Deformation and Strain
Deformation occurs in response to forces (stress)
No simple stress-deformation relations
Deformation: collective
displacements of points
in a body.
 distortion
 translation
 rotation
Strain: describes changes
displacements of points in
a body relative to one
another.
Components of deformation:
The displacement of the corner of the
square (rigid body) represent the
deformation field with strain ellipse.
Strain along axes x and y, which are
the principal strain axes.
The displacement
field is subdivided
into 3
components:
 Distortion
 Translation
 Rotation
What is the missing element
Change in shape
Deviatoric stress: unequal
shape changes
Hydrostatic component: equal
changes of volume (burial)
Dilation: area or volume
change
positive = volume increase
negative = volume decrease
Review of deformation
 Rigid body translation
 Rigid body rotation
 Strain (distortion)
 Volume change (or dilation)
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 strains Xf and Yf are the same in (a) and (b), but the strain
path is different.
Importance of incremental strain history of rocks and limitation of
finite strain analysis.
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
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
Strain Quantities
How much strain is
necessary to produce
these strains?
How do we determine
this?
Quantifying Strain
Line length change
Volume change
Angular change
Small fold with axial plane cleavage.
Longitudinal strain
Longitudinal strain is
expressed by elongation, e:
e = (lf – lo)/lo
Negative e = shortening
Positive e = extension
Maximum, e1
Minimum, e3
e x 100 = % extension or
shortening
Volumetric strain
Volumetric strain, D
D = Vf – Vo)/Vo or
D = dV/Vo
Negative D = V loss
Positive D = 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, y.
The tangent of y is shear
strain, g
g = tan y
D = dV/Vo
Other strain quantities
Quadratic elongation
l = (lf / lo)2 or (1 + e)2
l used in Mohr Circle for strain
Stretch, root of l
s = l ½ = lf / lo = 1 + e
Convenient measures. They describe the lengths of the
principal axes (X, Y, and Z) and the strain ellipsoid. change in
the angle, angular shear, y.
Mohr Circle for Strain
Unit square is shortened 50% and lengthened 100%
e1 = 1 and e3 = -.0.5
thus l1 = 4 and l3 = 0.25
Now get reciprocal values, thus l1 = 0.25 and l3 = 4
Plot on Mohr Strain Circle
Mohr Circle for Strain
For a deformed rock: we use the reciprocal values of the principal strain
a)Plotted in gl - space, where l = 1/l and g = gl
Where, l is quadratic elongation (l/lo)2 and
g is shear strain (tangent of angular shear)
Mohr Circle for Strain
Plot on Mohr Circle with l1 = 0.25 and l3 = 4
Easy to find longitudinal strain (e value) and angular strain for any line
oriented at angle  °) to strain axes.
Plot 2   ° and find corresponding strain values l = 1.1 and g = 1.5
Strain States
(a) General strain
(b) Axially symmetric
extension
(c) Axially symmetric
shortening
(d) Plane strain
(e) Simple shortening
Strain Representation
XY sectional ellipse of the finite
strain ellipsoid in Alps
and
Sectional ellipsoid on cross-section
Long axis of strain ellipse
lies in the direction of thrust
transport
Strain ratio increases with
depth
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
How much strain is required to
fold a rock?
Shortening rate in metamorphic
rocks or along brittle faults