Elasticity and Seismic Waves Elasticity and Seismic Waves
• Some mathematical basics
• Strain-displacement relation
Linear elasticity
Strain tensor – meaning of its elements
• Stress-strain relation (Hooke’s Law)
Stress tensor Symmetry
Elasticity tensor Lame’s parameters
• Equation of Motion
P and S waves
Plane wave solutions
Stress-strain regimes Stress-strain regimes
• Linear elasticity (teleseismic waves)
• rupture, breaking
• stable slip (aseismic)
• stick-slip (with sudden ruptures)
Linear deformation Stable slip Stick slip Breaking
Stress
Linear and non-linear stress and strain Linear and non-linear stress and strain
Stress vs. strain for a loading cycle with rock that breaks. For wave propagation problems assuming linear elasticity is usually sufficient.
Stress vs. strain for a loading cycle with rock that breaks. For wave propagation problems assuming linear elasticity is usually sufficient.
Linear stress-strain
Principal stress, hydrostatic stress Principal stress, hydrostatic stress
Horizontal stresses are influenced by tectonic forces (regional and local). This implies that usually there are two uneven horizontal principal stress directions.
Example: Cologne Basin Horizontal stresses are influenced by tectonic forces (regional and local). This implies that usually there are two uneven horizontal principal stress directions.
Example: Cologne Basin
When all three orthogonal principal stresses are equal we speak of hydrostatic stress.
When all three orthogonal principal stresses are equal
we speak of hydrostatic stress.
Plate velocities
Plate velocities
Elasticity Theory Elasticity Theory
A time-dependent perturbation of an elastic medium (e.g. a rupture, an earthquake, a meteorite impact, a nuclear explosion etc.) generates elastic waves
emanating from the source region. These
disturbances produce local changes in stress and strain.
To understand the propagation of elastic waves we need to describe kinematically the deformation of our medium and the resulting forces (stress). The relation between deformation and stress is governed by elastic constants.
The time-dependence of these disturbances will lead
us to the elastic wave equation as a consequence of
conservation of energy and momentum.
Some mathematical basics - Vectors Some mathematical basics - Vectors
The mathematical description of deformation processes heavily relies on vector analysis. We therefore review the fundamental concepts of vectors and tensors.
Usually vectors are written in boldface type, x is a scalar but y is a vector, y
iare the scalar components of a vector
a
b
c
c=a+b b=c-a a=c-b
q
Scalar or Dot Product
3 2 1
y y y
y
3 2 1
ay ay ay
ay
3 3
2 2
1 1
bx ay
bx ay
bx ay
b a y x
2 3 2
2 2
1
3 3 2
2 1
1
|
|
cos
|
||
| ) a
a a
a
b a b
a b
a b
a
b ( q
a
Vectors – Triple Product Vectors – Triple Product
The triple scalar product is defined as
a b
c bxc
z y
x
which is a scalar and represents the volume of the parallelepiped defined by a,b, and c.
It is also calculated like a determinant:
The vector cross product is defined as:
c) b a (
3 2
1
3 2
1
3 2
1
(
c c
c
b b
b
a a
a
b c) a
3 2
1
3 2
1
b b
b
a a
a
k j
i
b
a
Vectors – Gradient Vectors – Gradient
Assume that we have a scalar field F (x), we want to know how it changes with respect to the coordinate axes, this leads to a vector called the gradient of F
With the nabla operator and
The gradient is a vector that points in the direction of maximum rate of change of the scalar function F (x).
What happens if we have a vector field?
F
F
F
F
z y x
z y x
x
x
Vectors – Divergence + Curl Vectors – Divergence + Curl
The divergence is the scalar product of the nabla operator with a vector field V(x). The divergence of a vector field is a scalar!
Physically the divergence can be interpreted as the net flow out of a volume (or change in volume). E.g. the divergence of the seismic wavefield corresponds to compressional waves.
The curl is the vector product of the nabla operator with a vector field V(x). The curl of a vector field is a vector!
The curl of a vector field represents the rotational part of that field (e.g. shear waves in a seismic wavefield)
z z y
y x
x
V V V
V
x y y
x
z x x
z
y z z
y
V V
V V
V V
V V
V
x y zz y
x
k j
i
V
Deformation Deformation
Let us consider a point P
0at position r relative to some fixed origin and a second point Q
0displaced from P
0by dx
P
0
x y
Q
0 x
x u
r
u
P
1y Q
1v
Unstrained state:
Relative position of point P
0w.r.t. Q
0is x.
