rocks) are made below the [brittle–ductile transition](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-strength-envelope), where the fault continues downward as a band of ductile shear. For a normal [geotherm](https://one-course.com/books/earth/3/en/chapter/10-the-earths-heat-budget#def-b1-heat-budget-geotherm) and enough water, the transition between the two lies at $10\text{ to }15\,\mathrm{km}$, close to the $13.7\,\mathrm{km}$ of the envelope of the previous chapter, and the fault is strongest there. Across a large fault brought up from depth, the same order is found side by side: gouge zones near the surface, then [cataclasite](#def-b1-faults-folds-fabrics-fault-rocks), then [mylonite](#def-b1-faults-folds-fabrics-fault-rocks). The name [mylonite](#def-b1-faults-folds-fabrics-fault-rocks), from the Greek word for milling flour, records a mistake: it was thought to be ground rock. The Year 3 volume studies those deep shear zones and the mechanisms that make [mylonites](#def-b1-faults-folds-fabrics-fault-rocks).

![Left: a slickenside on a fault surface in chert, about 3\, m × 3\, m of a surface 10\, m high, Corona Heights, San Francisco; the polished face is striated along the slip. Photograph: Dextersinister, CC BY-SA 4.0. Right: fault breccia of the Keystone Thrust, Red Rock Canyon, Nevada: angular fragments of grey carbonate rock in a paler matrix; the lens cap at the top right is 5.8\, cm across. Photograph: Jstuby, public domain. Both from Wikimedia Commons.](https://one-course.com/images/onecourse/chapters/earth-3/b1-faults-folds-fabrics/img-2f2ec68c017b.jpg)

![Left: a slickenside on a fault surface in chert, about 3\, m × 3\, m of a surface 10\, m high, Corona Heights, San Francisco; the polished face is striated along the slip. Photograph: Dextersinister, CC BY-SA 4.0. Right: fault breccia of the Keystone Thrust, Red Rock Canyon, Nevada: angular fragments of grey carbonate rock in a paler matrix; the lens cap at the top right is 5.8\, cm across. Photograph: Jstuby, public domain. Both from Wikimedia Commons.](https://one-course.com/images/onecourse/chapters/earth-3/b1-faults-folds-fabrics/img-430ef85066e4.jpg)

*Left: a [slickenside](#def-b1-faults-folds-fabrics-fault-rocks) on a fault surface in chert, about $3\,\mathrm{m} \times 3\,\mathrm{m}$ of a surface $10\,\mathrm{m}$ high, Corona Heights, San Francisco; the polished face is striated along the slip. Photograph: Dextersinister, CC BY-SA 4.0. Right: [fault breccia](#def-b1-faults-folds-fabrics-fault-rocks) of the Keystone Thrust, Red Rock Canyon, Nevada: angular fragments of grey carbonate rock in a paler matrix; the lens cap at the top right is $5.8\,\mathrm{cm}$ across. Photograph: Jstuby, public domain. Both from Wikimedia Commons.*

**Method 17.6 (Measuring the slip on a fault).**

1. Measure the [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $\alpha/\delta$ of the fault surface and write its [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) , $\alpha - 90{}^{\circ}$ .
2. Measure the [slickenlines](#def-b1-faults-folds-fabrics-fault-rocks) : their [rake](#def-b1-faults-folds-fabrics-fault-elements) on the fault, or their [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) and [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) ; for a [rake](#def-b1-faults-folds-fabrics-fault-elements) $r$ , $\sin p = |\sin r|\sin\delta$ .
3. Find which way the [hanging wall](#def-b1-faults-folds-fabrics-fault-elements) moved, from markers offset across the fault, and give the [rake](#def-b1-faults-folds-fabrics-fault-elements) its sign: negative if the [hanging wall](#def-b1-faults-folds-fabrics-fault-elements) went down.
4. Find the [net slip](#def-b1-faults-folds-fabrics-fault-elements) $L$ from two points that were once one; if only the [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) $h$ of a horizontal marker is known, $L = h/(|\sin r|\sin\delta)$ .
5. Compute the components with [Proposition 17.2](#prop-b1-faults-folds-fabrics-slip-components) and name the fault with [Definition 17.3](#def-b1-faults-folds-fabrics-fault-types) .

**In the field — Measuring a fault surface.**

A geologist who finds a fault surface in an [outcrop](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-geological-map) measures it like any other plane, with a compass-clinometer laid on the surface, and then measures the [slickenlines](#def-b1-faults-folds-fabrics-fault-rocks) on it, either as a [rake](#def-b1-faults-folds-fabrics-fault-elements), with a protractor laid on the plane, or as a [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) and [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge). The two are checked against each other with the formula of step 2 of the method. The sense of movement is the hard part. Small steps on a [slickenside](#def-b1-faults-folds-fabrics-fault-rocks), and fibres of quartz or calcite that grew as the blocks moved apart, can show which way the missing block went, but some steps point the wrong way, so the sense is trusted only when it agrees with the offset of a marker bed or a vein. [Fault rocks](#def-b1-faults-folds-fabrics-fault-rocks) are described and sampled, and a fault zone too broad to measure in one place is mapped as a band. Each fault is recorded at a numbered station of the notebook, with its [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude), its [slickenlines](#def-b1-faults-folds-fabrics-fault-rocks), the sense of movement and how sure the geologist is of it.

## 17.2 Anderson’s theory of faulting

Why are most [normal faults](#def-b1-faults-folds-fabrics-fault-types) steep and most thrusts gentle? In 1905 the Scottish geologist Ernest Masson Anderson answered with two remarks: the surface of the Earth fixes the directions of the [principal stresses](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-principal), and the Coulomb criterion then fixes the angle of the faults.

**Proposition 17.7 (A free surface makes a principal plane).**

At the surface of the Earth, where it is horizontal, one principal stress is vertical and the other two are horizontal.

**Proof.** The air presses on the ground only with its pressure, normal to the surface: it exerts no shear stress. The stress on the horizontal plane at the surface is therefore normal to it. In three dimensions, the stress on a plane of unit normal $\mathbf n$ is the vector $\boldsymbol\sigma\mathbf n$ (physics books, Year 3); if it has no shear part, it is parallel to $\mathbf n$, so that $\boldsymbol\sigma\mathbf n = \lambda\mathbf n$ and $\mathbf n$ is an eigenvector of the symmetric matrix $\boldsymbol\sigma$: a principal direction ([Definition 16.4](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-principal)). The vertical is thus a principal direction, and the two others, perpendicular to it, are horizontal. ∎

Anderson assumed that the [principal stresses](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-principal) keep these directions down to the depths at which faults form. The assumption holds well under a gently sloping surface and in rocks without strong old weaknesses; it fails near steep relief and where old faults or layers guide the new ones. With it, only three cases are possible, according to which of $\sigma_1$, $\sigma_2$ and $\sigma_3$ is vertical.

**Theorem 17.8 (Anderson’s theory of faulting).**

Let one principal stress be vertical and let the rock break according to the Coulomb criterion, with [angle of internal friction](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-coulomb) $\phi$. New faults form on planes that contain the direction of $\sigma_2$ and make the angle $45{}^{\circ} - \phi/2$ with $\sigma_1$, and they slip perpendicular to $\sigma_2$, shortening the rock along $\sigma_1$. Hence

1. if $\sigma_1$ is vertical, [normal faults](#def-b1-faults-folds-fabrics-fault-types) form, striking along $\sigma_2$ and dipping $45{}^{\circ} + \phi/2$ ;
2. if $\sigma_3$ is vertical, [thrust faults](#def-b1-faults-folds-fabrics-fault-types) form, striking along $\sigma_2$ and dipping $45{}^{\circ} - \phi/2$ ;
3. if $\sigma_2$ is vertical, vertical [strike-slip faults](#def-b1-faults-folds-fabrics-fault-types) form, striking at $45{}^{\circ} - \phi/2$ on either side of $\sigma_1$ .

With $\phi = 30{}^{\circ}$, the [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) are $60{}^{\circ}$, $30{}^{\circ}$ and $90{}^{\circ}$.

