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Rheology, deformation mechanisms, and mylonites

Stress-strain relationships

Brittle and ductile deformation

This section is about the different ways in which rock materials respond to stress. Elastic responses and brittle fracture were covered in an earlier section.

For each material under a particular range of conditions, there is a relationship between stress and strain, which is called the constitutive law or flow law. In elastic deformation, strain and stress are proportional, a relationship expressed in Hooke's Law. In studying ductile deformation, the flow law is typically a relationship between stress and strain rate; strain continues to be acquired while stress is applied, but when the stress is removed, the strain is retained. We say that the deformation is non-recoverable.

Brittle-ductile transition

At high mean stress (pressure) there is a transition from brittle deformation to permanent (non-recoverable) deformation that is not associated with a loss of cohesion, termed continuous deformation, flow, ductile flow or creep.

This transition can be seen in stress tests at high confining pressure and high temperature.

In creep, it is the strain rate, not the finite strain, that is a function of the applied stress. Strain rate is represented by de/dt or by e ˙

Progressive strain and flow

Basics

In getting from its original state A to some final deformed state B, a rock passes through an infinite series of very small steps of deformation, involving translation, rotation, dilation and distortion. We refer to this as the strain history. There are infinitely many routes that could be followed that could result in the same finite strain.

In analysing deformation, it is sometimes helpful to look at a very small part of the strain history, which is referred to as the incremental strain. Incremental strains may be represented by symbols such as δe or δγ

In strain theory, we sometimes go one step further and consider an infinitesimally small increment of strain, which is referred to as instantaneous strain or infinitesimal strain, effectively the derivative of the finite strain.

For example, if finite strain is represented by an extension e or a shear strain γ then an instantaneous strain is de or .

The finite strain is the integral of the the infinitesimal strain, from the time of the formation of the rock to the present day.

de/dt and dγ/dt (also written e ˙ , γ ˙ ) are strain rates - measures of the speed at which rocks flow.

What are the units of strain and strain rates? Strains and strain rates are dimensionless quantities, because they are ratios of two lengths. Therefore the units of strain rate are just "per second" or s-1. Sometimes structural geologists will say "strains per second". Typical geological strain rates are of the order of 10-14 strains per second (which works out to about 0.3 strains per million years).

Describing flow

Defining the complete strain history is a much larger challenge than measuring the finite strain.  To keep thing simple at first we will continue to deal with only two dimensions - nothing flows in or out of our cross-section - 'plane strain'.

To describe progressive strain and flow, it's sometimes helpful to draw a field of small arrows representing the velocity field. We can also join these up to make longer arrows representing particle paths.

Particle paths are usually curved. Often there are one or two special particle paths that are straight. These have a special role in the generation of fabric in metamorphic rocks. They are known as flow asymptotes or flow apophyses. In linear algebra terms, they coincide with the eigenvectors of the deformation matrix.

At each moment in a strain history, the ongoing deformation is defined by instantaneous strain axes (or ISA: these are the instantaneous stretching and instantaneous shortening axes). Between them, there are usually lines of no instantaneous extension (LNIE).

All these lines may change orientation during the strain history. Except for special types of strain, the instantaneous directions of these lines do not coincide with their finite strain equivalents.

During deformation, material lines may move from the field of shortening to the field of extension or vice versa.

If rocks, and strain, were perfectly homogeneous, we would never be able to figure out strain histories because we would only see the end result, the finite strain. Fortunately, strain partitioning is common. In strain partitioning, different components of a rock record different parts of the strain history.

Coaxial flow, or progressive pure shear

In coaxial strain the strain axes remain along the same material lines throughout the strain history: ie the strain axes do not rotate relative to the rock. As a result, the finite strain axes coincide with the instantaneous strain axes, and with the flow apophyses.

Note that although the strain axes don't rotate, other lines tend to rotate towards the stretching direction and away from the shortening direction.

Every increment of strain is non-rotational or pure strain. The vorticity of the deformation is zero.

The deformation matrix at every stage is a symmetric matrix. The flow apophyses, which are the same as the strain axes, coincide with the eigenvectors of the deformation matrix.

A diagram will show the effect of progressive pure shear on a rock with competent layers oriented initially perpendicular to s1

As strain starts, competent layers buckle to produce folds. Then, with increasing strain, limbs of the folds rotate from the field of incremental shortening into the field of incremental extension. Initial folds may become unfolded and undergo boudinage.

Geometries like these are extremely common in folded rocks - limbs that are boudinaged while the hinges are tightly buckled.

Noncoaxial flow

In noncoaxial flow, the strain axes and the rock rotate relative to each other. The instantaneous strain axes do not lie along the same material lines througout deformation.

In noncoaxial flow the deformation matrix is not symmetric and the flow apophyses (the eigenvectors of the deformation matrix) are not perpendicular.

