P A R T B

B R I T T L E S T R U C T U R E S

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6 .1 IN T RODUCTION

Drop a glass on a tile floor and watch as it breaks intodozens of pieces; you have just witnessed an exampleof brittle deformation! Because you’ve probably bro-ken a glass or two (or a plate or a vase) and have seencracked buildings, sidewalks, or roads, you alreadyhave an intuitive feel for what brittle deformation is allabout. In the upper crust of the Earth, roughly 10 kmin depth, rocks primarily undergo brittle deformation,creating a myriad of geologic structures. To understandwhy these structures exist, how they form in rocks ofthe crust, and what they tell us about the conditions ofdeformation, we must first learn why and how brittledeformation takes place in materials in general. Ourpurpose in this chapter is to introduce the basic termi-nology used to describe brittle deformation, to explainthe processes by which brittle deformation takes place,and to describe the physical conditions that lead to

brittle deformation. This chapter, therefore, provides abasis for the discussion of brittle structures that wepresent in Chapters 7 and 8.

6 . 2 VO C A BU L A RY OF BRIT T LEDE FOR M ATION

Research in the last few decades has changed the waygeologists think about brittle deformation and, as aconsequence, the vocabulary of brittle deformation hasevolved. To avoid misunderstanding, therefore, webegin the chapter on brittle deformation by definingour terminology. Table 6.1 summarizes definitions ofterms that we use in discussing brittle deformation,and includes additional terms that you may comeacross when reading articles on this topic. Rememberthat some of the brittle deformation vocabulary is

114

Brittle Deformation

6.1 Introduction 1146.2 Vocabulary of Brittle Deformation 1146.3 What Is Brittle Deformation? 1176.4 Tensile Cracking 118

6.4.1 Stress Concentration and Griffith Cracks 1186.4.2 Exploring Tensile Crack Development 1216.4.3 Modes of Crack-Surface

Displacement 1226.5 Processes of Brittle Faulting 123

6.5.1 Slip by Growth of Fault-Parallel Veins 1236.5.2 Cataclasis and Cataclastic Flow 123

6.6 Formation of Shear Fractures 1246.7 Predicting Initiation of Brittle Deformation 126

6.7.1 Tensile Cracking Criteria 1266.7.2 Shear-Fracture Criteria

and Failure Envelopes 127

6.8 Frictional Sliding 1326.8.1 Frictional Sliding Criteria 1326.8.2 Will New Fractures Form or Will

Existing Fractures Slide? 1336.9 Effect of Environmental Factors in Failure 134

6.9.1 Effect of Fluids on Tensile Crack Growth 134

6.9.2 Effect of Dimensions on Tensile Strength 136

6.9.3 Effect of Pore Pressure on Shear Failure and Frictional Sliding 136

6.9.4 Effect of Intermediate Principal Stress on Shear Rupture 136

6.10 Closing Remarks 136Additional Reading 137

C H A P T E R S I X

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1156 . 2 V O C A B U L A R Y O F B R I T T L E D E F O R M A T I O N

T A B L E 6 . 1 T E R M I N O L O G Y O F B R I T T L E D E F O R M A T I O N

Brittle deformation The permanent change that occurs in a solid material due to the growth of fractures and/or due to sliding on fractures. Brittle deformation only occurs when stresses exceed a critical value, and thus only after a rock has already undergone some elastic and/or plasticbehavior.

Brittle fault zone A band of finite width in which slip is distributed among many smaller discrete brittle faults,and/or in which the fault surface is bordered by pervasively fractured rock.

Brittle fault A single surface on which movement occurs specifically by brittle deformation mechanisms.

Cataclasis A deformation process that involves distributed fracturing, crushing, and frictional sliding of grains or of rock fragments.

Crack Verb: to break or snap apart. Noun: a fracture whose displacement does not involve sheardisplacement (i.e., a joint or microjoint).

Fault Broad sense: a surface or zone across which there has been measurable sliding parallel to thesurface. Narrow sense: a brittle fault. The narrow definition emphasizes the distinctionsbetween faults, fault zones, and shear zones.

Fracture zone A band in which there are many parallel or subparallel fractures. If the fractures are wavy, theymay anastomose with one another. Note: The term has a somewhat different meaning in thecontext of ocean-floor tectonics.

Fracture A general term for a surface in a material across which there has been loss of continuity and,therefore, strength. Fractures range in size from grain-scale to continent-scale.

Healed microcrack A microcrack that has cemented back together. Under a microscope, it is defined by a planecontaining many fluid inclusions. (Fluid inclusions are tiny bubbles of gas or fluid embedded in a solid).

Joint A natural fracture which forms by tensile loading, i.e., the walls of the fracture move apart veryslightly as the joint develops. Note: A minority of geologists argue that joints can form due toshear loading.

Microfracture A very small fracture of any type. Microfractures range in size from the dimensions of a singlegrain to the dimensions of a thin section.

Microjoint A microscopic joint; microjoints range in size from the dimensions of a single grain to thedimensions of a hand-specimen. Synonymous with microcrack.

Shear fracture A macroscopic fracture that grows in association with a component of shear parallel to thefracture. Shear fracturing involves coalescence of microcracks.

Shear joint A surface that originated as a joint but later became a surface of sliding. Note: A minority of geologists consider a shear joint to be a joint that initially formed in response to shearloading.

Shear rupture A shear fracture.

Shear zone A region of finite width in which ductile shear strain is significantly greater than in thesurrounding rock. Movement in shear zones is a consequence of ductile deformationmechanisms (cataclasis, crystal plasticity, diffusion).

Vein A fracture filled with minerals precipitated from a water solution.

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controversial, and you will find that not all geologistsagree on the definitions that we provide.

Brittle deformation is simply the permanentchange that occurs in a solid material due to the growthof fractures and/or due to sliding on fractures oncethey have formed. By this definition, a fracture is anysurface of discontinuity, meaning a surface acrosswhich the material is no longer bonded (Figure 6.1). Ifa fracture fills with minerals precipitated out of ahydrous solution, it is a vein, and if it fills with(igneous or sedimentary) rock originating from else-where, it is a dike. A joint is a natural fracture in rockacross which there is no measurable shear displace-ment. Because of the lack of shear involved in jointformation, joints can also be called cracks or tensilefractures. Shear fractures, in contrast, are meso-scopic fractures across which there has been displace-ment. Sometimes geologists use the term “shear frac-ture” instead of “fault” when they wish to imply thatthe amount of shear displacement on the fracture is rel-atively small, and that the shear displacement accom-panied the formation of the fracture in once intact rock.In Chapter 7 we show several examples of fracturesand veins in natural rocks.

In a broad sense, a fault is a surface or zone onwhich there has been measurable displacement. In a

narrower sense, geologists restrict use of the term faultto a fracture surface on which there has been sliding.When using this narrow definition of fault, we applythe term fault zone to refer either to a band of finitewidth across which the displacement is partitionedamong many smaller faults, or to the zone of rock bor-dering the fault that has fractured during faulting.Chapter 8 shows many examples of natural faults. Weapply the term shear zone to a band of finite width inwhich the ductile shear strain is significantly greaterthan in the surrounding rock. Movement in shear zonescan be the consequence of cataclasis (distributed frac-turing, crushing, and frictional sliding of grains of rockor rock fragments), crystal-plastic deformation mecha-nisms (dislocation glide, dislocation climb), and diffu-sion. We’ll mention cataclastic shear zones in thischapter, but other types of shear zones are discussed inChapter 12, after we have introduced ductile deforma-tion (Chapter 9).

Regardless of type, a fracture does not extend infi-nitely in all directions (Figure 6.2). Some fracturesintersect the surface of a body of rock, whereas othersterminate within the body. The line representing theintersection of the fracture with the surface of a rockbody is the fracture trace, and the line that separatesthe region of the rock which has fractured from the

116 B R I T T L E D E F O R M A T I O N

F I G U R E 6 . 1 A geologist measuring fractures in an outcrop, near Tuross Point on thesoutheastern coast of Australia. The more intensely fractured rock is a fine-grained mafic intrusiveand the less fractured rock is a coarse-grained felsic intrusive.

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region that has not fractured is the fracture front. Thepoint at which the fracture trace terminates on the sur-face of the rock is the fracture tip. In three dimensions,some fractures have irregular surfaces whereas othershave geometries that roughly resemble coins or blades.

6 . 3 W H AT IS BRIT T LEDE FOR M ATION ?

To understand brittle deformation we need to look atthe atomic structure of materials. Solids are composedof atoms or ions that are connected to one another bychemical bonds that can be thought of as tiny springs.Each chemical bond has an equilibrium length, and theangle between any two chemical bonds connected tothe same atom has an equilibrium value (Figure 6.3a).During elastic strain, the bonds holding the atomstogether within the solid stretch, shorten, and/or bend,but they do not break (Figure 6.3b)! When the stress isremoved, the bonds return to their equilibrium condi-tions and the elastic strain disappears. In other words,elastic strain is fully recoverable. Recall from Chapter 5that this elastic property of solids explains the propa-gation of earthquake waves though Earth.

Rocks cannot accumulate large elastic strains; youcertainly cannot stretch a rock to twice its originallength and expect it to spring back to its original shape!

1176 . 3 W H A T I S B R I T T L E D E F O R M A T I O N ?

