Physical Processes in Earth and Environmental Sciences Phần 4 pdf

To understand the values ofthe axis of the strain ellipse imagine the homogeneousdeformation of a circle having a radius of magnitude 1, which will be the value of l0Fig.. 3.84 Pure shea

shows the object deformed by homogeneous flattening(Fig 3.83b), the sides of the square remain perpendicular

to each other, but notice that both diagonals of the square

(a and b in Fig 3.83) initially at 90 have experienced

deformation by shear strain moving to the positions a b

shear, the original perpendicular situation of both lines has

to be reconstructed and then the angle  can be measured.

In this case the line a has suffered a negative shear with respect to b The line a perpendicular to b

ted and the angle between a and a

shear The shear strain is calculated by the tangent of theangle  The same procedure can be followed to calculate

the strain angle between both lines plotting a line normal

to a angle between b and a

example (Fig 3.83c) the square has been deformed by

simple shear into a rhomboid, both the sides and the onals of the square have experienced shear strain

diag-3.14.5 Pure shear and simple shearPure shear and simple shear are examples of homogeneousstrain where a distortion is produced while maintainingthe original area (2D) or volume (3D) of the object Bothtypes of strain give parallelograms from original cubes

Pure shear or homogeneous flattening is a distortion which

converts an original reference square object into a gle when pressed from two opposite sides The shorteningproduced is compensated by a perpendicular lengthening(Fig 3.84a; see also Figs 3.81 and 3.83b) Any line in theobject orientated in the flattening direction or normal to itdoes not suffer angular shear strain, whereas any pair ofperpendicular lines in the object inclined respect to thesedirections suffer shear strain (like the diagonals or the rec-tangle in Fig 3.83b or the two normal to each other radii

rectan-in the circle rectan-in Fig 3.84a)

Simple shear is another kind of distortion that

trans-forms the initial shape of a square object into a rhomboid,

so that all the displacement vectors are parallel to eachother and also to two of the mutually parallel sides of therhomboid All vectors will be pointing in one direction,

known as shear direction All discrete surfaces which slide

with respect to each other in the shear direction are named

shear planes, as will happen in a deck of cards lying on a

table when the upper card is pushed with the hand(Fig 3.84c) The two sides of the rhomboid normal to thedisplacement vectors will suffer a rotation defining anangular shear  and will also suffer extension, whereas the

sides parallel to the shear planes will not rotate and willremain unaltered in length as the cards do when we dis-place them parallel to the table Note the difference withrespect to the rectangle formed by pure shear whose sides

do not suffer shear strain Note also that any circle sented inside the square is transformed into an ellipse inboth simple and pure shear To measure strain, fossils orother objects of regular shape and size can be used If theoriginal proportions and lengths of different parts in thebody of a particular species are known (Fig 3.85a), it ispossible to determine linear strain for the rocks in whichthey are contained Figure 3.85 shows an example ofhomogeneous deformation in trilobites (fossile arthropods)deformed by simple shear (Fig 3.85b) and pure shear(Fig 3.85c) Note how two originally perpendicular lines

repre-in the specimen, repre-in this case the cephalon (head) and thebilateral symmetry axis of the body, can be used to meas-ure the shear angle and to calculate shear strain

Fig 3.83 Examples of measuring the angular shear in a square object (a) deformed into a rectangle by pure shear (b) and a rhomboid (c)

b

a (3)

3.14.6 The strain ellipse and ellipsoid

We have seen earlier (Figs 3.80 and 3.84) that when

homo-geneous deformation occurs any circle is transformed into

a perfectly regular ellipse This ellipse describes the change

in length for any direction in the object after strain; it is

called the strain ellipse For instance, the major axis of the

ellipse, which is named S1(or e1), is the direction of

maxi-mum lengthening and so the circle is mostly enlarged in

this direction Any other lines having different positions on

the strained objects which are parallel to the major axis of

the ellipse suffer the maximum stretch or extension

Similarly the minor axis of the ellipse, which is known as S3

(or e3) is the direction where the lines have been shortened

most, and so the values of the extension e and the stretch S

are minimum The axis of the strain ellipse S and S are

known as the principal axis of the strain ellipse and are

mutually perpendicular The strain ellipse records not onlythe directions of maximum and minimum stretch or exten-sion but also the magnitudes and proportions of bothparameters in any direction To understand the values ofthe axis of the strain ellipse imagine the homogeneousdeformation of a circle having a radius of magnitude 1,

which will be the value of l0(Fig 3.86a) Now, if we apply

the simple equation of the stretch S (Equation 2; Fig 3.81) whereas for a given direction, the stretch e is the difference

in length between the radius of the ellipse and the initialundeformed circle of radius 1 it is easy to see that the major

axis of the ellipse will have the value of S1and the minor

axis the value of S3 An important property of the strainaxes is that they are mutually perpendicular lines whichwere also perpendicular before strain Thus the directions

Flattening direction (a)

(c)

(b)

Fig 3.84 Pure shear (a) and simple shear (b) are two examples of homogeneous strain Both consist of distortions (no area or volume changes are produced; (c) Simple shear has been classically compared to the shearing of a new card deck whose cards slide with respect to each other when pushed (or sheared) by hand in one direction.

