Force and Stress Earth Structure 2 nd Edition

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

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 3/2/2021 © Earth. Structure (2 nd ed) 2

Units and conversions Stress = Force/Area = (m. a)/Area = kg. m. s-2. m-2 = N. m-2 = Pa (Pascal) 1 E 5 = 0. 1 MPa = 1 bar ; 1 kbar = 100 MPa 3/2/2021 © Earth. Structure (2 nd ed) 3

Stress on a plane: Tractions Stress on a twodimensional plane is defined by a stress acting perpendicular to plane (normal stress) and a stress acting along plane (shear stress). 3/2/2021 © Earth. Structure (2 nd ed) 4

Relationship between Force and Stress F = s. 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. 3/2/2021 © Earth. Structure (2 nd ed) 5

3 D stress 3 rd Law: six independent components: 3/2/2021 © Earth. Structure (2 nd ed) 6

Infinitessimal stress and stress ellipsoid (a) two-dimensions (stress ellipse); (b) three dimensions (stress ellipsoid) Principal stresses: s 1 ≥ s 2 ≥ s 3 3/2/2021 © Earth. Structure (2 nd ed) 7

Interactive stress module 3/2/2021 © Earth. Structure (2 nd ed) 8

Normal and shear stress relationships 3/2/2021 © Earth. Structure (2 nd ed) 9

Deriving normal and shear stress F = s. area 3/2/2021 © Earth. Structure (2 nd ed) 10

Mohr diagram for stress Rearrange: Circle with radius r, centered on x-axis at distance a from origin Radius is 1/2(s 1 -s 3) = ss, or half the differential stress 3/2/2021 © Earth. Structure (2 nd ed) 11

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). 3/2/2021 © Earth. Structure (2 nd ed) 12

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: 3/2/2021 © Earth. Structure (2 nd ed) 13

Stress states 3/2/2021 © Earth. Structure (2 nd ed) 14

Isotropic and non-isotropic stress (a) volume change and (b) shape change, reflecting mean stress and deviatoric stresses, respectively: 3/2/2021 © Earth. Structure (2 nd ed) 15

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 σ11 oriented parallel to the 1 -axis and acting on a plane perpendicular to the 1 -axis, σ12 oriented parallel to the 1 -axis and acting on a plane perpendicular to the 2 -axis, and so on. Mean stress and deviatoric stress: 3/2/2021 © Earth. Structure (2 nd ed) 16

Stress measurement Fossen (2010) 3/2/2021 © Earth. Structure (2 nd ed) 17

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. Rel 2008, 2008: www. world-stress-map. org 3/2/2021 © Earth. Structure (2 nd ed) 18

World Stress Map www. world-stress-map. org Fossen (2010) 3/2/2021 © Earth. Structure (2 nd ed) 19

Global stress fields 3/2/2021 © Earth. Structure (2 nd ed) 20

Stress at depth P(litho) = ρ ⋅ g ⋅ h If ρ (density is 2700 kg/m 3, g (gravity) is 9. 8 m/s 2, 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 3/2/2021 © Earth. Structure (2 nd ed) 21

Lithospheric stress a) b) Cold lithosphere (cratons; Precambrian rocks) Hot lithosphere (orogenic belts, ocean floor; Cenozoic rocks) (s 1 -s 3 is differential stress) 3/2/2021 © Earth. Structure (2 nd ed) 22

Stress. Mohr 3/2/2021 © Earth. Structure (2 nd ed) 23

Extra 3/2/2021 © Earth. Structure (2 nd ed) 24

Stress trajectories 3/2/2021 © Earth. Structure (2 nd ed) 25