(a)

(b)

Futurefractureplane

Fracture front

“Blade-shaped” fracture

“Coin-shaped” fracture

Beddingplane

Fracture trace

Fracture front

Fracture tip

LIBERLIBERTYTY

19961996

IINN

GGOODD WWEE TTRRUUSSTT

0

F I G U R E 6 . 2 (a) A block diagram illustrating that a fracturesurface terminates within the limits of a rock body. The topsurface of the block is the ground surface; erosion exposes thefracture trace. Note that the trace of the fracture on the groundsurface is a line of finite length (with a fracture tip at each end).The arrows indicate that this particular fracture grew radiallyoutward from an origin, labeled O, the dot in the center of thefracture plane. (b) A blade fracture which has propagated in asedimentary layer and terminates at the bedding plane.

la' = distorted length

Elasticity

(a) (b) (c)

Break

θ

lb

l a

lb' lb

θ'

l a'

l a

Fracture

la' > la lb' < lb

la = undistorted length θ' < θ

F I G U R E 6 . 3 A sketchillustrating what is meant bystretching and breaking of atomicbonds. The chemical bonds arerepresented by springs and theatoms by spheres. (a) Fouratoms arranged in a lattice at equilibrium. (b) As aconsequence of stretching of the lattice, some bonds stretchand some shorten, and the anglebetween pairs of bondschanges. (c) If the bonds arestretched too far, they break,and elastic strain is released.

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At most, a rock can develop a few percent strain by elas-tic distortion. If the stress applied to a rock is greater thanthe stress that the rock can accommodate elastically, thenone of two changes can occur: the rock deforms in a duc-tile manner (strains without breaking), or the rockdeforms in a brittle manner (that is, it breaks).

What actually happens during brittle deformation?Basically, if the stress becomes large enough to stretchor bend chemical bonds so much that the atoms are toofar apart to attract one another, then the bonds break,resulting in either formation of a fracture (Figure 6.3c)or slip on a preexisting fracture. In contrast to elasticstrain, brittle deformation is nonrecoverable, meaningthat the distortion remains when the stress is removed.Again we will use earthquakes as an example; in thiscase, elastic strain is exceeded at the focus, resulting infailure and displacement. The pattern of breakage dur-ing brittle deformation depends on stress conditionsand on material properties of the rock, so brittle defor-mation does not involve just one process. For purposesof our discussion, we divide brittle deformationprocesses into four categories that are listed in Table 6.2 and illustrated in Figure 6.4.

6 .4 T E NSILE C R ACK ING

6.4.1 Stress Concentration and Griffith Cracks

In Figure 6.5 we illustrate a crack in rock on the atomicscale. One way to create such a crack would be for allthe chemical bonds across the crack surface to break atonce. In this case, the tensile stress necessary for thisto occur is equal to the strength of each chemical bond

multiplied by all the bonds that had once crossed thearea of the crack. If you know the strength of a singlechemical bond, then you can calculate the stress nec-essary to break all the bonds simply by multiplying thebond strength by the number of bonds. Using realisticvalues for the elasticity (Young’s modulus, E) andsmall strain (<10%), Equation 5.3 in Chapter 5 resultsin a theoretical strength of rock that is thousands ofmegapascals. Measurement of rock strength in theEarth’s crust shows that tensile cracking occurs atcrack-normal tensile stresses of less than about 10 MPa,when the confining pressure is low,1 a value that ishundreds of times less than the theoretical strength ofrock. Keeping the concept of theoretical strength inmind, we therefore face a paradox: How can naturalrocks fracture at stresses that are so much lower thantheir theoretical strength?

The first step toward resolving the strength para-dox came when engineers studying the theory of elas-ticity realized that the remote stress (stress due to aload applied at a distance from a region of interest)gets concentrated at the sides of flaws (e.g., holes)inside a material. For example, in the case of a circularhole in a vertical elastic sheet subjected to tensilestress at its ends (Figure 6.6a), the local stress (i.e.,stress at the point of interest) tangent to the sides of thehole is three times the remote stress magnitude (σr).The magnitude of the local tangential stress at the topand bottom of the hole equals the magnitude of theremote stress, but is opposite in sign (i.e., it is com-pressive). If the hole has the shape of an ellipse instead

118 B R I T T L E D E F O R M A T I O N

T A B L E 6 . 2 C A T E G O R I E S O F B R I T T L E D E F O R M A T I O N P R O C E S S E S

Cataclastic flow This type of brittle deformation refers to macroscopic ductile flow as a result of grain-scale fracturingand frictional sliding distributed over a band of finite width.

Frictional sliding This process refers to the occurrence of sliding on a preexisting fracture surface, without thesignificant involvement of plastic deformation mechanisms.

Shear rupture This type of brittle deformation results in the initiation of a macroscopic shear fracture at an acuteangle to the maximum principal stress when a rock is subjected to a triaxial compressive stress.Shear rupturing involves growth and linkage of microcracks.

Tensile cracking This type of brittle deformation involves propagation of cracks into previously unfractured materialwhen a rock is subjected to a tensile stress. If the stress field is homogeneous, tensile crackspropagate in their own plane and are perpendicular to the least principal stress (σ3).

1The failure strength of rocks under tension is much less than the failurestrength of rocks under compression; failure strength under compressiondepends on the confining pressure.

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1196 . 4 T E N S I L E C R A C K I N G

(c)(b)(a)

Shearfracture

Maximum principalcompressive stress, σ1

Intactrock

σ3

Minimum principal compressive stress

(or tensile stress), σ3

Joint(tensile crack)

(f)

Frictional sliding on shearfracture (brittle fault)

(d)

Frictional sliding on reactivatedjoint; reactivation occurs becauseof change in stress field orientationwith respect to crack

(e)

Cataclastic sliding on former joint

(g)

Cataclastic flow (incataclastic shear zone)

σ1

F I G U R E 6 . 4 Types of brittle deformation. (a) Orientation of the remote principal stress directions with respect to an intact rockbody. (b) A tensile crack, forming parallel to σ1 and perpendicular to σ3 (which may be tensile). (c) A shear fracture, forming at anangle of about 30° to the σ1 direction. (d) A tensile crack that has been reoriented with respect to the remote stresses and becomesa fault by undergoing frictional sliding. (e) A tensile crack which has been reactivated as a cataclastic shear zone. (f) A shear fracturethat has evolved into a fault. (g) A shear fracture that has evolved into a cataclastic shear zone.

Crack

F I G U R E 6 . 5 A cross-sectional sketch of a crystal lattice(balls are atoms and sticks are bonds) in which there is a crack.The crack is a plane of finite extent across which all atomicbonds are broken.

of a circle (Figure 6.6b), the amount of stress concen-tration, C, is equal to 2b/a + 1, where a and b are theshort and long axes of the ellipse, respectively. Thus,values for stress concentration at the ends of an ellipti-cal hole depend on the axial ratio of the hole: the largerthe axial ratio, the greater the stress concentration. Forexample, at the ends of an elliptical hole with an axialratio of 8:1, stress is concentrated by a factor of 17, andin a 1 µm × 0.02 µm crack the stress is magnified by afactor of ∼100!

With this understanding in mind, A. W. Griffith, inthe 1920s, took the next step toward resolving thestrength paradox when he applied the concept of stressconcentration at the ends of elliptical holes to fracturedevelopment. Griffith suggested that all materials con-tain preexisting microcracks or flaws at which stressconcentrations naturally develop, and that because ofthe stress concentrations that develop at the tips ofthese cracks, they propagate and become larger cracks

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even when the host rock is subjected to relatively lowremote stresses. He discovered that in a material withcracks of different axial ratios, the crack with thelargest axial ratio will most likely propagate first. Inother words, stress at the tips of preexisting cracks canbecome sufficiently large to rupture the chemicalbonds holding the minerals together at the tip andcause the crack to grow, even if the remote stress is rel-atively small. Preexisting microcracks and flaws in arock, which include grain-scale fractures, pores, andgrain boundaries, are now called Griffith cracks in hishonor. Thus we resolve the strength paradox by learn-ing that rocks in the crust are relatively weak becausethey contain Griffith cracks.

Griffith’s concept provided useful insight into thenature of cracking, but his theory did not adequatelyshow how factors such as crack shape, crack length,and crack orientation affect the cracking process. Insubsequent years, engineers developed a new approachto studying the problem. In this approach, called linearelastic fracture mechanics, we assume that cracks ina material have nearly infinite axial ratio (defined aslong axis/short axis), meaning that all cracks are verysharp. Linear elastic fracture mechanics theory pre-dicts that, if factors like shape and orientation are

equal, a longer crack will propagate before a shortercrack. We’ll see why later in this chapter, when we dis-cuss failure criteria.

We can examine how preexisting cracks affect themagnitude of stress necessary for tensile cracking in asimple experiment. Take a sheet of paper (Figure 6.7)and pull at both ends. You have to pull quite hard inorder for the paper to tear. Now make two cuts, onethat is ∼0.5 cm long and one that is ∼2 cm long, in theedge of the sheet near its center, and pull again. Thepull that you apply gets concentrated at the tip of the preexisting cuts, and at this tip the strength of

120 B R I T T L E D E F O R M A T I O N

(a) (b)

σr

σr

σr σr

3σr

Cσr

σr

a ab

F I G U R E 6 . 6 Stress concentration adjacent to a hole in an elastic sheet. If the sheet is subjected to a remote tensilestress at its ends (σr), then stress magnitudes at the sides of the holes are equal to Cσr, where C, the stress concentrationfactor, is (2b/a) + 1. (a) For a circular hole, C = 3. (b) For anelliptical hole, C > 3.