ψ

Fig 3.85 Homogeneous deformation in fossil trilobites: (a) nondeformed specimen; (b) deformed by simple shear Note how two originally perpendicular lines such as the cephalon base and the bilateral symmetry axis can be used to measure the shear angle and calculate shear strain (c) deformed by pure shear If the original size and proportions of three species is known, linear strain can be established.

of maximum and minimum extension or stretch spond to directions that do not experience (at that point)shear strain (note the analogy with the stress ellipse inwhich the principal stress axis are directions in which no

corre-shear stress is produced) Shear strain can be determined inthe ellipse by two originally perpendicular lines, radii R ofthe circle and the line tangent to a radius at the perimeter(Fig 3.87) In (a), before deformation, the tangent line to

the circle is perpendicular to the radius R In (b), after

deformation, the lines are no longer normal to each otherand so an angular shear  can be measured and the shear

strain calculated, as explained earlier

In strain analysis two different kinds of ellipse can be

defined, (i) the instantaneous strain ellipse which defines

the homogeneous strain state of an object in a small

incre-ment of deformation and (ii) the finite strain ellipse which

represents the final deformation state or the sum of all thephases and increments of instantaneous deformations thatthe object has gone through In 3D a regular ellipsoid willdevelop with three principal axes of the strain ellipsoid,

namely S1, S2, and S3, being S1S2S3.Now that we have introduced the concept of the strainellipse we can return to the previous examples of homoge-neous deformation and have a look at the behavior of thestrain axes In the example of Fig 3.84 the familiar square

is depicted again showing an inner circle (Fig 3.88) Twomutually perpendicular radius of the circle have beenmarked as decoration Note that a pure shear strain hasbeen produced in four different steps The circlehas become an ellipse that, as the radius of the circle has

a value of 1, will represent the strain ellipsoid, with two

principal axes S1and S3 Note that when a pure shear isproduced the orientation of the principal strain axisremains the same through all steps in deformation and so

it is called coaxial strain (Fig 3.88) This means that the

directions of maximum and minimum extension are served with successive stages of flattening A very differentsituation happens when simple shear occurs (Fig 3.89):the axes of the strain ellipsoid rotate in the shear direction

pre-Fig 3.86 The stress ellipse in 2D strain analysis reflects the state of strain of an object and represents the homogeneous deformation of

a circle of radius 1 transformed into an ellipsoid As I0is 1,

S1 I1/1 I1 which represents the stretch S of the long axis.

Similarly S3 I1 giving the stretch S of the short axis.

Fig 3.87 Shear strain in the strain ellipse In (a), before deformation, the tangent line to the circle is perpendicular to the radius R In (b), after

deformation, both lines are not normal to each other, the angular shear  can be obtained and the shear strain calculated by tracing a normal line

to the tangent to the circle at the point where R’ intercepts the circle, and measuring the angle  The shear strain can be calculated as y  tan .

After deformation, the circle has suffered strain and oped into a perfect ellipse by homogeneous flattening

devel-(Fig 3.90b) The original radius R of the circle, with length l0, has been elongated and will correspond to the

lengths, the extension, e (Equation 1; Fig 3.81) or the stretch, S (Equation 2; Fig 3.81), can be easily calculated.

The reciprocal quadratic elongation can be directlyobtained as 0/l1)2 The angular deformation can bemeasured by plotting the tangent to the ellipse at the point

p, where the radius intercepts the ellipse perimeter, then

plotting the normal to the tangent, and measuring the

angle with respect to the radius R

The Mohr circle strain diagram is a useful tool to ically represent and calculate strain parameters, following asimilar procedure that was used to calculate stress compo-nents In this case the ratio between the shear strain and thequadratic elongation (/) is represented on the vertical

graph-axis and the reciprocal quadratic elongation (horizontal axis (Fig 3.90c) The / ratio is an index of the relative importance of the angular deformation versus

the linear elongation When the ratio is very small, changes

in length dominate, in fact when the ratio equals zero,there is no shear strain, which coincides with the directions

of the principal strain axis In homogeneous strain of pure

and so the strain is noncoaxial The orientation of the axes

is not maintained, which means that the directions of

max-imum and minmax-imum extension rotate progressively with

time

3.14.7 The fundamental strain equations and

the Mohr circles for strain

For any strained body the shear strain and the stretch can

be calculated for any line forming an angle  with respect

to the principal strain axis S1if the orientation and values

of S1and S3 are known As in the case of stress analysis

the approach can be taken in 2D or 3D Although it is

important to remember that the physical meanings of

strain and stress are completely different, the equations

have the same mathematical form (Fig 3.90) and can be

derived using a similar approach The fundamental strain

equations allow the calculation of changes in length of

lines, defined by means of the reciprocal quadratic

elonga-tion(

respect to the direction of maximum stretch S1 To

illus-trate the use and significance of the Mohr circles for strain,

an original circle of radius R can be used (as in Fig 3.87).

