1. Ductile Deformation Processes – Lecture 19 – Spring 2016
Structural Geology
Ductile Deformation Processes – Lecture 19 – Spring 2016
Some materials flow without breaking. Ice is an excellent example, since the flow rates of ice are observable on human time frames. Glaciologists have studied ice, and its movement and effects, for many years.

2. Medial Moraines
Observation of medial moraines in many glaciers shows evidence of flow and deformation
Such flow represents permanent, nonrecoverable deformation
It is not elastic
Thus, more than bending or stretching of atomic bonds is involved. Understanding ductile deformation has been an important goal of structural geologists for several decades. It is another area whether application of principles from other scientific fields has paid benefits, and further illustrates why a solid grounding in fields such as physics and chemistry is important in geology.
We have previously seen that strain distributed over the body of a rock, as opposed to distribution over localized areas, is what distinguishes ductile and brittle behavior. We have seen that flow may be linear viscous, or Newtonian, or non-linear viscous (non-Newtonian). We will now see that the scale of observation makes a difference. In structural geology, the scale of observation ranges from the atomic (nanometers) to regions viewed from satellites (hundreds of kilometers). Ductile behavior has traditionally meant uniform flow down to the scale of a hand specimen, or mesoscopic processes. Because a large body of literature has been developed with this concept, we retain it, while realizing that processes that seem linear at hand-specimen scale may not be linear at smaller scales.
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3. Ductile Flow Mechanisms
There are three mechanisms that allow ductile flow in rocks:
Cataclastic flow
Diffusional mass transfer
Crystal plasticity

4. Parameters Affecting Ductile Flow
Ductile flow processes depend on several factors:
Temperature
Stress
Strain rate
Grain size
Rock composition
Fluid content

5. Temperature
Temperature plays a very important role, but the effect of temperature varies greatly from mineral to mineral
It is useful to define the homologous temperature, Th
Th = T/Tm
where T = absolute temperature (in Kelvin) and Tm is the melting point temperature of the material, again in Kelvin

6. Homologous Temperature
Since Th is a ratio of temperatures, it is dimensionless
Using Th, we can approximately define various conditions:
Low-temperature - 0 < Th < 0.3
Medium-temperature 0.3 < Th < 0.7
High-temperature < Th < 1.0

7. Bean Bag Analogy
If the bean bag is pulled through a hole smaller than the individual beans, they will break (brittle fracture)
The broken pieces then slide past each other through the hole (frictional sliding)
Cataclasis is mesoscopic ductile flow.
A strong pillow filled with beans can be distorted to various shapes.
The atomic equivalent is the breaking of bonds between grains (intergranular fracture) and across grains (intragranular fracture). If pressure is increased, the ability of atomic scale fracturing to occur is greatly reduced, since pressure pushes atoms together and increases bond strengths. We might expect, therefore, that cataclastic flow would be limited to low pressure regimes. Indeed, cataclastic flow is observed only in the upper few kilometers of the earths crust.
Figure 9.2 in text

8. Types of Crystal Defects
There are three types of defects:
Point defects
Line defects (dislocations)
Planar defects (stacking defects)
Although stacking defects are of interest to mineralogists, they affect the total bond strength across a plane very little, and thus play almost no role in ductile flow
At greater depths, where cataclasis isn’t possible, temperatures are elevated, as well as pressures. Ductile flow is found to depend on defects in crystals, which weaken the bonding. There are three types of defects:
Although stacking defects are of interest to mineralogists, they affect the total bond strength across a plane very little, and thus play almost no role in ductile flow.

9. Point Defects
Point defects occur because of some irregularity in the crystal structure
An ion may be missing (vacancy) (a in figure)
There are no bonds to the void, so the overall bond strength within the crystal is reduced
Figure 9.4 in text

10. Impurity Ions and Interstitial Defects
Figure 9.4 in text
An ion of the wrong type may be present, creating a substitutional or impurity defect
There will be bonds to the impurity ion, but they are often weaker than bonds to the correct ion would be
b in the drawing represents an impurity ion

11. Effect of Bonding to Interstitial Ions
Figure 9.4 in text
Finally, an ion may occur within the crystal structure where there isn’t supposed to be any atom or ion
This is an interstitial defect
Bonds to the interstitial ion weaken surrounding bonds, and often make the remaining structure weaker

12. Vacancy Migration Vacancies can exchange place with surrounding atoms
Figure 9.4 in text
Vacancies can exchange place with surrounding atoms
This allows the vacancy to migrate
If there are gradients present within the crystals, the atoms or the vacancies may migrate along the gradient, which is the process known as diffusion