(a)

(b)

(c)

Process zone

F I G U R E 6 . 7 Illustration of a home experiment to observethe importance of preexisting cracks in creating stressconcentrations. (a) An intact piece of paper is difficult to pullapart. (b) Two cuts, a large one and a small one, are made in thepaper. (c) The larger preexisting cut propagates. In the shadedarea, a region called the process zone, the plastic strength ofthe material is exceeded and deforms.

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the paper is exceeded. You will find that it takes muchless force to tear the paper, and that it tears apart bygrowth of the longer preexisting cut. The reason thatsharp cracks do not propagate under extremely smallstresses is that the tips of real cracks are blunted by acrack-tip process zone, in which the material deformsplastically (Figure 6.7c).

It is implicit in our description of crack propagationthat the total area of a crack does not form instanta-neously, but rather a crack initiates at a small flaw andthen grows outward. If you have ever walked out on thinice covering a pond, you are well aware of this fact. Asyou move away from shore, you suddenly hear a soundlike the echo of a gunshot; this is the sound of a fractureforming in the ice due to the stress applied by yourweight. If you have the presence of mind under suchprecarious circumstances to watch how the crack forms,you will notice that the crack initiates under your boot,and propagates outwards into intact ice at a finite veloc-ity. This means that at any instant, only chemical bondsat the crack tip are breaking. In other words, not all thebonds cut by a fracture are broken at once, and thus thebasis we used for calculating theoretical strength in the first place does not represent reality.

6.4.2 Exploring Tensile CrackDevelopment

Let’s consider what happens during a laboratory exper-iment in which we stretch a rock cylinder along its axisunder a relatively low confining pressure (Figure 6.8a),a process called axial stretching. As soon as the remotetensile stress is applied, preexisting microcracks in thesample open slightly, and the remote stress is magnifiedto create larger local stresses at the crack tips. Eventu-ally, the stress at the tip of a crack exceeds the strengthof the rock and the crack begins to grow. If the remotetensile stress stays the same after the crack begins topropagate, then the crack continues to grow, and mayeventually reach the sample’s margins. When this hap-pens, the sample fails, meaning it separates into twopieces that are no longer connected (Figure 6.8b).

We can also induce tensile fracturing by subjectinga rock cylinder to axial compression, under conditionsof low confining pressure. Under such stress condi-tions, mesoscopic tensile fractures develop parallel tothe cylinder axis (Figure 6.9a), a process known aslongitudinal splitting. Longitudinal splitting is simi-lar to tensile cracking except that, in uniaxial com-pression, the cracks that are not parallel to the σ1

direction are closed, whereas cracks that are parallelto the compression direction can open up. To picture

1216 . 4 T E N S I L E C R A C K I N G

(a) (b)

σt

Grips

Largestcrack,≈ 90°to σt

Throughgoingcrack

F I G U R E 6 . 8 Development of a throughgoing crack in ablock under tension. (a) When tensile stress (σt) is applied,Griffith cracks open up. (b) The largest, properly oriented crackspropagate to form a throughgoing crack.

(b)

σ1

(a)

F I G U R E 6 . 9 (a) A cross section showing a rock cylinderwith mesoscopic cracks formed by the process of longitudinalsplitting. (b) An “envelope” model of longitudinal splitting. If youpush down on the top of an envelope (whose ends have beencut off), the sides of the envelope will move apart.

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this, imagine an envelope standing on its edge. If youpush down on the top edge of the envelope, the sidesof the envelope pull apart, even if they were not sub-jected to a remote tensile stress (Figure 6.9b). Inrocks, as the compressive stress increases, the tensilestress at the tips of cracks exceeds the strength of therock, and the crack propagates parallel to the com-pressive stress direction.

In the compressive stress environment illustrated inFigure 6.9, the confining pressure required is very small;but tensile cracks can also be generated in a rock cylin-der when the remote stress is compressive under higherconfining pressure when adding fluid pressure in poresand cracks of the sample (i.e., the pore pressure; Fig-ure 6.10). The uniform, outward push of a fluid in amicrocrack can have the effect of creating a local tensilestress at crack tips, and thus can cause a crack to propa-gate. We call this process hydraulic fracturing. Assoon as the crack begins to grow, the volume of thecrack increases, so if no additional fluid enters the crack,the fluid pressure decreases. Crack propagation ceaseswhen pore pressure drops below the value necessary tocreate a sufficiently large tensile stress at the crack tip,and does not begin again until the pore pressure buildsup sufficiently. Therefore, tensile cracking driven by anincrease in pore pressure typically occurs in pulses.We’ll return to the role of hydraulic fracturing later inthis chapter and again in Chapter 8.

6.4.3 Modes of Crack-SurfaceDisplacement

Before leaving the subject of Griffith cracks, we need toaddress one more critical issue, namely, the direction inwhich an individual crack grows when it is loaded. So farwe have limited ourselves tocracks that are perpendicular to aremote stress. But what aboutcracks in other orientations withrespect to stress, and how do theypropagate? Materials scientistsidentify three configurations ofcrack loading. These configura-tions result in three differentmodes of crack-surface displace-ment (Figure 6.11). Note that the“displacement” we are referringto when describing crack propa-gation is only the infinitesimalmovement initiating propagationof the crack tip and is not measur-able mesoscopic displacement asin faults.

122 B R I T T L E D E F O R M A T I O N

Fluid-filledcrack

Confiningvessel

Sample

Pore fluid

(a)

Movingpiston

�1

�2 = �3 (b)

(a) (b) (c)

Mode I Mode II(sliding)

Mode III(tearing)

Shear-mode cracksTensile-mode cracks

F I G U R E 6 . 1 1 Block diagrams illustrating the three modes of crack surfacedisplacement: (a) Mode I, (b) Mode II, (c) Mode III. Mode I is a tensile crack, and Mode II and Mode III are shear-mode cracks.

F I G U R E 6 . 1 0 (a) Cross-sectional sketch illustrating a rockcylinder in a triaxial loading experiment. Fluid has access to therock cylinder and fills the cracks. (b) A fluid-filled crack that isbeing pushed apart from within by pore-fluid pressure.

During Mode I displacement, a crack opens veryslightly in the direction perpendicular to the crack sur-face, so Mode I cracks are tensile cracks. They form par-allel to the principal plane of stress that is perpendicularto the σ3 direction, and can grow in their plane withoutchanging orientation. During Mode II displacement(the sliding mode), rock on one side of the crack surfacemoves very slightly in the direction parallel to the frac-ture surface and perpendicular to the fracture front. Dur-ing Mode III displacement (the tearing mode), rock onone side of the crack slides very slightly parallel to thecrack but in a direction parallel to the fracture front.

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Although shear-mode cracks appear similar tomesoscopic faults, Mode II (and Mode III) cracks arenot simply microscopic equivalents of faults. We real-ize the difference when we examine the propagation ofshear-mode cracks. As they start growing, shear-modecracks immediately curve into the orientation of tensileor Mode I cracks, meaning that shear-mode cracks donot grow in their plane. Propagating shear-modecracks spawn new tensile cracks called wing cracks, aprocess illustrated in Figure 6.12.

6 . 5 PRO C E S S E S OF BRIT T LEFAU LTING

A brittle fault is a surface on which measurable slip hasdeveloped without much contribution by plastic defor-mation mechanisms. Brittle faulting happens inresponse to the application of a differential stress,because slip occurs in response to a shear stress paral-lel to the fault plane. In other words, for faulting to takeplace, σ1 cannot equal σ3, and the fault surface cannotparallel a principal plane of stress. Faulting causes achange in shape of the overall rock body that containsthe fault. Hence, faulting contributes to development ofregional strain; however, because a brittle fault is, bydefinition, a discontinuity in a rock body, the occur-rence of faulting does not require development of mea-surable ductile strain in the surrounding rock.

There are two basic ways to create a brittle fault (Fig-ure 6.4). The first is by shear rupturing of a previouslyintact body of rock. The second is by shear reactivationof a previously formed weak surface (for example, ajoint, a bedding surface, or a preexisting fault) in a body

of rock. A preexisting weak surface may slip before thedifferential stress magnitude reaches the failure strengthfor shear rupture of intact rock. Once formed, move-ment on brittle faults takes place either by frictional slid-ing, by the growth of fault-parallel veins, or by cataclasticflow. We’ll focus mostly on fault formation and friction,the central processes of brittle faulting, and only brieflymention the other processes here.

6.5.1 Slip by Growth of Fault-Parallel Veins

Not all faults that undergo displacement in the brittlefield move by frictional sliding. On some faults, theopposing surfaces are separated from one another bymineral crystals (e.g., quartz or calcite) precipitatedout of water solutions present along the fault duringmovement (Figure 6.13). Such fault-surface veinsmay be composed of mineral fibers (needle-like crys-tals), of blocky crystals, or of both.

The process by which fault-surface veins form is notwell understood. In some cases, they may form whenhigh fluid pressures cause a crack to develop along aweak fault surface. Immediately after cracking, one sideof the fault moves slightly with respect to the other; thecrack then seals by precipitation of vein material in thecrack (see also Chapter 7). In other cases, vein forma-tion may reflect gradual dissolution of steps (or asperi-ties) on the fault surface, followed by the transfer of ionsthrough fluid films to sites of lower stress, where min-eral precipitation takes place. This second process mayoccur without the formation of an actual discontinuity,across which the rock loses cohesion. Whether syn-slipveining or frictional sliding takes place on a fault sur-face probably depends on strain rate and on the presence

of water. Fault-surface veining is probably morecommon when water is present along the fault,and when movement occurs slowly.

