50 Basic Seismological Theory
σ ij = c ijkl u k,l .
(65)
Substituting this expression in Eqn 48 gives the equation of
motion in terms of the displacements:
σ ij,j (x, t) + f i (x, t) = (c ijkl u k,l ) , j (x, t) + f i (x, t) = ρ
∂
∂ 2
2 u
t
t
i ( , ) .
x
(66)
Thus the elastic moduli control how displacements evolve in
time and space in response to an applied force, and so, as we
will see in the next section, determine the velocity of seismic
waves.
The elastic moduli c ijkl form a more complicated tensor than
we have dealt with so far. It has four subscripts and relates
the stress and strain tensors, each of which have two subscripts. This situation is analogous to the way in which the
stress tensor, with two subscripts, relates the normal and traction vectors, each with one subscript. Because the subscripts
each range from 1 to 3, c ijkl has 3 4 , or 81, components. Fortunately, the number of independent components is reduced by
symmetry considerations. The stress and strain tensors are
symmetric
c ijkl = c jikl , c ijkl = c ijlk ,
(67)
so the number of independent components is 36 because there
are 6 independent components of the stress and strain tensors.
A further symmetry relation
c ijkl = c klij ,
(68)
based on the idea of strain energy, which we will discuss later,
reduces the number of independent components that characterize a general elastic medium to 21.
On a large scale, material within the earth has approximately the same physical properties regardless of orientation, a
condition known as isotropy. For an isotropic material, the c ijkl
have further symmetries, so there are only two independent
elastic moduli, which can be defined in various ways. One
useful pair are the Lamé constants λ and µ, which are defined
such that
c ijkl = λδ ij δ kl + µ(δ ik δ jl + δ il δ jk ).
(69)
In terms of the Lamé constants, the constitutive equation
(Eqn 64) for an isotropic material is written
σ ij = λe kk δ ij + 2µe ij = λθδ ij + 2µ e ij ,
(70)
where θ is the dilatation. So, for example, σ 11 = λθ + 2µe 11 , and
σ 12 = 2µe 12 . We will use this constitutive relation to study
seismic waves in the next section. We will also see that the
velocities of seismic waves depend on the elastic moduli, so
in an isotropic material the velocities of seismic waves do not
depend on the direction in which they propagate. Deviations
from isotropy occur in many parts of the earth, notably in the
oceanic lithosphere and at the base of the mantle (Section 3.7).
Although the c ijkl completely describe the behavior of an
elastic material, they are hard to visualize. This is also true for
the Lamé constant λ. 4 By contrast, µ, called the rigidity or shear
modulus, has a simple physical interpretation. Consider the
response of an isotropic elastic body to an applied shear stress
σ 12 . In this case, the term in the constitutive equation (Eqn 70)
involving the dilatation is zero (recall that δ 12 = 0), so only a
shear strain, e 12 = σ 12 /2µ, results. The response to shear is thus
described by the rigidity. µ must be nonnegative, so the sense
of strain is consistent with the applied stress (consider Fig.
2.3-12c). A material with large µ is quite rigid and responds to
a given stress with a small strain. By contrast, a given shear
stress produces a larger strain in a material with lower rigidity.
A material in which µ is zero cannot support shear stresses, and
corresponds to a perfect fluid, one with zero viscosity. In such a
fluid, the stress tensor is diagonal in any coordinate system, and
the pressure equals the negative of the mean stress. Although
perfect fluids do not exist,
5 the ocean can generally be treated
this way for seismic waves incident on the sea floor. Even more
surprisingly, the hot iron fluid thought to comprise the earth’s
outer core can be described as an ideal fluid for seismological
purposes.
Other elastic constants that can be defined in terms of simple
experiments are often useful. The incompressibility, or bulk
modulus, K, is defined by subjecting a body to a lithostatic
pressure dP, such that
dσ ij = −dPδ ij .
(71)
For an isotropic elastic body, the resulting strains, from Eqn 70,
are
−dPδ ij = λdθδ ij + 2µde ij .
(72)
Setting i = j and summing yields
−3dP = 3λdθ + 2µdθ,
(73)
because δ ii = 3. The bulk modulus is thus the ratio of the pressure applied to the fractional volume change that results:
K
dP
d
.
=
−
= +
θ
λ
µ
2
3
(74)
The term incompressibility is apt because the larger the value of
K, the smaller the volume change produced by a given pressure.
K is greater than zero, because otherwise objects would expand
4 Unfortunately, this Lamé constant is not only hard to interpret; it has no
common name and is denoted by the same symbol as is used for wavelength.
5 Perfect fluids have been called “dry water” to illustrate that no real fluid behaves
exactly this way.