13. Dislocations
A linear array of atoms that bounds an area in the crystal that has slipped relative to the rest of the crystal is called a dislocation
There are two types, edge and screw dislocations.
Figure 9.6 in text

14. Edge Dislocation
An edge dislocation results from an extra half-plane of atoms in the crystal structure (NOT lattice, the term used in the book)
The extra half-plane will terminate somewhere within the crystal, and the bonding at the end of the half-plane will be disrupted
Bonds will be much longer than usual, and therefore much weaker
Figure 9.7 in text

15. Screw Dislocations
Figure 9.7 in text
In screw dislocations, the atoms will be displaced in a rotational fashion
Since atoms all along the screw axis are out of position, bonding is disrupted, and therefore weak

16. Burgers Circuit Around Edge Dislocation
Figure illustrates the application of a Burgers circuit around the an edge dislocation
A circuit in an intact crystal structure would be complete, ending where it started
With dislocations, the circuit is incomplete
An arrow connecting the ends of the incomplete circuit is termed a Burgers vector (b). Burger vector magnitudes are typically on the order of a nanometer or so, or ten times the size of an atom.
Figure 9.8a in text

17. Burgers Circuit Around Screw Dislocation
Figure illustrates the application of a Burgers circuit around the a screw dislocation
Figure 9.8b in text

18. Location of Burgers Vectors
Edge dislocations have b z R, where R is the dislocation line
A screw dislocation has b 2 R
If a dislocation involves both edge and screw properties, it is known as a mixed dislocation

19. Distortion of Crystal Structure Around an Edge Dislocation
Figure 9.10a in text
Dislocations distort the crystal structure, which creates a local stress field around the dislocation
The edge dislocation has compression around the extra half-plane, and tension in the region below the dislocation
It mimics the effect of driving a wedge into wood, in order to split the wood

20. Distortion of Crystal Structure Around a Screw Dislocation
Around a screw dislocation, there is shear stress
The effect of the dislocations may be approximated by thinking of compression and tension as the poles of a magnet
Figure 9.10b in text

21. Attraction and Repulsion
Like poles repel, so compression repels compression and tension repels tension
Unlike poles attract, so compression attracts tension
If the dislocations are not too far apart, they “see” each other, and interact
Edge and screw dislocations are called end member configurations. They are perfect, because each involves a change of exactly one unit magnitude in the Burgers vector.
Figure 9.11 in text

22. Twinning in Calcite
Calcite is prone to twinning, and the twins are energetically more favorable than the fully dislocated structure
The process is termed dissociation
Vacancies and atoms do not move readily at low temperatures or in the absence of a fluid film. Yet crustal rocks have been observed to deform readily under these conditions, so another mechanism must be involved. That mechanism is the movement of dislocations. If the activation energy for movement is exceeded, a dislocation can move through a crystal structure. Deformation produced in this manner is called crystal plasticity. Dislocation movement may occur by glide, a combination of glide and climb called creep, or, in certain minerals, by twinning.
Dislocation Glide
Energy introduced into a system by deformation and temperature increase allows dislocations to move along what are known as glide planes. A glide plane contains both the Burgers vector, b, and the dislocation line, l. Two non-parallel lines define a single plane, so edge dislocations produce a single glide plane. Parallel lines define many planes, so screw dislocations, where b is parallel to l, have many glide planes. Movement is possible because only bonds along the dislocation line are broken, not all the bonds across a plane. In addition, bonds may break and reform, so that only a few bonds are broken at any given moment. Snakes and caterpillars move single segments of their bodies, but the net result is movement of the whole.
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23. Edge Dislocation Movement
Edge dislocations move when the unattached atoms at the bottom of the extra half-plane bond to the next lower and forward atoms, located directly below the glide plane
This moves the location of the half-plane as the dislocation moves
Edge dislocations move by successive bond-breakage under the influence of a minimum stress acting on the glide plane. The stress is called the critical resolved shear stress, or CRSS.
Figure 9.12a in text

24. Screw Dislocation Movement
Screw dislocations move by shearing one atomic distance forward
In both cases, a perfect crystal lattice is left behind
The process continues until the edge of the crystal is reached, and the crystal edge is offset. A stair-step structure produced on the crystal edge is known as a slip band. Dislocation movements introduce permanent strain in rocks, but cause no loss of cohesion.
Figure 9.12b in text