Fig 3.88 Pure shear is considered to be a coaxial strain since the orientation of the axes of the strain ellipse S1and S3remain with the same orientation through progressively more deformed situations.

distortion, where there is no change in volume or area,there are two directions that suffer no finite stretch, wherethe value of   1 Finally two directions of maximum shear

strain are present, corresponding to the lines forming anangle   45 with respect to S1 The Mohr circle for strainhas an obvious relation to the fundamental strain equations(Equations 4 and 5; Fig 3.90a) as shown in Fig 3.90d

To plot the circle in the coordinate axes, the reciprocalvalues 1and 3of 1and 3are first calculated and rep-resented along the horizontal axis The circle will have adiameter 3 1 and the center will have coordinates( 1 3)/2, 0 Note that as the expressions on the x-axis

are the reciprocal quadratic elongations, the maximum

value, at the right end of the circle, corresponds to 3andthe minimum, at the left end, to 1 Once the circle isplotted, it is possible to calculate the values of

(Fig 3.90d) for any line forming an angle  respect to the direction of the major principal strain axis S1 The line isplotted from the center of the circle at the angle 2 sub-

tended from 1 into the upper half of the circle if theangle is positive or into the lower half if it is negative Thecoordinates of the point of intersection between the line andthe circle have the values

 and then that of S can be calculated Knowing  it is

also possible to calculate  and finally the angle of shear

strain .

Fig 3.90 (a) The fundamental strain equations The Mohr circles for strain display graphically the relations between the / ratio and the reciprocal quadratic elongation

circle into an ellipse; (c) the Mohr circle strain diagram; and (d) Relation between the Mohr circle and the fundamental strain equations.

The fundamental strain equations Considering the quadratic elongation l = (1 + e)2 = S2 (1)

To define the strain equations, the reverse of the quadratic

elongation, or reciprocal quadratic elongation ( l’) is used:

l = 1/l

g/l g/l

(b)

(d)

(c) (a)

experiments: the study of strain–stress relations or how therocks or other materials respond to stress under certain con-

ditions is the concern of rheology Different kinds of

experi-ments are possible, generally undertaken on centimeter-scale

cylindrical rock samples Both tensional (the sample is

gen-erally pulled along the long axis) and compressive (sample

is pushed down the long axis) stresses can be applied, both

in laterally confined (axial or triaxial tests) or unconfined

conditions (uniaxial texts) Experiments involving the

application of a constant load to a rock sample and

observ-ing changes in strain with time are called creep tests.

Experimental results are analyzed graphically by plotting

stress, (), against strain, (), or strain rate (d/dt), the

latter obtained by dividing the strain by time (Fig 3.91)

Simple mathematical models can be developed for different

regimes of rheological behavior Stress is usually

repre-sented as the differential stress (1 3) Other important

variables are lithology, temperature, confining pressure,

and the presence of fluids in the interstitial pores

causing pore fluid pressures There are three different pure

rheological behavioral regimes: elastic, plastic, and viscous

(Fig 3.91) Elastic and plastic are characteristic of solids

whereas viscous behavior is characteristic of fluids Solids

under certain conditions, for example, under the effect of

permanent stresses, can behave in a viscous way Elastic,

plastic, and viscous are end members of a more complex

suite of behaviors Several combinations are possible, such

as visco-elastic, elastic–plastic, and so on.

3.15.2 Elastic model

Elastic deformation is characterized by a linear relationship

in stress–strain space This means that the relation betweenthe applied stress and the strain produced is proportional(Fig 3.91a) An instantaneous applied stress is followedinstantly by a certain level of strain The larger the stressthe larger the strain, up to a point at which the rock can bedistorted no further and it breaks This limit is called theelastic boundary and represents the maximum stress thatthe rock can suffer before fracturing If the stress isreleased before reaching the elastic limit such that no frac-tures are produced, elastic deformation disappears Inother words, elastic strained bodies recover their originalshape when forces are no longer applied The classical ana-log model is a spring (Fig 3.92a) The spring at reposerepresents the nondeformed elastic object When a load is

Fig 3.91 Strain/stress diagrams for different rheological behaviors

(a) Elastic solids show linear relations The slope of the straight line

is the Young’s modulus; (b) viscous behavior is characteristic of

flu-ids Fluids deform continuously at a constant rate for a certain stress

value The slope of the line is the viscosity (); (c) plastics will not

deform under a critical stress value or yield stress ( ).

a load by a flat surface with an initial resistance to slide.

Hea vy

added to the spring in one of the extremes (as adynamometer) or it is pulled by one of the edges, it willstretch by the action of the applied force The bigger theload, or the more the spring is pulled on the extremes, thelonger it becomes by stretching When the spring isreleased or liberated from the load in one of the extremesthe spring returns to the original length.

