Introduction to Strain and Borehole Strainmeter Data
Evelyn RoeloffsU. S. Geological Survey
Earthquake Science Center
March 28, 2016
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 1 / 28
Plate Boundary Observatory (PBO) BoreholeStrainmeter Network
Funded by NSF as partof the Earthscopeiniative
78 Gladwin TensorStrainmeters
Installed 2004-2013
Depths 500-800 feet(150-250 m)
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 2 / 28
UNAVCO Engineers Installing B201 (Mount St. Helens)
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 3 / 28
A typical PBO borehole strainmeter installation
Strainmeter is grouted into anuncased section of borehole
Every PBO borehole alsocontains a 3-componentseismometer
23 of the boreholes alsocontain pore pressure sensors
PBO boreholes at Mount St.Helens and Yellowstone alsocontain tiltmeters
Circular diagram shows BSMgauge orientations
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 4 / 28
Gladwin Tensor Strain Meter
Developed in Australia byMichael Gladwin
Four ”gauges” measure innerdiameter of steel housing
Three gauges (CH0, CH1, andCH2) are 120◦ apart aroundthe borehole axis
The fourth gauge (CH3) isperpendicular to CH1
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 5 / 28
Gladwin Tensor Strain Meter - Capacitive SensingElement
Gauge changes length inresponse to strain along axis
Gap d1 is fixed
Strain changes capacitanceacross gap d2
Capacitance changes aremeasured using a bridgecircuit whose other arms areat the surface
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 6 / 28
Strainmeters fill ”gap” between Seismology and GPS
Strain = spatial derivative of displacement
Seismometer: measures time derivative of displacement; needarray to measure strain
GPS: measures displacement; need array to measure strain
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 7 / 28
Strainmeter ”niche” is hours to days
7.5 yearsdaily averages
996 1000 1004 1008
07/18/09 07/25/09 08/01/09 08/08/09 08/15/09 08/22/09 08/29/09
B201 barometer (hPa)
0 4 8
12
mm B201 rainfall
-40-20 0 20 40
nano
stra
in
B201 CH3 N 48.0E cc.tb
-40-20 0 20 40 B201 CH2 N 78.0E c
c.tb
-40-20 0 20 40 B201 CH1 N138.0E c
c.tb
-40
0
40
80B201 coldwt201bwa2007 Coldwater Visitor Center Mount St Helens
B201 CH0 N 18.0E cc.tb
45 days30-minute samples
detrended
4 minutes20 samples per second
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 8 / 28
Basic GTSM data processing steps
7.5 yearsdaily averages 996
1000 1004 1008
07/18/09 07/25/09 08/01/09 08/08/09 08/15/09 08/22/09 08/29/09
B201 barometer (hPa)
0 4 8
12
mm B201 rainfall
-40-20 0 20 40
nano
stra
in
B201 CH3 N 48.0E cc.tb
-40-20 0 20 40 B201 CH2 N 78.0E c
c.tb
-40-20 0 20 40 B201 CH1 N138.0E c
c.tb
-40
0
40
80B201 coldwt201bwa2007 Coldwater Visitor Center Mount St Helens
B201 CH0 N 18.0E cc.tb
45 days30-minute samples
detrended
Applied to figures shown here:
”Clean” the dataRemove long-term trendsCorrect for atmosphericpressureRemove tidal variations
To be discussed:
Optionally, develop yourown calibrationsObtain strain components aslinear combinations of gaugeelongations
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 9 / 28
What is ”strain”?
”Strain” is a change in one or more dimensions of a solid body,relative to a reference state
Size may changeShape may change
We assume here that strains are small, so ”infinitesimal straintheory” applies
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 10 / 28
Coordinate Systems
Right-handed Cartesian coordinate system
Various sets of names for coordinate axes (examples above)
Strainmeters do not care about:
Curvature of the earthGeodetic reference frames
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 11 / 28
Displacements
Material at a point can move in three directions, e.g. (ux, uy, uz)
Various sets of names for components of displacement
e.g., 1,2,3 or x, y, zHorizontal axes will not always be East and North
Strain is a result of spatially varying displacement
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 12 / 28
Spatial derivatives of displacement
Displacements can vary in three coordinate directions
9 partial derivatives ∂ui∂xj
where i, j = 1, 2, 3
Strain components: εij = 12 [ ∂ui
∂xj+
∂uj
∂xi]
”Normal” strains have i = j: εii = ∂ui∂xi
(no summation implied)
”Shear” strains have i 6= j, note that εij = εji
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 13 / 28
Horizontal strain components
unstrainedmaterial
+x
+y
εxxε εyy εxy
contraction in the x-direction
(a negative strain)extension
in the y-direction(a positive strain)
xy shear(a positive strain becausey-displacement increases
with increasing x)
εxx = ∂ux∂x εyy =
∂uy
∂y εxy = 12 [∂ux
∂y +∂uy
∂x ]
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 14 / 28
”Units” of strain
Strain is dimensionless but often referred to as if it had units:
εxx = 0.01 might be called ”1% strain” or ”10,000 microstrain” or”10,000 ppm strain”1 mm shortening of a 1-km baseline is a strain of −10−6 = -1microstrain = -1 ppm strain1 mm lengthening of a 1000-km baseline is a strain of 10−9 = 1nanostrain = 1 ppb strain
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 15 / 28
Sign conventions
The following sign conventions are used here to minimizemathematical confusion:
