VII Stress

Material for this section is in three chapters in Twiss and Moores. The theory of stress is dealt with in great detail in Ch 8. We will follow a similar presentation, but will emphasize the more compact vector and matrix formulas where possible, merely stating the principal results rather than proving them from first principles.

Chapter 9 deals with the mechanics of fracturing. We will follow a somewhat simplified presentation, that assumes some basic knowledge from EAS 321.

Chapter 10 deals with states of stress in the Earth's crust. The first part deals with the determination of in situ stresses in the crust. The second part deals with predictions that we can make for stresses in portions of the crust under idealized conditions.

Following Twiss & Moores we will defer the treatment of ductile responses to stress until we have explored the subject of strain more fully in the next section.

Stress on a plane, stress vector

Forces

Distinguish in Geology between body forces, like weight, that increase with the volume of material (ie the cube of length), and surface forces, that increase as the square of length.

Traction and Stress

For surface forces, useful to represent them as tractions - force per unit area or force/area

More strictly we might want to recognize that force can vary over a surface, so we would define traction at a point in terms of an infinitesimally small area, as dF/dA

Traction on a given surface has a magnitude and a direction. Magnitude is always positive. In general, we can resolve a traction on a surface into normal and shear traction components. Direction is given in xyz components and magnitude is always positive.

In general every force has an equal and opposite reaction force, so tractions exist in pairs. A pair of oppositely directed tractions acting on the two sides of a surface is called a stress. So concept is very similar to a traction (in fact, most structural texts fail to make this distinction at all), but helps in definition of sign conventions. Thus defined stress components are positive if they are directed towards the surface (compression) and negative if away from the surface (tension).

Units of stress

Newtons/m2 or Pascal (Pa).

Normal and shear stress

Stress acting on a plane has a magnitude and a direction - a vector quantity.

The component parallel to the plane is the shear stress; component perpendicular is the normal stress.

Stress in 3-D

Stress tensor

Worm's eye view of an infinitesimal cube of rock (why do we do this - so that we see all three surfaces that face toward the origin. The tractions acting on these three surfaces are equal to the principal stresses. (Note if we show other surfaces then some signs change). We designate the face perpendicular to axis 1 as face 1, etc..

These nine stresses define the stress tensor

Note that t12 signifies the stress on face 1 that acts parallel to axis 2.

Note that we have used t to distinguish shear stresses but this was unnecessary. Many presentations use s for both, and just rely on the subscripts to distinguish theshear stresses from normal stresses.

For an infinitesimal cube 12 and 21 tend to rotate the cube in opposite directions. We can argue that for an infinitesimal cube the turning moment of the forces acting should be zero. Hence these two shear stress components must be equal. Likewise the others. Hence the stress tensor is symmetric.

Coordinate transformation

In general we can choose any three sets of mutually perpendicular axes for our coordinates, and formulae exist for converting one to another.

In fact we can make a matrix that converts from one coordinate system to another.

If direction cosines of new axis i, relative to old axis j are nij then the coordinate transformation matrix n is given by

Then to transform a stress tensor to new coordinate system we do
' = n nT or

Principal stresses and stress axes

There is always a coordinate system that can be found such that in the new coordinates the stress tensor is reduced to..

Note that in this one coordinate system there are no shear stresses parallel to the axes. Also the three normal stresses represent a mimimum a maximum and a 'minimax'. These coordinates are the stress axes

Mohr circle for stress in 2D and 3D

Stress on a plane

If we represent a plane by its pole, a unit vector with components (direction cosines) l = (l1 l2 l3)

Then if the state of stress is S the shear stress on any plane is given by S.l

This can be decomposed into two components, a normal stress

N = 1l12 + 2l22 + 3l32

and a shear stress

= (1-2)2l12l22 + (2-3)2l22l32 + (2-3)2l22l32

Mohr Circle for stress in 2D

Mohr circle construction

For the 2D case, the above simplify to

N = [(1 + 3)/2 + (1 - 3)/2] cos 2

and = [(1 - 3)/2] sin 2

Which are the equations of a circle.

Diag - Mohr circle for stress

Twiss & Moores fig 8.7

Note the angles on the Mohr circle are doubled with respect to the angles in real space. Thus the principal stresses lie at opposite points 180° apart on the horizontal diameter of the Mohr Circle.

Any two planes oriented 90°apart in real space have their states of stress located 180° apart on the Mohr circle.

If we use the convention that compressive stresses are positive, and we plot angles in the same sense (clockwise vs counterclockwise) on the Mohr diagram, then shear stresses show up as positive for counterclockwise shear couples, and negative for clockwise.

Maximum shear stress is on two planes at 45° to the principal stresses, called the conjugate planes of maximum shear stress.

Scalar invariants of the stress in 2D

Mean stress (1 + 3)/2 and radius of the circle (1 - 3)/2 which is also the maximum shear stress. In geology we often find it more useful to speak of double this quantity or (1 - 3) which is called the differential stress.

Mohr circle for stress in 3D

The equations for 2D actually work for any plane that is perpendicular to a principal stress. So two smaller circles fit inside the Mohr circle for 1 & 3. All 3 principal stresses plot on the horizontal axis.

Twiss & Moores fig 8.4.2

Special states of stress

Hydrostatic pressure 1 = 2 =3 = Pressure

Mean stress = (1 + 2 + 3)/3

Deviatoric stress

Uniaxial crompression 1 >0 2 = 3 = 0

Uniaxial tension 1 = 2 = 0 3 < 0

Axial compression 1 > 2 = 3 > 0

Axial extensional stress 1 = 2 > 3

Triaxial stress 1 > 2 > 3

Pure shear stress 1 = - 3 2 = 0

Measurements of present-day stresses

Stress relief measurements.

Depend on elastic properties of material.