25. Pile-Ups
Point defects can stop a dislocation from moving to the edge of a crystal, as can unfavorable stress fields created by the dislocations.
Short of supplying more energy to the system, the only way around a pile-up is for the dislocation to move to a different plane. The processes involved are called climb and cross-slip. These processes require more energy than dislocation movement alone. Screw dislocations, which are not confined to a single plane, have a large advantage over edge dislocations in this regard.
Image:
One analogy is the backup created by an accident on a freeway, which restricts traffic to one lane
The result of many immobile dislocations is therefore called a “pile-up”

26. Cross-Slip
Figure 9.13a in text
Screw dislocations can leave one glide plane and move to another in a process called cross-slip
The process of leaving one glide plane and moving to another is known as cross-slip (9.13a). Usually, the original glide plane has a short Burger vector, and is therefore more favorable, than the alternative glide plane. The CRSS for the alternative glide plane must increase. It is also possible that temperature increases, since increasing temperature weakens bonds, allowing cross-slip to occur more easily.

27. Displacement of Edge Dislocation
Figure 9.13b in text
Climb is a diffusive process. Vacancy creation increases as temperature does, so the efficiency of climb is a temperature-dependent process. Thus, both climb and cross-slip occur more rapidly at higher temperatures, corresponding to greater depths within the earth. Typically, temperatures in excess of 300̊C are required for quarzitic or carbonaceous rocks, and temperatures in excess of 500̊C for minerals like dolomite, feldspar, and olivine. Glide and climb combined are often called dislocation creep.
Edge dislocations are confined to a single glide plane
They can climb to a parallel plane, if a vacancy exists to accept the lowest atom of the extra half-plane

28. Dislocation Annihilation
Figure 9.14a in text
Dislocation annihilation is one method for reducing strain within a crystal
If two edge dislocations lying in the same glide plane, one with a positive (upwards) dislocation and the other with a negative (downwards) dislocation, they annihilate
Screw dislocations with opposing Burger vectors which combine will also annihilate.

29. Partial Annihilation
Figure 9.14b
in text
Two screw dislocations of opposite sign on different glide planes may attract each other
The result is partial annihilation, but with a defect left behind
The processes of climb and cross-slip increase the probability of dislocation annihilation, so this process is temperature dependent
Deformational Twinning
Crystals develop twins as a result of stress. Two major types of twins are growth twins and deformational (or mechanical) twins. Growth twins experience stress when growing from an aqueous solution or in a magma. As crystals become larger, they impinge on each other, and create stress. This stress is localized within the rock, and says nothing about external stress, so growth twins are of little interest to structural geologists. Deformational twinning is a type of crystal plasticity. The twinning process involves glide of partial dislocations. The two halves of the twinned crystal are separated by the twin boundary, which may or may not be planar. The twins may be related by a twin plane, which means that the twin are mirror images of each other. Not all twins are mirror images, however. Since the twin plane cannot be a mirror plane in the symmetry of the mineral in question, twin planes are most commonly found in low symmetry minerals.
Deformational twins are the result of the resolved shear stress exceeding a critical value along the future twin surface. The lattice of the twin crystal rotates in a direction that produces the shortest movement of atoms. This somewhat resembles dislocation glide, but there are differences:
Common examples are calcite, dolomite, and the triclinic feldspars.

30. Calcite Twins Calcite twins in marble from southern Ontario
Figure 9.15 in text
Calcite twins in marble from southern Ontario
Twinning is visible when viewed in cross-polarized light
Width of field is ~ 4 mm
In dislocation glides, the atoms move an integral number of atomic distances; during rotation, they move a fraction of an atomic distance.

31. Twinning Creates Mirror Images
In many cases, the twinned part is a mirror image of the original lattice
Figure 9.16a in text

32. Dislocation Glide Does Not Change Orientation
In dislocation guide, the two lattices have the same orientation
Figure 9.16b in text

33. Calcite Twinning
Figure 9.17 in text
Calcite crystals have twins which involve specific crystallographic planes, with rotation through a specific angle, which allows calcite to be used as a measure of finite strain
Figure illustrates twinning in calcite

34. Calcite as a Strain-Gauge
Figure 9.18 in text
Figure 9.18 shows how calcite may be used as a strain gauge
An original grain with a single twin of thickness t is shown on the left
A grain with multiple twins is on the right

35. Shear Strain for Twinned Grain
γ = tan ψ = q/T,
where q = displacement distance and T is the grain thickness perpendicular to the twin plane
For a single twin, q = p (which is the total displacement), so:
γ = tan (2t tan(φ/2))/T,
where t is the twin thickness and φ is the rotation angle
Ψ (psi)
φ (phi)

36. Shear Strain for Multiple Twins
For multiple twins, shear strain can be obtained by summation over the individual strains of each twin:
where n is the number of twins in the grain

37. Equation Simplification
For calcite, φ is about 38º, and the equation becomes:
Measurement of the grain thickness perpendicular to the twin plane and the total width of the twin grains we can obtain the total shear strain for a single twinned grain. In rocks with multiple twinned calcite grains, the shear strains will vary due to differing crystallographic orientation of individual crystals relative to the bulk strain ellipsoid. Some crystals will show zero shear strain, while others will have a maximum value. These crystals can be used to determine the principal strain axes, using a method known as the calcite strain-gauge method.