**Proof.** On a plane that contains the direction of $\sigma_2$, the normal lies in the plane of $\sigma_1$ and $\sigma_3$, and the stress on the plane is that of the two-dimensional problem of [Chapter 16](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#ch-b1-stress-strain-rheology): its point lies on the Mohr circle of $\sigma_1$ and $\sigma_3$, and its shear part lies in the plane of $\sigma_1$ and $\sigma_3$, perpendicular to $\sigma_2$. On any other plane, the point $(\sigma_n, \tau)$ lies inside that circle, which is the largest of the three Mohr circles of a three-dimensional state of stress (admitted from the three-dimensional theory). As the stresses grow, the Coulomb line is therefore first touched by planes containing $\sigma_2$, and among them, by [Theorem 16.14](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#thm-b1-stress-strain-rheology-failure-angle), by those at $45{}^{\circ} - \phi/2$ from $\sigma_1$. Slip on such a plane, perpendicular to $\sigma_2$, brings closer the two blocks along the direction of greatest compression and moves them apart along the least, as it must to relieve the stress. In case (a) the plane makes the angle $45{}^{\circ} - \phi/2$ with the vertical, so it [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $45{}^{\circ} + \phi/2$; it [strikes](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) along the horizontal $\sigma_2$; the slip lies in the vertical plane of the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude), and the block above goes down, lengthening the rock horizontally: a [normal fault](#def-b1-faults-folds-fabrics-fault-types). In case (b) the plane makes $45{}^{\circ} - \phi/2 < 45{}^{\circ}$ with the horizontal $\sigma_1$, and the slip shortens the rock horizontally, pushing the [hanging wall](#def-b1-faults-folds-fabrics-fault-elements) up: a [thrust fault](#def-b1-faults-folds-fabrics-fault-types). In case (c) the planes contain the vertical, so they are vertical; the slip is horizontal, perpendicular to $\sigma_2$: [strike-slip faults](#def-b1-faults-folds-fabrics-fault-types), at $45{}^{\circ} - \phi/2$ on either side of the horizontal $\sigma_1$. ∎

In each regime two sets of faults are possible, symmetric about $\sigma_1$: conjugate faults, meeting at $90{}^{\circ} - \phi$, about $60{}^{\circ}$, with $\sigma_1$ in the acute angle between them. Conversely, a pair of conjugate faults measured in the field gives the directions of the stresses that made them: $\sigma_1$ bisects their acute angle, $\sigma_2$ is their line of intersection.

![Anderson’s three regimes, with the faults at the dips of the theorem for = 30: 60, 30 and vertical. Thick red arrows: _1; thin blue arrows: _3; _2 runs along the strike of the faults in (a) and (b) and is vertical in (c). A marker bed (orange) is offset by the throw in (a) and (b); in (c) a line on the ground is offset horizontally, and the fault is dextral: seen from either block, the other has moved to the right. The black arrows give the slip.](https://one-course.com/images/onecourse/chapters/earth-3/b1-faults-folds-fabrics/fig-9a665bfbb8f0.svg)

*Anderson’s three regimes, with the faults at the [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of the theorem for $\phi = 30{}^{\circ}$: $60{}^{\circ}$, $30{}^{\circ}$ and vertical. Thick red arrows: $\sigma_1$; thin blue arrows: $\sigma_3$; $\sigma_2$ runs along the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of the faults in (a) and (b) and is vertical in (c). A marker bed (orange) is offset by the [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) in (a) and (b); in (c) a line on the ground is offset horizontally, and the fault is [dextral](#def-b1-faults-folds-fabrics-fault-types): seen from either block, the other has moved to the right. The black arrows give the slip.*

The three regimes match the three kinds of plate boundary of Chapters [13](https://one-course.com/books/earth/3/en/chapter/13-rifts-ridges-and-passive-margins#ch-b1-rifts-ridges-margins) and [14](https://one-course.com/books/earth/3/en/chapter/14-active-margins#ch-b1-active-margins). Where the [lithosphere](https://one-course.com/books/earth/3/en/chapter/9-the-layered-earth#def-b1-layered-earth-lithosphere) is stretched, at rifts and ridges, the horizontal stress falls below the vertical one and [normal faults](#def-b1-faults-folds-fabrics-fault-types) form; where it is squeezed, at convergent margins and in mountain belts, the horizontal stress exceeds the weight of the rock and thrusts form; along [transform faults](https://one-course.com/books/earth/3/en/chapter/13-rifts-ridges-and-passive-margins#def-b1-rifts-ridges-margins-transform) the two horizontal stresses straddle the vertical one. The [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) fit the observations in a rough way: the [normal faults](#def-b1-faults-folds-fabrics-fault-types) of rifts [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) around $60{}^{\circ}$ near the surface, and thrusts are low-angle [reverse faults](#def-b1-faults-folds-fabrics-fault-types). The exceptions are informative. Many [normal faults](#def-b1-faults-folds-fabrics-fault-types) flatten with depth, and some cross-sections of rifts show them reaching a nearly flat zone of shear: these low-angle [normal faults](#def-b1-faults-folds-fabrics-fault-types), or detachments, cut the [crust](https://one-course.com/books/earth/3/en/chapter/9-the-layered-earth#def-b1-layered-earth-layers) at well under $45{}^{\circ}$. Anderson’s theory cannot explain them as new faults, and [Exercise 17.10](#exo-b1-faults-folds-fabrics-10) shows that Byerlee’s friction cannot make them slip under Andersonian stresses either: they need weak [fault rocks](#def-b1-faults-folds-fabrics-fault-rocks), high pore pressure or rotated stresses. A fault that already exists slips at the friction of [Byerlee’s law](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#prop-b1-stress-strain-rheology-byerlee), $\mu = 0.85$, $\phi = 40.4{}^{\circ}$; the best-oriented [normal faults](#def-b1-faults-folds-fabrics-fault-types) then [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $65.2{}^{\circ}$, as the problem of [Chapter 16](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#ch-b1-stress-strain-rheology) found.

**History — Anderson’s faults.**

Ernest Masson Anderson, a Scottish geologist, published his theory of the three kinds of faults in 1905, in a short paper on the dynamics of faulting, and gave its full form, with its application to the dykes and faults of Britain, in his book *The Dynamics of Faulting and Dyke Formation*. A small number of papers and that book still shape structural geology, seismology and rock mechanics: his three regimes, normal, thrust and strike-slip, are the usual way of sorting states of stress in the [crust](https://one-course.com/books/earth/3/en/chapter/9-the-layered-earth#def-b1-layered-earth-layers) and the faults they make, and of reading the stress from the faults.

## 17.3 Fold geometry

[Chapter 4](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#ch-b1-geological-maps) named [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) and read their patterns on maps. In three dimensions a [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) needs more names: where it bends, how sharply, and how it is set in space.

**Definition 17.9 (Hinge, hinge line, limb, axial plane, interlimb angle, cylindrical fold).**

On a folded surface, the *hinge* is the zone of greatest curvature and the *hinge line* the line that joins its points of greatest curvature; the *limbs* are the less curved parts on either side. The axial surface of a [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is the surface that joins the hinge lines of its successive folded surfaces; when it is a plane, it is the *axial plane*. The *interlimb angle* $\iota$ is the angle between the two limbs, measured in the profile, the plane perpendicular to the hinge line, between the tangents to the limbs at their points of inflection; $0 \le \iota \le 180{}^{\circ}$. A [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is a *cylindrical fold* when its surfaces are generated by a straight line moved parallel to itself; the direction of that line is the [fold axis](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) of [Definition 4.14](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold), and every hinge line of a cylindrical [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is a straight line parallel to it.

![The elements of a cylindrical anticline, here with a horizontal hinge line: the hinge, where the bed bends most; the two limbs; the axial plane through the hinge lines of the top and the base of the bed; and the interlimb angle between the tangents to the limbs, measured in the profile at the front, here 80, drawn on the base of the bed. The grey lines on the top of the bed are parallel to the fold axis.](https://one-course.com/images/onecourse/chapters/earth-3/b1-faults-folds-fabrics/fig-0b079b709556.svg)

*The elements of a cylindrical [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold), here with a horizontal [hinge line](#def-b1-faults-folds-fabrics-fold-elements): the [hinge](#def-b1-faults-folds-fabrics-fold-elements), where the bed bends most; the two [limbs](#def-b1-faults-folds-fabrics-fold-elements); the [axial plane](#def-b1-faults-folds-fabrics-fold-elements) through the [hinge lines](#def-b1-faults-folds-fabrics-fold-elements) of the top and the base of the bed; and the [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) $\iota$ between the tangents to the [limbs](#def-b1-faults-folds-fabrics-fold-elements), measured in the profile at the front, here $80{}^{\circ}$, drawn on the base of the bed. The grey lines on the top of the bed are parallel to the [fold axis](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold).*

When the [limbs](#def-b1-faults-folds-fabrics-fold-elements) are planes, as in the chevron [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) below and in the Varden Hills, the geometry of a [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) follows from the [attitudes](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of its [limbs](#def-b1-faults-folds-fabrics-fold-elements). The upward unit normal of a plane of [dip direction](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $\alpha$ and [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $\delta$ is $\mathbf n = (\sin\delta\sin\alpha, \sin\delta\cos\alpha, \cos\delta)$, in east, north and up components.

