42 Basic Seismological Theory
T′ = σ ′4′,
(16)
so, by Eqns 14 and 15,
T′ = AT = Aσ 4 = AσA
T 4′.
(17)
Comparison of Eqn 16 and the last term in Eqn 17 shows that
σ ′ = AσA T .
(18)
This equation defines a tensor in Cartesian coordinates. Recall
that what makes a vector more than a set of three numbers is its
transformation properties: the numerical values of the components that describe it transform between coordinate systems
in a way that preserves the vector as an entity independent of
coordinate system. Similarly, a matrix of numbers is a tensor
only if it transforms between coordinate systems according
to Eqn 18. We derived this transformation by assuming that a
tensor, in this case stress, is an operator relating two vectors, in
this case the normal and traction, in a specific way regardless of
coordinate system. The tensor’s components transform between
coordinate systems, so the tensor as an entity does not change.
Because one application of the transformation matrix transforms a vector, two applications transform a tensor that relates
two vectors. Unfortunately, tensors are harder to visualize than
vectors. Although the stress tensor may seem puzzling, it is one
of the easier tensors to interpret physically.
To illustrate these ideas, we consider an example of how a
stress tensor’s components change between coordinate systems.
Assume that a block of material, with faces perpendicular to
the x 1 and x 2 axes, is subject only to normal stresses σ 1 and σ 2
(Fig. 2.3-6), so the stress tensor is diagonal,
σ
σ
σ
.
=
⎛
⎝
⎜
⎜ ⎜
⎞
⎠
⎟
⎟ ⎟
1
2
0 0
0
0
0 0 0
(19)
x 2
x 1
x′ 2
11
σ
x ′
1
θ
22
σ
22
σ
′
21
σ
′
12
σ
′
21
σ
′
12
σ
11
σ
Fig. 2.3-6 An example of the stress tensor’s different components in
different coordinate systems. In the x 1 , x 2 axis coordinate system, the
stress tensor is diagonal. In contrast, shear stresses act on a volume with
faces normal to the x′ 1 and x′ 2 coordinate axes, which are rotated by θ
with respect to the x 1 , x 2 axes.
Now, consider the stress acting on a smaller block, with faces
of a different orientation, within the larger one. To find the
tractions on the second block’s sides, we define a second
coordinate system in which the x′ 1 and x′ 2 axes are normal to the
faces and rotated by θ with respect to the x 1 and x 2 axes,
whereas the x 3 and x′ 3 axes coincide. Although the stress is
the same in both blocks, the components of the stress tensor
expressed in the two coordinate systems differ. The relation
between the components is given by
σ ′ = AσA T
= −
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
cos
sin
sin
cos
cos
sin
sin
cos
θ
θ
θ
θ
σ
σ
θ
θ
θ
θ
0
0
0
0
1
0 0
0
0
0 0 0
0
0
0
0
1
1
2
=
+
−
−
+
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
cos
sin
(
)sin cos
(
) sin cos
sin
cos
.
σ
θ σ
θ
σ
σ
θ
θ
σ
σ
θ
θ
σ
θ σ
θ
1
2
2
2
2
1
2
1
1
2
2
2
0
0
0
0
0
(20)
For example, if σ 1 = 1, σ 2 = −1, and θ = 45°,
σ ′
.
=
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
0 1 0
1 0 0
0 0 0
(21)
Thus, although the large block is oriented such that the stress
tensor causes only normal tractions, giving compression along
the x 2 axis and tension along the x 1 axis, only shear tractions
act on the smaller block because its sides are oriented differently. The negative shear stress values yield tractions in the −x 2
′
direction on the face with normal ê 1 ′, and in the x 2
′ direction on
the opposite face with normal −ê 1 ′, consistent with what we expect from the normal tractions on the larger block. Although the
components of the stress tensor in the two coordinate systems
differ, they represent the same entity, the physical state of stress.
2.3.4 Principal stresses
For a given state of stress, the traction vector acting on most
surfaces within a material has components both normal to the
surface and tangential to it. There are, however, some surfaces
oriented such that the shear tractions on them vanish. These
surfaces can be characterized by their normal vectors, called
principal stress axes; the normal stresses on these surfaces are
called principal stresses. The concept of principal stress axes is
important for discussion of earthquake source mechanisms
(Section 4.2).
To find the principal stresses, we use the concepts of
eigenvalues and eigenvectors (Section A.5.2). The shear components of the traction will be zero if the traction and normal
vectors are parallel, such that they differ only by a multiplicative constant, λ,
T i = σ ij n j = λn i .
