Today: Back to stress stress in the earth Strain Measuring Strain Strain Examples Today: Back to stress stress in the earth Strain Measuring Strain Strain Examples

Present day stress Difficult to measure  EQ focal mechanisms  Bore-hole breakouts  in situ measurements Measuring Stress in situ borehole measurements of  d (  1 –  3 ) with depth.

World stress map and topography showing maximum horizontal stress. Stress in the Earth

Generalized pattern based on stress trajectories for individual plates. Stress in the Earth

Strength – the ability of a material to support differential stress Maximum stress before a rock fails Strength curves: differential stress magnitude versus depth. Stress and strength at depth

Strength – the ability of a material to support differential stress Maximum stress before a rock fails Strength curves: differential stress magnitude versus depth. Stress and strength at depth A.Regional with normal geothermal gradient B.Regional low geothermal gradients Give some geologic examples? This is important and will be on exam 1!

Strain Deformation: collective displacements of points in a body.  distortion  translation  rotation

Strain and Stress Deformation occurs in response to stress We know some simple stress-strain relations; Elastic behaviour Viscous behaviour This field of study is called Rheology (Ch 5)

The deformation field is subdivided into 3 components:  ?? Components of deformation:

State of stress and Change in shape Deviatoric stress=> shape changes, Hydrostatic stress=> changes of volume (for example, burial) Dilation: area or volume change As a geologist, this is hard to measure; often can’t include in strain analysis.

Homogenous strain and the strain ellipsoid Homogeneous strain  Original straight lines remain straight  Origin parallel lines remain parallel  Circles become ellipses, in 3-D, spheres becomes ellipsoids Homogenous strain (2-D): two material lines that do not rotate relative to one another

Homogeneous strain describes the transformation of a square to a rectangle or a circle to an ellipse.  Two material lines that remain perpendicular before and after strain are the principal axes of the strain ellipse  The dashed lines are material lines that do not remain perpendicular after strain, the rotate towards the long axes of the strain ellipse.

Homogeneous strain  Note the two material lines that form the ellipse change length.  In 3-D, the three material lines that remain perpendicular also change length from initial to final stages.   The lines that are perpendicular before and after strain are called Principal Strain Axes.

Finite Strain vs Incremental Strain These two circles have undergone identical finite strain. But, how did they get there? Initial State Final State Finite Strain: The total, final strain Typically what we measure as geologists

Finite strains X f and Y f are the same in (a) and (b), but the strain path is different. They underwent different incremental strain evolutions.. Very different tectonic histories…. Importance of incremental strain history of rocks and limitation of finite strain analysis

Coaxial strain accumulation  The principal incremental strain axes remain perpendicular to the finite strain axes.  The magnitude of strain axes change with each step in both types of strain Coaxial and non-coaxial strain accumulation

Non-coaxial strain accumulation  All material lines except those that remain perpendicular before and after strain, rotate relative to one another.  Here the principal incremental strain axes rotate relative to the finite strain axes. Coaxial and non-coaxial strain accumulation

Small fold with axial plane cleavage. Strain Quantities How much strain has occured? How do we determine this? Quantifying Strain  Line length change  Volume change  Angular change

Longitudinal strain Longitudinal strain is expressed by elongation, e : e = (l f – l o )/l o Negative e = shortening Positive e = extension Maximum, e 1 Minimum, e 3 e x 100 = % extension or shortening

Volumetric strain Volumetric strain,   = (V f – V o )/V o or  =  V/V o Negative  = V loss Positive  = V gain

Angular strain Angular strain is the change in angle between 2 lines that were initially perpendicular. The change in the angle, angular shear, . The tangent of  is shear strain,   = tan 

Strain States (a) General strain (b) Axially symmetric extension (c) Axially symmetric shortening (d) Plane strain (e) Simple shortening

What do we learn? Measure strain magnitudes across a region, an outcrop or hand specimen Quantify strain history Sharp increase of strain may define a region of high strain = ductile shear zone Shortening rate in metamorphic rocks or along brittle faults

Quantifying strain examples: Longitudinal strain is expressed by elongation (or extension), e : where e = (l f – l o )/l o Angular strain is expressed by The change in the angle: angular shear, . and: shear strain,  Where  = tan 

Quantifying strain examples: