1. Force and Stress Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm © WW Norton, unless noted otherwise

2. © EarthStructure (2 nd ed)2 5/28/2016 Mechanics (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

3. © EarthStructure (2 nd ed)3 5/28/2016 Units and conversions Stress = Force/Area = (m. a)/Area = kg.m.s -2.m -2 = N.m -2 = Pa (Pascal) 1E5 = 0.1MPa = 1bar ; 1kbar = 100 MPa

4. © EarthStructure (2 nd ed)4 5/28/2016 Stress on a plane: Tractions Stress on a two- dimensional plane is defined by a stress acting perpendicular to plane (normal stress) and a stress acting along plane (shear stress).

5. © EarthStructure (2 nd ed)5 5/28/2016 Relationship between Force and Stress F = . area (c) normalized values of Fn and σn on plane with angle θ; (d) normalized values of Fs and σs on a plane with angle θ.

6. © EarthStructure (2 nd ed)6 5/28/2016 3D stress 3 rd Law: six independent components:

7. © EarthStructure (2 nd ed)7 5/28/2016 Infinitessimal stress and stress ellipsoid (a) two-dimensions (stress ellipse); (b) three dimensions (stress ellipsoid) Principal stresses:  ≥  ≥ 

8. © EarthStructure (2 nd ed)8 5/28/2016 Interactive stress module

9. © EarthStructure (2 nd ed)9 5/28/2016 Normal and shear stress relationships

10. © EarthStructure (2 nd ed)10 5/28/2016 Deriving normal and shear stress F = . area

11. © EarthStructure (2 nd ed)11 5/28/2016 Mohr diagram for stress Rearrange: Circle with radius r, centered on x-axis at distance a from origin Radius is 1/2(  1-  3) =  s, or half the differential stress

12. © EarthStructure (2 nd ed)12 5/28/2016 Planes in stress space For each value of the shear stress and the normal stress there are two corresponding planes, as shown in the Mohr diagram (a). The corresponding planes in σ1 – σ3 space are shown in (b).

13. © EarthStructure (2 nd ed)13 5/28/2016 Homework To estimate the normal and shear stresses on the 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 σn and σs:

14. © EarthStructure (2 nd ed)14 5/28/2016 Stress states

15. © EarthStructure (2 nd ed)15 5/28/2016 Isotropic and non-isotropic stress (a) volume change and (b) shape change, reflecting mean stress and deviatoric stresses, respectively:

16. © EarthStructure (2 nd ed)16 5/28/2016 Stress tensor 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 stress:

17. © EarthStructure (2 nd ed)17 5/28/2016 Stress measurement Fossen (2010)

18. © EarthStructure (2 nd ed)18 5/28/2016 World Stress Map (2008) 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

19. World Stress Map © EarthStructure (2 nd ed)19 5/28/2016 Fossen (2010) www.world-stress-map.org

20. © EarthStructure (2 nd ed)20 5/28/2016 Global stress fields

21. © EarthStructure (2 nd ed)21 5/28/2016 Stress at depth P(litho) = ρ ⋅ g ⋅ h If ρ (density is 2700 kg/m3, g (gravity) is 9.8 m/s2, and h (depth) is 3000 m: P(litho) = 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 30 MPa. Differential stress increases to a few hundred MPa until Brittle-Plastic transition

22. © EarthStructure (2 nd ed)22 5/28/2016 Lithospheric stress a) Cold lithosphere (cratons; Precambrian rocks) b) Hot lithosphere (orogenic belts, ocean floor; Cenozoic rocks)  1 -  3 is differential stress)

23. © EarthStructure (2 nd ed)23 5/28/2016 StressMohr

24. Extra © EarthStructure (2 nd ed)24 5/28/2016

25. © EarthStructure (2 nd ed)25 5/28/2016 Stress trajectories