6.5.2 Cataclasis and CataclasticFlow

Cataclasis refers to movement on a fault by acombination of microcracking, frictional slidingof fragments past one another, and rotation andtransport of grains. To picture the process of cat-aclasis, imagine what happens to corn passingbetween two old-fashioned mill stones. Themillstones slide past one another and, in theprocess, transform the corn into cornmeal. Cata-clasis, if affecting a relatively broad band of rock,results in mesoscopic ductile strain, in which caseit is also called cataclastic flow (Chapter 9),

1236 . 5 P R O C E S S E S O F B R I T T L E F A U L T I N G

�1

�3 �3

(a) (b)

Mode IIcrack

Tensile stressconcentration

�1

Mode I stresscracks

F I G U R E 6 . 1 2 Propagating shear-mode crack and the formation ofwing cracks. (a) A tensile stress concentration occurs at the ends of aMode II crack that is being loaded. (b) Mode I wing cracks form in thezones of tensile-crack concentration.

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because the rock over the width of the band effectivelyflows. To picture cataclastic flow, think of how thecornmeal that we just produced, when poured fromone container to another, behaves much like a fluid,even though the individual grains certainly are solid.Movement on a fault that involves development of azone in which cataclastic flow occurs is often referredto as a cataclastic shear zone.

6 . 6 FOR M ATION OF S H E A RF R ACT U R E S

Shear fractures differ markedly from tensile cracks. Ashear fracture is a surface across which a rock losescontinuity when the shear stress parallel to the surfaceis sufficiently large. Shear fractures are initiated inlaboratory rock cylinders at a typical angle of about30° to σ1 under conditions of confining pressure (σ1 > σ2 = σ3). Because there is a component of nor-mal stress acting on the fracture in addition to shearstress, friction resists sliding on the fracture during itsformation. If the shear stress acting on the fracturecontinues to exceed the frictional resistance to sliding,the fracture grows and displacement accumulates.Shear fractures (or faults) are therefore not simplylarge shear-mode cracks, because, as we have seen,shear-mode cracks cannot grow in their own plane.

This conceptual differ-ence is very important.

So how do shear frac-tures form? We can gaininsight into the process ofshear-fracture formationby generating shear rup-tures during a laboratorytriaxial loading experi-ment, using a rock cylinderunder confining pressure.So to begin our search foran answer to this question,we first describe such anexperiment.

In a confined-compres-sion triaxial-loading exper-iment, we take a cylinderof rock, jacket it in copperor rubber, surround it witha confining fluid in a pres-sure chamber, and squeezeit between two hydraulic

pistons. In the experiment shown in Figure 6.14, therock itself stays dry. During the experiment, we applya confining pressure (σ2 = σ3) to the sides of the cylin-der by increasing the pressure in the surrounding fluid,and an axial load (σ1) to the ends of the cylinder by mov-ing the pistons together at a constant rate. By keeping thevalue of σ3 constant while σ1 gradually increases, weincrease the differential stress (σd = σ1 – σ3). In thisexperiment we measure the magnitude of σd, thechange in length of the cylinder (which is the axialstrain, ea), and the change in volume (∆) of the cylinder.

A graph of σd versus ea (Figure 6.14a) shows thatthe experiment has four stages. In Stage I, we find thatas σd increases, ea also increases and that the relation-ship between these two quantities is a concave-upcurve. In Stage II of the experiment, the relationshipbetween σd and ea is a straight line with a positiveslope. During Stage I and most of Stage II, the volumeof the sample decreases slightly. In Stage III of theexperiment, the slope of the line showing the relationbetween σd and ea decreases. The stress at which thecurve changes slope is called the yield strength. Dur-ing the latter part of Stage II and all of Stage III, weobserve a slight increase in volume, a phenomenonknown as dilatancy, and if we had a very sensitivemicrophone attached to the sample during this time,we would hear lots of popping sounds that reflect theformation and growth of microcracks. Suddenly, whenσd equals the failure stress (σf), a shear rupture surfacedevelops at an angle of about 30° to the cylinder axis,

124 B R I T T L E D E F O R M A T I O N

F I G U R E 6 . 1 3 Photograph of stepped calcite slip fibers on a fault surface; pencil indicatesdisplacement direction.

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and there is a stress drop. A stress drop in this contextmeans that the axial stress supported by the specimensuddenly decreases and large strain accumulates at alower stress. To picture this stress drop, imagine thatyou’re pushing a car that is stuck in a ditch. You haveto push hard until the tires come out of the ditch, atwhich time you have to stop pushing so hard, or youwill fall down as the car rolls away. The value of σd atthe instant that the shear rupture forms and the stressdrops is called the failure strength for shear rupture.Once failure has occurred, the sample is no longer

intact and frictional resistance to sliding on the fracturesurface determines its further behavior.

What physically happened during this experiment?During Stage I, preexisting open microcracks under-went closure. During Stage II, the sample underwentelastic shortening parallel to the axis, and because ofthe Poisson effect expanded slightly in the directionperpendicular to the axis (Figure 6.14d). The Poissoneffect refers to the phenomenon in which a rock that isundergoing elastic shortening in one direction extendsin the direction at right angles to the shortening

1256 . 6 F O R M A T I O N O F S H E A R F R A C T U R E S

(a) (b)

(f)

(e)

(c)

0(d)

II Elastic compression

Closure of existing cracks and some pores

III New cracking

IV Crack coalescenceFailure stress

Yield stress

�d

�2 = �3

�V/V

�1

ea ea

(d) (e) (f)

I

F I G U R E 6 . 1 4 Fracture formation. (a) Stress–strain plot (differential stress versus axialshortening) showing the stages (I–IV) in a confined compression experiment. The labels indicatethe process that accounts for the slope of the curve. (b) The changes in volume accompanying theaxial shortening illustrate the phenomenon of dilatancy; left of the dashed line, the sample volumedecreases, whereas to the right of the dashed line the sample volume increases. (c–f) Schematiccross sections showing the behavior of rock cylinders during the successive stages of a confinedcompression experiment and accompanying stress–strain plot, emphasizing the behavior ofGriffith cracks (cracks shown are much larger than real dimensions). (c) Pre-deformation state,showing open Griffith cracks. (d) Compression begins and volume decreases due to crack closure.(e) Crack propagation and dilatancy (volume increase). (f) Merging of cracks along the futurethroughgoing shear fracture, followed by loss of cohesion of the sample (mesoscopic failure).

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direction. The ratio between the amount of shorteningand the amount of extension is called Poisson’s ratio,ν.2 At the start of Stage III, tensile microcracks beginto grow throughout the sample, and wing cracks growat the tips of shear-mode cracks. The initiation andgrowth of these cracks causes the observed slightincrease in volume, and accounts for the poppingnoises (Figure 6.14e). During Stage III, the tensilecracking intensifies along a narrow band that cutsacross the sample at an angle of about 30° to the axialstress. Failure occurs in Stage IV when the cracks self-organize to form a throughgoing surface along whichthe sample loses continuity, so that the rock on one sidecan frictionally slide relative to the rock on the otherside (Figure 6.14f). As a consequence, the cylindersmove together more easily and stress abruptly drops.

The fracture development scenario described aboveshows that the failure strength for shear fracture is nota definition of the stress state at which a single crackpropagates, but rather it is the stress state at which amultitude of small cracks coalesce to form a through-going rupture. Also note that two ruptures form insome experiments, both at ∼30° to the axial stress. Theangle between these conjugate fractures is ∼60°, andthe acute bisector is parallel to the maximum principalstress. With continued displacement, however, it isimpossible for both fractures to remain active, becausedisplacement on one fracture will offset the other.Thus, typically only one fracture will evolve into athroughgoing fault (see next section).

6 .7 PR E DI CTING INITI ATION OF BRIT T LE DE FOR M ATION

We have seen that brittle structures develop when rockis subjected to stress, but so far we have been rathervague about defining the stress states in which brittledeformation occurs. Clearly, an understanding of thestress state at which brittle deformation begins is valu-able, not only to geologists, who want to know when,where, and why brittle geologic structures (joints,faults, veins, and dikes) develop in the Earth, but also toengineers, who must be able to estimate the magnitudeof stress that a building or bridge can sustain before itcollapses. When we discuss brittle deformation, we arereally talking about three phenomena: tensile crack

growth, shear fracture development, and frictional slid-ing. In this section, we examine the stress conditionsnecessary for each of these phenomena to occur.

6.7.1 Tensile Cracking CriteriaA tensile cracking criterion is a mathematical state-ment that predicts the stress state in which a crackbegins to propagate. All tensile cracking criteria arebased on the assumption that macroscopic cracks growfrom preexisting flaws (Griffith cracks) in the rock,because preexisting flaws cause stress concentrations.Griffith was one of the first researchers to propose atensile cracking criterion. He did so by looking at howenergy was utilized during cracking. Griffith envi-sioned that a material in which a crack forms can bemodeled as a thermodynamic system consisting of anelastic plate containing a preexisting elliptical crack. Ifa load is applied to the ends of the sheet so that itstretches and the crack propagates, the total energy ofthis system can be defined by the following equation(known as the Griffith energy balance):3

dUT = dUs – dWr + dUE Eq. 6.1

where dUT is the change in total energy of the system,dUs is the change in surface energy due to growth ofthe crack (this term arises because crack formationbreaks bonds, and energy stored in a broken bond isgreater than the energy stored in a satisfied bond), dWr

is the work done by the load in deforming the plate,and dUE is the change in the strain energy stored in theplate (strain energy is the energy stored by chemicalbonds that have been stretched out of their equilibriumlength or angle). Griffith pointed out that because thesystem starts with the load already in place, formationof the crack does not change the total energy of thesystem. Thus, the change in total energy for an incre-ment of crack growth (dc) equals 0 (in equation form,dUT/dc = 0 for an equilibrium condition). With thispoint in mind, Equation 6.1 can be rewritten as

dWr = dUs + dUE Eq. 6.2

In words, Equation 6.2 means that the work done on asystem is divided between creation of new elasticstrain energy in the sheet and creation of new cracksurface by propagation of the crack. Using this con-

126 B R I T T L E D E F O R M A T I O N

2Since Poisson’s ratio has the units length/length, it is dimensionless; a typ-ical value of ν for rocks is 0.25.

