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Geodynamics

Forces and stresses Lecture 3.4 - Stresses in 3D

Lecturer: David Whipp david.whipp@helsinki.fi

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Goal of this lecture

• Present stresses in 3D and some common terminology for referring to stress

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Stress in three dimensions

• In three dimensions, we consider forces acting on all six faces of an infinitesimal cube of dimension 𝛿𝑥 ⨉ 𝛿𝑦 ⨉ 𝛿𝑧

• Normal stresses: 𝜎𝑥𝑥, 𝜎𝑦𝑦, 𝜎𝑧𝑧

• Shear stresses: 𝜎𝑥𝑦, 𝜎𝑦𝑥, 𝜎𝑥𝑧, 𝜎𝑧𝑥, 𝜎𝑦𝑧, 𝜎𝑧𝑦

• At equilibrium we can state 𝜎𝑥𝑦 = 𝜎𝑦𝑥, 𝜎𝑥𝑧 = 𝜎𝑧𝑥, 𝜎𝑦𝑧 = 𝜎𝑧𝑦

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Fig. 2.15, Turcotte and Schubert, 2014

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Stress in three dimensions

• As before, we can also determine the principal stresses in three dimensions

• The convention is that 𝜎1 ≥ 𝜎2 ≥ 𝜎3

• Isotropic stress is when 𝜎1 = 𝜎2 = 𝜎3 = 𝑝

• The hydrostatic state of stress is when the normal stresses equal 𝑝 and there are no shear stresses

• The lithostatic stress is the hydrostatic state of stress where stress increases in proportion to the density of the overlying rock

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Fig. 2.15, Turcotte and Schubert, 2014

www.helsinki.fi/yliopistoGeodynamics

Stress in three dimensions

• As before, we can also determine the principal stresses in three dimensions

• The convention is that 𝜎1 ≥ 𝜎2 ≥ 𝜎3

• Isotropic stress is when 𝜎1 = 𝜎2 = 𝜎3 = 𝑝

• The hydrostatic state of stress is when the normal stresses equal 𝑝 and there are no shear stresses

• The lithostatic stress is the hydrostatic state of stress where stress increases in proportion to the density of the overlying rock

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Fig. 2.15, Turcotte and Schubert, 2014

www.helsinki.fi/yliopistoGeodynamics

Stress in three dimensions

• As before, we can also determine the principal stresses in three dimensions

• The convention is that 𝜎1 ≥ 𝜎2 ≥ 𝜎3

• Isotropic stress is when 𝜎1 = 𝜎2 = 𝜎3 = 𝑝

• The hydrostatic state of stress is when the normal stresses equal 𝑝 and there are no shear stresses

• The lithostatic stress is the hydrostatic state of stress where stress increases in proportion to the density of the overlying rock

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Fig. 2.15, Turcotte and Schubert, 2014

www.helsinki.fi/yliopistoGeodynamics

Stress in three dimensions

• As before, we can also determine the principal stresses in three dimensions

• The convention is that 𝜎1 ≥ 𝜎2 ≥ 𝜎3

• Isotropic stress is when 𝜎1 = 𝜎2 = 𝜎3 = 𝑝

• The hydrostatic state of stress is when the normal stresses equal 𝑝 and there are no shear stresses

• The lithostatic stress is the hydrostatic state of stress where stress increases in proportion to the density of the overlying rock

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Fig. 2.15, Turcotte and Schubert, 2014

www.helsinki.fi/yliopistoGeodynamics

Stress in three dimensions

• When the principal stresses are not equal

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p =1

3(�1 + �2 + �3)

=1

3(�

xx

+ �yy

+ �zz

)

Fig. 2.15, Turcotte and Schubert, 2014

www.helsinki.fi/yliopistoGeodynamics

Deviatoric stresses

• We often subtract pressure from normal stresses to determine the deviatoric stresses (indicated by primes)

• The same can be done for the principal stresses

• What is the sum of the deviatoric principal stresses?

• For rock deforming by viscous flow, all deformation is the result of a nonzero deviatoric stress

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�0xx

= �xx

� p

�0xy

= �xy

�0yy

= �yy

� p

�0xz

= �xz

�0zz = �zz � p

�0yz = �yz

�01 = �1 � p �0

2 = �2 � p �03 = �3 � p

www.helsinki.fi/yliopistoGeodynamics

Deviatoric stresses

• We often subtract pressure from normal stresses to determine the deviatoric stresses (indicated by primes)

• The same can be done for the principal stresses

• What is the sum of the deviatoric principal stresses?

• For rock deforming by viscous flow, all deformation is the result of a nonzero deviatoric stress

10

�0xx

= �xx

� p

�0xy

= �xy

�0yy

= �yy

� p

�0xz

= �xz

�0zz = �zz � p

�0yz = �yz

�01 = �1 � p �0

2 = �2 � p �03 = �3 � p

www.helsinki.fi/yliopistoGeodynamics

Deviatoric stresses

• We often subtract pressure from normal stresses to determine the deviatoric stresses (indicated by primes)

• The same can be done for the principal stresses

• What is the average of the deviatoric principal stresses?

• For rock deforming by viscous flow, all deformation is the result of a nonzero deviatoric stress

11

�0xx

= �xx

� p

�0xy

= �xy

�0yy

= �yy

� p

�0xz

= �xz

�0zz = �zz � p

�0yz = �yz

�01 = �1 � p �0

2 = �2 � p �03 = �3 � p

www.helsinki.fi/yliopistoGeodynamics

Deviatoric stresses

• We often subtract pressure from normal stresses to determine the deviatoric stresses (indicated by primes)

• The same can be done for the principal stresses

• What is the average of the deviatoric principal stresses?

• For rock deforming by viscous flow, all deformation is the result of a nonzero deviatoric stress

12

�0xx

= �xx

� p

�0xy

= �xy

�0yy

= �yy

� p

�0xz

= �xz

�0zz = �zz � p

�0yz = �yz

�01 = �1 � p �0

2 = �2 � p �03 = �3 � p

Let’s see what you’ve learned…

• If you’re watching this lecture in Moodle, you will now be automatically directed to the quiz!

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