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Theory of Seismic waves
I. Elasticity
Theory of elasticity
• Seismic waves are stress (mechanical) waves that are generated as a response to acting on a material by a force.
• The force that generates this stress comes from a source of seismic energy such artificial (Vibroseis, dynamite, ... etc) or natural earthquakes.
• The stress will produce strain (deformation) in the material relating to elasticity theory.
• Therefore, we need to study a little bit of elasticity theory in order to better understand the theory of seismic waves.
Stress
• Stress, denoted by , is force per unit area, with units of pressure such as Pascal (N/m2).
• xx denotes a stress produced by a force that is parallel to the x-axis acting upon a surface (YZ plane) which is perpendicular to the x-axis.
• xy denotes a stress produced by a force that is parallel to the x-axis acting upon a surface (XZ plane) which is perpendicular to the y-axis.
Stress
• There should be a maximum of 9 stress components associated with every possible combination of the coordinate system axes (xx, xy, xz, yx, yy, yz, zx, zy, zz).
• According to equilibrium (body is not moving but only deformed as a result of stress application): ij = ji, meaning that xy = yx, yz = zy, and zx = xz.
• If the force is perpendicular to the surface, we have a normal stress (xx, yy, zz); while if it’s tangential to the surface, we have a shearing stress (xy, yz, xz).
Stress
• The stress matrix composed of nine components of the stress:
zzzyzx
yzyyyx
xzxyxx
Strain
• Strain, denoted by , is the fractional change in a length, area, or volume of a body due to the application of stress.
• For example, if a rod of length L is stretched by an amount L, the strain is L/L.
• As a matter of fact, strain is dimensionless.
Strain
• To extend this analysis to three dimensional case, consider a body with dimensions of X, Y, and Z along the x-, y-, and z-axes respectively.
• To extend this analysis to three dimensional case, consider a body with dimensions of X, Y, and Z along the x-, y-, and z-axes respectively.
• If the body is subjected to stress, then generally X will change by an amount of u(x,y,z), Y by an amount of v(x,y,z), and Z by an amount of w(x,y,z)
Strain
• There are generally 9 strain components corresponding to the 9 stress components (xx, xy, xz, yx, yy, yz, zx, zy, zz) because of equilibrium: ij = ji, meaning that xy = yx, yz = zy, and zx = xz.
• We can define the following strains:
– Normal strains ( )
– Shear strains ( )
z
wzz
y
vyy
x
uxx
,,
x
w
z
uzx
z
v
y
wyz
y
u
x
vxy
,,
Strain
• Dilatation () is known as the change in volume (V) per unit volume (V):
• The strain matrix composed of the nine components of strain:
z
w
y
v
x
uzzyyxx
V
V
zzzyzx
yzyyyx
xzxyxx
Components of stress and strain
• If a stretching force is acting in the x-y plane and the corresponding motion is only occurred in the direction of x- axis, we will have the situation depicted in the corresponded figure.
• The point P moves a distance u to point P’ after stretching while point Q moves a distance ux+ux to point Q’.
y
x
P QP’ Q’
ux
x
xx
uu x
x
Normal Strain
As we know that normal strain in x- direction is know as the ratio between the change of length of QP to the original length of QP
y
x
P QP’ Q’
ux
x
CoordinatesP(x,y)Q(x+x,y)P’(x+u,y)
),(' yxxx
uux Q
xx
QP
QPPQ
QP of length original
QP of length in changexx
''
uxxxx
uuxPQ x
''
xxxQP
x
u xxx
Ask students to do similar processing for yy and zz.
xx
uu x
x
Shear Strain• If a stretching force is acting in the
x-y plane and the corresponding motion is induced either in the direction of x- axis and y-axis, we will have the situation depicted in the corresponded figure.
• The infinitesimal rectangular PQRS will have displaced and deformed into the diamond P’Q’R’S’.
• After stretching, points P, Q, S and R move to P’, Q’, S’, and R’ with coordinates.
y
x
P QP’ Q’ux
xx
x
uu
xx
S R
S’R’
xx
uu
yy
uyy
yyxu
xu
Shear Strain
• The deformation in y coordinates in relative to x-axis is given by
CoordinatesP(x,y) P’(x+ux,y+uy)
Q(x+x,y)
S(x,y+y)
),(' xx
uuyxx
x
uux Q
yy
xx
x
yPyPyQyQ
length-x original
length-x to relativey in changexy
)'()'(
Ask students to substitute the coordinates of points P, Q, P’, and Q’ to get the shear-strain component in
the x-y plane
),(' yyxu
xuyyyyxu
uxx S
y
x
P QP’
Q’
ux
xx
x
uu
xx
S R
S’R’
xx
uu
yy
uyy
Hook’s law
• It states that the strain is directly proportional to the stress producing it.