Elasticity in rocks is defined by several parameters; the

most commonly used being Young’s modulus (E) and the Poisson coefficient ( ) Young’s modulus is a measure of

the resistance to elastic deformation which is reflected in thelinear relation between the stress () and the strain (): E 

/ (Fig 3.93a) This linear relation, which was observed

initially by Hooke in the mid-seventeenth century by ing tensile stresses to a rod and measuring the extension, iscommonly known as Hooke’s Law Considering that allparameters used to measure strain (stretch, extension, orquadratic elongation) are dimensionless, the Young’s modu-lus is measured in stress units (N m2, MPa) and has negativevalues of the order of 104or 105 The reason why thevalues are negative is because the applied stress is extensionaland hence has a negative value, and the strain produced is alengthening, which is conventionally considered positive

apply-Not all rocks follow Hooke’s Law; some deviations occur

but they are small enough so a characteristic value of E can

be defined for most rock types (Fig 3.93a) A high absolute

value for the Young’s modulus means that the level of strainproduced is small for the amount of stress applied, whereaslow values indicate higher deformation levels for a certainamount of stress Rigid solids produce high Young’s modu-lus values as they are very reluctant to change shape or vol-

ume Rigid materials experience brittle deformation when

their mechanical resistance is exceeded by the applied stresslevel at the elastic boundary

When applying uniaxial compressional tests to rock samples, vertical shortening may be accompanied by somehorizontal expansion The Poisson coefficient () shows

the relation between the lateral dilation or barreling of arock sample and the longitudinal shortening produced byloading: thus  lateral/longitudinaland it can be seen thatPoisson’s coefficient is dimensionless (Fig 3.93b) Whenstresses are applied, if there is no volume loss, the samplehas to thicken sideways to account for the vertical shorten-ing Typically, the sample should develop a barrel form(nonhomogeneous deformation) or increase its surfacearea as it expands laterally For perfect, incompressible,isotropic, and homogeneous materials which compensatethe shortening by lateral dilation without volume loss, thePoisson’s coefficient is 0.5; although values for naturalmaterials are generally smaller (Fig 3.93b) In very rigidrock bodies, the lateral expansion may be very limited

or not occur at all; in this case there is a volume loss and

E = s/e (Hooke’s Law)

E a > E b

Marble Limestone Granite Shale Quartzite Diorite

–4.8 –5.3 –5.6 –6.8 –7.9 –8.4

Rock type E (10 4 MPa)

Schist, biotite Shale, calcareous Diorite Granite Aplite Siltstone Dolerite

0.01 0.02 0.05 0.11 0.20 0.25 0.28 Rock type n

Final state Original

Uniaxial compression

n = ed/ec

c

d

ec ed

Fig 3.93 Elastic parameters (a) The Young’s modulus describes the slope of the stress/strain straight line, being a measure of the rock

resistance to elastic deformation Line a has a higher value of Young’s modulus (E a ) being more rigid than line b (Young modulus E b) (i.e it is less strained for the same stress values); (b) Poisson’s coefficient relates the proportion in which the rock deforms laterally when it is compressed vertically Comparing the original and final lengths before and after deformation strain  can be calculated and the Poisson’s ratio established.

elastic stresses have to be accumulated somehow Rock

samples will fragment at the elastic limit after experiencing

very little lateral strain when the Poisson’s ratio is very

small (close to zero) The reciprocal to the Poisson’s

coef-ficient is called the Poisson’s number m  1/ This

num-ber is also constant for any material, and so the relation

between the longitudinal and lateral strains have a linear

relation Nonetheless, as in the case of Young’s modulus

there may be slight variations in the linear trend of

Poisson’s coefficient (Fig 3.94) It is important to

remem-ber that experiments to establish elasticity relationships

under unconfined uniaxial stress conditions allow the rock

samples to expand laterally In the crust, any cube of rock

that we can define is not only subject to a vertical load due

to gravity but also due to adjacent cubes of rock in every

direction and is not free to expand laterally; in such cases

complex stress/strain relations can develop

Other elastic parameters are the rigidity modulus (G) and the bulk modulus (K) The rigidity modulus or shear

modulus is the ratio between the shear stress ( ) and the

shear strain () in a cube of isotropic material subjected to

simple shear: G  / (Fig 3.95) G is another measure of

the resistance to deformation by shear stress, in a way

equivalent to the viscosity in fluids The bulk modulus (K)

relates the change in hydrostatic pressure (P) in a block of

isotropic material and the change in volume (V) that it

experiences consequently: K  dP/dV The reverse to the

bulk modulus is the compressibility (1/K).

Viscous deformation occurs in fluids (Sections 3.9 and 3.10);

fluids have no shear strength and will flow when shear

stresses, even infinitesimal, are applied One of the chiefdifferences between an elastic solid and a viscous fluid isthat when a shear stress is applied to a piece of elastic mate-rial it causes an increment of strain proportional to thestress, if the same level of stress is maintained no furtherdeformation is achieved (Fig 3.96a) In fluids when ashear stress () is applied the material suffers certain

amount of strain but the fluid keeps deforming with timeeven when the stress is maintained with the same value(Fig 3.96b) In this case a level of stress gives way to astrain rate (d/dt), not a simple increment of strain as in

the elastic solids Higher stress values will give way tohigher strain rates, so the fluid will deform at more speed

As in elastic materials there is no initial resistance todeformation even when stresses acting are very small, butthe deformations are permanent in the viscous fluid case(Fig 3.97a,b)

As we have seen earlier (Sections 3.9 and 3.10) the

parame-ter relating stress to strain rate is the coefficient of dynamic cosity or simply viscosity ( ):   /(d/dt), which is

Fig 3.94 Longitudinal and lateral strain experienced by a rock

sample when an uniaxial compression is applied The relation

between both strains may not be linear as in this case, and the

Poisson’s ratio is not constant, it varies slightly for different stress

values.