3This is a plane stress criterion; for plane strain conditions, E is replaced byE/(1 – ν)2, where ν is Poisson’s ratio.

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cept, along with several theorems from elasticity the-ory, Griffith devised a tensile cracking criterion, whichwe present without derivation:

σt = (2Eγ/πc)1/2 Eq. 6.3

where σt = critical remote tensile stress (tensile stressat which the weakest Griffith crack begins to grow),E = Young’s modulus, γ = energy used to create newcrack surface, and c = half-length of the preexistingcrack. Reading this equation, we see that the criticalremote tensile stress for a rock sample is proportionalto material properties of the sample and the length ofthe crack. Note that as crack length increases, the valueof critical remote tensile stress (σt) decreases.

Subsequently, researchers have utilized conceptsfrom the engineering study of linear elastic fracturemechanics to develop tensile cracking criteria. Accord-ing to this work, the following equation defines condi-tions at which Mode I cracks propagate:

KI = σt Y(πc)1/2 Eq. 6.4

where KI (pronounced “kay one,” where the “one” rep-resents a Mode I crack) is the stress intensity factor,σt is the remote tensile stress, Y is a dimensionlessnumber that takes into account the geometry of thecrack (e.g., whether it is penny-shaped, blade-shaped, ortunnel-shaped), and c is half of the crack’s length. Inthis analysis, all cracks in the body are assumed tohave very large ellipticity; that is, cracks are assumedto be very sharp at their tips. Note that cracks withsmaller ellipticity require higher stresses to propagate.

Equation 6.4 says that the value of KI increaseswhen σ increases. A crack in the sample begins togrow when KI attains a value of KIc, which is the crit-ical stress intensity factor or the fracture toughness(a measure of tensile strength). The fracture toughnessis constant for a given material. When KI reaches KIc,the value of σ reaches σt, where σt is the criticalremote tensile stress at the instant the crack starts togrow. We can rewrite Equation 6.4 to create an equa-tion that more directly defines the value of σt at theinstant the crack grows:

σt = KIc/[Y(πc)1/2] Eq. 6.5

Note that, in this equation, the remote stress necessaryfor cracking depends on the fracture toughness, thecrack shape, and the length of the crack. If other fac-tors are equal, a longer crack generally propagatesbefore a shorter crack. Similarly, if other factors areequal, crack shape determines which crack propagates

first. Because c increases as the crack grows, crackpropagation typically leads to sample failure. Thisrelationship has another interesting consequence fornatural rocks. Because crack length depends on thegrain size of a sample, fine-grained rocks should bestronger than coarse-grained rocks.

Equations like Equations 6.4 and 6.5 can also bewritten for Mode II and Mode III cracks. By compar-ing equations for the three different modes of cracking,you will find that, other factors being equal, a Mode Icrack (i.e., a crack perpendicular to σ3) propagatesbefore a Mode II or Mode III crack. However, sinceother factors like crack shape and length come intoplay, Mode II or Mode III cracks sometimes propagatebefore Mode I cracks in a real material. Remember thatthe instant they propagate, Mode II and III crackseither bend and become Mode I cracks, or theydevelop wing cracks at their tips; shear cracks cannotpropagate significantly in their own plane.

In summary, we see that the stress necessary to ini-tiate the propagation of a crack depends on the ellip-ticity, the length, the shape, and the orientation of apreexisting crack. Study of crack-propagation criteriais a very active research area, and further details con-cerning this complex subject (such as subcritical crackgrowth under long-term loading or corrosive condi-tions) are beyond the scope of this book (see the read-ing list).

6.7.2 Shear-Fracture Criteria and Failure Envelopes

A shear-fracture criterion is an expression thatdescribes the stress state at which a shear ruptureforms and separates a sample into two pieces. Becauseshear-fracture initiation in a laboratory sampleinevitably leads to failure of the sample, meaning thatafter rupture the sample can no longer support a loadthat exceeds the frictional resistance to sliding on thefracture surface, shear-rupture criteria are also com-monly known as shear-failure criteria.

Charles Coulomb4 was one of the first to propose ashear-fracture criterion. He suggested that if all theprincipal stresses are compressive, as is the case in aconfined compression experiment, a material fails bythe formation of a shear fracture, and that the shearstress parallel to the fracture surface, at the instant offailure, is related to the normal stress by the equation

σs = C + µσn Eq. 6.6

1276 . 7 P R E D I C T I N G I N I T I A T I O N O F B R I T T L E D E F O R M A T I O N

4An eighteenth-century French naturalist.

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where σs is the shear stress parallel to the fracture sur-face at failure; C is the cohesion of the rock, a constantthat specifies the shear stress necessary to cause failureif the normal stress across the potential fracture planeequals zero (note that this C is not the same as the c inEquations 6.3–6.5); σn is the normal stress across theshear fracture at the instant of failure; and µ is a con-stant traditionally known as the coefficient of internalfriction. The name for µ originally came from studiesof friction between grains in unconsolidated sand andof the control that such friction has on slope angles ofsand piles, so the name is essentially meaningless inthe context of shear failure of a solid rock; µ should beviewed simply as a constant of proportionality. Equa-tion 6.6, also known as Coulomb’s failure criterion,basically states that the shear stress necessary to initi-ate a shear fracture is proportional to the normal stressacross the fracture surface.

The Coulomb criterion plots as a straight line on aMohr diagram5 (Figure 6.15). To see this, let’s plot theresults of four triaxial loading experiments in whichwe increase the axial load on a confined granite cylin-der until it ruptures. In the first experiment, we set theconfining pressure (σ2 = σ3) at a relatively low value,increase the axial load (σ1) until the sample fails, andthen plot the Mohr circle representing this criticalstress state, meaning the stress state at the instant offailure, on the Mohr diagram. When we repeat theexperiment, using a new cylinder, and starting at ahigher confining pressure, we find that as σ3 increases,the differential stress (σ1 – σ3) at the instant of failurealso increases. Thus, the Mohr circle representing thesecond experiment has a larger diameter and lies to theright of the first circle. When we repeat the experimenttwo more times and plot the four circles on the dia-gram, we find that they are all tangent to a straight linewith a slope of µ (i.e., tan φ) and a y-intercept of C, andthis straight line is the Coulomb criterion. Note that wecan also draw a straight line representing the criterionin the region of the Mohr diagram below the σn-axis.

A line drawn from the center of a Mohr circle to thepoint of its tangency with the Coulomb criterion defines2θ, where θ is the angle between the σ3 direction and theplane of shear fracture (typically about 30°). Because

the Coulomb criterion is a straight line, this angle isconstant for the range of confining pressures for whichthe criterion is valid. The reason for the 30° anglebecomes evident in a graph plotting normal stressmagnitude and shear stress magnitude as a function ofthe angle between the plane and the σ1 direction (Fig-ure 6.16). Notice that the minimum normal stress doesnot occur in the same plane as the maximum shearstress. Shear stress is at its highest on a potential fail-ure plane oriented at 45° to σ1, but the normal stressacross this potential plane is still too large to permitshear fracturing in planes of this orientation. The shearstress is a bit lower across a plane oriented at 30° to σ1,but is still fairly high. However, the normal stressacross the 30° plane is substantially lower, favoringshear-fracture formation.

Coulomb’s criterion is an empirical relation, mean-ing that it is based on experimental observation alone,not on theoretical principles or knowledge of atomic-scale or crystal-scale mechanisms. This failure criteriondoes not relate the stress state at failure to physicalparameters, as does the Griffith criterion, nor does itdefine the state of stress in which the microcracks,which eventually coalesce to form the shear rupture,begin to propagate. The Coulomb criterion does notpredict whether the fractures that form will dip to theright or to the left with respect to the axis of the rockcylinder in a triaxial loading experiment. In fact, as

128 B R I T T L E D E F O R M A T I O N

–σs = – (C + σ

n tanφ)

σn

σs

σ s = C + σ n ta

nφ

φ

C

F I G U R E 6 . 1 5 Mohr diagram showing a Coulomb failureenvelope based on a set of experiments with increasingdifferential stress. The circles represent differential stressstates at the instant of shear failure. The envelope isrepresented by two straight lines, on which the dots representfailure planes.

5Recall that on the Mohr diagram, normal stresses (σn) are plotted on the x-axis and shear stresses (σs) are plotted on the y-axis. A Mohr circle repre-sents the stress state by indicating the values of σn and σs acting on a planeoriented at θ° to the σ3 direction. The circle intersects the x-axis at σ1 andσ3 (both of which are normal stresses, because they are principal stresses),and the angle between the x-axis and a radius from the center of the circleto a point on the circle defines the angle 2θ.

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mentioned earlier, conjugate shear fractures, onewith a right-lateral shear sense and one with a left-lateral shear sense, may develop (Figure 6.17). Thetwo fractures, typically separated by an angle of ∼60°,correspond to the tangency points of the circle repre-senting the stress state at failure with the Coulomb fail-ure envelope.

