Stress and Strain
Stress - force per area (Pa or kg/m-s2)
Strain - deformation of the solid due to stress (dimensionless)
Body Forces and Surface Forces
Body force acts throughout the volume of a solid (e.g., gravity)
Surface force acts on a surface bounding a volume (e.g., lithostatic stress or weight of overburden)
Stress in the Lithosphere - isostatic equilibrium (or Archimedes' principle). In the mantle below the lithosphere, pressure must be equal. Thus, the weight of the overlying rocks must be equal. The thick low density crust floats higher on the asthenosphere. The continental crust is under tension because the force at any given depth in the crust is higher than in the adjacent mantle. The low density crust wants to spread out over the mantle.
Thrust Sheets - Shear stress results from the frictional resistance to motion on the fault. Shear stress is related to the weight of overburden and the coefficient of friction.
Stress in Two Dimensions - State of stress in 2D can be defined by two normal stresses and 1 shear stress (the two shear stresses must be equal or the body would rotate).
Normal and shear stress on any arbitrary plane can be determined from the two normal stresses (x- and y-axes) and one shear stress and the angle between the normal to the plane and the x-axis.
Principal axes of stress are normal to planes of zero shear stress. The principal axes are orthogonal.
Maximum shear stress is 45° from the principal stress direction. Maximum shear stress is one half the difference of the principal stresses.
Stress in Three Dimensions - 9 components of stress but only 6 are independent. Three normal stresses and 3 shear stresses.
Maximum shear stress is one half the difference between the maximum and minimum principal stress.
Hydrostatic state of stress - all 3 principal stresses are equal to the pressure. No shear stresses
Lithostatic state of stress - Hydrostatic state of stress with pressure increasing with depth.
Pressure is equal to their mean value of the 3 principal stresses.
Deviatoric normal stress - normal stress minus pressure
Stress Measurement
Overcoring - drill a hole and install strain meters, then drill an annulus around the original hole and measure the strain induced by relieving the stresses.
Hydrofracture - in an isolated section raise fluid pressure until the rock breaks (fluid pressure drops). This is the breakdown pressure. Then raise pressure back up until it levels off. This is the instantaneous shut in pressure (pressure needed to keep the fractures open). ISIP is equal to the minimum horizontal principal stress.
Basic Ideas about Strain - stresses produce changes in the distances separating neighboring small elements of the solid.
Normal Strains - changes dimensions but not shape. It is the change in length, area, or volume divided by the original length, area, or volume. Dilatation is the volume change divided by original volume. In terms of displacement, w, normal strain is equal to the derivative of the displacement, ¶wx/¶x.
Shear Strains - changes in shape (and possibly dimensions). Usually measured in terms of change in angles between sizes of a rectangle as it is distorted. In terms of displacement, w, shear strain is equal to the average of the cross derivative of the displacements, 0.5*(¶wx/¶y + ¶wy/¶x). Shear strains may also produce rigid body rotations if the cross derivatives of displacement components are not equal. The solid body rotation is equal to 0.5((¶wy/¶x - ¶wx/¶y).
Pure Shear - Changes in angles are the same ¶wx/¶y = ¶wy/¶x. No solid-body rotation
Simple Shear - Only one of the angles is non-zero. This is equivalent to shear plus a solid body rotation. Strike-slip faults usually involve simple shear.
Principal Strains - Shear strain component is zero.
Average Normal Strain - average of the principal strains. Invariant to coordinate system.
Dilatation - change in volume divided by original volume or sum of prinicpal strains. Invariant to coordinate system.
Deviatoric Strains - principal strains minus average normal strain.
Strain Measurments - strains are very small so precise techniques are required. Strain rates are 0.000001 per year or less.
Geodetic Network - triangulation is used. If the distance between two points is known and the angles between the line connecting those two points and a third point are measured. The location of the third point can be determined. A theodolite is used to measure angles between points (monuments). Changes in the angle with time are recorded. If the distance between the points is known, then the deformation rate (velocity) can be determine in the same fashion as rotation about an Euler pole.