Increasing length (”extension”) is positive strain; decreasing length(”contraction”) is negative strainIncreasing area or volume (”expansion”) is positive strainStresses that produce positive strains are positive (ie., tension ispositive)Stresses that produce negative strains are negative (ie., compressionis negative)
...but note that some publications do not use these signconventions:
In geotechnical literature, contraction and compressional stress arereferred to as positivePublished work on volumetric strainmeter data describescontraction as positive
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 16 / 28
Example: Locked vs. creeping strike-slip fault
Map view
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 17 / 28
GTSM gauge output is proportional to inner diameterchange of housing
Empty Borehole
Goal: Measure the strain of
the formation
Deforms much more than formation
Borehole filledwith rigid material
Does not deform at all
Strainmeter Groutedinto Borehole
Deforms more than formation but less
than empty boreholeDeformation depends on relative moduli of
strainmeter, grout, and formationEvelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 18 / 28
Elongation of a single ideal gauge in response to strain
Gauge elongation, ei, is a linear combination of strainparallel and perpendicular to the gauge
ex = Aεxx −BεyyA and B are positive scalars with A > B
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 19 / 28
Areal strain, differential extension, engineering shear
unstrainedmaterial
+x
+y
xxε yy+ε2εxy
areal contraction (a negative strain)
differential extension(a positive strain) engineering shear
(a positive strain becausey-displacement increases
with increasing x)
xxε yy−ε
Areal strain εxx + εyy does not change if axes are rotatedDifferential extension (εxx − εyy) and engineering shear 2εxy areshear strain components
Neither shear strain component changes areaShear strains depend on coordinate system
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 20 / 28
Elongation of ideal gauge: Areal strain and differentialextension
ex = Aεxx −Bεyy can be re-written as linear combination of arealstrain and differential extension
Rearrange: ex = 0.5(A−B)(εxx + εyy) + 0.5(A+B)(εxx − εyy)
Define C = 0.5(A−B) and D = 0.5(A+B)
ex = C(εxx + εyy) +D(εxx − εyy)
NOTE: εxy does not change length for ideal gauge parallel to x or y
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 21 / 28
PBO 4-component GTSM: Gauge configuration
CH0
CH1
CH2
CH3
North
φ0
60o
60o
30o
This is the azimuthon the PBO web pagegiven CW from North
=e1
=e3
x
y
CH0,CH1, and CH2 are equally spacedCH3 is perpendicular to CH1
Blue dots: end of gauge whose azimuth is givenRed circles: ends of CH2 and CH0 that are -120o and +120o from CH1
1
1
CH2 =e0
CH0 =e2
East
θ
x1 and y1 are Cartesiancoordinates parallel andperpendicular to CH1
Azimuths are clockwise fromNorth; Polar coordinate anglesare counterclockwise from East
Polar coordinates (r, θ) areused for math
The polar angle of y1 is +90from the polar angle of x1
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 22 / 28
2 strain components from 2 perpendicular ideal gauges
x, y = directions ofperpendicular gauges CH1and CH3e1 = C(εxx+εyy)+D(εxx−εyy)e3 = C(εxx+εyy)−D(εxx−εyy)
Solve for areal strain and differential extension:(εxx + εyy) = (e1 + e3)/2C(εxx − εyy) = (e1 − e3)/2DAreal strain is proportional to average of gauge elongations
Differential extension is proportional to difference between gaugeelongations
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 23 / 28
Transforming horizontal strains to rotated coordinates
Horizontal strain tensor canbe expressed in a coordinatesystem rotated about thevertical axis
εx′x′ + εy′y′
εx′x′ − εy′y′2εx′y′
=
1 0 00 cos 2θ sin 2θ0 − sin 2θ cos 2θ
εxx + εyyεxx − εyy
2εxy
Areal strain is invariant under rotation:
εx′x′ + εy′y′ = εxx + εyy for any value of θ
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 24 / 28
Expressing gauge elongations in rotated coordinates
Each gauge has its owngauge-parallel coordinates, e.g.:e1 = C(εxx + εyy) +D(εxx − εyy)e0 = C(εx′x′ +εy′y′)+D(εx′x′−εy′y′)
Express CH2 elongation in CH1-parallel coordinates:
εx′x′ + εy′y′ = εxx + εyyεx′x′ − εy′y′ = cos 2θ(εxx − εyy) + sin 2θ(2εxy)
e0 = C(εxx + εyy) +D cos 2θ(εxx − εyy) +D sin 2θ(2εxy)e0 = C(εxx + εyy)− 0.5D(εxx − εyy) + 0.866D(2εxy) for θ = 60◦
NOTE: e0 does not respond to 2εx′y′ , but does respond to 2εxy
Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 25 / 28
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)
Express elongations in CH1-parallel coordinates:e0 = C(εxx + εyy) +Dcos(−240◦)(εxx − εyy) +Dsin(−240◦)(2εxy)e1 = C(εxx + εyy) +D(εxx − εyy)e2 = C(εxx + εyy) +Dcos(240◦)(εxx − εyy) +Dsin(240◦)(2εxy)
Solve for strain components:(εxx + εyy) = (e0 + e1 + e2)/3C(εxx − εyy) = [(e1 − e0) + (e1 − e2)]/3D2εxy = (e0 − e2)/[2(0.866D)]
Areal strain is proportional to average of outputs from equallyspaced gauges
Shear strains are proportional to differences among gauge outputs
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Gauge elongations to strain components: Example
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Any questions?
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