Earthquake Seismology:
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The stress tensor
Equation of motion
Hooke’s law
The elastodynamic equation of motion
P-waves and S-waves
Follows mainly on Lay and Wallace, Modern global seismology, Academic Press.
THE STRESS TENSOR
STRESS = FORCE/AREA
Pascal=Newton/m2
Why stress (and not force)?
Fig. from Twiss and Moores
THE STRESS TENSOR
• Normal stress: The component of stress acting perpendicular to the
plane
• Shear stress: The component of stress acting parallel to the plane
THE STRESS TENSOR
Consider a small cubic element of rock extracted from the Earth. The
stress acting on this element may be visualized as follows:
• The first index indicates the plane in question
• The second index indicates the direction at which the stress acts
THE STRESS TENSOR
The stress tensor in 3D:
æ s
s 12 s 13
11
ç
s ij = ç s 21 s 22 s 23
çç
è s 31 s 32 s 33
and 2D (that is, when s 13 = s 31 = s 23 = s 32 = s 33 = 0 ) :
æ s
s 12
11
s ij = ç
ç s 21 s 22
è
ö
÷
÷
ø
ö
÷
÷
÷÷
ø
THE STRESS TENSOR
The stress tensor is symmetric
s xy = s yx , s xz = s zx and s yz = s zy
EQUATION OF MOTION
What’s the sum of forces acting at the x1 direction on x1 plane is?
EQUATION OF MOTION
Sum of forces acting at the x1 direction on x1 plane is:
éæ
¶s 11 d x1 ö æ
¶s 11 d x1 öù
¶s 11
s
+
s
d
x
d
x
=
d x1d x2d x3
êç 11
÷ ç 11
÷ú 2 3
¶x1 2 ø è
¶x1 2 øû
¶x1
ëè
Sum of forces acting at the x1 direction on x2 plane is:
éæ
¶s 21 d x2 ö æ
¶s 21 d x2 öù
¶s 21
s
d
x
d
x
=
d x1d x2d x3
êçs 21 +
ú
÷ ç 21
÷ 1 3
¶x2 2 ø è
¶x2 2 øû
¶x2
ëè
Sum of forces acting at the x1 direction on x3 plane is:
éæ
¶s 31 d x3 ö æ
¶s 31 d x3 öù
¶s
êçs 31 +
÷ - çs 31 ÷úd x1d x2 = 31 d x1d x2d x3
¶x3 2 ø è
¶x3 2 øû
¶x3
ëè
What’s the sum of forces acting at the x1 direction?
EQUATION OF MOTION
The stress balance at the x1 direction is:
æ ¶s 11 ¶s 21 ¶s 31 ö
¶2u1
+
+
ç
÷ + r g1 = r 2
¶x3 ø
¶t
è ¶x1 ¶x2
The stress balance at the x2 direction is:
æ ¶s 12 ¶s 22 ¶s 32 ö
¶2u2
+
+
ç
÷ + r g2 = r 2
¶x2
¶x3 ø
¶t
è ¶x1
The stress balance at the x3 direction is:
æ ¶s 13 ¶s 23 ¶s 33 ö
¶2u3
+
+
ç
÷ + r g3 = r 2
¶x2
¶x3 ø
¶t
è ¶x1
Or more compactly:
¶s ij
¶2 ui
+ r gi = r 2
¶x j
¶t
This is the equation of
motion
EQUATION OF MOTION
For static problems:
¶s ji
+ r gi = 0
¶xi
HOOKE’S LAW
In general, stress-strain relations are material, pressure and temperature dependent. Yet, under
very small strains (such as those occur during the passage of the seismic waves), most materials
exhibit linear elastic behavior.
HOOKE’S LAW
Rocks subject to infinitesimal strain of short duration, deform reversibly and the strain is related
(linearly) to strain according to Hooke’s law:
s ij = Cijklekl
OR:
eij = Sijkls kl
Where C and S are the forth order stiffness and compliance tensors, respectively.
(The former formalism is convenient for problems in which the strains components are known
and the stress components are the dependent variables, whereas the latter formalism is
convenient for problems in which the stress components are known and the strain components
are the dependent variables.)