The German engineer Otto Mohr conducted furtherstudies of shear-fracture criteria and found thatCoulomb’s straight-line relationship only works for alimited range of confining pressures. He noted that atlower confining pressure, the line representing thestress state at failure curved with a steeper slope, andthat at higher confining pressure, the line curved witha shallower slope (Figure 6.18). Mohr concluded that

over a range of confining pressure, the failure criteriafor shear rupture resembles a portion of a parabolalying on its side, and this curve represents the Mohr-Coulomb criterion for shear fracturing. Notice thatthis criterion is also empirical. Unlike Coulomb’sstraight-line relation, the change in slope of the Mohr-Coulomb failure envelope indicates that the anglebetween the shear fracture plane and σ1 actually doesdepend on the stress state. At lower confining pres-sures, the angle is smaller, and at high confining pres-sures, the angle is steeper.

The Mohr-Coulomb criterion (both for positive andnegative values of σs) defines a failure envelope on theMohr diagram. A failure envelope separates the fieldon the diagram in which stress states are “stable” from

1296 . 7 P R E D I C T I N G I N I T I A T I O N O F B R I T T L E D E F O R M A T I O N

∆σn

σn

σs

σnσ1

σs

α

∆σs

Plane

θ = 90 – α

Str

ess

mag

nitu

de

0 15 30 45 60 75 90

θ

1

2

F I G U R E 6 . 1 6 The change in magnitudes of the normaland shear components of stress acting on a plane as a functionof the angle α between the plane and the σ1 direction; theangle θ = 90 – α is plotted for comparison with other diagrams.At point 1 (α = θ = 45°), shear stress is a maximum, but thenormal stress across the plane is quite large. At point 2 (θ = 60°, α = 30°), the shear stress is still quite high, but thenormal stress is much lower.

(a) (b)

60°

F I G U R E 6 . 1 7 Cross-sectional sketch showing how onlyone of a pair of conjugate shear fractures (a) evolves into afault with measurable displacement (b).

σn

σs

2α12α2

2α3

F I G U R E 6 . 1 8 Mohr failure envelope. Note that the slope ofthe envelope steepens toward the σs-axis. Therefore, the valueof α (the angle between fault and σ1) is not constant (compare2α1, 2α2, and 2α3).

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the field in which stress states are “unstable” (Fig-ure 6.19). By this definition, a stable stress state isone that a sample can withstand without undergoingbrittle failure. An unstable stress state is an impossiblecondition to achieve, for the sample will have failed by fracturing before such a stress state is reached (Fig-ure 6.19). In other words, a stress state represented bya Mohr circle that lies entirely within the envelope isstable, and will not cause the sample to develop a shearrupture. A circle that is tangent to the envelope speci-fies the stress state at which brittle failure occurs.Stress states defined by circles that extend beyond theenvelope are unstable, and are therefore impossiblewithin the particular rock being studied.

Can we define a failure envelope representing thecritical stress at failure for very high confining pres-sures, very low confining pressures, or for conditionswhere one of the principal stresses is tensile? Theanswer to this question is controversial. We’ll look ateach of these conditions separately.

At high confining pressures, samples may begin todeform plastically. Under such conditions, we are nolonger really talking about brittle deformation, so theconcept of a “failure” envelope no longer reallyapplies. However, we can approximately represent the“yield” envelope, meaning the stress state at which thesample begins to yield plastically, on a Mohr diagramby a pair of lines that parallel the σn-axis (Fig-ure 6.20a). This yield criterion, known as Von Misescriterion, indicates that plastic yielding is effectivelyindependent of the differential stress, once the yieldstress has been achieved.

If the tensile stress is large enough, the sample failsby developing a throughgoing tensile crack. The ten-sile stress necessary to induce tensile failure may berepresented by a point, T0, the tensile strength, along

the σn-axis to the left of the σs-axis (Figure 6.20b). Aswe have seen, however, the position of this pointdepends on the size of the flaws in the sample. Thus,even for the same rock type, experiments show that thetensile strength is very variable and that it is best rep-resented by a range of points along the σn-axis.

130 B R I T T L E D E F O R M A T I O N

σs σs σs σs

σn σn σn σn

IMPOSSIBLE

(a) (b) (c) (d)

F I G U R E 6 . 1 9 (a) A brittle failure envelope as depicted on a Mohr diagram. Within theenvelope (shaded area), stress states are stable, but outside the envelope, stress states areunstable. (b) A stress state that is stable, because the Mohr circle, which passes through valuesfor σ1 and σ3 and defines the stress state, falls entirely inside the envelope. (c) A stress state atthe instant of failure. The Mohr circle touches the envelope. (d) A stress state that is impossible.

“Transitionaltensile”

Mohrenvelope

Tensile-strengthrange

σs

σs

σn

σn

(a)

(b)

von Mises

F I G U R E 6 . 2 0 (a) Mohr diagram illustrating the Von Misesyield criterion. Note that the criterion is represented by twolines that parallel the σn-axis. (b) The extrapolation of a Mohrenvelope to its intercept with the σn-axis, illustrating the“transitional-tensile” regime, and the tensile strength (T0). Note that the tensile strength has a range of values, because it depends on the dimensions of preexisting flaws in thedeforming sample.

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There are competing views as to the nature of fail-ure for rocks subjected to tensile stresses that are lessthan the tensile strength. Some geologists have sug-gested that failure occurs under such conditions by theformation of fractures that are a hybrid between tensilecracks and shear ruptures, and have called these frac-tures transitional-tensile fractures or hybrid shearfractures. The failure envelope representing the con-ditions for initiating transitional-tensile fractures is thesteeply sloping portion of the parabolic failure enve-lope (Figure 6.20b). Most fracture specialists, how-ever, claim that transitional-tensile fractures do notoccur in nature, and point out that no experiments haveyet clearly produced transitional-tensile fractures inthe lab. We’ll discuss this issue further in Chapter 8.

Taking all of the above empirical criteria intoaccount, we can construct a composite failure enve-lope that represents the boundary between stable andunstable stress states for a wide range of confiningpressures and for conditions for which one of the prin-cipal stresses is tensile (Figure 6.21). The enveloperoughly resembles a cross section of a cup lying on itsside. The various parts of the curve are labeled. Start-ing at the right side of the diagram, we have Von Misescriteria, represented by horizontal lines. (Rememberthat the Von Mises portion of the envelope is really aplastic yield criterion, not a brittle failure criterion.)The portion of the curve where the lines begin to slopeeffectively represents the brittle–plastic transition. Tothe left of the brittle–plastic transition, the envelopeconsists of two straight sloping lines, representingCoulomb’s criterion for shear rupturing. For failureassociated with the Coulomb criterion, remember thatthe angle between the shear rupture and the σ1 direc-tion is independent of the confining pressure. Closer tothe σs-axis, the slope of the envelope steepens, and theenvelope resembles a portion of a parabola. This para-bolic part of the curve represents Mohr’s criterion, andfor failure in this region, the decrease in the anglebetween the fracture and the σ1 direction depends onhow far to the left the Mohr circle touches the failurecurve. The part of the parabolic envelope with steepslopes specifies failure criteria for supposed transitional-tensile fractures formed at a very small angle to σ1, butas we discussed, the existence of such fracturesremains controversial. The point where the envelopecrosses the σn-axis represents the failure criterion fortensile cracking, but as we have discussed, this criterionreally shouldn’t be specified by a point, for the tensilestrength of a material depends on the dimensions of theflaws it contains. Note that for a circle tangent to thecomposite envelope at T0, 2θ = 180 (or, α = 0), so

the fracture that forms is parallel to σ1! Also, note thatthere is no unique value of differential stress needed tocause tensile failure, as long as the magnitude of thedifferential stress (the diameter of the Mohr circle) isless than about 4T0, for this is the circle whose curva-ture is the same as that of the apex of the parabola.

1316 . 7 P R E D I C T I N G I N I T I A T I O N O F B R I T T L E D E F O R M A T I O N

σ3

σ3

σ3

σ3

σ3

σ1

σ1

σ1

σ1

σ1

σs

σn

(a)

AB

CD E

Composite failure envelope

A: Tensile failure criterionB: Mohr (parabolic) failure criterionC: Coulomb (straight-line) failure criterionD: Brittle-plastic transitionE: von Mises plastic yield criterion

Tensile fracture

“Transitional-tensile”fracture

Coulomb shearfracture

Brittle-plastictransition

45°

Plasticyielding

Increasing confining pressure

(b)

F I G U R E 6 . 2 1 (a) A representative composite failureenvelope on a Mohr diagram. The different parts of the envelope arelabeled, and are discussed in the text. (b) Sketches of the fracturegeometries that form during failure. Note that the geometrydepends on the part of the failure envelope that represents failureconditions, because the slope of the envelope is not constant.

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6 . 8 F RI CTION A L S LIDING

Friction is the resistance to sliding on a surface. Fric-tional sliding refers to the movement on a surface thattakes place when shear stress parallel to the surfaceexceeds the frictional resistance to sliding. The princi-ples of frictional sliding were formulated hundreds ofyears ago. We can do a simple experiment that pro-duces, at first, counterintuitive behavior (Figure 6.22).Attach a spring to a beam of wood that is placed on atable with the flat side down. Pull the spring until thebeam slides. Now place the beam on its narrow sideand repeat the experiment. Surprisingly, the springextends by the same amount before beam slidingoccurs, irrespective of the area of contact. Similarexperiments led to what are often called Amontons’slaws6 of friction:

• Frictional force is a function of normal force.• Frictional force is independent of the (apparent) area

of contact (in his words, “the resistance caused byrubbing only increases or diminishes in proportion to greater or lesser pressure [load] and not accordingto the greater or lesser extent of the surfaces”).