Strained state:
Relative position of point P
0has been displaced a distance u to P
1and point Q
0a distance v to Q
1.Relative positive of point P
1w.r.t. Q
1is y= x+ u. The change in
relative position between Q and P is
just u .
Linear Elasticity Linear Elasticity
The relative displacement in the unstrained state is u(r). The relative displacement in the strained state is v=u(r+ x).
So finally we arrive at expressing the relative displacement due to strain:
u=u ( r+ x ) -u ( r )
We now apply Taylor’s
theorem in 3-D to arrive at:
P
0
Q
0 x
x u
u
P
1y Q
1v
What does this equation mean?
k k
i
i x
x
u u
Linear Elasticity – symmetric part Linear Elasticity – symmetric part
The partial derivatives of the vector components
P
0
Q
0 x
x u
u
P
1y Q
1v
• symmetric
• deformation
represent a second-rank tensor which can be resolved into a symmetric and anti-symmetric part:
• antisymmetric
• pure rotation
k i
x u
k k
i i
k k
i k k
i
i x
x u x
x u x
u x
u u
( )
2 ) 1
2 ( 1
Linear Elasticity – deformation tensor Linear Elasticity – deformation tensor
The symmetric part is called the deformation tensor
P
0
Q
0 x
x u
u
P
1y Q
1v
and describes the relation between deformation and displacement in linear elasticity. In 2-D this tensor looks like
) 2 (
1
i j j
i
ij x
u x
u
y u x
u y
u
x u y
u x
u
y x y
x y x
ij
) 2 (
1
) 2 (
1
Deformation tensor – its elements Deformation tensor – its elements
Through eigenvector analysis the meaning of the elements of the deformation tensor can be clarified:
The deformation tensor assigns each point – represented by position vector y a new position with vector u (summation over repeated
indices applies):
The eigenvectors of the deformation tensor are those y’s for which the tensor is a scalar, the eigenvalues l :
The eigenvalues l can be obtained solving the system (Exercise):
j ij
i y
u
i
i y
u l
0
ij
ij l
Deformation tensor – its elements Deformation tensor – its elements
Thus
... in other words ...
the eigenvalues are the relative change of length along the three coordinate axes
In arbitrary coordinates the diagonal elements are the relative change of length
along the coordinate axes and the off- diagonal elements are the infinitesimal
shear angles.
shear angle 1
1
1 y
u l
1 1
1 y
u l
2 2
2 y
u l u 3 l 3 y 3
Deformation tensor – trace Deformation tensor – trace
The trace of a tensor is defined as the sum over the diagonal elements. Thus:
This trace is linked to the volumetric change after deformation.
Before deformation the volume was V
0.. Because the diagonal elements are the relative change of lengths along each direction, the
new volume after deformation is
... and neglecting higher-order terms ...
) 1
)(
1 )(
1
( xx yy zz
V
zz yy
xx
ii
ii
ii V
V 1 0
u x u
u x
u x
u x
u V
V i
ii
1 2 3 div
Deformation tensor – applications Deformation tensor – applications
The fact that we have linearised the strain-displacement relation is quite severe. It means that the elements of the strain tensor should be <<1. Is this the case in seismology?
Let’s consider an example. The 1999 Taiwan earthquake (M=7.6) was recorded in FFB. The maximum ground displacement was 1.5mm
measured for surface waves of approx. 30s period. Let us assume a phase velocity of 5km/s. How big is the strain at the Earth’s surface, give an estimate (EXERCISE) !
The answer is that would be on the order of 10
-7<<1. This is typical for global seismology if we are far away from the source, so that the assumption of infinitesimal displacements is acceptable.
For displacements closer to the source this assumption is not valid.
There we need a finite strain theory. Strong motion seismology is an
own field in seismology concentrating on effects close to the seismic
source.
Pinon Flat Observatory
Pinon Flat Observatory
Strainmeter
Strainmeter
Strain vs. Translation
Strain vs. Translation
Real data: strain vs. translation
Real data: strain vs. translation
Exercise Exercise
Download DATA at station FUR, correct to displacement. Estimate strain!
