Force and Stress Processes in Structural Geology & Tectonics Ben van der Pluijm WW Norton+Authors, unless noted otherwise 1/9/2017 12:35 PM
We Discuss Force and Stress Force and Units (Trigonometry) Newtonian and Continuum Mechanics 2D and 3D Stress Normal Stress and Shear Stress Mohr Construction of Stress Stress States Measurement of Stress At surface At depth Lithospheric Stresses Modern Geologic Provinces Crustal Stress Trajectories Force&Stress PSG&T 2
Mechanics: Force and Displacements (a) Newtonian mechanics: displacements between bodies 1 st Law: No force on object means constant velocity (inertia law) 2 nd Law: F = m.a 3 rd Law: F = -F (b) Continuum mechanics: displacement between and within bodies Force&Stress PSG&T 3
Stress and Tractions Stress = Force/Area (or, stress is intensity of force ) Stress on a plane in space is defined by stress acting perpendicular to plane (normal stress, s n ) and stress acting along plane (shear stress, s s ). These resolved stresses are also called tractions. Force&Stress PSG&T 4
Units and Conversions Stress or Pressure = Force/Area = (m.a)/area = kg.m.s -2.m -2 = N.m -2 = Pa (Pascal) 1 bar (atmospheric conditions) = 1.10 5 Pa (1E5 Pa) = 0.1 MPa 1 kbar (geologic conditions) = 100 MPa Force&Stress PSG&T 5
Trigonometry In a right triangle: θ sin θ = a/c cos θ = b/c tan θ = a/b https://www.khanacademy.org/math/trigonometry Force&Stress PSG&T 6
Relationship between Force and Stress (2D analysis) F = (σ. Area); Here (2D): F = (σ. Length) F n = F cos θ = (σ AB) cos θ = σ EF cos 2 θ (AB = EF cos θ ) F s = F sin θ = (σ AB) sin θ = σ EF cos θ sin θ = σ EF ½(sin 2θ) Thus, corresponding stresses are: σ n = F n /EF = σ cos 2 θ σ s = F s /EF = σ ½(sin2θ) c) normalized values of F n and σ n on plane with angle θ; d) normalized values of F s and σ s on a plane with angle θ. Force&Stress PSG&T 7
Another Explanation σ n = F n /area F n decreases from finite value to 0, while area increases to infinite. σ s = F s /area F s increases from 0 to finite value, while area increases faster to infinite. F, σ Force&Stress PSG&T 8
3D stress 3 rd Law: non-moving object, so, balancing tractions define six independent components: Force&Stress PSG&T 9
Infinitessimal Stress and Stress Ellipsoid Shrink cube to a point (a) two-dimensions: stress ellipse (b) three dimensions: stress ellipsoid Ellipsoid axes are called principal stresses, with s 1 s 2 s 3 Force&Stress PSG&T 10
Normal and Shear Stress Relationships Force&Stress PSG&T 11
Deriving s normal and s shear from Principal Stresses s normal s shear F = s. Area Here (2D): F = s. Length Force&Stress PSG&T 12
Graphical Solution: Mohr Diagram for Stress ϴ is 2 x angle with s 3 Rearrange for s d, and square: Circle with radius r, centered on x-axis at distance a from origin Radius is 1/2(s 1 -s 3 ) = s s, Diameter is (s 1 -s 3 ) = s d So, shear stress is half differential stress Force&Stress PSG&T 13
Planes in Stress and Real Space (a) For each value of shear stress and normal stress there are two corresponding planes, shown in Mohr diagram. Note that this is stress solution space, not real space. (b) Corresponding planes in (σ 1 σ 3 ) space, which is real space. Force&Stress PSG&T 14
Homework: Stress To estimate normal and shear stresses on six planes shown in (a) apply the Mohr construction in the graph (b). The principal stresses and angles θ are given. You should check your estimates from the construction by using the derived Equations for σ n and σ s : Name: Date: Force&Stress PSG&T 15
MohrPlotter Note: MohrPlotter uses poles to planes, meaning angles are complements. So, 30 o angle with s 3 become 60 o and 60 o angle with s 1 becomes 30 o. See stereonet labs. Confusing, I know. Windows: http://www.geo.cornell.edu/geology/faculty/rwa/programs/ mohrplotterwin.zip Mac: http://www.geo.cornell.edu/geology/faculty/rwa/programs/ mohrplottermac.zip R Allmendinger Force&Stress PSG&T 17
3D Stress states Mohr circle construction allows 3D stress ellipsoid solution in 2D plot space a) General triaxial stress: σ 1 > σ 2 > σ 3 0 b) Biaxial (or plane) stress: one axis = 0 (e.g., σ 1 > σ 2 > 0) c) Uniaxial compression: σ 1 > 0; σ 2 = σ 3 = 0 (and Uniaxial tension: σ 1 = σ 2 = 0; σ 3 < 0) d) Pressure (or Hydrostatic/Lithostatic stress): σ 1 = σ 2 = σ 3 Force&Stress PSG&T 18
Isotropic and Non-isotropic Stress (a) volume change, reflects mean stress, s m (b) shape change, reflects deviatoric stresses, s dev s m + s dev Matrix algebra Force&Stress PSG&T 19
