Geol 244 – structural geology

Lecture 4 – Introduction to Mohr circles and stress in the lithosphere

Memorial to Otto Mohr, the NetherlandsPhoto by Carolyn Boulton

Shear and normal stress and the inclination of the plane

• As plane steepens past 45°, area increases dramatically – stress magnitudes drop

• Normal stress is at max when θ = 0°• Shear stress is at max when θ = 45°• Ratio of shear stress to normal stress is

greatest when θ ~60°θ is the angle between σ1 and the normal to the plane.

0 20 40 60 80 1001

10

100

1000

Dip of plane "theta" (degrees)

area

bel

ow u

nit s

quar

e

Lets plot this relationship!

• Mohr circle is the locus of points that represent the state of stress on individual planes at all their orientations

Constructing a Mohr circle

3.75 MPa2.25 MPa4.5 MPa

Reference states of stress in the lithosphere

• General triaxial stress: σ1>σ2>σ3≠0• Represented by the strain ellipse • Common stress regime in the

lithosphere

Reference states of stress in the lithosphere

• Hydrostatic stress (pressure): σ1=σ2=σ3• Also called confining stress• Newly solidified magma chambers• Young, undeformed sediments• Stress ellipsoid is a sphere

HS

Reference states of stress in the lithosphere

• Biaxial (plane) stress: one axis =0• (e.g.,σ1>0 >σ3)• Implicit in many 2d

cross-sections• Imagine two

immovable sides

• Uniaxial compressional stress: 2 axes = 0• σ1>0; σ2=σ3=0

• Condition of mine pillars

Reference states of stress in the lithosphere

If we want to understand these rocks…• We first need to understand some concepts…

Factor Importance

Inputs

Force/stress Dynamic input

Elasticity Mechanical and Rheological (time-dependent) properties of the materials being subjected to stress. These, in conjunction with the timescales at which stress is applied, control the material behaviour

Plasticity

Viscosity

Strength

Time Controls material behaviour in conjunction with rheology

Outputs

Strain/deformation/kinematics where Y: deformation; m: rheology (time dependent); and x: force/stressMaterial behaviours

Brittle, ductile, elastic, plastic and viscous

Describe the behaviour of given materials in response to force and stress over given time-scales

Concept 3 – Deformation and strain

• So far we have talked about:• Force – created by changes in velocity of tectonic plates

(even small changes

• Stress – the intensity at which force is applied• Depends on the angle of plane relative to applied force• Depends on the area over which forces are applied

• Now we will talk about strain, and how to measure it.

http://www.geology.sdsu.edu/visualstructure/download.htm

Have a look at Visualizing Stress (Mohr circle plotting) and Visualizing Strain at this website

Strain• What is rock strain?• change of shape and size under loading

• The load is the external force that causes the deformation• Applying a load leads to strain

• Style of strain depends on • the area over which load is applied• Orientation of rocks/planes relative to stresses

• I.e. applying a tectonic stress leads to strain

Stress – σStrain - ε

•

Deformation ≠ strain

Components of deformation

• Translation• change in the

position of a body

• Distortion • change in shape of

a body• Includes dilation

• Change in volume

• Rotation• Change in position

by pivoting around a fixed axis

Strain versus deformation• Strain describes how points within

a body have changed relative to each other – ie distortion = strain

• Deformation describes how points in a body have moved relative to an external reference frame

Strain can be homogenous or inhomogenous

Particle movements during strain• Strain can occur due to • Pure shear• Simple shear• General shear

• Combination of Pure and Simple shear

• Particles flow along predictable paths during progressive strain accumulation

Pure shear/coaxial strain General shear Simple shear – Non coaxial strain