**Proposition 17.10 (A fold from its limbs).**

Let a [cylindrical fold](#def-b1-faults-folds-fabrics-fold-elements) have planar [limbs](#def-b1-faults-folds-fabrics-fold-elements), neither overturned, with upward unit normals $\mathbf n_1$ and $\mathbf n_2$. Its [hinge line](#def-b1-faults-folds-fabrics-fold-elements) is parallel to $\mathbf n_1 \times \mathbf n_2$, and its [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) is

$$
\iota = 180{}^{\circ} - \arccos(\mathbf n_1\cdot\mathbf n_2) .
$$

**Proof.** Each limb is generated by lines parallel to the [fold axis](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold), so it contains the direction of the axis, which is therefore perpendicular to both normals and parallel to their cross product; the [hinge line](#def-b1-faults-folds-fabrics-fold-elements) is parallel to the axis. The profile is perpendicular to the axis, so it contains both normals; in it, the two [limbs](#def-b1-faults-folds-fabrics-fold-elements) appear as two lines perpendicular to $\mathbf n_1$ and $\mathbf n_2$, and the angle between the lines equals the angle $\arccos(\mathbf n_1\cdot\mathbf n_2)$ between the normals or its supplement. When the layers are not folded the two normals coincide and the [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) is $180{}^{\circ}$; as the [limbs](#def-b1-faults-folds-fabrics-fold-elements) turn, the angle between the normals grows from $0$ while the [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) decreases from $180{}^{\circ}$, so that the two add up to $180{}^{\circ}$ as long as both normals point upward. ∎

For an [upright fold](#def-b1-faults-folds-fabrics-fold-classes) whose two [limbs](#def-b1-faults-folds-fabrics-fold-elements) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $\delta$ on either side, the normals make the angle $2\delta$ and $\iota = 180{}^{\circ} - 2\delta$.

**Definition 17.11 (Classes of folds).**

By its [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements), a [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is a *gentle fold* ($\iota$ from $180{}^{\circ}$ to $170{}^{\circ}$), an *open fold* ($170{}^{\circ}$ to $90{}^{\circ}$), a *tight fold* ($90{}^{\circ}$ to $10{}^{\circ}$) or, when its [limbs](#def-b1-faults-folds-fabrics-fold-elements) are nearly parallel ($\iota$ from $10{}^{\circ}$ to $0$), an *isoclinal fold*. By the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of its [axial plane](#def-b1-faults-folds-fabrics-fold-elements), a [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is an *upright fold* when the [axial plane](#def-b1-faults-folds-fabrics-fold-elements) [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) more than $80{}^{\circ}$, a *recumbent fold* when it [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) less than $10{}^{\circ}$, and an *inclined fold* in between. A [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) one of whose [limbs](#def-b1-faults-folds-fabrics-fold-elements) has turned past the vertical, so that both [limbs](#def-b1-faults-folds-fabrics-fold-elements) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) the same way and the beds of that limb are upside down, is an *overturned fold*. A [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) whose [hinge line](#def-b1-faults-folds-fabrics-fold-elements) is not horizontal is a *plunging fold*.

Other limits are in use for the classes of tightness; this book keeps the four above.

![Fold classes, in profile. Top: by interlimb angle, with a vertical axial plane (dashed). Bottom: by the dip of the axial plane, for folds of interlimb angle 40; the inclined fold drawn is also overturned, both its limbs dipping to the left.](https://one-course.com/images/onecourse/chapters/earth-3/b1-faults-folds-fabrics/fig-8e30560a11cd.svg)

*[Fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) classes, in profile. Top: by [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements), with a vertical [axial plane](#def-b1-faults-folds-fabrics-fold-elements) (dashed). Bottom: by the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of the [axial plane](#def-b1-faults-folds-fabrics-fold-elements), for [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) of [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) $40{}^{\circ}$; the [inclined fold](#def-b1-faults-folds-fabrics-fold-classes) drawn is also overturned, both its [limbs](#def-b1-faults-folds-fabrics-fold-elements) dipping to the left.*

The thickness of the layers distinguishes two styles. In parallel, or concentric, [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) each layer keeps its thickness measured perpendicular to it, as a stack of sheets of paper does when it is bent; such [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) form by buckling of strong layers interbedded with very weak ones. In similar [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) every surface has the same shape, the layers keep their thickness measured parallel to the [axial plane](#def-b1-faults-folds-fabrics-fold-elements), and they thin on the [limbs](#def-b1-faults-folds-fabrics-fold-elements) and thicken in the [hinges](#def-b1-faults-folds-fabrics-fold-elements), as the profiles drawn above do; they form when all the layers are weak, so that they play no mechanical part. Chevron [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) have straight [limbs](#def-b1-faults-folds-fabrics-fold-elements) and narrow [hinges](#def-b1-faults-folds-fabrics-fold-elements), with an [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) near $60{}^{\circ}$; they are parallel [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) of strong layers set close together, which slide over each other as they bend.

![Folded beds of sandstone and shale in the cliffs at Millook, North Cornwall: chevron folds with nearly straight limbs and sharp hinges, their axial planes close to vertical. Photograph: Smalljim, CC BY 3.0 (Wikimedia Commons).](https://one-course.com/images/onecourse/chapters/earth-3/b1-faults-folds-fabrics/img-44acd108fe9e.jpg)

*Folded beds of sandstone and shale in the cliffs at Millook, North Cornwall: chevron [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) with nearly straight [limbs](#def-b1-faults-folds-fabrics-fold-elements) and sharp [hinges](#def-b1-faults-folds-fabrics-fold-elements), their [axial planes](#def-b1-faults-folds-fabrics-fold-elements) close to vertical. Photograph: Smalljim, CC BY 3.0 (Wikimedia Commons).*

## 17.4 Fabrics

A deformed rock is often deformed throughout, not only on a few surfaces: its grains are flattened, turned or regrown into a common orientation, which gives it a fabric. A planar fabric makes the rock split, a linear one makes it break into rods.

**Definition 17.12 (Foliation, schistosity, slaty cleavage, axial-planar cleavage).**

A *foliation* is a planar fabric that pervades a rock: a set of closely spaced parallel surfaces, made by aligned platy or flattened grains or by thin layers, along which the rock splits most easily. *Schistosity* is the foliation of a schist, made by aligned grains coarser than about $1\,\mathrm{mm}$. A *cleavage* is a foliation of a fine-grained rock made by its deformation; *slaty cleavage* is the cleavage of a slate, made by sheet silicates too small to be seen with the naked eye, and so closely set that the rock can be split almost anywhere into thin sheets. An *axial-planar cleavage* is a cleavage parallel, or nearly parallel, to the [axial planes](#def-b1-faults-folds-fabrics-fold-elements) of the [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) in the same rocks.

A [foliation](#def-b1-faults-folds-fabrics-foliation) forms roughly perpendicular to the direction of greatest shortening, by the turning of platy grains, the growth of new [micas](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), and the dissolution of grains on the faces that are pressed hardest. The [cleavage](#def-b1-faults-folds-fabrics-foliation) of a slate therefore has nothing to do with the bedding of the mud: it records the squeezing. When [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) and [cleavage](#def-b1-faults-folds-fabrics-foliation) grow in the same shortening, the [cleavage](#def-b1-faults-folds-fabrics-foliation) lies close to the [axial planes](#def-b1-faults-folds-fabrics-fold-elements) of the [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold), and it fans slightly across them, converging or diverging according to the strength of each layer, as the photograph below shows.

![A small anticline in alternating sandstone and shale of Carboniferous age, near Potes, Cantabria. The brown sandstone beds arch over the hinge in the lower middle of the picture; in the dark shale above them the cleavage, a set of steep, closely spaced fractures crossing the beds, is axial-planar: nearly parallel to the axial plane of the fold, it fans across it, the fan changing from layer to layer. The handle of the walking stick is about 4\, cm across. Photograph: Mikenorton, CC BY-SA 3.0 (Wikimedia Commons).](https://one-course.com/images/onecourse/chapters/earth-3/b1-faults-folds-fabrics/img-097236cc2e9e.jpg)

*A small [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) in alternating sandstone and shale of Carboniferous age, near Potes, Cantabria. The brown sandstone beds arch over the [hinge](#def-b1-faults-folds-fabrics-fold-elements) in the lower middle of the picture; in the dark shale above them the [cleavage](#def-b1-faults-folds-fabrics-foliation), a set of steep, closely spaced fractures crossing the beds, is axial-planar: nearly parallel to the [axial plane](#def-b1-faults-folds-fabrics-fold-elements) of the [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold), it fans across it, the fan changing from layer to layer. The handle of the walking stick is about $4\,\mathrm{cm}$ across. Photograph: Mikenorton, CC BY-SA 3.0 (Wikimedia Commons).*

**Definition 17.13 (Lineation, intersection lineation, stretching lineation).**

A *lineation* is a linear fabric that pervades a rock, measured by its [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) and [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge). An *intersection lineation* is the set of lines in which two planar fabrics cut each other, such as the bedding and a [cleavage](#def-b1-faults-folds-fabrics-foliation); a *stretching lineation* is made of grains or pebbles drawn out along the direction of greatest [stretch](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-strain).