σ ij = c ijkl u k,l .
(65)
Substituting this expression in Eqn 48 gives the equation of
motion in terms of the displacements:
σ ij,j (x, t) + f i (x, t) = (c ijkl u k,l ) , j (x, t) + f i (x, t) = ρ
∂
∂ 2
2 u
t
t
i ( , ) .
x
(66)
Thus the elastic moduli control how displacements evolve in
time and space in response to an applied force, and so, as we
will see in the next section, determine the velocity of seismic
waves.
The elastic moduli c ijkl form a more complicated tensor than
we have dealt with so far. It has four subscripts and relates
the stress and strain tensors, each of which have two subscripts. This situation is analogous to the way in which the
stress tensor, with two subscripts, relates the normal and traction vectors, each with one subscript. Because the subscripts
each range from 1 to 3, c ijkl has 3 4 , or 81, components. Fortunately, the number of independent components is reduced by
symmetry considerations. The stress and strain tensors are
symmetric
c ijkl = c jikl , c ijkl = c ijlk ,
(67)
so the number of independent components is 36 because there
are 6 independent components of the stress and strain tensors.
A further symmetry relation
c ijkl = c klij ,
(68)
based on the idea of strain energy, which we will discuss later,
reduces the number of independent components that characterize a general elastic medium to 21.
On a large scale, material within the earth has approximately the same physical properties regardless of orientation, a
condition known as isotropy. For an isotropic material, the c ijkl
have further symmetries, so there are only two independent
elastic moduli, which can be defined in various ways. One
useful pair are the Lamé constants λ and µ, which are defined
such that
c ijkl = λδ ij δ kl + µ(δ ik δ jl + δ il δ jk ).
(69)
In terms of the Lamé constants, the constitutive equation
(Eqn 64) for an isotropic material is written
σ ij = λe kk δ ij + 2µe ij = λθδ ij + 2µ e ij ,
(70)
where θ is the dilatation. So, for example, σ 11 = λθ + 2µe 11 , and
σ 12 = 2µe 12 . We will use this constitutive relation to study
seismic waves in the next section. We will also see that the
velocities of seismic waves depend on the elastic moduli, so
in an isotropic material the velocities of seismic waves do not
depend on the direction in which they propagate. Deviations
from isotropy occur in many parts of the earth, notably in the
oceanic lithosphere and at the base of the mantle (Section 3.7).
Although the c ijkl completely describe the behavior of an
elastic material, they are hard to visualize. This is also true for
the Lamé constant λ. 4 By contrast, µ, called the rigidity or shear
modulus, has a simple physical interpretation. Consider the
response of an isotropic elastic body to an applied shear stress
σ 12 . In this case, the term in the constitutive equation (Eqn 70)
involving the dilatation is zero (recall that δ 12 = 0), so only a
shear strain, e 12 = σ 12 /2µ, results. The response to shear is thus
described by the rigidity. µ must be nonnegative, so the sense
of strain is consistent with the applied stress (consider Fig.
2.3-12c). A material with large µ is quite rigid and responds to
a given stress with a small strain. By contrast, a given shear
stress produces a larger strain in a material with lower rigidity.
A material in which µ is zero cannot support shear stresses, and
corresponds to a perfect fluid, one with zero viscosity. In such a
fluid, the stress tensor is diagonal in any coordinate system, and
the pressure equals the negative of the mean stress. Although
perfect fluids do not exist,
5 the ocean can generally be treated
this way for seismic waves incident on the sea floor. Even more
surprisingly, the hot iron fluid thought to comprise the earth’s
outer core can be described as an ideal fluid for seismological
purposes.
Other elastic constants that can be defined in terms of simple
experiments are often useful. The incompressibility, or bulk
modulus, K, is defined by subjecting a body to a lithostatic
pressure dP, such that
dσ ij = −dPδ ij .
(71)
For an isotropic elastic body, the resulting strains, from Eqn 70,
are
−dPδ ij = λdθδ ij + 2µde ij .
(72)
Setting i = j and summing yields
−3dP = 3λdθ + 2µdθ,
(73)
because δ ii = 3. The bulk modulus is thus the ratio of the pressure applied to the fractional volume change that results:
K
dP
d
.
=
−
= +
θ
λ
µ
2
3
(74)
The term incompressibility is apt because the larger the value of
K, the smaller the volume change produced by a given pressure.
K is greater than zero, because otherwise objects would expand
4 Unfortunately, this Lamé constant is not only hard to interpret; it has no
common name and is denoted by the same symbol as is used for wavelength.
5 Perfect fluids have been called “dry water” to illustrate that no real fluid behaves
exactly this way.