(22)
T′ = σ ′4′,
(16)
so, by Eqns 14 and 15,
T′ = AT = Aσ 4 = AσA
T 4′.
(17)
Comparison of Eqn 16 and the last term in Eqn 17 shows that
σ ′ = AσA T .
(18)
This equation defines a tensor in Cartesian coordinates. Recall
that what makes a vector more than a set of three numbers is its
transformation properties: the numerical values of the components that describe it transform between coordinate systems
in a way that preserves the vector as an entity independent of
coordinate system. Similarly, a matrix of numbers is a tensor
only if it transforms between coordinate systems according
to Eqn 18. We derived this transformation by assuming that a
tensor, in this case stress, is an operator relating two vectors, in
this case the normal and traction, in a specific way regardless of
coordinate system. The tensor’s components transform between
coordinate systems, so the tensor as an entity does not change.
Because one application of the transformation matrix transforms a vector, two applications transform a tensor that relates
two vectors. Unfortunately, tensors are harder to visualize than
vectors. Although the stress tensor may seem puzzling, it is one
of the easier tensors to interpret physically.
To illustrate these ideas, we consider an example of how a
stress tensor’s components change between coordinate systems.
Assume that a block of material, with faces perpendicular to
the x 1 and x 2 axes, is subject only to normal stresses σ 1 and σ 2
(Fig. 2.3-6), so the stress tensor is diagonal,
σ
σ
σ
.
=
⎛
⎝
⎜
⎜ ⎜
⎞
⎠
⎟
⎟ ⎟
1
2
0 0
0
0
0 0 0
(19)
x 2
x 1
x′ 2
11
σ
x ′
1
θ
22
σ
22
σ
′
21
σ
′
12
σ
′
21
σ
′
12
σ
11
σ
Fig. 2.3-6 An example of the stress tensor’s different components in
different coordinate systems. In the x 1 , x 2 axis coordinate system, the
stress tensor is diagonal. In contrast, shear stresses act on a volume with
faces normal to the x′ 1 and x′ 2 coordinate axes, which are rotated by θ
with respect to the x 1 , x 2 axes.
Now, consider the stress acting on a smaller block, with faces
of a different orientation, within the larger one. To find the
tractions on the second block’s sides, we define a second
coordinate system in which the x′ 1 and x′ 2 axes are normal to the
faces and rotated by θ with respect to the x 1 and x 2 axes,
whereas the x 3 and x′ 3 axes coincide. Although the stress is
the same in both blocks, the components of the stress tensor
expressed in the two coordinate systems differ. The relation
between the components is given by
σ ′ = AσA T
= −
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
cos
sin
sin
cos
cos
sin
sin
cos
θ
θ
θ
θ
σ
σ
θ
θ
θ
θ
0
0
0
0
1
0 0
0
0
0 0 0
0
0
0
0
1
1
2
=
+
−
−
+
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
cos
sin
(
)sin cos
(
) sin cos
sin
cos
.
σ
θ σ
θ
σ
σ
θ
θ
σ
σ
θ
θ
σ
θ σ
θ
1
2
2
2
2
1
2
1
1
2
2
2
0
0
0
0
0
(20)
For example, if σ 1 = 1, σ 2 = −1, and θ = 45°,
σ ′
.
=
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
0 1 0
1 0 0
0 0 0
(21)
Thus, although the large block is oriented such that the stress
tensor causes only normal tractions, giving compression along
the x 2 axis and tension along the x 1 axis, only shear tractions
act on the smaller block because its sides are oriented differently. The negative shear stress values yield tractions in the −x 2
′
direction on the face with normal ê 1 ′, and in the x 2
′ direction on
the opposite face with normal −ê 1 ′, consistent with what we expect from the normal tractions on the larger block. Although the
components of the stress tensor in the two coordinate systems
differ, they represent the same entity, the physical state of stress.
2.3.4 Principal stresses
For a given state of stress, the traction vector acting on most
surfaces within a material has components both normal to the
surface and tangential to it. There are, however, some surfaces
oriented such that the shear tractions on them vanish. These
surfaces can be characterized by their normal vectors, called
principal stress axes; the normal stresses on these surfaces are
called principal stresses. The concept of principal stress axes is
important for discussion of earthquake source mechanisms
(Section 4.2).
To find the principal stresses, we use the concepts of
eigenvalues and eigenvectors (Section A.5.2). The shear components of the traction will be zero if the traction and normal
vectors are parallel, such that they differ only by a multiplicative constant, λ,
T i = σ ij n j = λn i .
(22)