**Proposition 17.14 (The intersection lineation of a fold).**

In a [cylindrical fold](#def-b1-faults-folds-fabrics-fold-elements) with an [axial-planar cleavage](#def-b1-faults-folds-fabrics-foliation), the [intersection lineation](#def-b1-faults-folds-fabrics-lineation) of the bedding and the [cleavage](#def-b1-faults-folds-fabrics-foliation) is parallel to the [hinge line](#def-b1-faults-folds-fabrics-fold-elements), everywhere on the [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold).

**Proof.** Every bedding surface of a [cylindrical fold](#def-b1-faults-folds-fabrics-fold-elements) contains the direction of its axis. The [axial plane](#def-b1-faults-folds-fabrics-fold-elements) contains the [hinge lines](#def-b1-faults-folds-fabrics-fold-elements), which are parallel to the axis, so it contains that direction too, and so does every [cleavage](#def-b1-faults-folds-fabrics-foliation) plane parallel to it. Two planes that are not parallel and both contain the same direction meet along a line of that direction. ∎

The proposition is the field geologist’s way to the [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) of a [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) too large to be seen whole: on any [outcrop](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-geological-map) of either limb, the lines in which the [cleavage](#def-b1-faults-folds-fabrics-foliation) cuts the bedding give the [hinge line](#def-b1-faults-folds-fabrics-fold-elements). The bedding and the [cleavage](#def-b1-faults-folds-fabrics-foliation) together say more: where the [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is, and which way up the beds are.

**Definition 17.15 (Vergence).**

The *vergence* of an asymmetric [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is the horizontal direction toward which its upper part has moved relative to its lower part: the direction in which it is overturned or, for an [inclined fold](#def-b1-faults-folds-fabrics-fold-classes), toward which it leans.

**Proposition 17.16 (Bedding and cleavage on the limbs of a fold).**

Consider a [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) with straight [limbs](#def-b1-faults-folds-fabrics-fold-elements) and an [axial-planar cleavage](#def-b1-faults-folds-fabrics-foliation), in rocks that were the right way up before folding, and in which both [limbs](#def-b1-faults-folds-fabrics-fold-elements) descend from each [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) [hinge](#def-b1-faults-folds-fabrics-fold-elements). Seen in the profile:

1. on a limb where the bedding and the [cleavage](#def-b1-faults-folds-fabrics-foliation) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) the same way, the bedding is the steeper of the two exactly when the limb is overturned;
2. the acute rotation that turns the [cleavage](#def-b1-faults-folds-fabrics-foliation) onto the bedding has opposite senses on the two [limbs](#def-b1-faults-folds-fabrics-fold-elements) of a [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) ; on a limb that is not overturned, it moves the top of the [cleavage](#def-b1-faults-folds-fabrics-foliation) horizontally toward the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) [hinge](#def-b1-faults-folds-fabrics-fold-elements) at the top of that limb.

**Proof.** Look along the [hinge](#def-b1-faults-folds-fabrics-fold-elements) with the [axial plane](#def-b1-faults-folds-fabrics-fold-elements) dipping to the left at the angle $\gamma$, $0 < \gamma \le 90{}^{\circ}$, and count angles anticlockwise from the horizontal pointing right. The [cleavage](#def-b1-faults-folds-fabrics-foliation), parallel to the [axial plane](#def-b1-faults-folds-fabrics-fold-elements), runs upward at the angle $\gamma$. At an [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) [hinge](#def-b1-faults-folds-fabrics-fold-elements) the two [limbs](#def-b1-faults-folds-fabrics-fold-elements) leave downward at the angles $180{}^{\circ} + \gamma - \iota/2$ (limb 1) and $180{}^{\circ} + \gamma + \iota/2$ (limb 2), on either side of the [axial plane](#def-b1-faults-folds-fabrics-fold-elements), with $\iota/2 \le \gamma$ since both descend; every limb of a train of [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is one of these two for the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) at its top. The rocks young outward, away from the core of the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold), along the normal to the limb that points away from the [axial plane](#def-b1-faults-folds-fabrics-fold-elements).

Limb 1 goes down to the left: it [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) to the left at $\gamma - \iota/2$, less steeply than the [cleavage](#def-b1-faults-folds-fabrics-foliation). Its outward normal, at $90{}^{\circ} + \gamma -
\iota/2$, points upward: it is the right way up. Running upward, it makes the angle $\gamma - \iota/2$, so the acute rotation from the [cleavage](#def-b1-faults-folds-fabrics-foliation) to the bedding is clockwise, through $\iota/2$: the top of the [cleavage](#def-b1-faults-folds-fabrics-foliation) moves to the right, where the [hinge](#def-b1-faults-folds-fabrics-fold-elements) is, since the limb leaves the [hinge](#def-b1-faults-folds-fabrics-fold-elements) toward the left.

Limb 2 makes the angle $\gamma + \iota/2$ with the horizontal, upward. If $\gamma + \iota/2 > 90{}^{\circ}$, it goes down to the right and [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) to the right, opposite to the [cleavage](#def-b1-faults-folds-fabrics-foliation); its outward normal, at $270{}^{\circ} + \gamma +
\iota/2$, has the upward component $-\cos(\gamma + \iota/2) > 0$: it is the right way up. If $\gamma + \iota/2 < 90{}^{\circ}$, it goes down to the left, dipping to the left at $\gamma + \iota/2$, more steeply than the [cleavage](#def-b1-faults-folds-fabrics-foliation), and the same component is negative: the limb is overturned. In both cases the acute rotation from the [cleavage](#def-b1-faults-folds-fabrics-foliation) to the bedding is anticlockwise, through $\iota/2$, and moves the top of the [cleavage](#def-b1-faults-folds-fabrics-foliation) to the left; when the limb is the right way up it leaves the [hinge](#def-b1-faults-folds-fabrics-fold-elements) toward the right, so the [hinge](#def-b1-faults-folds-fabrics-fold-elements) is to the left. [Limbs](#def-b1-faults-folds-fabrics-fold-elements) that [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) the same way as the [cleavage](#def-b1-faults-folds-fabrics-foliation) are thus limb 1, less steep, and an overturned limb 2, steeper, which is (a); the senses are clockwise on limb 1 and anticlockwise on limb 2, which is (b). ∎

The sense of the rotation thus changes at every [hinge](#def-b1-faults-folds-fabrics-fold-elements), [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) or [syncline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold). A geologist who records it along a traverse finds the [hinges](#def-b1-faults-folds-fabrics-fold-elements) of [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) too large to see in one [outcrop](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-geological-map) where the sense changes, and the overturned [limbs](#def-b1-faults-folds-fabrics-fold-elements) where the bedding is steeper than the [cleavage](#def-b1-faults-folds-fabrics-foliation); the word [vergence](#def-b1-faults-folds-fabrics-vergence) is also used for this sense of rotation, and the reversals of [minor-fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) [vergence](#def-b1-faults-folds-fabrics-vergence) are mapped in the same way to find the [hinges](#def-b1-faults-folds-fabrics-fold-elements) of major [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold): the small [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) on the two [limbs](#def-b1-faults-folds-fabrics-fold-elements) of a large one have opposite asymmetries, and its axial trace runs between them.