Extra: Stress tensor and Matrix Algebra The transformation of point P defined by coordinates P(x, y, z) to point P (x, y, z ). We describe the transformation of the three coordinates of P as a function of P by The tensor that describes the transformation from P to P is the matrix: In matrix notation, the nine components of a stress tensor are: with σ 11 oriented parallel to the 1-axis and acting on a plane perpendicular to the 1-axis, σ 12 oriented parallel to the 1-axis and acting on a plane perpendicular to the 2-axis, and so on. Mean stress and deviatoric stresses: Force&Stress PSG&T 20
Isotropic Stress: Lithostatic Pressure (P l ) Mean stress is hydrostatic (water) pressure, P h, or lithostatic (rock) pressure, P l. Consider rock at depth of 3 km, lithostatic pressure is : P l = ρ g h If ρ (density) is 2700 kg/m 3, g (gravity) is 9.8 m/s 2, h (depth) is 3000 m, we get: P l = 2700 9.8 3000 = 79.4 106 Pa 80 MPa (or 800 bars) For every kilometer in Earth s crust, lithostatic pressure increases by approximately 27 MPa (and more with higher density rocks) -- or, ~1kbar per 3.3km. Corresponding P h = ρ g h = 1000 9.8 3000 29 Mpa (or 290 bars) Note that deviatoric stresses arising form plate movement are much smaller (see later)! Force&Stress PSG&T 21
Modern Stress Measurements Borehole breakouts - Shape of a borehole changes after drilling in response to stresses in host rock. Hole becomes elliptical with long axis of ellipse parallel to minimum horizontal principal stress (σ s, hor). Hydrofracture - If water is pumped under sufficient pressure into a well that is sealed off, host rock will fracture. Fractures are parallel to maximum principal stress (σ 1 ), because water pressure necessary to open fractures is equal to minimum principal stress. Strain release - A strain gauge, consisting of tiny electrical resistors in a thin plastic sheet, is glued to bottom of borehole. The hole is drilled deeper with a hollow drill bit (called overcoring), thereby separating core to which strain gauge is connected from wall of hole. Inner core expands (by elastic relaxation), which is measured by the strain gauge. The direction of maximum elongation is parallel to direction of maximum compressive stress and its magnitude is proportional to stress according to Hooke s Law. Fault-plane solutions - When an earthquake occurs, global seismometer records of first motion divide area into two sectors of compression (white) and two sectors of tension (gray), separated by two perpendicular planes. One of these planes is fault plane on which earthquake occurred, and from distribution of compressive and tensile sectors, sense of slip on the fault is determined. Bisector of two planes in tensile sector (T) represents minimum principal stress (σ3) and bisector in compressive field (P) parallels maximum compressive stress (σ 1 ). USGS Force&Stress PSG&T 22
Heidbach, O., Tingay, M., Barth, A., Reinecker, J., Kurfeß, D., and Müller, B., The World Stress Map database release 2008 doi:10.1594/gfz.wsm.rel2008, 2008: www.world-stress-map.org http://www.world-stress-map.org/ World Stress Map (2008) Fossen (2010) 23 Force&Stress PSG&T
Differential Stress with Depth KTB, Germany (9100m) P(litho) = ρ g h ρ (density) is 2700 kg/m 3, g (gravity) is 9.8 m/s 2, and h (depth) is 1000 m, P(litho) 27 MPa (or 800 bars) For every kilometer in Earth s crust, isotropic lithostatic pressure increases by approximately 27MPa (or 100Mpa=1kbar per ~3.3 km). Instead, non-isotropic differential stress (s d = s 1 -s 3 ) is smaller, increasing to a few hundred MPa until dropping at Frictional-Plastic Transition (see later). Force&Stress PSG&T 24
Differential Stress and Geologic Provinces Peanut butter sandwich model (s 1 -s 3 ) is differential stress Cold lithosphere (cratons; Precambrian rocks). Hot lithosphere (orogens, ocean floor; Cenozoic rocks). Force&Stress PSG&T 25
Crustal Stress Trajectories and Failure Stress trajectories of σ 1 (full lines) and σ 3 (dashed lines) in a crustal block that is pushed from left, resisted by frictional forces at its base. Red dashes are predicted failure surfaces (Coulomb failure criterion; see later). Angle between maximum principal stress (σ 1 ) and fault surface (around 30, Coulomb failure criterion; see later) predicts orientation and curvature of (reverse) faults as common in nature. Force&Stress PSG&T 26