Overcoring: Drill hole; install strain gauge; then drill an annular hole all around to remove stress, and measure strain that results. Not simple, however because the initial hole affedcts the stress.

Flat jack: Measure a distance in rock between marker pins. Then cut a slot that relieves the stress locally and changes the distance between the markers. Seal the slot with a metal liner. Then pump fluid in until the original distance is restored. This measures the normal stress that existed prior to cutting the slot.

Other methods

Hydraulic fracturing: Pump fluid into borehole, until fracturing of wall occurs. Fracturing will occur at a pressure equal to the sum of the minimum compressive stress and the tensile strength of the rock. If we then reduce the pressure, the crack seals itself again at the shut-in pressure which is assumed to be the minimum horizontal compressive stress. If we can use the orientation of the crack (with a downhole camera or impression made of the wall) then we can determine the orientation and magnitude of 3 . The vertical normal stress can be estimated from the depth and average density of rock above. Hence we can get a reasonable picture of the state of stress.

Breakouts are naturally occurring conjugate fractures that occur after a borehole is left empty for a while. They occur along conjugate fractures that result in the borehole's becoming elliptical, and can be oriented with calipers or preferably with downhole cameras. Give the orientation but not magnitude of the stresses in the horizontal plane.

Earthquake first motion studies: give orientations but not magnitudes.

Elastic and brittle responses to stress

Method

Typical experimental setup for testing response of materials to stress

Diagram and photo (If you are viewing this on the web go to GCTS Inc)

Typically the presses are moved a given distance and stress is measured. So plots typically show the strain as the independent variable and the stress as dependent. In 'real life' it's much easier to think of stress as cause and strain as the effect.

Elastic response

Recoverable strain is called elastic. Typically in uniaxial compression the extension is proportional to the applied stress.

(Recall that extension is fractional change in length e=(l-L)/L where L is original length and l is new length.)

so = Ee or e = /E

E is called Young's modulus. Note that because a compression (+ve ) produces a shortening (-ve e) then E must be negative. For stronger materials E is more negative.

For most rocks E ranges from -0.5 x 105 MPa to -1.5 x 105 MPa

When rock cylinder in a press shortens it also thickens. If the volume remained exactly the same them the thickening would be close to -0.5e. Actually it isn't because the work done by the stress is divided between changing the shape and reducing the volume. A measure of the tendency of the material to change shape is Poisson's ratio v = |enormal/eparallel |

An incompressible liquid has a v = 0.5

Brittle response

With increased stress, failure occurs at a stress that is called the 'strength' of the material. Manifested by a sudden stress drop in experiments

Extension fractures form perpendicular to 3. This is called longitudinal splitting when the fracture is parallel to the axis of the machine (as it is in most experiments). This is also called a Mode I fracture.

Shear fractures form in confined compression tests, typically in conjugate pairs on either side of 1 at angles of less than 45° to it. If the state of stress is triaxial, then 2 is parallel to the intersection of the fracture planes.

A failure criterion for extensional fracture

Twiss & Moores 9.3

The failure envelope specifies a critical negative value of n, beyond which the rock fails. Critical normal stress n* = T0

A failure criterion for shear fracture

Twiss * Moores Fig 9.4

Shows a pair of linear boundaries between stable and unstable states of stress.

For a given normal stress we can define the critical shear stress as

|s*| = c + n (Coulomb fracture relationship)

Relationship of to slope of failure envelope : =tan

is called the coefficient of internal friction and is called the angle of internal friction.

Relationship of to orientation of fractures:

On the Mohr diagram, the radius of the Mohr circle is at 90° to the envelope. So the angle between the pole to the failure plane and the 1 axis is given by 2= 90 + or = 45 + /2 For most rocks is around 30° so is around 60°. Hence the angle between the plane itself and 1 is about 30°

More realistic failure envelopes

Twiss & Moores Figure 9.9 shows more realistic failure envelope (still schematic).

We recognize:

An infinitesimal vertical part of the curve at the point of tensile fracture

A parabolic section of envelope in the tensile field.

An approximately straight region where the material follows the Coulomb criterion for brittle failure.

Another concave region marking the brittle-ductile transition

A subhorizontal section of curve where normal stress has no influence on the failure of the material. This marks fully ductile behaviour and is called the Von Mises criterion

Effect of pore fluid pressure

Effect of pore fluid pressure is to lower the effective normal stresses, while keeping the shear stresses the same.

En = n - pf

We can re-write the Coulomb criterion

|s*| = c + (n - pf)

Twiss & Moores fig 9.13

Frictional response of already fractured material

Frictional sliding criterion resembles the Coulomb criterion except that (i) there is no cohesion and (ii) the true coefficient of friction is higher than the angle of internal friction.

|s*| = fn

This results in a crossing of the frictional sliding envelope and the Mohr failure envelope.

Twiss & Moores Fig 9.11

Above the crossing point, fracture is easier than sliding. This results in multiple fracturing producing cataclastic flow.

For many rocks, the coefficient of friction is about 0.85 up to about 2 kilobars, corresponding to an angle of sliding friction of about 40°.

Above 2 kilobars the anlge of sliding friction is 30 - 35°

These relationships were combined by Byerlee in 'Byerlee's Law'

|s*| = 0.85n (n < 200 MPa)

|s*| = 0.5 + 0.6n (n > 200 MPa)

(Diag Davis & Reynolds fig 5.47)

States of stress associated with particular models

Standard state - block with stress entirely due to its own weight (T& M p 203)

Horizontal compression plus lithostatic stress (T&M p 203-4)

Horizontal compression of block with frictional resistance at base (T&M p204-205,

Subhorizontal detachment fault block (Twiss & Moores p 158-164)

Gravity sliding of a block

Horizontal compression of a wedge