![A train of chevron folds overturned to the right, in profile, with an axial-planar cleavage (thin blue lines), drawn to scale: axial planes dipping 60 to the left, interlimb angle 40. The long limbs dip 40 to the left, less steeply than the cleavage, and are the right way up; the short limbs dip 80 to the left, more steeply than the cleavage, and are overturned. Turning the cleavage onto the bedding through the acute angle is clockwise on every long limb and anticlockwise on every short limb. The folds verge to the right.](https://one-course.com/images/onecourse/chapters/earth-3/b1-faults-folds-fabrics/fig-e5f78be1d610.svg)

*A train of chevron [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) overturned to the right, in profile, with an [axial-planar cleavage](#def-b1-faults-folds-fabrics-foliation) (thin blue lines), drawn to scale: [axial planes](#def-b1-faults-folds-fabrics-fold-elements) dipping $60{}^{\circ}$ to the left, [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) $40{}^{\circ}$. The long [limbs](#def-b1-faults-folds-fabrics-fold-elements) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $40{}^{\circ}$ to the left, less steeply than the [cleavage](#def-b1-faults-folds-fabrics-foliation), and are the right way up; the short [limbs](#def-b1-faults-folds-fabrics-fold-elements) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $80{}^{\circ}$ to the left, more steeply than the [cleavage](#def-b1-faults-folds-fabrics-foliation), and are overturned. Turning the [cleavage](#def-b1-faults-folds-fabrics-foliation) onto the bedding through the acute angle is clockwise on every long limb and anticlockwise on every short limb. The [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) verge to the right.*

## 17.5 Microstructures

Under the microscope, the deformation of a rock shows in its grains. A grain of quartz bent by the slow gliding of its crystal lattice no longer goes into extinction all at once between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope): as the stage turns, a dark band sweeps across it, because its parts point in slightly different directions. This undulose extinction ([Chapter 8](https://one-course.com/books/earth/3/en/chapter/8-optical-mineralogy#ch-b1-optical-mineralogy)) grows stronger as a shear zone is approached, because defects pile up in the crystal lattice. Where water is present, grains dissolve on the faces where they are pressed hardest and the dissolved matter is deposited where the stress is lower, or carried away: this pressure solution shortens rocks at low temperature and builds many [cleavages](#def-b1-faults-folds-fabrics-foliation). At higher temperature, the strained grains are replaced, while the deformation goes on, by new, smaller and unstrained ones, first along the old grain boundaries: dynamic recrystallisation, which shrinks the grain size and makes [mylonites](#def-b1-faults-folds-fabrics-fault-rocks). The shapes of deformed grains and of the [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) that grow in their shadows also show which way a shear zone moved; these shear-sense indicators, and the mechanisms themselves, belong to the Year 3 volume.

## 17.6 Exercises

**Exercise 17.1 ★.**

Name each fault from the [rake](#def-b1-faults-folds-fabrics-fault-elements) of its [slip vector](#def-b1-faults-folds-fabrics-fault-elements), the plane dipping $60{}^{\circ}$: (a) $r = -90{}^{\circ}$; (b) $r = 85{}^{\circ}$; (c) $r = 0$; (d) $r = 180{}^{\circ}$; (e) $r = -40{}^{\circ}$. (f) What would (b) be called if its [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) were $25{}^{\circ}$?

**Solution of Exercise 17.1.**

(a) A [normal fault](#def-b1-faults-folds-fabrics-fault-types). (b) A [reverse fault](#def-b1-faults-folds-fabrics-fault-types), nearly pure dip-slip. (c) A [sinistral](#def-b1-faults-folds-fabrics-fault-types) [strike-slip fault](#def-b1-faults-folds-fabrics-fault-types). (d) A [dextral](#def-b1-faults-folds-fabrics-fault-types) [strike-slip fault](#def-b1-faults-folds-fabrics-fault-types). (e) An [oblique-slip fault](#def-b1-faults-folds-fabrics-fault-types): the [hanging wall](#def-b1-faults-folds-fabrics-fault-elements) moved down and along the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude), to the left; its strike-slip component, $\cos40{}^{\circ} = 0.77$ of the [net slip](#def-b1-faults-folds-fabrics-fault-elements), is larger than its dip-slip component, $0.64$, so it is called a [sinistral](#def-b1-faults-folds-fabrics-fault-types) fault with a normal component. (f) With a [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of $25{}^{\circ}$, less than $45{}^{\circ}$, the [reverse fault](#def-b1-faults-folds-fabrics-fault-types) is a [thrust fault](#def-b1-faults-folds-fabrics-fault-types).

**Exercise 17.2 ★.**

A [normal fault](#def-b1-faults-folds-fabrics-fault-types) [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $55{}^{\circ}$; its slip is pure dip-slip, with a [net slip](#def-b1-faults-folds-fabrics-fault-elements) of $300\,\mathrm{m}$. Compute its [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) and its [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw).

**Solution of Exercise 17.2.**

With $|\sin r| = 1$, the [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) is $300\sin55{}^{\circ} = 245.7\,\mathrm{m}$ and the [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) $300\cos55{}^{\circ} = 172.1\,\mathrm{m}$.

**Exercise 17.3 ★.**

Give the [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) and the class of each [upright fold](#def-b1-faults-folds-fabrics-fold-classes), whose [limbs](#def-b1-faults-folds-fabrics-fold-elements) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) in opposite directions: (a) [limbs](#def-b1-faults-folds-fabrics-fold-elements) dipping $35{}^{\circ}$ and $35{}^{\circ}$; (b) $70{}^{\circ}$ and $75{}^{\circ}$; (c) $85{}^{\circ}$ and $88{}^{\circ}$.

**Solution of Exercise 17.3.**

For an [upright fold](#def-b1-faults-folds-fabrics-fold-classes) whose [limbs](#def-b1-faults-folds-fabrics-fold-elements) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $\delta_1$ and $\delta_2$ in opposite directions, $\iota = 180{}^{\circ} - \delta_1 - \delta_2$. (a) $110{}^{\circ}$: open. (b) $35{}^{\circ}$: tight. (c) $7{}^{\circ}$: isoclinal.

**Exercise 17.4 ★.**

A sandstone has the [angle of internal friction](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-coulomb) $35{}^{\circ}$. Following Anderson’s theory, at what angles do its [normal faults](#def-b1-faults-folds-fabrics-fault-types) and its [thrust faults](#def-b1-faults-folds-fabrics-fault-types) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude)? What angle do two conjugate [strike-slip faults](#def-b1-faults-folds-fabrics-fault-types) make with each other?

**Solution of Exercise 17.4.**

[Normal faults](#def-b1-faults-folds-fabrics-fault-types) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $45{}^{\circ} + 35{}^{\circ}/2 = 62.5{}^{\circ}$, thrusts $45{}^{\circ} - 17.5{}^{\circ} = 27.5{}^{\circ}$. Conjugate [strike-slip faults](#def-b1-faults-folds-fabrics-fault-types) each make $27.5{}^{\circ}$ with $\sigma_1$, on either side of it, so they meet at $55{}^{\circ}$.

**Exercise 17.5 ★★.**

A fault has the [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $090/50$, and [slickenlines](#def-b1-faults-folds-fabrics-fault-rocks) of [rake](#def-b1-faults-folds-fabrics-fault-elements) $-30{}^{\circ}$ on it; the [net slip](#def-b1-faults-folds-fabrics-fault-elements) is $400\,\mathrm{m}$. (a) Compute the strike-slip and dip-slip components, the [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) and the [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw). (b) Write the [slip vector](#def-b1-faults-folds-fabrics-fault-elements) in east, north and up components, and give its [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) and [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge). (c) Name the fault.

**Solution of Exercise 17.5.**

(a) The strike-slip component is $400\cos30{}^{\circ} = 346.4\,\mathrm{m}$ along the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude), the dip-slip component $400\sin-30{}^{\circ} = -200\,\mathrm{m}$, that is $200\,\mathrm{m}$ down the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude); the [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) is $200\sin50{}^{\circ} = 153.2\,\mathrm{m}$ and the [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) $200\cos50{}^{\circ} = 128.6\,\mathrm{m}$. (b) The [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) is $000$, so $\mathbf s = (0, 1, 0)$; $\mathbf u = (-\cos50{}^{\circ}, 0, \sin50{}^{\circ}) =
(-0.643, 0, 0.766)$. Then $\overrightarrow{AA'} = 400(0.866\,\mathbf s - 0.5\,\mathbf u) =
(128.6, 346.4, -153.2)$ in metres. Its [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) is $\arctan(128.6/346.4) =
20.4{}^{\circ}$ and its [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $\arcsin(153.2/400) = 22.5{}^{\circ}$, which is also $\arcsin(\sin30{}^{\circ}\sin50{}^{\circ})$. (c) An [oblique-slip fault](#def-b1-faults-folds-fabrics-fault-types): the [rake](#def-b1-faults-folds-fabrics-fault-elements) lies between $0$ and $-90{}^{\circ}$, closer to $0$: a [sinistral](#def-b1-faults-folds-fabrics-fault-types) fault with a normal component.

