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Amanda Sherman & Hunter Philson, 2011
Edited by Katherine Guns and Ben Paulus, 2013
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.
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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
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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:
e1e2e3 & s1s2s3
Figure: Principal axes of strain
Area strain describes the two-dimensional 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)
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.
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.
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 symmetric about the diagonal (like the stress tensor). If we are using the principal strain coordinate system, the off-diagonal 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
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Amanda Sherman & Hunter Philson, 2011
Edited by Katherine Guns and Ben Paulus, 2013