Dynamics: Rheology and deformation mechanisms

Stress-strain relationships

In these sections we examine the different ways in which rock materials respond to stress. A variety of laboratory rigs exist for stress testing materials.

VDPM 05.08

VDPM 05.10

Different types of materials respond to stress in different ways. Typically, for each material under a particular range of conditions, there is a relationship between stress and strain, or between stress and strain rate, which is called the constitutive law for that material.

Elastic strain

The simplest response is one in which the finite strain is proportional to the applied stress. In this case, if the stress decreases, the strain decreases too, so we say the strain is recoverable, or elastic. All rocks respond elastically to some degree, and this type of strain is very important for understanding seismic waves. However, natural elastic strains in rocks are very small and are not preserved when rocks are exposed at the surface and stress is removed.

In elastic deformation, note that the finite strain is related to stress in a proportional way.

Longitudinal Stress-strain relationship

The stress-strain relationship in one dimension for an elastic material is

normal stress σn = E.e

where E is Young's modulus of elasticity.

Length-width relationship: Poisson's ratio

Most rocks when subjected to a compressive stress in one direction will tend to expand at right angles to this direction. This tendency is expressed by Poisson's ratio ν, the ratio of thickening to shortening when stressed in this way.

Poisson's ratioν= - e1 / e3

Where e1 is the extension parallel to σ1 and e3 is the extension parallel to σ3. For a sample that retains constant volume, the Poisson's ratio will be 0.5. Most rocks show a negative change in volume when uniaxially stressed, and therefore show Poisson's ratio between 0.5 and 0.

For an isotropic elastic material, it turns out that Young's modulus and Poisson's ratio are all that's necessary to describe the elastic properties, though other moduli are sometimes used.

Other elastic moduli

Shear stress-strain relationship

For example, for shear strain we can write

shear stress σs = G.γ

where the constant G is the shear modulus of elasticity. It turns out that the shear modulus can be derived from Young's modulus and Poisson's ratio:

G = E / (2+2ν)

Bulk modulus

The bulk modulus K relates volume change to an axial stress.

mean stress σm = K Δ

As you might expect it can also be derived from Young's modulus and Poisson's ratio.

K = E / (3-6ν)

Brittle failure

Most rocks near the Earth's surface, subjected to increasing differential stress, will eventually fail by brittle fracture. When failure occurs, the strength of the rock falls to zero across the plane of fracture. This is called loss of cohesion.

Fracture propagation

Fractures actually propagate over a period of time (may be very fast), exploiting pre-exisiting microfractures. At any instant a fault is bounded by a tip line.

VDPM 06.08

Fractures may propagate as

Failure criteria

The condition for failure can be represented on the Mohr diagram for stress by a failure envelope. Several different formulas for the failure envelope have been proposed.

Coulomb failure envelope VDPM 06.15

Mohr failure envelope VDPM 06.18

These failure criteria predict conjugate faults VDPM

Pre-existing fractures

Byerlee's law for movement on pre-existing fractures VDPM 06.24

Notice how under Byerlee's law conditions the failure envelope does not extend into the tensile field on the left of the diagram. This means that the rock cannot sustain any tensile stress - it has no cohesion.

Effect of fluid pressure on failure

Fluid pressure encourages fracture by reducing the effective normal stress, while allowing shear stresses to remain the same. Effectively the Mohr circle is shifted left, leading to hydraulic fracturing

VDPM 06.27

Faults vs. Cataclastic flow

Many brittle fractures occur as discrete planes, but in some rock units there are multiple anastomosing fractures, that create the appearance of continuous deformation. Such a deformed unit may appear ductile at outcrop scale but is brittle at microscopic scale. This type of process is called cataclastic flow.

VDPM 06.04

Brittle-ductile transition

At high mean stress there is a transition from brittle deformation to continuous deformation or creep.

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

VDPM 06.21