**Exercise 17.6 ★★.**

Two vertical [strike-slip faults](#def-b1-faults-folds-fabrics-fault-types) of the same age [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $020$ and $080$. (a) Following Anderson, give the directions of $\sigma_1$, $\sigma_2$ and $\sigma_3$, and the [angle of internal friction](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-coulomb) of the rock. (b) Which fault is [dextral](#def-b1-faults-folds-fabrics-fault-types) and which [sinistral](#def-b1-faults-folds-fabrics-fault-types)? Argue from the shortening along $\sigma_1$.

**Solution of Exercise 17.6.**

(a) The two faults make $60{}^{\circ}$ with each other; $\sigma_1$ bisects the acute angle, so it is horizontal along $050$ (and $230$); $\sigma_3$ is horizontal along $140$; $\sigma_2$ is vertical, along the line in which the two vertical faults meet. Each fault makes $30{}^{\circ}$ with $\sigma_1$, so $45{}^{\circ} - \phi/2 =
30{}^{\circ}$ and $\phi = 30{}^{\circ}$. (b) The two faults cut the ground into four wedges; the wedges that contain $\sigma_1$, toward $050$ and toward $230$, move toward each other, and those that contain $\sigma_3$ move apart, toward $140$ and $320$. Take the fault striking $020$, between the wedge toward $050$ (to its east) and the wedge toward $320$ (to its west): relative to the western wedge, the eastern one moves toward $230$ and away from $320$, both of which have a component toward $200$ along the fault. An observer on the western wedge, facing east, sees the eastern one move toward $200$, to the right: the fault striking $020$ is [dextral](#def-b1-faults-folds-fabrics-fault-types). The same argument, or the symmetry about $\sigma_1$, makes the fault striking $080$ [sinistral](#def-b1-faults-folds-fabrics-fault-types).

**Exercise 17.7 ★★.**

The two [limbs](#def-b1-faults-folds-fabrics-fold-elements) of a [cylindrical fold](#def-b1-faults-folds-fabrics-fold-elements) have the [attitudes](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $120/40$ and $260/30$. With [Proposition 17.10](#prop-b1-faults-folds-fabrics-hinge), find the [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) and [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) of its [hinge line](#def-b1-faults-folds-fabrics-fold-elements), its [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) and its class.

**Solution of Exercise 17.7.**

The upward normals are

$$
\begin{align*}
  \mathbf n_1 &= (\sin40{}^{\circ}\sin120{}^{\circ}, \sin40{}^{\circ}\cos120{}^{\circ}, \cos40{}^{\circ}) = (0.557, -0.321, 0.766),\\
  \mathbf n_2 &= (\sin30{}^{\circ}\sin260{}^{\circ}, \sin30{}^{\circ}\cos260{}^{\circ}, \cos30{}^{\circ}) = (-0.492, -0.087, 0.866).
\end{align*}
$$

Their cross product is $(-0.211, -0.859, -0.207)$, which points downward: the [hinge line](#def-b1-faults-folds-fabrics-fold-elements) [plunges](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $\arcsin 0.207/0.909 = 13.1{}^{\circ}$ toward $180{}^{\circ} + \arctan(0.211/0.859) =
193.8{}^{\circ}$. $\mathbf n_1\cdot\mathbf n_2 = 0.417$, the angle between the normals is $65.3{}^{\circ}$, and $\iota = 114.7{}^{\circ}$: an [open fold](#def-b1-faults-folds-fabrics-fold-classes).

**Exercise 17.8 ★★.**

At an [outcrop](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-geological-map), the bedding has the [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $040/30$ and an [axial-planar cleavage](#def-b1-faults-folds-fabrics-foliation) the [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $160/80$. Find the [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) and [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) of the [intersection lineation](#def-b1-faults-folds-fabrics-lineation). What does it tell about the [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold)?

**Solution of Exercise 17.8.**

$\mathbf n_1 = (0.321, 0.383, 0.866)$ for the bedding and $\mathbf n_2 = (0.337, -0.925, 0.174)$ for the [cleavage](#def-b1-faults-folds-fabrics-foliation). Their cross product, taken downward, gives a line that [plunges](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $25.4{}^{\circ}$ toward $074.8{}^{\circ}$. By [Proposition 17.14](#prop-b1-faults-folds-fabrics-intersection), this is the direction of the [hinge line](#def-b1-faults-folds-fabrics-fold-elements): the [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) [plunges](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) about $25{}^{\circ}$ toward the east-north-east, which any other [outcrop](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-geological-map) of either limb should confirm.

**Exercise 17.9 ★★.**

A vertical fault [strikes](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) east–west. A vertical dyke striking north–south crosses it; north of the fault the dyke is found $350\,\mathrm{m}$ east of where it is south of the fault. (a) Is the fault [dextral](#def-b1-faults-folds-fabrics-fault-types) or [sinistral](#def-b1-faults-folds-fabrics-fault-types)? (b) If the [slickenlines](#def-b1-faults-folds-fabrics-fault-rocks) are horizontal, what is the [net slip](#def-b1-faults-folds-fabrics-fault-elements)? (c) If instead they [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $20{}^{\circ}$ to the east, what are the [net slip](#def-b1-faults-folds-fabrics-fault-elements) and the vertical component of the slip, and which block went down?

**Solution of Exercise 17.9.**

(a) An observer south of the fault, facing north, sees the northern part of the dyke displaced to the east, to the right: the fault is [dextral](#def-b1-faults-folds-fabrics-fault-types). (b) The dyke is a vertical plane perpendicular to the fault, so its offset along the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) is the strike-slip component: $350\,\mathrm{m}$, which is the whole slip if the [slickenlines](#def-b1-faults-folds-fabrics-fault-rocks) are horizontal. (c) The slip then makes $20{}^{\circ}$ with the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) in the vertical fault plane: its horizontal component is still $350\,\mathrm{m}$, so the [net slip](#def-b1-faults-folds-fabrics-fault-elements) is $350/\cos20{}^{\circ} = 372.5\,\mathrm{m}$ and its vertical component $350\tan20{}^{\circ} = 127.4\,\mathrm{m}$. The northern block moved east, and the [slickenlines](#def-b1-faults-folds-fabrics-fault-rocks) descend toward the east, so it also went down, by $127.4\,\mathrm{m}$. A vertical dyke cannot show that vertical movement, since sliding it along itself leaves its trace unchanged.

**Exercise 17.10 ★★★.**

A fault with no [cohesion](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-coulomb) and the friction coefficient $\mu$ makes the angle $\theta$ with $\sigma_1$. (a) Show that it slips when the effective [principal stresses](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-principal) reach the ratio

$$
\frac{\sigma_1'}{\sigma_3'} = \frac{1 + \mu\cot\theta}{1 - \mu\tan\theta} .
$$

(b) With $\mu = 0.6$ and $\sigma_1$ vertical, compute the ratio for [normal faults](#def-b1-faults-folds-fabrics-fault-types) dipping $60{}^{\circ}$, $45{}^{\circ}$ and $35{}^{\circ}$. (c) Show that a [normal fault](#def-b1-faults-folds-fabrics-fault-types) dipping less than a certain angle can never slip, and compute that angle. (d) For $\mu = 0.6$ and $\mu = 0.85$, give the [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of the best-oriented normal and [thrust faults](#def-b1-faults-folds-fabrics-fault-types).

**Solution of Exercise 17.10.**

(a) The normal to the fault makes $90{}^{\circ} - \theta$ with $\sigma_1$, so by [Proposition 16.5](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#prop-b1-stress-strain-rheology-mohr), $\sigma_n' = \sigma_1'\sin^2\theta + \sigma_3'\cos^2\theta$ and $\tau = (\sigma_1' - \sigma_3')\sin\theta\cos\theta$. Slip starts when $\tau = \mu\sigma_n'$; dividing by $\sigma_3'\cos^2\theta$ and writing $R = \sigma_1'/\sigma_3'$, $(R - 1)\tan\theta = \mu(R\tan^2\theta + 1)$, so $R\tan\theta(1 - \mu\tan\theta) = \tan\theta + \mu$, which is the formula. (b) With $\sigma_1$ vertical, $\theta$ is $90{}^{\circ}$ minus the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude): $\theta = 30{}^{\circ}$ gives $R = 3.12$; $\theta = 45{}^{\circ}$, $R = 4.00$; $\theta = 55{}^{\circ}$, $R = 9.92$. (c) The denominator vanishes when $\tan\theta = 1/\mu$, $\theta = 59.0{}^{\circ}$; beyond it, $R$ would be negative: no ratio of positive effective stresses can make the fault slip, however large $\sigma_1'$ becomes. [Normal faults](#def-b1-faults-folds-fabrics-fault-types) dipping less than $31.0{}^{\circ}$ cannot slip under a vertical $\sigma_1$ with $\mu = 0.6$. (d) The best-oriented plane is at $45{}^{\circ} - \phi/2$ from $\sigma_1$. For $\mu = 0.6$, $\phi = 31.0{}^{\circ}$: [normal faults](#def-b1-faults-folds-fabrics-fault-types) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $60.5{}^{\circ}$, thrusts $29.5{}^{\circ}$. For $\mu = 0.85$, $\phi = 40.4{}^{\circ}$: $65.2{}^{\circ}$ and $24.8{}^{\circ}$.

