### Introduction to Strain, Evelyn Roeloffs

```Introduction to strain and borehole
strainmeter data
Evelyn Roeloffs
USGS
3 March 2014
Strains are spatial gradients of displacement
Reid’s Elastic Rebound Theory
• Strain near a strike-slip fault
– “At start”: no displacement, no strain
– “Before” earthquake: displacement varies with distance from fault;
area near fault undergoes strain
– After earthquake: elastic rebound reduces strain, leaves offset
Strain, tilt, and stress: Basic math and mechanics
• Basic assumptions
– 1) "small" region:
• The region is small enough that displacement throughout the region is
adequately approximated using displacement at a single point and its
spatial derivatives
– 2) "small" strains:
• Generally we will be speaking of strains in the range 10-10 (0.1
nanostrain) to 10-4 (100 microstrain).
– 3) Only changes matter
• For example, we will consider strain changes caused by atmospheric
pressure fluctuations, but we will not be concerned with the more or
less constant overburden pressure.
Coordinates
• Right-handed coordinate system
• Various sets of names for coordinate axes will be used, for
example:
• Curvature of earth and reference frame distinctions are
unimportant to the way a strainmeter works
Displacements
– Displacement of a point is a vector consisting of 3 scalar
displacements, one in each coordinate direction.
– The scalar displacements can be referred to in various ways:
Strain in 1
dimension
• Rod is of length
and force F stretches it by
• Strain
is the dimensionless quantity
–
•
is positive because the rod is getting longer
depends only on the length change of the rod
– it doesn't matter which end is fixed or free
• The strain is uniform along the entire rod
•
is the only strain component in this 1-D example
"Units" of strain and sign conventions
• Strain is dimensionless but often referred to as if it had units:
– 1% strain is a strain of 0.01 = 10,000 microstrain = 10,000 ppm
– 1 mm change in a 1-km baseline is a strain of 10-6= 1 microstrain=1ppm
– 0.001 mm change in a 1-km baseline is a strain of 10-9= 1 nanostrain = 1 ppb
• Sign conventions that minimize mathematical confusion:
– Increases of length, area, or volume (expansion) are positive strains.
– Shear strains are positive for displacement increasing in the relevant
coordinate direction
• In some publications, contractional strains are described as positive
– In geotechnical literature contraction (and compressional stress) are referred to
as positive.
– Published work on volumetric strainmeter data describes contraction as
positive.
Example:
Transition
from Locked to
Creeping on a
Strike-Slip
Fault
•
•
•
•
•
Relative strike-slip displacement uy>0 for x <0 , uy>0 for x >0.
Creeping at plate rate: steep displacement gradient at fault.
Creeping below plate rate: negative shear strain near fault
Locked fault: shear strain is distributed over a wide area.
uy decreases from plate rate to zero with increasing y
– yy stretches material where x <0 and contracts it where x >0.
Strain
Matrices
• Strain components in 3D as a 3x3 symmetric
matrix:
• Simpler form with no vertical shear strain:
• Simpler form if earth’s surface is a stress-free
boundary:
– zz = - (xxyy)
Response of one PBO strainmeter gauge to
horizontal strain
• A strainmeter gauge measures change of housing’s inner
diameter
• x and y are parallel and perpendicular to the gauge.
The gauge output does not simply represent strain along
the gauge's azimuth.
Response of one gauge, continued
• The gauge's output ex is proportional to L/L:
ex = Axx - Byy
– A and B are positive scalars with A > B.
• Rearrange:
ex = 0.5 (A- B)xx +yy+0.5 (A+B)xx -yy
• Define C = 0.5(A- B) and D = 0.5(A+B) so C<D :
ex = Cxx +yy+ Dxx -yy
• xx +yy is "areal strain" ; xx -yy is "differential extension".