Shortening Dilation

Longitudinal strain

Lateral strain

a strain rate That is why strain rate is used in rheological plots instead of strain as in solids The fluid body will remain deformed permanently once the force is removed.

Solid elastic body

Fluid viscous body

measured in Pascals Fluids that show a linear relation betweenthe stress and the strain rate, and so have a constant viscosity,are called Newtonian Fluids, whose viscosity changes with thelevel of stress are called non-Newtonian (Fig 3.98) Viscousbehavior is generally compared to a piston or a dashpot con-taining some hydraulic fluid (Fig 3.92b) The fluid is pressed

by the piston (creating a stress or loading) and the fluid moves

up and down a cylinder, producing permanent deformation;

the quicker the piston moves the more rapid the fluid deforms

or flows up and down The viscosity can be described as theresistance of the fluid to movement High viscosity fluids aremore difficult to displace by the piston up and down the cylin-der For non-Newtonian fluids (Fig 3.98) as the piston ispushed more and more strongly in equal increments of addedstress the rate of movement or strain rate rapidly increases in anon-linear fashion

3.15.4 Plastic model

Plastic deformation is characteristic of materials which do

not deform immediately when a stress is applied A certain

Fig 3.97 Strain of different materials with time (stages T 1 to T 5)

applying increasing levels of stress: (a) Elastic solids show discrete strain increments with increasing stress levels (linear relation);

strain is reversible once the stress is removed (T 5); (b) Viscous

fluids flow faster (higher strain rates) with increasing stress; the deformation is permanent once the stress is released; (c) Plastic solids will not deform until a critical threshold or yield stress is

overpassed (at T4 in this case) Deformation is nonreversible (at T 5).

Stress released Final state Increassing stress applied

level of stress is required to start deformation, as the rial has an initial resistance to deformation This stressvalue is called yield stress  y(Fig 3.91c) After the yieldstress is reached the body of material will be deformed abig deal instantaneously, and the deformation will be per-manent and without a loss of internal coherence So, twoimportant differences with respect to elastic behavior arethat the strain is not directly proportional to the stress, asthere is an initial resistance, and that the strain is notreversible as in elastic behavior (Fig 3.97) An analogicalmodel for plastic deformation is that of a heavy load rest-ing on the floor (Fig 3.92c) If the force used to slide theload along a surface is not big enough, the load will notbudge This would depend on the frictional resistanceexerted by the surface Once the frictional resistance, and

mate-so the yield stress, is exceeded, the load will slide easily andthe movement can be maintained indefinitely as long asthe force is sustained at the same level over the criticalthreshold or yield stress The load will not go back on itsown! So the deformation is not reversible (Fig 3.92c)

3.15.5 Combined rheological modelsElastic, viscous, and plastic models correspond to simplemathematical relationships which apply to materials under

Fig 3.98 Viscosity is the resistance of a fluid to deform or flow: it is the slope of the curve stress/strain rate Fluids showing linear relations (constant viscosity) are Newtonian Fluids with nonlinear relation (

0.8  10 –3 0.08

10 2

10 8

10 16

10 22 Fluid h (Pa)

(h constant)

(h variable)

ideal conditions; they are considered homogeneous (the

rock has the same composition in all its volume) and

isotropic (the rock has the same physical properties in all

directions) Rocks are rarely completely homogeneous or

isotropic due to their granular/crystalline nature and

because of the presence of defects and irregularities in the

crystalline structure, as well as layers, foliations, fractures,

and so on Nevertheless, although such aberrations would

be important in small samples, on a large scale, when large

volumes are being considered, rocks can be sometimes

regarded as homogeneous Usually, however, natural

rhe-ological behavior corresponds to a combination of two or

even three different simple models, such as elastic–plastic,

visco-elastic, visco-plastic, or elastic–visco-plastic Also

materials can respond to stress differently depending on

the time of application (as in instantaneous loads versus

long-term loads)

A well-known example of a combined rheological model

is the elastic–plastic (Prandtl material) (Fig 3.99); it shows

an initial elastic field of behavior where the strain is erable, but once a yield stress ( y) value is reached thematerial behaves in a plastic way The analogical model is aspring (elastic) attached to a heavy load (plastic) movingover a rough surface (Fig 3.99b) The spring will deforminstantly whereas the load remains in place until the yieldstress is reached, then the load will move; after releasingthe force, the spring will recover the original shape but thelongitudinal translation is not recoverable Elastic–plasticmaterials thus recover part of the strain (initial elastic) butpartly remain under permanent strain (plastic) Rememberthat in a pure elastic material, permanent strain does notoccur and after the elastic limit is reached the rock breaks

recov-(b, Fig 3.99c; line I) whereas in a Prandtl material there is

a nonreversible strain (c, Fig 3.99c, line II) Once the plastic

limit is reached, the material can then break but only after

suffering some permanent barreling (d, Fig 3.99c, line II).

