Introduction to strain and borehole strainmeter data Evelyn Roeloffs USGS 3 March 2014

Strains are spatial gradients of displacement Reid’s Elastic Rebound Theory This picture, taken near Bolinas in Marin County by G.K. Gilbert, shows a fence that was offset about 8.5 feet along the trace of the fault (from Steinbrugge Collection of the UC Berkeley Earthquake Engineering Research Center). 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 done

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 Basically done but would be good to enlarge text in figure

"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. Done

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 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 = 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 Steps on Feb 9 2014 caused by creep events on San Andreas fault – see notes for a plot

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, σ 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 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: 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