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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

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## Introduction to strain and borehole strainmeter data

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**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 astrain 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 forx >0. • Creeping at plate rate: steep displacement gradientat fault. • Creeping below plate rate: negative shear strain near fault • Locked fault: shear strain is distributed over a wide area. • uy decreases fromplate rate to zero with increasing y • yy stretches material where x <0 and contracts it where x >0.**StrainMatrices**• 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 = - (xxyy)**Response of one PBO strainmeter gauge to horizontal strain**• A strainmeter gauge measures change of housing’s inner diameter • x and y areparallel 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 = Axx-Byy • A andBare 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= Cxx+yy+ Dxx-yy • xx+yyis "areal strain" ; xx-yy is "differential extension".**2 strain components from 2 gauges**• For gauge along the x-axis, elongation is: ex = Cxx+yy+ Dxx-yy • For a gauge aligned along the y -axis, with same response coefficients C and D, the gauge elongation is ey = Cxx+yy- Dxx-yy • Cansolve for arealstrain and differentialextension: xx+yy=0.5(1/ C) ex + ey xx-yy=0.5(1/ D) ex - ey • To obtain engineering shear, need a thirdgauge…**Gauge configuration of PBO 4-component 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 = Cxx+yy+ Dcosxx-yy + Dsinxy e1 = Cxx+yy+ Dxx-yy e2 = Cxx+yy+ D cosxx-yy+ D sinxy Solve for straincomponents: 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.**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 everyinterior surface. • Thej-th component of internal force acting on a plane whose normal is in thexi direction is theij-component of the Cauchy stress tensor, σij. • 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 .**Stress as a matrix (tensor)**• 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: σij = σji.**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: • Gis the shear modulus units of force per unit area); isthe 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

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