**Exercise 17.11 ★★★.**

Along an east–west traverse across chevron [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) whose [hinges](#def-b1-faults-folds-fabrics-fold-elements) [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) north, an [axial-planar cleavage](#def-b1-faults-folds-fabrics-foliation) [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $60{}^{\circ}$ to the west everywhere. The bedding [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $40{}^{\circ}$ W at station A, $80{}^{\circ}$ W at station B and $40{}^{\circ}$ W at station C, from west to east, in rocks that were the right way up before folding. (a) Which station is on an overturned limb? (b) In which sense does the [cleavage](#def-b1-faults-folds-fabrics-foliation) turn onto the bedding at each station, seen looking north? (c) Where are the [hinges](#def-b1-faults-folds-fabrics-fold-elements), and which are [anticlines](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold)? (d) Find the [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) and the [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of the [axial planes](#def-b1-faults-folds-fabrics-fold-elements), and say which way the [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) verge.

**Solution of Exercise 17.11.**

Looking north, east is to the right, and planes dipping west rise toward the right. (a) At B the bedding [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) the same way as the [cleavage](#def-b1-faults-folds-fabrics-foliation), more steeply: by [Proposition 17.16](#prop-b1-faults-folds-fabrics-cleavage-bedding)(a), B is on an overturned limb; A and C, where the bedding is less steep, are the right way up. (b) The [cleavage](#def-b1-faults-folds-fabrics-foliation) rises at $60{}^{\circ}$; at A and C the bedding rises at $40{}^{\circ}$, and the acute rotation from one to the other is clockwise, through $20{}^{\circ}$, moving the top of the [cleavage](#def-b1-faults-folds-fabrics-foliation) to the east; at B the bedding rises at $80{}^{\circ}$ and the rotation is anticlockwise, through $20{}^{\circ}$, moving it to the west. (c) The sense changes between A and B and again between B and C: there is a [hinge](#def-b1-faults-folds-fabrics-fold-elements) in each interval. At A, which is the right way up, the top of the [cleavage](#def-b1-faults-folds-fabrics-foliation) moves east, toward the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) at the top of the limb: the [hinge](#def-b1-faults-folds-fabrics-fold-elements) between A and B is an [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold), and the one between B and C a [syncline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold). (d) In the notation of the proof, the right-way-up [limbs](#def-b1-faults-folds-fabrics-fold-elements) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $\gamma - \iota/2 = 40{}^{\circ}$ and the overturned [limbs](#def-b1-faults-folds-fabrics-fold-elements) $\gamma + \iota/2 =
80{}^{\circ}$, so $\gamma = 60{}^{\circ}$, as the [cleavage](#def-b1-faults-folds-fabrics-foliation) says, and $\iota =
40{}^{\circ}$: [tight folds](#def-b1-faults-folds-fabrics-fold-classes). The [axial planes](#def-b1-faults-folds-fabrics-fold-elements) [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) north and [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $60{}^{\circ}$ to the west, [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $270/60$. The short [limbs](#def-b1-faults-folds-fabrics-fold-elements) have turned past the vertical with their tops toward the east: the [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) verge to the east, as in the figure of the chapter.

**Exercise 17.12 ★★★.**

A layer of length $L_0$ is folded into a train of symmetric chevron [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) of [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) $\iota$, with its length unchanged. (a) Show that its length measured across the [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) becomes $L_0\sin(\iota/2)$. (b) What shortening do chevron [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) of [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) $60{}^{\circ}$ represent? (c) What [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) corresponds to a shortening of $30\,\%$? (d) A train of [upright folds](#def-b1-faults-folds-fabrics-fold-classes) with [limbs](#def-b1-faults-folds-fabrics-fold-elements) dipping $12{}^{\circ}$: what shortening?

**Solution of Exercise 17.12.**

(a) In a symmetric chevron [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) each limb makes the angle $\iota/2$ with the [axial plane](#def-b1-faults-folds-fabrics-fold-elements), which is perpendicular to the direction of shortening; a limb of length $l$ therefore spans $l\sin(\iota/2)$ across the [folds](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold), and the whole layer $L_0\sin(\iota/2)$. (b) $\sin30{}^{\circ} = 0.5$: a shortening of $50\,\%$. (c) $\sin(\iota/2) = 0.7$ gives $\iota = 88.9{}^{\circ}$. (d) $\iota = 156{}^{\circ}$ and $\sin78{}^{\circ} = 0.978$: a shortening of $2.2\,\%$, which is also $1 - \cos12{}^{\circ}$.

## 17.7 Problem: The Varden Hills’ structures, measured

**Problem 17.1.**

Weekend problem — the anticline and the normal fault of the Varden Hills, from the attitudes of their planes to the slip on the fault

The Varden Hills are folded into an [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold), whose axis runs north–south at $x = 3.5\,\mathrm{km}$, and a [syncline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) $4\,\mathrm{km}$ to the east of it; both have planar [limbs](#def-b1-faults-folds-fabrics-fold-elements). A field survey measured the bedding on the east limb of the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) as $080.7/12.2$ and on its west limb as $279.3/12.2$. The cover is cut by a [normal fault](#def-b1-faults-folds-fabrics-fault-types) of [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $120/60$, whose [hanging wall](#def-b1-faults-folds-fabrics-fault-elements) moved down the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude), with a [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) of $200\,\mathrm{m}$. Give angles to $0.1{}^{\circ}$ and lengths to $0.1\,\mathrm{m}$.

**Part I — The limbs.**

1. Write the upward unit normals $\mathbf n_1$ (east limb) and $\mathbf n_2$ (west limb) in east, north and up components.
2. Compute $\mathbf n_1\cdot\mathbf n_2$ and the angle between the normals.
3. Deduce the [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) of the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) .
4. Classify the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) by its [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) .
5. The two [limbs](#def-b1-faults-folds-fabrics-fold-elements) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $12.2{}^{\circ}$ . Why is the [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) not exactly $180{}^{\circ} - 2\times12.2{}^{\circ}$ ? Compute the [apparent dip](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-apparent-dip) of either limb in an east–west vertical section.
6. The [syncline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) shares the east limb of the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) , and its own east limb is parallel to the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) ’s west limb. What is its [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) ?

**Part II — The hinge line and the axial plane.**

7. Compute $\mathbf n_1\times\mathbf n_2$ , and the [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) and [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) of the [hinge line](#def-b1-faults-folds-fabrics-fold-elements) .
8. Give the [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of the [axial plane](#def-b1-faults-folds-fabrics-fold-elements) . Is the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) upright, inclined or recumbent?
9. Give the full description of the [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) : tightness, [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of the [axial plane](#def-b1-faults-folds-fabrics-fold-elements) , [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) .
10. On the [geological map](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-geological-map) of [Chapter 4](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#ch-b1-geological-maps) , which way do the [outcrops](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-geological-map) of the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) close, and what crops out in its core?
11. The shale of the cover carries an [axial-planar cleavage](#def-b1-faults-folds-fabrics-foliation) . Give its [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) , and the [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) and [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) of its [intersection lineation](#def-b1-faults-folds-fabrics-lineation) on the east limb.
12. On the east limb, in which sense does the [cleavage](#def-b1-faults-folds-fabrics-foliation) turn onto the bedding, seen looking north, and on which side is the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) [hinge](#def-b1-faults-folds-fabrics-fold-elements) ?

**Part III — The fault and the stresses.**

13. Give the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of the fault by the right-hand rule, and say on which side of its trace the [hanging wall](#def-b1-faults-folds-fabrics-fault-elements) lies.
14. Which principal stress was vertical when the fault formed, according to Anderson?
15. Give the directions of the two horizontal [principal stresses](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-principal) .
16. What [angle of internal friction](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-coulomb) does the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) imply? What [coefficient of internal friction](https://one-course.com/books/earth/3/en/chapter/16-stress-strain-and-rheology#def-b1-stress-strain-rheology-coulomb) ?
17. Give the [attitude](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) of the conjugate set of faults that the same stresses could have made, and the angle between the two sets.
18. Would the fault be best oriented to slip again under the same regime, with Byerlee’s friction of 0.85?