38. Applications of Calcite Strain-Gauge Method
The calcite strain-gauge method has proven very useful in studies of limestones subjected to small stresses
Applications include:
Kinematics of folding
Formation of veins
Early deformational history of fold-and-thrust belts
Strain patterns in continental interiors
Strain-Producing and Rate-Controlling Mechanisms
Dislocation glide produces a change of shape in the grains in which it occurs. This in turn introduces strain into the crystal. It is the major strain-producing mechanism of crystal plasticity. Cross-slip and climb allow a dislocation to move around an impurity, but neither produces much strain. However, they are slower than dislocation glide, and are therefore the rate-controlling mechanisms of crystal plasticity. Since climb occurs at higher temperatures than glide in the same mineral, the term low-temperature creep is sometimes applied to glide or twinning and high-temperature creep to glide plus climb.
Generation of Dislocations
We have seen that dislocations tend to move to the edge of the grain, leaving a perfect lattice behind. Yet crystal plasticity requires that more dislocations be present, so there must be a way in which to generate dislocations in-situ.

39. Frank-Read Source One mechanism is the Frank-Read source
Figure 9.19a+b
One mechanism is the Frank-Read source
A dislocation becomes pinned, and bows outward (b)

40. Dislocation Pinning With time, bowing increases
Finally, the bowing forms a loop (e and f)
The Burgers vectors at a and b point in opposite directions, and annihilate
Figure 9.19c-f

41. Restoration of Original Dislocation
The loop closes, forming a perfect circle of slipped crystal, and the original dislocation is restored
The process then repeats, creating a steady supply of dislocations
Figure 9.19e-h

42. Diffusional Mass Transfer
Three diffusion-related processes are possible in natural rocks:
Pressure solution
Grain-boundary diffusion
Volume diffusion
Diffusion is a migration process. Either an atom or point defect can migrate. All diffusion processes are temperature dependent, occurring faster as temperature rises. Increasing temperature means that the stretching and bending of chemical bonds is increased, weakening them. Atoms vibrate more violently, and can jump from site to site. Near the melting point of iron, the jump frequency (Γ, gamma) of vacancies is about 1010/sec. The jump distance, r, is the diameter of an atom, or 0.1 nm. The average distance traveled is given by Einstein’s equation:

43. Einstein’s Equation R2 = Γtr2 where t is time
If t = 1 second, and Th = 1 (Fe metal) then
R2 = 1010/sec ∙ 1 second ∙ (1 x m)2 = 0.1 mm2
This is a small time, but geologists have plenty of time

44. Geologic Time Say 1 million years
R = (1010/sec ∙ 3.1 x 1013 seconds ∙ 1 x m)0.5 = 5.6 x 106 m
Diffusion is not a straight line process. Atoms are as likely to jump one way as another, unless a gradient is present which causes them to move in one direction. In an isotropic-field, diffusion is a random-walk process (sometimes called a drunkard’s walk). In addition, we are not near the melting point in most rocks, so the real diffusion distance in on the order of cm to m/million years.

45. Diffusion Coefficient
If a concentration gradient is present, we can define a diffusion coefficient:
D = (Γ/6) r2
Where Γ is the jump frequency of vacancies
r is the jump distance (the diameter of an atom)

46. Temperature Dependent Diffusion Coefficient
That can be rewritten in a temperature dependent form:
D = D0 exp(-Eº/RT)
where D0 is an empirically determined material constant,
Eº is the activation energy for migration expressed in kJmol-1,
R is the gas constant Jmol-1K-1 and
T is the temperature in K
Volume Diffusion and Grain-Boundary Diffusion
A crystal under stress will experience both volume diffusion and grain-boundary diffusion. Vacancies will migrate toward areas of high-stress, and atoms will migrate toward regions of low-stress.

47. Diffusion Diagram
Figure 9.20 in text
Volume diffusion occurs through the body of the crystal, whereas grain-boundary diffusion is a surface phenomenon
Volume diffusion occurs through the body of the crystal, whereas grain-boundary diffusion is a surface phenomenon. Volume-diffusion is also called Nabarro-Herring creep, and grain-boundary diffusion is sometimes called Coble creep. The movement of mass results in a change in mass distribution. In a nonisotropic stress field, diffusion becomes a directional process.