• Frictional force is mostly independent of the mate-rial used (in his words, “the resistance caused byrubbing is more or less the same for iron, lead, cop-per and wood in any combination if the surfaces arecoated with pork fat”).

Let’s explore the reason for this behavior, which hasimportant consequences for natural faulting processesand earthquake mechanics. Note that we should distin-guish between static friction, which is associated withfirst motion (associated with Coulomb failure), anddynamic friction, which is associated with continuedmotion.

Friction exists because no real surface in nature, nomatter how finely polished, is perfectly smooth. Thebumps and irregularities which protrude from a roughsurface are called asperities (Figure 6.23a). When twosurfaces are in contact, they touch only at the asperi-ties, and the asperities of one surface may indent orsink into the face of the opposing surface (Figure 6.23band c). The cumulative area of the asperities that con-tact the opposing face is the real area of contact (Ar inFigure 6.23). In essence, asperities act like an anchorholding a ship in place. In order for the ship to drift,

either the anchor chain must break, or the anchor mustdrag along the sea floor. Similarly, in order to initiatesliding of one rock surface past another, it is necessaryeither for asperities to break off, or for them to plow afurrow or groove into the opposing surface.

The stress necessary to break off an asperity or tocause it to plow depends on the real area of contact, so,as the real area of contact increases, the frictional resis-tance to sliding (that is, the force necessary to causesliding) increases. Again, considering our ship analogy,it takes less wind to cause a ship with a small anchor todrift than it does to cause the same-sized ship with alarge anchor to drift. Thus, the frictional resistance tosliding is proportional to the normal force componentacross the surface, because of the relation between realarea of contact and friction. An increase in the normalforce (load) pushes asperities into the opposing wallmore deeply, causing an increase in the real area of con-tact. Returning to our earlier experiment with a slidingbeam, we can now understand that the object’s mass,rather than the (apparent) area of contact (the side ofthe beam), determines the ability to slide.

6.8.1 Frictional Sliding CriteriaBecause of friction, a certain critical shear stress mustbe achieved in a rock before frictional sliding is initi-ated on a preexisting fracture, and a relation definingthis critical stress is the failure criterion for frictionalsliding. Experimental work shows that failure criteriafor frictional sliding, just like the Coulomb failure cri-terion for intact rock, plot as sloping straight lines on aMohr diagram. A compilation of friction data from alarge number of experiments, using a great variety ofrocks (Figure 6.24), shows that the failure criterion forfrictional sliding is basically independent of rock type:

σs/σn = constant Eq. 6.7

132 B R I T T L E D E F O R M A T I O N

Ff

Ff

Ff = f sin �

Fn = f cos �

Ff /Fn = sin �/cos � = �

Fn F

�Fn F

F I G U R E 6 . 2 2 Frictional sliding of objects with same mass,but with different (apparent) contact areas. The frictioncoefficients and, therefore, sliding forces (Ff) are equal for both objects, regardless of (apparent) contact area.

6Named after the seventeenth-century French physicist Guillaume Amon-tons; Leonardo da Vinci earlier described similar relationships in the fif-teenth century.

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The empirical relationship between normal and shearstress that best fits the observations, known as Byerlee’slaw,7 depends on the value of σn. For σn < 200 MPa, thebest-fitting criterion is a line described by the relation-ship σs = 0.85σn, whereas for 200 MPa < σn < 2000 MPa,the best-fitting criterion is a line described by the equa-tion σs = 50 MPa + 0.6σn. The proportionality betweennormal and shear stress is, as before, called the fric-tion coefficient.

6.8.2 Will New Fractures Form or WillExisting Fractures Slide?

Failure envelopes allow us to quickly determine whetherit is more likely that an existing shear rupture will slip ina sample, or that a new shear rupture will form (Figure6.25). For example, look at Figure 6.25b, which showsboth the Byerlee’s frictional sliding envelope and theCoulomb shear fracture envelope for Blair dolomite.Note that the slope and intercept of the two envelopes

1336 . 8 F R I C T I O N A L S L I D I N G

(b)

(a)

RAC

Void

A = (apparent) areaAr = real area

A

(c)

Ar

Ar

Ar

Ar

Fn

Ff F

F I G U R E 6 . 2 3 Concept of asperities and the real area of contact (Ar). (a) Schematic cross-sectional close-up showing the irregularity of a fracture surface and the presence of voids andasperities along the surface. (b) Idealized asperity showing the consequence of changing the load(normal force) on the real area of contact. (c) Map of a fracture surface; the shaded areas are realareas of contact.

200

100

0 100 200 300

�n(MPa)

�s = 0.85 �n

�s = 50 + 0.6 �n

�s(

MP

a)

Byerlee'slaw

F I G U R E 6 . 2 4 Graph of shear stress and normal stressvalues at the initiation of sliding on preexisting fractures in avariety of rock types. The best-fit line defines Byerlee’s law,which is defined for two regimes.

7After the geophysicist J. Byerlee, who first proposed the equations in the1970s.

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are different, so that for a specific range of preexistingfracture orientations, the Mohr circle representing thestress state at failure touches the frictional envelopebefore it touches the fracture envelope, meaning the pre-existing fracture slides before a new fracture forms.

However, preexisting fractures do not always slidebefore a new fracture is initiated. Confined compres-sion experiments indicate that if the preexisting fractureis oriented at a high angle to the σ1 direction (generally> 75°; plane E in Figure 6.25), the normal stress com-ponent across the discontinuity is so high that frictionresists sliding, and it is actually easier to initiate a newshear fracture at a smaller angle to σ1 (plane B in Fig-ure 6.25). Sliding then occurs on the new fracture. If apreexisting fracture is at a very small angle to σ1 (gen-erally < 15°; plane A in Figure 6.25), the shear stress onthe surface is relatively low, so again it ends up beingeasier to initiate a new shear fracture than to cause slid-ing on a preexisting weak surface. Thus, preexistingplanes whose angles to σ1 are between 15° and 75°probably will be reactivated before new fractures form.

6 . 9 E F FECT OFE N V IRON M E N TA L FACTOR S IN FA ILU R E

The occurrence and character of brittle deformation at agiven location in the earth depends on environmentalconditions (confining pressure, temperature, and fluid

pressure) present at that loca-tion, and on the strain rate (seeChapter 5). Conditions con-ducive to the occurrence ofbrittle deformation are morecommon in the upper 10–15km of the Earth’s crust. How-ever, at slow strain rates or inparticularly weak rocks, duc-tile deformation mechanismscan also occur in this region,as evident by the developmentof folds at shallow depths inthe crust. Below 10–15 km,plastic deformation mecha-nisms dominate. However, atparticularly high fluid pres-sures or at very rapid strainrates, brittle deformation canstill occur at these depths.

In this chapter, we havedescribed brittle deformation

without considering how it is affected by environmentalfactors. Not surprisingly, temperature, fluid pressure,strain rate, and rock anisotropy play significant roles inthe stress state at failure and/or in the orientation of thefractures that form when failure occurs. Most of thesefactors have already been discussed in Chapter 5, so weclose this chapter on brittle deformation by focusing onthe effect of fluids.

6.9.1 Effect of Fluids on Tensile Crack Growth

All rocks contain pores and cracks—we’ve alreadyseen how important these are in the process of brittlefailure. In the upper crust of the earth below the watertable, these spaces, which constitute the porosity ofrock, are filled with fluid. This fluid is most commonlywater, though in some places it is oil or gas.

If there is a high degree of permeability in the rock,meaning that water can flow relatively easily frompore to pore and/or in and out of the rock layer, thenthe pressure in a volume of pore water at a location inthe crust is roughly hydrostatic, meaning that the pres-sure reflects the weight of the overlying water column(Figure 6.26). Hydrostatic (fluid) pressure is definedby the relationship Pf = ρ ⋅ g ⋅ h, where ρ is the densityof water (1000 kg/m3), g is the gravitational constant(9.8 m/s2), and h is the depth. Pore pressure, which isthe fluid pressure exerted by fluid within the pores ofa rock, may exceed the hydrostatic pressure if perme-ability is restricted. For example, the fluid trapped in a

134 B R I T T L E D E F O R M A T I O N

500

0.6

0.85

0.4

0.2

100

800100 200 300 400 500 600 700

200

300

400

�s = 45 + 1.0 �n

�3

�1�

s (M

Pa)

�n (MPa)

~22¡ A

~56¡C ~66¡

D

~78¡E

~48¡B

A

BCDE

�

Coulombenvelope

F I G U R E 6 . 2 5 (a) Mohr diagram based on experiments with Blair dolomite, showing how asingle stress state (Mohr circle) would contact the frictional sliding envelope before it wouldcontact the Coulomb envelope (heavy line). Surface A in (b) is the Coulomb shear fracture thatwould form in an intact rock. However, sliding would occur on surfaces between intersectionswith the friction envelope (marked by shaded area for friction envelope µ = 0.85) before newfracture initiation. Preexisting surfaces B to E are surfaces that will slide with decreasingfriction coefficients. Consider the geologic relevance of decreasing friction coefficients forstress state, failure, and fracture orientation.