2 strain components from 2 gauges
• For gauge along the x-axis, elongation is:
ex = Cxx +yy+ Dxx -yy
• For a gauge aligned along the y -axis, with same response
coefficients C and D, the gauge elongation is
ey = Cxx +yy- Dxx -yy
• Can solve for areal strain and differential extension:
xx +yy=0.5(1/ C)
ex + ey
xx -yy=0.5(1/ D) ex - ey
• To obtain engineering shear, need a third gauge…
Gauge
configuration of
PBO 4component BSM
• Azimuths are
measured CW
from North.
• Polar coordinate
angles are
measured CCW
• Recommend
polar coordinates
for math.
3 gauge elongations to 3 strain components:
•
•
x, y are parallel and perpendicular to CH1= e1
3 identical gauges 120° apart (CH2,CH1,CH0)=(e0, e1, e2)
e0 = Cxx +yy+ D cosxx -yy + D sinxy
e1 = Cxx +yy+ Dxx -yy
e2 = Cxx +yy+ D cosxx -yy+ D sinxy
Solve for strain components:
exx +eyy =(e0 + e1+ e2 )/3C
(exx -eyy =[(e1 - e0) + (e1 - e2)]/3D
exy =(e0 - e2)/(2 × 0.866 D)
• Areal strain = average of outputs from equally spaced gauges.
• Shear strains= differences among gauge outputs.
From gauge elongations to strain: Example
Stress
• Stresses arise from spatial variation of force
– A force with no spatial variation causes only rigid body motion
– External forces on a body at rest lead to internal forces ("tractions")
acting on every interior surface.
– The j-th component of internal force acting on a plane whose normal is in
the xi direction is the ij-component of the Cauchy stress tensor,  ij.
– The 3 stress components with two equal subscripts are called “normal
stresses”. They apply tension or compression in a specified coordinate
direction. They act parallel to the normal to the face of a cube of
material.
– Stresses with i≠ j are shear stresses. They act parallel to the faces of
Stress as a matrix (tensor)
és xx s xy s xz ù
ê
ú
s 3´3 = ês xy s yy s yz ú
êës xz s yz s zz úû
– The 3 stress components with two equal subscripts are called “normal
stresses”.
• They apply tension or compression in a specified coordinate direction.
• They act parallel to the normal to the face of a cube of material.
– Stresses with i≠ j are shear stresses.
• They act parallel to the faces of the cube.
• Shear stresses are often denoted with a  instead of a
.
– To balance moments acting on internal volumes, shear stresses must be
symmetric: ij =ji .
Stress-strain equations: Isotropic elastic medium
• “Constitutive equations” describe coupling of stress and strain
• For a linearly elastic medium, constitutive relations say that
strain is proportional to stress, in 3 dimensions.
– An isotropic material has equal mechanical properties in all directions.
• Constitutive equations in an isotropic linearly elastic material:
•
G is the shear modulus units of force per unit area);  is the
Poisson ratio (dimensionless).
Elastic moduli and relationships among them
• Only two independent material properties are needed to
relate stress and strain in an isotropic elastic material, but
there are many equivalent alternative pairs of properties.
• K, E, and G are “moduli”( dimensions of force/unit area).
• The Poisson ratio couples extension in one direction to
contraction in the perpendicular directions.
– It is always >0 and <0.5, taking on the upper limit of 0.5 for liquids.
– The Poisson ratio is dimensionless and is not a modulus.
Borehole
strainmeters
as elastic
inclusions the need for in
situ
calibration
• If 2 identical strainmeters in formations with different elastic
moduli are subject to the same in situ stress state, the
strainmeter in the stiffer formation will deform more.
• To convert the strainmeter output to a measurement of the
strain that would have occurred in the formation (before the
strainmeter was installed), the strainmeter's response to a
known strain must be used to "calibrate" the strainmeter.
– The solid earth tidal strain is usually used as this "known" strain.
Strains accompanying seismic waves can also be used.
Topics for later presentations:
•
•
•
•
Removal of atmospheric pressure and earth tide effects
Removal of long-term trends
Rotating strains to different coordinate systems
Seasonal signals
```