Visco-elastic models correspond to solids (called

Maxwell materials) which have no initial resistance to

Fig 3.99 (a) Elastic–plastic material shows an initial elastic field characterized by recoverable deformation strain followed by a plastic field in which the strain is permanent The boundary between both fields is the elastic limit located at the yield stress value (y); (b) The analog model

is a load attached to a spring; (c) Part of the strain is recovered (the length of the spring) and part is not (the displacement of the load).

elastic field

plastic field

sy

sy(b)

F

strain as in both elastic and viscous models (Fig 3.100a).

Part of the strain will recover following an elastic behaviorbut part will remain permanently deformed Maxwellsolids behave elastically when the stresses are short lived,like a ball of silicon putty that bounces elastically on thefloor when thrown with some force; but will accumulatepermanent deformations at a constant rate if the stress orload (like the proper weight of the material) is applied for

a longer time Visco-elastic models can be represented by

a spring attached longitudinally to a dashpot (Fig 3.100b)

The spring will provide the recoverable strain whereas thedashpot will supply the nonrecoverable strain when apulling force is applied parallel to the system

Visco-plastic materials (called Bingham plastics) only

behave like viscous fluids after reaching a yield stress, thestrain rate subsequently being proportional to the stress;

initially the material does not respond to the applied stress

as for plastic solids (Fig 3.100c) The analogy will be inthis case a dashpot attached in parallel to a load sliding on

a surface with an initial resistance to movement; once theload is in motion it behaves in viscous fashion

3.15.6 Ductile and brittle deformationFrom the different rheological models discussed above itcan be concluded that there are several kinds of deforma-tion First, strain produced when loads are applied can bereversible; this is characteristic of elastic behavior as in the

elastic curves or elastic–plastic materials (a, Fig 3.99c)

when small stress increments are applied Deformations canalso be nonreversible, which means that once the load isreleased the rock will be deformed permanently

Deformation is said to be ductile when rocks or other solids

are strained permanently without fracturing, which pens in plastic or elastic–plastic materials once the elasticlimit or yield strength (stress value which separates the elas-

hap-tic and plashap-tic fields) is reached (as c in Fig 3.99c).

Fig 3.100 (a) Visco-elastic or Maxwell materials have a recoverable strain part belonging to the elastic component and a permanent strain due to the viscous behavior like a spring attached to a dashpot (b); (c) visco-plastic or Bingham materials behave in a viscous way but after reaching a critical stress value or yield stress (y) like a dashpot linked to a load moving on a rough surface (d).

Nonetheless, ductile is a general, descriptive term that does

not involve a specific rheological behavior or strain

mecha-nism It is not a synonymous term for plastic, which is a very

well-defined and particular rheological behavior Strains

pro-duced during plastic deformations are larger in magnitude

than those produced in the elastic field and are generally

formed by dislocations of the crystalline lattices and/or

dif-fusive processes Ductile deformations are also called ductile

flows as the material deforms or flows in a solid state (as a

gla-cier sliding downslope does, Section 6.7.5) Examples of

ductile deformation in rocks are the formation of folds and

salt diapirs Rocks have a limited ability to change their shape

or volume, which also depends on such external parameters

as the temperature, confining pressure, and so on

Brittle deformation happens when the internal strength

of rocks is exceeded by stresses; they bust, so internal

cohesion is lost in well-defined surfaces or fractures Brittle

deformation can occur after the elastic limit is exceeded

not only in pure elastic bodies (b, Fig 3.99c) but also

when the stresses reach the plastic limit after some ductile

deformation has taken place Such samples will be

perma-nently deformed and also fractured (d, Fig 3.99c).

3.15.7 Parameters controlling rock deformation

Lithology (rock type) is a variable which may cause diverse

modes of stress–strain behavior Different rocks or

sub-stances may need different rheological models with which

to describe their deformation Competency is a qualitative

term used to describe rocks in terms of their inner strength

or capacity for deformation Rocks which deform easily

and generally in a ductile way are described as incompetent,

such as salts, shale, mudstone, or marble Strong or

compe-tent rocks are those which are more difficult to deform,

such as quartzite, granite, quartz sandstones, or fresh

basalts Competent rocks are stiffer and deform generally

in a brittle way Nevertheless, competency depends not

only on lithology but also on temperature, confining

pres-sure, pore prespres-sure, strain rate, time of application of the

stress, etc To compare competencies of different kinds of

rocks, experiments must take place at equal temperatures

and confining pressures

Temperature has particularly important effects in

rheo-logical behavior (Fig 3.101) Comparing several

experi-ments on samples of the same lithology under the same

conditions of confining pressure, it is possible to compare

stress–strain relations at different temperatures At higher

temperatures, rocks behave in a more ductile way, so

com-petence is reduced and fractures are more difficult to

pro-duce For rocks that are elastic at low temperatures a

plastic field can develop In elastic–plastic materials, perature lowers the elastic limit, which is thus reached atlower stress levels Rocks may also behave in a viscous way

