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Force and Stress
Earth Structure (2nd 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
1st Law: No force on object means constant velocity (inertia law)
2nd Law: F = m .a
3rd Law: F = -F
(b) Continuum mechanics: displacement between and WITHIN bodies
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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
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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).
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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.
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3D stress
3rd Law: six independent components:
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Infinitessimal stress and stress ellipsoid
(a) two-dimensions (stress ellipse); (b) three dimensions (stress ellipsoid)
Principal stresses: s1 ≥ s2 ≥ s3
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Interactive stress module
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Normal and shear stress relationships
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Deriving normal and shear stress
F = s . area
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Mohr diagram for stress
Rearrange:
Circle with radius r, centered on
x-axis at distance a from origin
Radius is 1/2(s1-s3) = ss,
or half the differential stress
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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).
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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:
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Stress states
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Isotropic and non-isotropic stress
(a) volume change and (b) shape change, reflecting mean stress and deviatoric
stresses, respectively:
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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:
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Stress measurement
Fossen (2010)
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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
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World Stress Map
www.world-stress-map.org
Fossen (2010)
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Global stress fields
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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
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Lithospheric stress
a)
b)
Cold lithosphere (cratons; Precambrian rocks)
Hot lithosphere (orogenic belts, ocean floor; Cenozoic rocks)
(s1-s3 is differential stress)
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StressMohr
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Extra
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Stress trajectories
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