HOOKE’S LAW
In general, the stiffness matrix consists of 81 independent
entries
s 11 = C1111e11 + C1112e 12 + C1113e 13 + C1121e 21 + C1122e 22 + C1123e 23 + C1131e 31 + C1132e 32 + C1133e 33
s 12 = C1211e 11 + C1212e 12 + C1213e 13 + C1221e 21 + C1222e 22 + C1223e 23 + C1231e 31 + C1232e 32 + C1233e 33
s 13 = C1311e 11 + C1312e 12 + C1313e 13 + C1321e 21 + C1322e 22 + C1323e 23 + C1331e 31 + C1332e 32 + C1333e 33
s 21 = C2111e 11 + C2112e 12 + C2113e 13 + C2121e 21 + C2122e 22 + C2123e 23 + C2131e 31 + C2132e 32 + C2133e 33
s 22 = C2211e11 + C2212e 12 + C2213e 13 + C2221e 21 + C2222e 22 + C2223e 23 + C2231e 31 + C2232e 32 + C2233e 33
s 23 = C2311e11 + C2312e 12 + C2313e 13 + C2321e 21 + C2322e 22 + C2323e 23 + C2331e 31 + C2332e 32 + C2333e 33
s 31 = C3111e11 + C3112e 12 + C3113e 13 + C3121e 21 + C3122e 22 + C3123e 23 + C3131e 31 + C3132e 32 + C3133e 33
s 32 = C3211e 11 + C3212e 12 + C3213e 13 + C3221e 21 + C3222e 22 + C3223e 23 + C3231e 31 + C3232e 32 + C3233e 33
s 33 = C3311e 11 + C3312e 12 + C3313e 13 + C3321e 21 + C3322e 22 + C3323e 23 + C3331e 31 + C3332e 32 + C3333e 33
HOOKE’S LAW
Thanks to the symmetry of the stress tensor, the number of
independent elastic constants is reduced to 54
s 11 = C1111e11 + C1112e 12 + C1113e 13 + C1121e 21 + C1122e 22 + C1123e 23 + C1131e 31 + C1132e 32 + C1133e 33
s 22 = C2211e11 + C2212e 12 + C2213e 13 + C2221e 21 + C2222e 22 + C2223e 23 + C2231e 31 + C2232e 32 + C2233e 33
s 33 = C3311e 11 + C3312e 12 + C3313e 13 + C3321e 21 + C3322e 22 + C3323e 23 + C3331e 31 + C3332e 32 + C3333e 33
s 12 = C1211e 11 + C1212e 12 + C1213e 13 + C1221e 21 + C1222e 22 + C1223e 23 + C1231e 31 + C1232e 32 + C1233e 33
s 13 = C1311e 11 + C1312e 12 + C1313e 13 + C1321e 21 + C1322e 22 + C1323e 23 + C1331e 31 + C1332e 32 + C1333e 33
s 23 = C2311e11 + C2312e 12 + C2313e 13 + C2321e 21 + C2322e 22 + C2323e 23 + C2331e 31 + C2332e 32 + C2333e 33
HOOKE’S LAW
Thanks to the symmetry of the strain tensor, the number of
independent elastic constants is further reduced to 36
s 11 = C1111e11 + C1112e 12 + C1113e 13 + C1122e 22 + C1123e 23 + C1133e 33
s 22 = C2211e11 + C2212e 12 + C2213e 13 + C2222e 22 + C2223e 23 + C2233e 33
s 33 = C3311e 11 + C3312e 12 + C3313e 13 + C3322e 22 + C3323e 23 + C3333e 33
s 12 = C1211e 11 + C1212e 12 + C1213e 13 + C1222e 22 + C1223e 23 + C1233e 33
s 13 = C1311e 11 + C1312e 12 + C1313e 13 + C1322e 22 + C1323e 23 + C1333e 33
s 23 = C2311e11 + C2312e 12 + C2313e 13 + C2322e 22 + C2323e 23 + C2333e 33
A further symmetry relation (Cijkl=Cklij) follows from consideration of a strain energy density
function (see Malvern, 1969), leaving 21 elastic moduli in the most general elastic material.
HOOKE’S LAW
The stress-strain relation of an isotropic elastic material may
be described by 2 independent elastic constants, known as
Lame constants, l and m, and:
Cijkl = ldijdkl + m (dikd jl + dild jk )
And one finds:
s 11 = l (e11 + e22 + e33 ) + 2me11 = lq + 2me11
s 22 = l (e11 + e22 + e33 ) + 2me22 = lq + 2me 22
s 33 = l (e11 + e22 + e33 ) + 2me33 = lq + 2me33
s 12 = 2me12
s 13 = 2me13
s 23 = 2me 23
HOOKE’S LAW
The 5 elastic constants are expressible in terms of 2 constants. Here is a conversion table:
HOOKE’S LAW
For many Earth materials, m » l and when they are exactly equal the material is called Poisson
solid, for which n = 0.25 .
THE ELASTODYNAMIC EQUATION
The following equations are combined to get the equation of motion of
isotropic linear elastic medium (with no body forces):
1. The homogeneous equation of motion
2. Hooke’s law
¶s ij
¶2 ui
+ r gi = r 2
¶x j
¶t
s ij = lqdij + 2meij
3. The strain-displacement relation
1 æ ¶ui ¶u j ö
÷÷
eij = çç
+
2 è ¶x j ¶xi ø
THE ELASTODYNAMIC EQUATION
We consider the x1 direction first:
æ
¶u1 ö æ æ ¶u1 ¶u2 öö æ æ ¶u1 ¶u3 öö
+
+
¶ ç lq + 2m
÷÷
÷÷ ¶ ç m ç
÷ ¶ç m ç
2
¶x
¶x
¶x
¶x
¶
s
¶x1 ø è è 2
¶u
¶s
¶s
¶s
è
è è 3
1 øø
1 øø
r 21 = 1 j = 11 + 12 + 13 =
+
+
¶t
¶x j
¶x1
¶x2
¶x3
¶x1
¶x2
¶x3
For isotropic material, the above equation can be rearranged to get:
¶2u1 l + m ¶q m 2
=
+ Ñ u1
2
¶t
r ¶x1 r
Similarly, for the x2 and x3 directions, we get:
¶2u2 l + m ¶q m 2
=
+ Ñ u2
2
¶t
r ¶x2 r
and
¶2u3 l + m ¶q m 2
=
+ Ñ u3
2
¶t
r ¶x3 r
THE ELASTODYNAMIC EQUATION
In summary, the 3D propagation of waves in isotropic homogeneous elastic medium is
described by:
2
¶
u1 l + m ¶q m 2
1.
=
+ Ñ u1
¶t 2
r ¶x1 r
2.
¶2u2 l + m ¶q m 2
=
+ Ñ u2
¶t 2
r ¶x2 r
3.
¶2u3 l + m ¶q m 2
=
+ Ñ u3
2
¶t
r ¶x3 r
Adding the x1 derivative of (1), the x2 derivative of (2) and the x3 derivative of (3)
gives:
2
1 ¶q
l + 2m
2
=
Ñ
q
with
a
=
a 2 ¶t 2
r
This is the wave equation in terms of volumetric strain. Such equations fall into the
category of hyperbolic partial differential equations, whose solution is of the form:
q = Aexpi(wt±kx) with w / k = a
and the speed at which volumetric strain propagates is equal to:
l + 2m
r
THE ELASTODYNAMIC EQUATION
The solution to the hyperbolic partial differential is equal to the real part
of a harmonic equation of the form:
q = Aexpi(wt±kx)
where w and k are the frequency and the wave number, respectively,
and the velocity at which the wave propagates is equal to:
v =w / k
The solution to q = Aexpi(wt-kx) describes a wave propagating at the +x
direction, whereas the solution to q = Aexpi(wt+kx) describes a wave
propagating at the –x direction.
THE ELASTODYNAMIC EQUATION
This is how the solution to q = Aexpi(wt-kx) looks like:
• The the value of q is constant
when the phase (wt-kx) is
constant.