tem-at high tempertem-atures if the applied stresses are long lasting

Confining pressure (lithostatic or hydrostatic pressure

acting on all sides of a rock volume) can be simulated inlaboratory experiments by introducing some fluid thatexerts a certain amount of pressure in the sample (triaxialtests) in addition to that provided by the compressive load,and by isolating the sample in a constraining metal jacket

to discriminate and separate the effects of the pore sure in the rock Experiments carried out on samples of thesame lithology and at the same temperature show thathigher confining pressures increase the yield strength in arock, and also the plastic field, so fracturing, if it happens,occurs after more intense straining (Fig 3.102) Thismeans that rocks became more ductile at higher levels ofconfining pressure

pres-When there is fluid trapped in the rock pores, it exerts anadditional hydrostatic pressure which has the effect ofcounteracting the confining pressure by the same value ofthe fluid pressure in the pores The state of stress is lowered

and an effective stress tensor can be defined by subtracting

the values of the fluid stresses from those of the solid normal stresses (Fig 3.103) The Mohr circle moves

toward lower values by an amount equal to the pore pressure (pf) sustained by the fluid Thus, when fluids are present inthe pores the effect is the same as lowering the confining

Fig 3.101 Effect of temperature in the strain–stress diagram for basalts under the same confining pressure (5 kbars).

pressure in the rocks, so that ductility decreases and tures are produced more easily Being hydrostatic in nature,the effectiveness of the normal stresses is lowered but theshear stresses remain unaltered The control of pore pres-sure in the rocks is of key importance in fracture formationand will be discussed in some more detail in Section 4.14.

frac-Other important factors are the time of application of thestresses: the instantaneous or long-term application of acertain level of stress may cause different rheological behav-iors, like the case of the silicon putty discussed earlier Rock

strength decreases when the stresses are applied for longtimes under small differential stresses (creep experiments).Also in relation to time, the rates of loading (velocity ofincreased loading in the experiments) also have importantimplications for the production of strain In a single exper-iment, the rate of strain is generally maintained constantbut the rates of strain can be changed from one experiment

to another When changes in strain are produced rapidly(high loading rates) the rock samples become ductile andbreak at higher stress levels

Fig 3.102 (a) Strain–stress diagram showing several curves corresponding to limestone samples of the same composition at different confining pressures (in MPa); (b) Differences in confining pressure give way to different fracturing or deformation modes Confining pressure from samples (from 0.1 to 35 MPa in the fractured samples and 100 MPa for the ductile flow).

80

130 140

Fig 3.103When there is some pressurized fluid in the rock pores, part of the stress is absorbed The state of stress is lowered and an effective stress tensor can be defined subtracting the values of the normal stresses from those of the fluid The Mohr circle moves toward lower values by an

amount equal to the pore pressure (Pf) sustained by the fluid.

t

sn

E s1 E s3

Applied stress Effective stress

(rock)

P.M Fishbane et al.’s Physics for Scientists and Engineers:

Extended Version (Prentice-Hall, 1993) is again invaluable.

Many good things of oceanographic interest can be found

in the exceptionally clear work of S Pond and G.L Pickard

– Introductory Dynamical Oceanography (Pergamon,

1983), while R McIlveen’s Fundamentals of Weather and

Climate (Stanley Thornes, 1998) is good on the

atmos-pheric side A more advanced text is D.J Furbish’s Fluid

Physics in Geology (Oxford, 1997) G.V Middeton and P.R.

Wilcox’s Mechanics in the Earth and Environmental Sciences

has a broad appeal at intermediate level and is very ough The best introduction to solid stress and strain is in

thor-G.H Davies and S.J Reynolds’s Structural Geology of Rocks and Regions (Wiley, 1996); R.J Twise and E.M Moores’s Structural Geology (1992) and J.G Ramsay and M Huber’s The Techniques of Modern Structural Geology, vol 1: Strain Analysis (Academic Press, 1993) are classics on structural

geology for advanced studies on solid stress W.D Means’s

Stress and Strain (Springer-Verlag, 1976) takes a careful and

rigorous course through the basics of the subject

Further reading

Earth is a busy planet: what are the origins of all thismotion? Generally, we know the answer from Newton’sFirst Law that objects will move uniformly or remain sta-tionary unless some external force is applied The uniformmotion of fluids must therefore involve a balance of forces inwhatever fluid we are dealing with In order to try to predictthe magnitude of the motion we must solve the equations ofmotion that we discussed previously (Section 3.12) Bulkflow (in the continuum sense, ignoring random molecularmovement) involves motion of discrete fluid masses fromplace to place; the masses must therefore transport energy:

mechanical energy as fluid momentum and thermal energy

as fluid heat There will also be energy transfers between thetwo processes, via the principle of the mechanical equivalent

of heat energy and the First Law of Thermodynamics(Section 2.2, conservation of energy) For the moment weshall ignore the transport of heat energy (seeSections 4.18–4.20) since radiation and conduction intro-duce the very molecular-scale motions that we wish toignore for initial simplicity and generality of approach