If we think of non-coaxial flow as being broken down into an infinite number of tiny increments, we can mathematically consider it as consisting of separate increments of distortion (pure strain) and increments of rigid-body rotation

In dealing with non-coaxial deformations, it would be helpful to have a measure of the amount of rotation compared with distortion. To do this we define a quantity called vorticity. For a non-dilational deformation, the kinematic vorticity number is the ratio of the rotation rate (in radians per second) to the distortion rate (in strains per second).

It's possible to distinguish different categories of noncoaxial flow, depending on different amounts of vorticity.

Progressive pure shear is characterized by a kinematic vorticity number of zero.

Spectrum of flow types:

Simple shear

Simple shear is a special case of noncoaxial flow, but is important because an ideal shear zone develops by progressive simple shear.

In simple shear:

However, as finite strain increases:

Subsimple shear

Subsimple shear combines distortion with gradual rotation of the strain axes, and is intermediate between pure shear and simple shear. There are two flow asymptotes or apophyses, which coincide with the eigenvectors of the deformation matrix, but they are not at right angles and do not coincide with the strain axes.

The kinematic vorticity number equal to the cosine of the angle between the flow apophyses, and is therefore between zero and 1.

Super-simple shear

If the rate of rotation exceeds the rate of extension, everything rotates. No particle paths are straight and there are no flow apophyses. The deformation matrix has no real eigenvectors.

The vorticity number is greater than 1.

Supersimple shear describes the behaviour of relatively rigid particles in a shear zone, which tend to cause strain partitioning. The rigid particles capture the vorticity while the softer matrix captures the distortion.

Supersimple shear tends to produce features like delta porphyroclasts and helicitic (snowball) porphyroblasts.

Rotation and strain in 3D

(We may skip this section if time is short; if this happens, this will not be covered in the final test)

Rotational deformation is much more complex in 3D than in 2D.

In 2D progressive deformation, rotation is about an axis perpendicular to the plane of section; any rotation axis is perpendicular to two strain axes.

In 3D, the rotational component of deformation can be perpendicular to two of the strain axes and therefore parallel to the third. When this is the case, the deformation is described as monoclinic. The 'before' and 'after' shapes of the strain ellipsoid have the same symmetry as a monoclinic crystal.

However, the rotation axis can be different from any of the strain axes. If this is the case, the strain symmetry is very low and the deformation is triclinic. Progressive triclinic strains (where rotation and distortion occur concurrently but on different axes) are extremely complicated to work with and require techniqes beyond the scope of this course.

The logarithmic Flynn plot, otherwise known as the Ramsay plot, comes into its own in dealing with 3D strain histories. It turns out that is a constant incremental strain is applied, the finite strain follows a straight line path on the Ramsay plot.

Mechanisms of continuous deformation or creep

Crystal defects

It's possible to calculate bond strengths in crystals from chemical data.  In most cases, the strengths indicate that it's impossible, with geologically reasonable stresses, to simultaneously break all the bonds in a crystal, so as to (for example) shear the crystal.

Nevertheless, we see clear evidence for distortion of crystals in deformed rocks.

It turns out that crystal defects - departures from the ideal lattice structure - are critically important in allowing crystals to deform.

In general, defects affect either points, lines, or planes within the crystal lattice.

Point defects and diffusion-based deformation mechanisms

Point defects

Point defects include:

Point defects can move through crystals by diffusion (random motion of atoms). Diffusion takes place in all crystalline materials, but the rate of diffusion increases with temperature. Vacancies are particularly important in allowing diffusion to take place

Diffusion-based deformation

Volume diffusion or Herring Nabarro Creep

Crystal changes shape by migration of point defects - particularly vacancies - through the crystal.

Grain-boundary diffusion or Coble Creep

Crystal changes shape by migration of atoms along grain boundaries. This works better at small grain sizes because the area of grain boundary is large.

Solution creep or pressure solution

Rate depends on amount, pressure, and chemistry of fluid present. Leads to development of stylolites and pressure solution cleavage

Superplasticity (Diffusion-accommodated grain-boundary sliding)

Grains slide past each other; corners and other irregularities that would normally prevent grain-boundary-sliding can be diffused out of the way leaving behind a fine-grained aggregate that does not show strong CPO.

Strain rate

Crystals undergoing deformation by solid-state diffusion typically show a strain rate that is proportional to the differential stress.

This type of stress-strain relationship is known as viscous or Newtonian.

The equation for viscous flow is

strain rate is proportional to differential stress

or Viscous

where η is the viscosity and σd is the differential stress σ1 - σ3

Strain rates for superplasticity are different. Superplastic typically follows a power law relationship in which strain is proportional to stress raised to some power n typically between 1.4 and 2

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Line defects and dislocation-based deformation mechanisms

Edge dislocations

Line defects are also known as dislocations.

The nature of the defect is defined by the Burger's vector b.

To define the Burger's vector, describe a closed loop in an undeformed section of crystal.  Then follow a similar loop around the dislocation; the gap between the start and end point is the Burger's vector b.