• Lines of constant phase
require that x increase when t
increase.
• The speed at which the
constant q line propagates is
equal to w/k.
• Only part of the solution is
physically viable – wave cone.
THE ELASTODYNAMIC EQUATION
Here are the solutions at a fixed position (top) and a fixed time
(bottom):
• The position-fixed solution
corresponds to a seismogram.
• Period=2p/w
• Wavelength=2p/k
P and S waves
According to Helmholtz’s theorem, any vector field u can be represented in terms of a vector
potential Y and a scalar potential f (see box at the end):
u = Ñf + Ñ ´ Y if Ñ´ f = 0 and Ñ× Y = 0
This means:
• A scalar potential field, f, that is curl-free, i.e. no shearing.
• A vector potential field, Y, that is divergent-free, i.e. no volume change.
Using Helmholtz’s theorem, the displacement resolved at the x1 direction is:
u1 =
¶f ¶Y 3 ¶Y 2
+
¶x1 ¶x2 ¶x3
One can now substitute this expression into the x1 component of the elastodynamic equation:
To get:
¶2 u1
¶q
r 2 = ( l + m ) + mÑ2u1
¶t
¶x1
æ
¶ ¶2f
¶ ¶2 Y 3
¶ ¶2 Y 2
¶ 2
¶Y 3 ¶Y 2 ö
2 ¶f
r
+r
-r
= (l + m)
Ñ f + mÑ ç +
÷
2
2
2
¶x1 ¶t
¶x2 ¶t
¶x3 ¶t
¶x1
è ¶x1 ¶x2 ¶x3 ø
P and S waves
æ
¶ ¶2f
¶ ¶2 Y 3
¶ ¶2 Y 2
¶ 2
¶Y 3 ¶Y 2 ö
2 ¶f
r
+r
-r
= (l + m)
Ñ f + mÑ ç +
÷
2
2
2
¶x1 ¶t
¶x2 ¶t
¶x3 ¶t
¶x1
è ¶x1 ¶x2 ¶x3 ø
¶ ¶2f
¶ ¶2 Y 3
¶ ¶2 Y 2
¶ 2
¶ 2
¶ 2
r
+
r
r
=
l
+
2
m
Ñ
f
+
m
Ñ
Y
m
Ñ Y2
(
)
3
2
2
2
¶x1 ¶t
¶x2 ¶t
¶x3 ¶t
¶x1
¶x2
¶x3
¶2f
2 2
=
a
Ñ f where a =
2
¶t
( l + 2m )
r
¶2 Y 2
m
2 2
=
b
Ñ
Y
where
b
=
2
¶t 2
r
¶2 Y 3
m
2 2
=
b
Ñ
Y
where
b
=
3
¶t 2
r
This indicates that elastodynamic disturbance within an unbounded body consists of three
different waves; a dilatational strain propagates at a speed equal to a and two perpendicular
shear waves that propagate at a speed that is equal to b, with a>b.
P and S waves
P and S waves
Teleseismic
Geometrical spreading for surface waves:
particle velocity µ r -1/2 .
and for body waves:
particle velocity µ r -1 .
The amplitude of
surface waves decays
slower with distance
than that of body
waves.
Consequently, after a
large earthquakes,
surface waves can
circle the Earth many
times.
Note the faster attenuation of
higher frequencies.
Note the reinforcement of the
surface waves near the antipodes.
In general, the wave amplitude
decreases with increasing
distance from the source.
Note the aftershock.
From Lay and Wallace, 1995
THE ELASTODYNAMIC EQUATION
The 3 elastodynamic equations may be written in equivalent vector form as:
1.
ru = ( l + m ) Ñ ( Ñ× u) + mÑ2u
The Laplacian of the displacement vector may be expressed as (see box at the end):
2.
Ñ2u = Ñ ( Ñ× u) - ( Ñ ´ Ñ ´ u)
Next, substituting (2) into (1) gives:
3.
ru = ( l + 2m ) Ñ ( Ñ× u) - (mÑ ´ Ñ ´ u)