Assume a realistic phase velocity! Do the values make sense?
Borehole breakout Borehole breakout
Source: www.fracom.fi
Stress - traction Stress - traction
In an elastic body there are restoring forces if deformation takes place.
These forces can be seen as acting on planes inside the body. Forces divided by an areas are called stresses.
In order for the deformed body to remain deformed these forces have to compensate each other. We will see that the relationship between the stress and the deformation (strain) is linear and can be described by tensors.
The tractions t
kalong axis k are
t
kt
1t
2t
... and along an arbitrary direction
... which – using the summation convention yields ..
3 2 1
k k k k
t t t t
i i
n t t
3 3 2
2 1
1
n t n t n
t
t
Stress tensor Stress tensor
... in components we can write this as
where
ijist the stress tensor and n
jis a surface normal.
The stress tensor describes the
forces acting on planes within a body.
Due to the symmetry condition
there are only six independent elements.
The vector normal to the corresponding surface
The direction of the force vector acting on that surface 22
23 21
1 3
2 j
ij
i n
t
ji
ij
ij
Stress - Glossary Stress - Glossary
Stress units bars (10
6dyn/cm
2), 1N=10
5dyn (cm g/s
2) 10
6Pa=1MPa=10bars
1 Pa=1 N/m
2At sea level p=1bar At depth 3km p=1kbar maximum
compressive stress
the direction perpendicular to the minimum compressive stress, near the surface mostly in horizontal direction, linked to tectonic processes.
principle stress axes
the direction of the eigenvectors of the
stress tensor
Stresses and faults
Stresses and faults
Stress-strain relation Stress-strain relation
The relation between stress and strain in general is described by the tensor of elastic constants c
ijklFrom the symmetry of the stress and strain tensor and a
thermodynamic condition if follows that the maximum number if independent constants of c
ijklis 21. In an isotropic body, where the properties do not depend on direction the relation reduces to
where l and m are the Lame parameters, q is the dilatation and
ijis the Kronecker delta.
Generalised Hooke’s Law
Hooke’s Law kl
ijkl
ij c
ij ij
ij l m
2
xx yy zz ij
ij kk
ij
Stress-strain relation Stress-strain relation
The complete stress tensor looks like
There are several other possibilities to describe elasticity:
E elasticity, Poisson’s ratio, K bulk modulus
For Poisson’s ratio we have 0< <0.5. A useful approximation is lm , then =0.25. For fluids 0.5 m0.
) (
) 2 (
2 2
2 )
( )
2 (
2
2 2
) (
) 2 (
yy xx
zz zy
zx
yz zz
xx yy
yx
xz xy
zz yy
xx ij
l
m l
m
m
m
l
m l
m
m
m
l
m l
m l
m l
m
( 3 2 )
E 2 ( l m )
l
l m
3
2
K )
2 1 )(
1
(
l
E
) 1
(
2
m E
Stress-strain - significance Stress-strain - significance
As in the case of deformation the stress-strain relation can be interpreted in simple geometric terms:
Remember that these relations are a generalization of Hooke’s Law:
l u
g l
u
F= D s
D being the spring constant and s the elongation.
mg
12
l
E
u 22
iiVV
K
K
P
Seismic wave velocities: P-waves Seismic wave velocities: P-waves
Material
Vp (km/s)Unconsolidated material
Sand (dry) 0.2-1.0
Sand (wet) 1.5-2.0
Sediments
Sandstones 2.0-6.0
Limestones 2.0-6.0
Igneous rocks
Granite 5.5-6.0
Gabbro 6.5-8.5
Pore fluids
Air 0.3
Water 1.4-1.5
Oil 1.3-1.4
Other material
Elastic anisotropy Elastic anisotropy
What is seismic anisotropy?
Seismic wave propagation in anisotropic media is quite different from isotropic media:
• There are in general 21 independent elastic constants (instead of in the isotropic case) 2
• there is shear wave splitting (analogous to optical birefringence)
• waves travel at different speeds depending in the direction of propagation
• The polarization of compressional and shear waves may not be perpendicular or parallel to the wavefront, resp.
kl ijkl
ij c
Shear-wave splitting
Shear-wave splitting
Anisotropic wave fronts Anisotropic wave fronts
From Brietzke, Diplomarbeit