Fault planes
3
σ
(a)
Normal
faulting
Fault planes
(b)
Reverse
or thrust
faulting
(c)
Strike-slip faulting
3
σ
1
σ
Fault
planes
Side view
Map view
1
σ
1
σ
33 =
σ
3
σ
33 =
σ
2
σ
33 =
σ
Fig. 2.3-9 Stress fields associated with three types of faulting, assuming
that the earthquake occurred on a plane of maximum shear stress. Normal
(a), reverse (b), and strike-slip (c) faulting involve different orientations of
the principal stresses.
2 Seismologists sometimes use the terms reverse and thrust fault interchangeably,
whereas structural geologists reserve the term thrust for a shallow-dipping reverse
fault.
2.3 Stress and strain 45
about 25°, rather than 45°, from the maximum principal stress
direction.
For simplicity, however, assume that faults in the earth form
on the planes of maximum shear stress. We will see (Section
2.3.10) that the earth’s surface is a free surface, where tractions
must be zero. Hence, at the surface one principal stress axis
must be vertical, and the other two must be parallel to the
surface. The three basic fault geometries — strike-slip, normal,
and thrust — are related to the stress axes (Fig. 2.3-9). If the vertical principal stress is the most compressive, the fault dips at
45°, and normal faulting occurs. If, instead, the vertical principal stress is the least compressive, the fault geometry is the
same, but reverse or thrust faulting occurs. 2 When the vertical
principal stress is the intermediate principal stress, strike-slip
motion occurs on a fault plane 45° from the maximum principal stress. Thus the geometry of faults, which can be mapped
geologically or inferred from seismograms of earthquakes, can
be used to study stress orientations. This model is subject
to limitations, especially because earthquakes often occur on
preexisting faults (Section 5.7.2). Nonetheless, the approach
is useful, especially when integrated with other methods of
estimating stress directions.
2.3.6 Deviatoric stresses
Large compressive stresses occur at depth within the earth
due to the weight of the overlying rock. It is convenient in many
applications to remove the effect of the overall compressive
stress and consider only the deviations from it. We thus define
the mean stress
M = (σ 11 + σ 22 + σ 33 )/3 = σ ii /3
(37)
as B of the sum of the normal stresses, the trace of the stress
tensor. The mean stress can be related to the principal stresses,
because the trace of the stress tensor is independent of the
coordinate system.
To see that the trace does not change, we write the transformation of the stress tensor between two coordinate systems
(Eqn 18) in terms of the components, using the summation
convention (Section A.3.5)
σ ′
ij = A ik σ kl A T
lj = A ik σ kl A jl .
(38)
The trace can be written
σ ′
ii = σ ′
ij δ ij = A ik σ kl A il = δ kl σ kl = σ kk ,
(39)
because A is an orthogonal matrix, so that A ik A il = δ kl . Thus the
trace is invariant under an orthogonal transformation, and so
is known as the first invariant of a tensor. The other two invariants (Eqn 25) are also preserved by such transformations.
The mean stress can thus be written in terms of the trace of
the diagonalized stress tensor (Eqn 29)
M = (σ 1 + σ 2 + σ 3 )/3
(40)
as B of the sum of the principal stresses. The deviatoric stress
tensor is defined by removing the effect of the mean stress
D ij = σ ij − Mδ ij
D
M
M
M
.
=
−
−
−
⎛
⎝
⎜
⎜ ⎜
⎞
⎠
⎟
⎟ ⎟
σ
σ
σ
σ
σ
σ
σ
σ
σ
11
12
13
21
22
23
31
32
33
(41)
Thus, when the principal stresses are large and nearly equal,
the deviatoric stress tensor removes their effect and indicates
the remaining stress state. The deviatoric stress tensor can be
diagonalized and has the same principal stress axes as the stress
tensor.