4.1.1 Very general questions

1 How does fluid flow originate on, above, and within theEarth? For example, atmospheric winds and ocean currentsoriginate somewhere and flow from place to place for certainreasons This raises the question of “start-up,” or the begin-nings of action and reaction

2 If fluid flow occurs from place A to place B, what pens to the fluid that was previously at place A? For exam-ple, the arrival of an air mass must displace the air masspreviously present This introduces the concept of anambient medium within which all flows must occur

hap-3 How does moving fluid interact with stationary or ing ambient fluid? For example, does the flow mix at all

mov-with the ambient medium? If so, at what rate? How doesthe interaction look physically?

4 What is the origin and role of variation in flow velocitywith time (unsteadiness problem)? It is to be expected thataccelerations will be very much greater in the atmospherethan in the oceans and of negligible account in the solidearth (discounting volcanic eruptions and earthquakes).Why is this?

4.1.2 Horizontal pressure gradients and flowStatic pressure at a point in a fluid is equal in all directions(Section 3.5) and equals the local pressure due to theweight of fluid above Notwithstanding the universal truth

of Pascal’s law, we saw in Section 3.5.3 that horizontalgradients in fluid pressure occur in both water and air.These cause flow at all scales when a suitable gradientexists The simplest case to consider is flow from a fluidreservoir from orifices at different levels (Fig 4.1) Herethe flow occurs across the increasingly large pressure gra-dient with depth between hydrostatic reservoir pressureand the adjacent atmosphere

The gradient of pressure in moving water (Fig 4.1) is

termed the hydraulic gradient, and the flow of subsurface

water leads to the principle of artesian flow and the basis ofour understanding of groundwater flow through the oper-ation of Darcy’s law (developed from the Bernoulliapproach in Sections 4.13 and 6.7) The flow of a liquiddown a sloping surface channel is also down the hydraulicgradient

Similar principles inform our understanding of the slowflow of water through the upper part of the Earth’s crust.Here, pressures may also be hydrostatic, despite the fluidheld in rock being present in void space between solid rockparticles and crystals (Fig 4.2); this occurs when the rocks

4 Flow, deformation,

and transport

4.1 The origin of large-scale fluid flow

Geostatic gradient

Hydrostatic gradient

rwater = 1,000 kg m rrock = 2,380 kg m–3

Flow, deformation, and transport 103

Fig 4.1 Flows induced by hydrostatic pressure.

In the hydrostatic condition all liquid levels are equal

p0 = Atmospheric p0 = Atmospheric

There is no change to this principle when the fluid occupies void space that has continuous connection

Hydrostatic gradient

Low atmospheric pressure and water “set-up” on lee-shore

Sloping isobars

High Low

Subsurface flow down horizontal hydrostatic pressure gradient (modified by Coriolis force in 3D)

Fig 4.4 Barotropic flow due to a horizontal gradient in hydrostatic pressure caused and maintained by atmospheric dynamics The spatial

gradients in atmospheric pressure and wind shear may act together or separately In both cases hydrostatic pressures above B are greater than hydrostatic pressures at all equivalent heights above A, by a constant gradient given by the water surface slope.

are porous to the extent that all adjacent pores

communi-cate, as is commonly the case in sands or gravels Severe

lateral and vertical gradients arise when pores are closed by

compaction, as in clayey rock; the hydrostatic condition

now changes to the geostatic condition when pore

pres-sures are greater due to the increased weight of overlying

rock compared to a column of pore water (Fig 4.3)

Interlayering of porous and nonporous rock then leads tohigh local pressure gradients down which subsurface fluidsmay move In passage down an oil or gas exploration well,pressure may jump quickly from a hydrostatic trend toward

lithostatic, causing potentially disastrous consequences forthe drill rig and possible “blowout.” The regionalhydraulic gradient drives the direction of migration ofsubsurface fluids like water and hydrocarbon Pressures in

partially molten rocks of the Earth’s upper crust in crustalmagma chambers (Section 5.1) may also vary betweenhydrostatic and geostatic values, with obvious implicationsfor the forces occurring during volcanic eruptions

Ambient fluid

r1

lockbox fluid r2

Conditions r1 < r2 ∆r = +ve

lockbox liquid r2

Conditions r1 > r2

∆r = –ve

lockbox liquid r2

Conditions r1 < r2

∆r = +ve

lockbox liquid r2

Rising plume (thermal)

Descending plume (open ocean cold convection, ice meltout) Wall jet

descending flow (bottom water production, turbidity currents)

Catabatic wind

Thunderstorm downdraught Sea breeze front Cold front

Wall jet (turbidity flow, thermohaline flow)

Ocean and lakes (unstratified)

Fig 4.5 Buoyancy-driven flows.