(a) (b)

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sandstone lens surrounded by impermeable shale can-not escape, so the pore pressure in the sandstone canapproach or even equal lithostatic pressure (Pl),meaning that the pressure approaches the weight of theoverlying column of rock (i.e., Pf = Pl = ρ ⋅ g ⋅ h, whereρ = 2000–3000 kg/m3). Note that rock, on average, istwo to three times denser than water. When the fluidpressure in pore water exceeds hydrostatic pressure,we say that the fluid is overpressured.

How does pore pressure affect the tensile failurestrength of rock? The pore pressure is an outward pushthat opposes inward compression from the rock, so thefluid supports part of the applied load. If pore pressureexceeds the least compressive stress (σ3) in the rock,tensile stresses at the tips of cracks oriented perpen-dicularly to the σ3 direction become sufficient for thecrack to propagate. In other words, pore pressure in arock can cause tensile cracks to propagate, even ifnone of the remote stresses are tensile, because porepressure can induce a crack-tip tensile stress thatexceeds the magnitude of σ3. This process is calledhydraulic fracturing. On a Mohr diagram, it can berepresented by movement of Mohr’s circle to the left(Figure 6.27). Note that rocks do not have to be over-pressured in order for natural hydraulic fracturing tooccur, but Pf must equal or exceed the magnitude of σ3.

Another effect of fluids comes from the chemicalreaction of the fluids with the minerals comprising arock. Reaction with fluids may lower the tensile stressneeded to cause a crack to propagate, even if the pore-fluid pressure is low. Water, for example, reacts with

quartz to bring about substitution of OH molecules forO atoms in quartz lattice at a crack tip (Figure 6.28).Since the bond between adjacent OH groups is not asstrong as the bond between oxygen atoms, it breaksmore easily, so it takes less remote tensile stress tocause the crack to propagate. This phenomenon iscalled subcritical crack growth, because crack prop-agation occurs at stresses less than the critical stressnecessary to cause a crack to propagate in dry rock.

1356 . 9 E F F E C T O F E N V I R O N M E N T A L F A C T O R S I N F A I L U R E

100 200 300 40016

14

12

10

8

6

4

2

Pressure (MPa)

Dep

th (

km)

0.1 0.2 0.3 0.4

Pressure (GPa)

Lithostatic gradientPfluid ≈Prock

Pfluid ≈1/3 Prock

Hydrostaticgradient

F I G U R E 6 . 2 6 Graph of lithostatic versus hydrostaticpressure as a function of depth in the Earth’s crust.

σs

σn

σt = (σ3 – Pf )(σ1 – Pf )

(σmean – Pf )

σ3

σ1σmean

σd = constant

F I G U R E 6 . 2 7 A Mohr diagram showing how an increase in pore pressure moves the Mohr circle toward the origin. Theincrease in pore pressure decreases the mean stress (σmean), but does not change the magnitude of differential stress (σ1 – σ3). In other words, the diameter of the Mohr circleremains constant, but its center moves to the left.

Si

O

Si

Si

Si

OH

OH

(a) (b)

F I G U R E 6 . 2 8 Sketch of the atoms at the tip of a crack thatillustrates the principle of hydrolytic weakening. (a) Si atomsare bonded to an O atom at the tip of a dry crack. (b) At the tipof a wet crack, Si atoms are bonded to two OH molecules thatreplace the O (same charge). The Si—O—Si bonding is strongerthan the Si—OH to OH—Si bonding and the Si—O—Si bondlength is less than that of Si—OH—OH—Si.

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6.9.2 Effect of Dimensions on Tensile Strength

Rock tensile strength is not independent of scale, in thatlarger rock samples are inherently weaker than smallerrock samples. Why? Because larger samples are morelikely to contain appropriately oriented and larger Grif-fith cracks that will begin to propagate when a stress isapplied, thereby nucleating the throughgoing cracksthat result in failure of the whole sample. Equation 6.2emphasizes this point, because the tensile stress at fail-ure is inversely proportional to the crack half-length.You can imagine that if a sample is so small that it con-sists only of a piece of perfect crystal lattice, it will bevery strong indeed. In fact, the reason that turbineblades in modern jet engines are so strong is that theyare grown as relatively flawless single crystals.

6.9.3 Effect of Pore Pressure on ShearFailure and Frictional Sliding

We can observe the effects of pore pressure on shearfracturing by running a confined compression experi-ment in which we pump fluid into the sample througha hole in one of the pistons, thereby creating a fluidpressure, Pf, in pores of the sample (Figure 6.10). Thefluid creating the confining pressure acting on the sam-ple is different from and is not connected to the fluidinside the sample. The magnitude of Pf decreases theconfining pressure (σ3) and σ1 by the same amount. So if the pore pressure increases in the sample, themean stress decreases but the differential stressremains the same. This effect can be represented by theCoulomb failure criterion equation; Pf decreases themagnitude of σn on the right side of the equation

σs = C + µ(σn – Pf) Eq. 6.8

The term (σn – Pf) is commonly labeled σn*, and iscalled the effective stress.

From a Mohr diagram, we can easily see the effectof increasing the Pf in this experiment. When Pf isincreased, the whole Mohr circle moves to the left butits diameter remains unchanged (Figure 6.27), andwhen the circle touches the failure envelope, shear fail-ure occurs, even if the relative values of σ1 and σ3 areunchanged. In other words, a differential stress that isinsufficient to break a dry rock, may break a wet rock,if the fluid in the wet rock is under sufficient pressure.Thus, an increase in pore pressure effectively weakensa rock. In the case of forming a shear fracture in intact

rock, pore pressure plays a role by pushing open micro-cracks, which coalesce to form a rupture at smallerremote stresses.

Similarly, an increase in pore pressure decreases theshear stress necessary to initiate frictional sliding on apreexisting fracture, for the pore pressure effectivelydecreases the normal stress across the fracture surface.Thus, as we discuss further in Chapter 8, fluids play animportant role in controlling the conditions underwhich faulting occurs.

6.9.4 Effect of Intermediate PrincipalStress on Shear Rupture

Fractures form parallel to σ2, and so the value of σ2

does not affect values of normal stress (σn) and shearstress (σs) across potential shear rupture planes. There-fore, in this chapter we have assumed that the value ofσ2 does not have a major effect on the shear failurestrength of rock, and we considered failure criteriaonly in terms of σ1 and σ3. In reality, however, σ2 doeshave a relatively small effect on rock strength. Specif-ically, rock is stronger in confined compression whenthe magnitude of σ2 is closer to the magnitude of σ1,than when the magnitude of σ2 is closer to the magni-tude of σ3. An increase in σ2 has the same effect as anincrease in confining pressure.

6 .10 CLOSING R E M A R K S

Why study brittle deformation? For starters, sincerocks deform in a brittle manner under the range ofpressures and temperatures found at or near theEarth’s surface (down to 10–15 km), fractures pervaderocks of the upper crust. In fact every rock outcropthat you will ever see contains fractures at some scale(Figure 6.29). Because they are so widespread, frac-tures play a major role in determining the permeabil-ity and strength of rock, and the resistance of rock toerosion. Therefore, fractures affect the velocity anddirection of toxic waste transport, the location of anore deposit, the durability of a foundation, the stabil-ity of a slope, the suitability of a reservoir, the safetyof a mine shaft, the form of a landscape, and so on.Moreover, fracturing is the underlying cause of earth-quakes, and contributes to the evolution of regionaltectonic features. This chapter has provided an intro-duction to the complex and rapidly evolving subject ofbrittle deformation and fracture mechanics. We have

136 B R I T T L E D E F O R M A T I O N

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tried to describe what fractures are, how they form,and under what conditions. In the next two chapters,we apply this information to developing an under-standing of the major types of brittle structures: joints,veins, and faults.

A DDITION A L R E A DING

Atkinson, B. K., ed., 1987. Fracture mechanics ofrock. London: Academic Press.

Bowden, F. P., and Tabor, D., 1950. The friction andlubrication of solids. Clarendon Press: Oxford.

Brace, W. F., and Bombolakis, E. G., 1963. A note onbrittle crack growth in compression. Journal ofGeophysical Research, 68, 3709–3713.

Brace, W. F., and Byerlee, J. D., 1966. Stick-slip as amechanism for earthquakes. Science, 153, 990–992.

Engelder, T., 1993. Stress regimes in the lithosphere.Princeton University Press.

Hancock, P. L., 1985. Brittle microtectonics: princi-ples and practice. Journal of Structural Geology, 7,437–457.

Jaeger, J. C., and Cook, N. G. W., 1979. Fundamentalsof rock mechanics (3rd ed.). Chapman and Hall:London.

Lockner, D. A., 1991. A multiple-crack model of brit-tle fracture. Journal of Geophysical Research, 96,19,623–19,642.

Paterson, M. S., 1978. Experimental rock deforma-tion—the brittle field. Springer-Verlag: New York.

Pollard, D. D., and Aydin, A., 1988. Progress in under-standing jointing over the past century. GeologicalSociety of America Bulletin, 100, 1181–1204.

Price, N. J., 1966. Fault and joint development in brit-tle and semi-brittle rock. Pergamon Press: London.

Scholz, C. H., 2002. The mechanics of earthquakesand faulting (2nd edition). Cambridge UniversityPress: Cambridge.

Scholz, C. H., 1998. Earthquakes and friction laws.Nature, 391, 37–42.

Secor, D. T., 1965. Role of fluid pressure in jointing.American Journal of Science, 263, 633–646.

137A D D I T I O N A L R E A D I N G

F I G U R E 6 . 2 9 Aerial view of the San Andreas Fault, north ofLanders, California.

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