This concept is important in discussing processes in the
earth, because the deviatoric stresses result from tectonic forces
and cause earthquake faulting and seismic wave propagation
effects like anisotropy. At depths greater than a few kilometers,
we often assume that a lithostatic state of stress exists, where
the normal stresses are equal to minus the pressure of the overlying material and the deviatoric stresses are zero. Because
3
σ
(a)
Normal
faulting
Fault planes
(b)
Reverse
or thrust
faulting
(c)
Strike-slip faulting
3
σ
1
σ
Fault
planes
Side view
Map view
1
σ
1
σ
33 =
σ
3
σ
33 =
σ
2
σ
33 =
σ
Fig. 2.3-9 Stress fields associated with three types of faulting, assuming
that the earthquake occurred on a plane of maximum shear stress. Normal
(a), reverse (b), and strike-slip (c) faulting involve different orientations of
the principal stresses.
2 Seismologists sometimes use the terms reverse and thrust fault interchangeably,
whereas structural geologists reserve the term thrust for a shallow-dipping reverse
fault.
2.3 Stress and strain 45
about 25°, rather than 45°, from the maximum principal stress
direction.
For simplicity, however, assume that faults in the earth form
on the planes of maximum shear stress. We will see (Section
2.3.10) that the earth’s surface is a free surface, where tractions
must be zero. Hence, at the surface one principal stress axis
must be vertical, and the other two must be parallel to the
surface. The three basic fault geometries — strike-slip, normal,
and thrust — are related to the stress axes (Fig. 2.3-9). If the vertical principal stress is the most compressive, the fault dips at
45°, and normal faulting occurs. If, instead, the vertical principal stress is the least compressive, the fault geometry is the
same, but reverse or thrust faulting occurs. 2 When the vertical
principal stress is the intermediate principal stress, strike-slip
motion occurs on a fault plane 45° from the maximum principal stress. Thus the geometry of faults, which can be mapped
geologically or inferred from seismograms of earthquakes, can
be used to study stress orientations. This model is subject
to limitations, especially because earthquakes often occur on
preexisting faults (Section 5.7.2). Nonetheless, the approach
is useful, especially when integrated with other methods of
estimating stress directions.
2.3.6 Deviatoric stresses
Large compressive stresses occur at depth within the earth
due to the weight of the overlying rock. It is convenient in many
applications to remove the effect of the overall compressive
stress and consider only the deviations from it. We thus define
the mean stress
M = (σ 11 + σ 22 + σ 33 )/3 = σ ii /3
(37)
as B of the sum of the normal stresses, the trace of the stress
tensor. The mean stress can be related to the principal stresses,
because the trace of the stress tensor is independent of the
coordinate system.
To see that the trace does not change, we write the transformation of the stress tensor between two coordinate systems
(Eqn 18) in terms of the components, using the summation
convention (Section A.3.5)
σ ′
ij = A ik σ kl A T
lj = A ik σ kl A jl .
(38)
The trace can be written
σ ′
ii = σ ′
ij δ ij = A ik σ kl A il = δ kl σ kl = σ kk ,
(39)
because A is an orthogonal matrix, so that A ik A il = δ kl . Thus the
trace is invariant under an orthogonal transformation, and so
is known as the first invariant of a tensor. The other two invariants (Eqn 25) are also preserved by such transformations.
The mean stress can thus be written in terms of the trace of
the diagonalized stress tensor (Eqn 29)
M = (σ 1 + σ 2 + σ 3 )/3
(40)
as B of the sum of the principal stresses. The deviatoric stress
tensor is defined by removing the effect of the mean stress
D ij = σ ij − Mδ ij
D
M
M
M
.
=
−
−
−
⎛
⎝
⎜
⎜ ⎜
⎞
⎠
⎟
⎟ ⎟
σ
σ
σ
σ
σ
σ
σ
σ
σ
11
12
13
21
22
23
31
32
33
(41)
Thus, when the principal stresses are large and nearly equal,
the deviatoric stress tensor removes their effect and indicates
the remaining stress state. The deviatoric stress tensor can be
diagonalized and has the same principal stress axes as the stress
tensor.
This concept is important in discussing processes in the
earth, because the deviatoric stresses result from tectonic forces
and cause earthquake faulting and seismic wave propagation
effects like anisotropy. At depths greater than a few kilometers,
we often assume that a lithostatic state of stress exists, where
the normal stresses are equal to minus the pressure of the overlying material and the deviatoric stresses are zero. Because