Homogeneous Strain Geometry
Strain records the history of
ductile deformation of a rock.
The mathematical formulation
of strain is similar to stress.
There are principal axes of
strain where shear strains are
equal to zero.
The strain
tensor is symmetric with
volumetric
(diagonal)
and
shear (off-diagonal) elements.
The strain ellipse is the strain’s
analog to stress’s Mohr Circle.
12.1 Measures of Strain
Strain is the change of length and shape
of a body during deformation.
Plane strain occurs only in two
dimensions.
Strain is homogenous when the strain
experienced by any small part is
geometrically the same as that of the
whole.
--Linear strain describes lengthening or
shortening along a line.
Stretch (sn) is the ratio of the deformed
length (x) to the undeformed length (L).
sn=x/L
Extension (en) is the ratio of the change
in length (ΔL) over the undeformed length
en= ΔL/L= (x-L)/L=sn-1
--Shear strain (es) is a change in shape as
described by a change in angle (i.e. square
to parallelogram, circle to ellipse).
es=0.5tan()
Shear angle () is shown in Figures 12.2
and 12.3 of Twiss and Moores.
12.2 The State of Strain
Homogenous strain always deforms
circles and spheres into strain ellipses
and strain ellipsoids, respectively. The
strain ellipsoid completely defines the
state of strain.
The axes of the strain ellipsoid are called
the principal axes of strain.
Principal extensions (ê1, ê2 ,ê3) and
principal stretches (s1, s2, s3) are parallel
to the principal axes of strain.
They are defined such that:
e1>e2>e3 & s1>s2>s3
Figure: Principal axes of strain
Area strain
describes the twodimensional change in area caused by
deformation.
Volume strain describes the change in
volume caused by deformation.
The inverse strain ellipse completely
describes the deformation similarly to the
strain ellipse. It uses an undeformed
ellipse that becomes a circle after
deformation (rather than an undeformed
circle to a deformed ellipse).
12.3 Special States of Strain
--- General Strains--Triaxial strain: deformation along all 3
principal strain axes
Plane strain: strain in 2D; no
deformation parallel to intermediate axis
Constant-volume strain: volume stays
the same; (sv=1) (ev=0)
---Homogeneous Strains--Pure strain: principal axes do not change
orientation (no rotational component)
Uniaxial strain: non-zero strain in only
one dimension
Pure shear: constant-volume plane strain
without rotation (T&M Figure 12.9)
Simple shear: constant-volume plane
strain where displacement is strictly
parallel to the shear plane (shears like a
deck of cards)
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Uniform dilation: pure volumetric strain
with no change in shape (e.g. a small cube
to a larger cube)
Flattening: pure triaxial strain that
deforms a cube into a plate-like
rectangular prism (or sphere to pancake)
Simple flattening: flattening for which
the stretches in the two principal
lengthening directions are equal
Constriction: pure triaxial strain where
one dimension is lengthened and the other
two are shortened
Simple constriction: same as above with
the two axes equally shortened
12.4 Progressive Deformation
Progressive strain (deformation) is the
non-rigid motion of a body that results in
the final deformed state.
A strain path is defined by the strain
states through which the body passes
during the deformation process.
The incremental strain ellipse represents
the amount of strain for an increment of
time (very small increments result in
instantaneous strain ellipses).
If the instantaneous strain ellipse is the
same for every increment in time, the
motion is called steady motion.
In irrotational progressive deformation,
the principle axes of strain do not rotate
with respect to the coordinate system.
However, they do rotate in rotational
progressive deformation.
If the principle axes of finite strain are
always parallel to the principle axes of
instantaneous strain, it is coaxial
progressive deformation. Non-coaxial is
when the two axes are not always parallel.
12.5 Progressive Stretch of Material
Lines
A feature (plane, line, etc.) is called a
material object if it is always defined by
the same particles.
The strain ellipse can be divided into
different sectors based on whether the
material object has been lengthened or
shortened. These different sectors move
during the progressive strain and some
areas can experience shortening and then
lengthening or vice versa.
Amanda Sherman & Hunter Philson, 2011
Edited by Katherine Guns and Ben Paulus, 2013
12.6 Homogeneous and Inhomogeneous
Deformation
Homogeneity is dependent on scale.
Deformation can be homogeneous on a
small scale but inhomogeneous on a
larger one, or vice versa. For something to
be locally homogenous the area
considered must be small compared to
large-scale inhomogeneities, but large
compared to tiny inhomogeneities that we
are not interested in.
symmetric about the diagonal (like the
stress tensor).
If we are using the
principal strain coordinate system, the offdiagonal elements are zero (also like the
stress tensor).
The deformation tensor is an alternate
matrix representation of strain where the
diagonal components are given in
stretches instead of extensions.
References & Resources
Robert J. Twiss, Eldridge M. Moores,
Structural Geology 2nd edition, (W. H.
Freeman), p. 319-355, 2006
12.7 Representation of Strain in 3D
Figure 12.21: Flinn diagram
Using ratios of the principle stretches, a
Flinn diagram depicts states of triaxial
strain for constant volume deformation.
The diagram shows only shape changes.
This is not a problem because volume
changes are difficult to determine in the
field anyway.
k= (a-1)/(b-1) where a and b are defined
on the diagram as ratios of principle
stretches.
12.8 Tensor Representations of Strain
Figure 12.39: The strain tensor
The strain tensor is a matrix depiction of
strain where the diagonal elements
represent the volumetric extensions and
the off-diagonal elements represent the
shear strains. The strain tensor is always
Amanda Sherman & Hunter Philson, 2011
Edited by Katherine Guns and Ben Paulus, 2013
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