48. Strain-Rate for Creep
Both Nabarro-Herring and Coble creep involve diffusion of vacancies
Strain rates for each mechanism are a function of the diffusion coefficients (Dv and Db) and the grain size (d):
Thus, the larger the grain, the smaller the strain rate. The activation energy for grain-boundary diffusion is less than for volume diffusion, and the grain-size dependence of volume diffusion is larger. Therefore, Coble creep is more important the Nabarro-Herring creep in the crust. Only in the mantle, at very high temperatures, or for very small crystals is Nabarro-Herring important.

49. Pressure Solution
Pressure solution is the third diffusional mass transfer process that occurs naturally in rocks.
A fluid film must coat the grain boundaries
When the film is present, this process can occur at much lower temperatures than vacancy diffusion
The film acts to dissolve the crystal, with the dissolved ions moving along a chemical gradient to regions of precipitation
Nonisotropic stresses create differential solubilities, which allows this process to work. This seems contradictory, since fluids cannot support shear stress. But fluid films adhere to the surface through chemical bonds, and these bonds provide sufficient attachment to support some shear stress. Evidence for this comes from the lack of fluid movement as shear stress is applied. Areas perpendicular to the maximum principal stress will experience dissolution. The stress has partially broken the bonds within the crystal, and bonds to the liquid further weaken the attachment of the outermost ions. This produces a migration of ions, very similar to that previously discussed. However, it can occur at low enough temperatures to be active at the earth’s surface, as well as lower, in the upper crust. The required fluid is mostly likely to be present in upper-crustal rocks, associated with sediments or sedimentary rocks. This process is known as fluid-assisted diffusion, while Nabarro-Herring and Coble creeps are solid-state diffusion.
Fluid-assisted diffusion can occur over substantial distances. The ions, once dissolved, can enter the pore fluid of the rock, which allows much more rapid diffusion. In addition, the pore fluid can be moving, since it is not attached to a grain surface, and groundwater flow can be many orders of magnitude faster than solid state diffusion. Movement of pore fluids can entirely remove ions from a rock, eventually resulting in volume loss. The dissolved ions may precipitate as vein fillings. A process called differentiation can lead to the deposition of alternating layers of mineral, such as quartz and mica. Quartz dissolves more easily than the mica, which has a complicated crystal structure. Quartz dissolves in high pressure areas, and migrates to low-pressure zones, like the hinge of a fold. Mica minerals are left on the fold limbs. This process can be so complete that it produces a compositional microlayering, with a completely different orientation than the orientation of preexisting foliation.
It is often associated with crenelation cleavage, a shortening of evenly spaced foliation to form sharply defined micro- to mesoscale folds (crenulations).

50. Crenulation Cleavage
Differentiation is often associated with crenulation cleavage, a shortening of evenly spaced foliation to form sharply defined micro- to mesoscale folds (crenulations)
Image:
The vertical foliation in this muscovite-biotite -garnet schist from New Mexico is a crenulation cleavage, and developed after the horizontal foliation.

51. Strain-Rate for Fluid-Assisted Diffusion
Widespread evidence for fluid-assisted diffusion can be found in rocks, so this is a more important process than many geologists realize
The strain rate associated with fluid-assisted diffusion is:
where Df is the diffusion coefficient of the phase in the fluid, and d is the grain size
Dislocation movement is a function of differential stress, ambient temperature, and the activation energy for breaking bonds. The rate at which strain occurs by dislocation movement is therefore a function of the same parameters. We can write this as the constitutive equation, aka flow law:

52. Flow Law ė = A f(σd) exp(-E*/RT) where A = material constant
f(σd) = differential stress function
E* = activation energy
R = gas constant
T= absolute temperature
The stress function for dislocation glide or low-temperature creep is exponential, so:

53. Low-temperature Creep
ė = A exp(σd) exp(-E*/RT)
For dislocation glide and glide (high-temperature creep), typical of deep crustal and mantle rocks, the stress is raised to a power n, so:

54. High-temperature Creep
ė = A σdn exp(-E*/RT)
n is the stress exponent, and this law is called power-law creep
The motion of individual defects or atoms is also a function of differential stress:

55. Motion of Individual Defects
ė = A σd exp(-E*/RT) d-r
Here, the stress exponent is 1, so diffusion is linearly related to strain
The d-r term relates to grain size, so diffusion is non-linearly related to grain size