An edge dislocation has b perpendicular to the dislocation line.  It marks the edge of an extra half-plane of atoms.

Large numbers of dislocations produce a curved crystal lattice

This shows up in thin section as undulatory or undulose extinction

Screw dislocations

A screw dislocation has b parallel to the dislocation line.  Atoms are connected in a helix ("spiral staircase") around the screw dislocation.

In practice, a given dislocation can change orientation from parallel to perpendicular to b, and all orientations in between.

Dislocation glide

Dislocation glide is the fundamental process of deformation at high strain rates, by which mylonites flow

The constitutive law for dislocation glide is typically exponential creep.

strain rate is proportional to exponential function of differential stress

Dislocation glide leads to huge densities of dislocations within crystals. Dislocation densities are often stated in units of cm-2. Highly strained quartz may have a dislocation density of 1011 cm-2. (That means every cubic centimetre of quartz contains a mind-boggling hundred thousand kilometres of dislocations!) Eventually this leads to strain hardening because the dislocations get tangled - they need good, intact crystal to migrate, and so are held up by other dislocations

Dislocation creep

However, at slower strain rates, there are other processes involving diffusion that can assist with dislocation motion.

Deformation involving a combination of diffusion with dislocation glide are known as dislocation creep. In metallurgy, diffusion-assisted reorganization of dislocations is known as recovery.

Several processes are involved in dislocation creep and recovery:

Dislocation creep typically follows a power law relationship in which strain is proportional to stress raised to some power between 1 and 5, typically around 3

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Dynamic recrystallization

Notice that the crystal lattice is distorted in the vicinity of a dislocation wherever one occurs. This results in availability of energy to break bonds, and thus there is a tendency for a crystal containing a lot of dislocations to rebuild itself into numerous small crystals. This is the process of dynamic recrystallization that produces mylonite.

Dynamic recrystallization produces a strong crystal preferred orientation (CPO) or lattice preferred orientation (LPO).

Mechanisms based on plane defects

Planar defects can also play a role in deformation

Twin gliding

Typically, across a twin plane, the structure of a crystal is reflected. (Rotations are also possible).  Calcite and plagioclase feldspar are two minerals that typically show abundant twin planes.

Strain twinning (twin gliding) is a major mechanism of deformation in carbonates

Twin gliding seems to follow a flow law that is exponential, similar to dislocation glide.

strain rate is proportional to exponential function of differential stress

Grain boundary migration

Grain boundaries can be thought of as major planar defects, and are associated with extra energy (surface energy) that can be released by the transfer of atoms across grain boundaries

Different conditions: different mechanisms

Deformation mechanism maps

For most of these types of deformation it's now possible to determine a theoretical relationship between stress and strain, based on knowledge of the kinetic behaviour of atoms at various temperatures, and on the bond strengths for different types of bonds (from chemistry).

Hence we can determine which mechanism gives the fastest strain rate at a given set of conditions (temperature and differential stress).

This leads to the idea of a deformation map - typically a plot with temperature and differential stress on the axes, divided into areas according to which mechanism is fastest.

These maps show the fastest deformation mechanism under given conditions of T and differential stress. Contours are drawn within the diagrams showing the strain rate predicted under those given conditions. (Note: the lines are numbered with the log of the strain rate: where it says '-9' this means 10-9).

In general, only the highest strain rates can be experimentally verified: the lower strain-rate parts of the diagram are predicted based on chemically determined diffusion constants and bond strengths.

Strength of the lithosphere

We can use these diagrams to make predictions about the strength of the lithosphere at various depths, by assuming a geothermal gradient (rate of temperature increase with depth).

In order to do this we have to bring brittle deformation into the picture. Brittle deformation is strongly dependent on pressure. The Mohr-Coulomb failure relationship or 'Byerlee's Law' allows us to predict the increasing differential stress required for fracture with increasing pressure. Hence the strength depth relationship shows an initial more or less linear increase of strength with depth.

(The relationship is a little different depending on whether the lithosphere is being extended or shortened. In general if the maximum compressive stress is vertical, a rather lower differential stress is required than if the maximum compressive stress is horizontal. This follows from the Mohr construction for brittle failure.)

At some depth, the brittle deformation curve intersects the strength-depth curve for one of the ductile mechanisms. This point marks the brittle-ductile transistion. Below this point, creep requires less differential stress than fracture, so rocks flow.

If we are going to make deductions about the strength of the lithosphere we need to assume some compositions: in the diagrams we often assume that quartz is the strength controlling-mineral in the crust, and that olivine controls strength in the mantle. For an average continental geothermal gradient this predicts a strong mid-upper crust, a weak lower crust, and a strong upper mantle.

We can use somewhat more sophisticated models. E.g. a crust that has a wet-quartz-dominated 'granitic' upper layer, a dry-quartz-dominated mid crust, a feldspar-dominated 'granulite' lower crustal layer, and an olivine-dominated peridotite lithospheric mantle.