**Part IV — The slip.**

19. Give the [rake](#def-b1-faults-folds-fabrics-fault-elements) of the [slip vector](#def-b1-faults-folds-fabrics-fault-elements) , and its [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) and [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) .
20. Write the unit vector of the slip in east, north and up components.
21. Compute the [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) of the fault.
22. Compute the east and north components of the horizontal displacement of the [hanging wall](#def-b1-faults-folds-fabrics-fault-elements) .
23. Check that the [net slip](#def-b1-faults-folds-fabrics-fault-elements) $L$ , the [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) $h$ and the [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) $e$ satisfy $L^2 = h^2 + e^2$ , and explain why.
24. To the nearest metre, what is the [net slip](#def-b1-faults-folds-fabrics-fault-elements) on the Varden fault?

**Solution of Problem 17.1.**

**1.** $\mathbf n = (\sin\delta\sin\alpha, \sin\delta\cos\alpha,
\cos\delta)$ gives

$$
\mathbf n_1 = (0.20855, 0.03415, 0.97742), \qquad
  \mathbf n_2 = (-0.20855, 0.03415, 0.97742).
$$

**2.** $\mathbf n_1\cdot\mathbf n_2 = -0.04349 + 0.00117 + 0.95535 =
0.91302$, and the angle between the normals is $24.1{}^{\circ}$.

**3.** By [Proposition 17.10](#prop-b1-faults-folds-fabrics-hinge), $\iota = 180{}^{\circ} - 24.1{}^{\circ} = 155.9{}^{\circ}$.

**4.** Between $170{}^{\circ}$ and $90{}^{\circ}$: an [open fold](#def-b1-faults-folds-fabrics-fold-classes).

**5.** The [limbs](#def-b1-faults-folds-fabrics-fold-elements) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) toward $080.7$ and $279.3$, not due east and west, because the [fold](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) [plunges](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge): their steepest lines are not in the profile, which shows only an [apparent dip](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-apparent-dip). In an east–west section, by [Theorem 5.4](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#thm-b1-cross-sections-apparent-dip), $\tan\beta = \tan12.2{}^{\circ}
\cos9.3{}^{\circ} = 0.2134$, so $\beta = 12.0{}^{\circ}$, and $180{}^{\circ} -
2\times12.0{}^{\circ} = 155.9{}^{\circ}$ (the profile is tilted $2{}^{\circ}$ from that section, which changes nothing to this precision).

**6.** Its [limbs](#def-b1-faults-folds-fabrics-fold-elements) are parallel to those of the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold), so its [interlimb angle](#def-b1-faults-folds-fabrics-fold-elements) is also $155.9{}^{\circ}$.

**7.** $\mathbf n_1\times\mathbf n_2 = (0.03415\times0.97742 -
0.97742\times0.03415,\; 0.97742\times(-0.20855) - 0.20855\times0.97742,\;
0.20855\times0.03415 + 0.03415\times0.20855) = (0, -0.40768, 0.01424)$. Taken downward, $(0, 0.40768, -0.01424)$: the [hinge line](#def-b1-faults-folds-fabrics-fold-elements) [trends](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $000$ and [plunges](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $\arctan(0.01424/0.40768) = 2.0{}^{\circ}$ to the north.

**8.** The [axial plane](#def-b1-faults-folds-fabrics-fold-elements) contains the [hinge line](#def-b1-faults-folds-fabrics-fold-elements) and bisects the two [limbs](#def-b1-faults-folds-fabrics-fold-elements); its normal is along $\mathbf n_1 - \mathbf n_2 = (0.41710, 0, 0)$, horizontal and east–west. It is vertical and [strikes](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) north–south: $090/90$. The [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) is upright.

**9.** An open, [upright fold](#def-b1-faults-folds-fabrics-fold-classes), plunging $2.0{}^{\circ}$ to the north.

**10.** By [Proposition 4.15](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#prop-b1-geological-maps-folds), the [outcrops](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-geological-map) of an [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) close in the direction of [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge): to the north. Its core holds the oldest rocks: the sandstone at the base of the cover and, where the river has cut deepest, the basement of schist and granite beneath it.

**11.** Axial-planar, the [cleavage](#def-b1-faults-folds-fabrics-foliation) is vertical and [strikes](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) north–south, $090/90$. By [Proposition 17.14](#prop-b1-faults-folds-fabrics-intersection), its [intersection lineation](#def-b1-faults-folds-fabrics-lineation) with the bedding is parallel to the [hinge line](#def-b1-faults-folds-fabrics-fold-elements): [trend](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $000$, [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $2.0{}^{\circ}$. Directly: the plane $080.7/12.2$ meets the vertical plane striking north along a line of [plunge](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $\arctan(\tan12.2{}^{\circ}
\cos80.7{}^{\circ}) = 2.0{}^{\circ}$ toward $000$.

**12.** Looking north, the east limb rises toward the left at $12.0{}^{\circ}$ and the [cleavage](#def-b1-faults-folds-fabrics-foliation) is vertical: the acute rotation from the [cleavage](#def-b1-faults-folds-fabrics-foliation) to the bedding, through $78.0{}^{\circ}$, is anticlockwise and moves the top of the [cleavage](#def-b1-faults-folds-fabrics-foliation) to the west. The limb is the right way up, so by [Proposition 17.16](#prop-b1-faults-folds-fabrics-cleavage-bedding)(b) the [anticline](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-fold) [hinge](#def-b1-faults-folds-fabrics-fold-elements) is to the west, at $x = 3.5\,\mathrm{km}$, as the map shows.

**13.** The [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) is $120 - 90 = 030$; the plane [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) toward $120$, so the [hanging wall](#def-b1-faults-folds-fabrics-fault-elements) lies on the east-south-east side of the trace.

**14.** The fault is normal: $\sigma_1$ was vertical.

**15.** $\sigma_2$ is horizontal along the [strike](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude), $030$–$210$; $\sigma_3$ horizontal across it, $120$–$300$.

**16.** $45{}^{\circ} + \phi/2 = 60{}^{\circ}$ gives $\phi = 30{}^{\circ}$, and $\mu = \tan30{}^{\circ} = 0.577$.

**17.** The conjugate set [strikes](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $030$ as well and [dips](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $60{}^{\circ}$ the other way: $300/60$. The two sets meet at $180{}^{\circ} - 2\times60{}^{\circ} = 60{}^{\circ}$, with the vertical $\sigma_1$ bisecting the acute angle.

**18.** With $\mu = 0.85$, $\phi = 40.4{}^{\circ}$ and the best-oriented [normal faults](#def-b1-faults-folds-fabrics-fault-types) [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude) $65.2{}^{\circ}$; the Varden fault, at $60{}^{\circ}$, is $5.2{}^{\circ}$ off. By [Exercise 17.10](#exo-b1-faults-folds-fabrics-10), it slips at the ratio $\sigma_1'/\sigma_3' = (1 + 0.85\cot30{}^{\circ})/(1 - 0.85\tan30{}^{\circ}) = 4.85$, against $4.68$ for the best plane: it is nearly, but not exactly, the best oriented, and would slip again at a stress ratio only $4\,\%$ higher.

**19.** The slip is down the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude): $r = -90{}^{\circ}$; the [slip vector](#def-b1-faults-folds-fabrics-fault-elements) [trends](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $120$ and [plunges](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-trend-plunge) $60{}^{\circ}$.

**20.** $(\cos60{}^{\circ}\sin120{}^{\circ}, \cos60{}^{\circ}\cos120{}^{\circ},
-\sin60{}^{\circ}) = (0.4330, -0.2500, -0.8660)$.

**21.** $e = 200/\tan60{}^{\circ} = 115.5\,\mathrm{m}$.

**22.** East: $115.47\sin120{}^{\circ} = 100.0\,\mathrm{m}$; north: $115.47\cos120{}^{\circ} = -57.7\,\mathrm{m}$, that is $57.7\,\mathrm{m}$ to the south.

**23.** $L = 200/\sin60{}^{\circ} = 230.9\,\mathrm{m}$, and $200^2 + 115.47^2 = 40\,000 + 13\,333 = 53\,333 = 230.94^2$. The [throw](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) and the [heave](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-throw) are the vertical and horizontal components of the slip, which lies in the vertical plane of the [dip](https://one-course.com/books/earth/3/en/chapter/4-geological-maps#def-b1-geological-maps-attitude): they are the sides of a right triangle whose hypotenuse is the slip.

**24.** To the nearest metre, the [net slip](#def-b1-faults-folds-fabrics-fault-elements) on the Varden fault is **$231\,\mathrm{m}$**.