• The strains produced by the energy released due to the sudden brittle failure.
• The energy is released in the form of of seismic waves in earth materials are such that Hooke’s law is always satisfied.
Hook’s law
• A deformation is the change in size or shape of an object.
• An elastic object is one that returns to its original size and shape after the act forces have been removed.
• If the forces acting on the object are too large, the object can be permanently distorted based on its physical properties.
Hook’s law
• Mathematically, Hooke’s law can be expressed as:
• where is the stress matrix, is the strain matrix, and C is the elastic-constants tensor, which is a fourth-order tensor consisting of 81 elastic constants (Cxxxx to Czzzz).
z)y,x,lk,j,(i, klijklCij
Hook’s law
• For example:
zzxyzzCzyxyzyCzxxyzxCyzxyyzCyyxyyyC
yxxyyxCxzxyxzCxyxyxyCxxxyxxC
z
xk
z
xl
klxyklCxy
Hook’s law
• In general, the stiffness matrix consists of 81 independent entries
Hook’s law
• Because σij = σji, there are only 6 independent components in the stress and strain matrices. This means that the elastic-constant tensor decreases to 54 elastic constants.
Hook’s law
• Because εij = εji, there are only 6 independent components in the stress and strain matrices. This means that the elastic-constant tensor decreases to 36 elastic constants.
• Moreover, because of the symmetry relations giving Cij = Cji, only 21 independent elastic constants that can exist in the most general elastic material.
Hook’s law
• The stress-strain relation of an isotropic elastic material may be described by 2 independent elastic constants, known as Lame constants, and , and:
• The stress components can be defined as:
23223
13213
12212
33233233221133
22222233221122
11211233221111
Note that V
V
Hook’s law
• In isotropic media, Hooke’s law takes the following form:
23
13
1233
22
11
00000
00000
00000
0002
0002
0002
23
13
1233
22
11
Hook’s law
• Hooke’s law in an isotropic medium is given by the following index equations:
• These equations are sometimes called the constitutive equations.
• Students should review Elastic constants in isotropic media (e.g. Young’s modulus, Bulk modulus, Poisson’s ratio, ....., etc.)
),,,,(2
),,(2
zyxjiji ij ij
zyxi ij ij
One dimensional wave equation
• To get the wave equation, we will develop Newton’s second law towards our goal of expressing an equation of motion.
• Newton’s second law simply states:
• The applicable force have one of two categories:– Body Forces: forces such as gravity that work equally well
on all particles within the mass- the net force is proportional (essentially) to the volume of the body.
– Surface Forces: forces that act on the surface of a body-the net force is proportional to the surface area over which the force acts.
amF
Equation of motion
• Using constitutive equations and Newton’s second law, students are asked to derive the wave equation in one dimension.
• In order to obtain the equations of motion for an elastic medium we consider the variation in stresses across a small parallelpiped.
dxxxx
xx
z
y
x dyy
xyxy
dx
dzdy
dzz
xzxz
xz
xyxx
Equation of Motion
• Stresses acting on the surface of a small parallelepiped parallel to the x-axis.
• Stresses acting on the front face do not balance those acting on the back face.
• The parallelepiped is not in equilibrium and motion is possible.
• If we first consider the forces acting in the x-direction, hence the forces will be acting on:– Normal to back- and front faces, – Tangential to the left- and right-
hand faces, and – Tangential to the bottom and top
faces.
26
dxxxx
xx
z
y
x dyy
xyxy
dx
dzdy
dzz
xzxz
xz
xyxx
Equation of Motion
• Normal force acting on the back face
force = stress x area
• Normal force acting on the front face
• The difference between two forces is given the final normal force acting on the sample in the x-direction
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dz dyxx
dz dy dxxxx
xx )(
dz dy dxxxxdz dy xxdz dy dx
xxx
xx
)(
Equation of Motion
• Tangential force acting on left-hand face
• Tangential force acting on right-hand face
• The difference between two forces is given one of tangential forces acting on the sample in the x-direction
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dz dx xy
dz dx dyy
xyxy )(
dz dy dx y
dz dx dz dx dyy
xyxy
xyxy
)(
Ask students to get the other tangential force acting on the sample in the x-direction
Equation of Motion
• The normal force can be balanced by the mass times the acceleration of the cube, as given by Newton's law:
• where • dxdydz is the mass. • Cancelling out the volume term on each side, the equation can
be written in the following form
2
2
t
xudz dy dxdz dy dx
xxx
2
2
t
xu
xxx
Equation of Motion
• Now we may use Hooke's law to replace stress with displacement:
• Now, substituting for xx, and remembering that the medium is uniform so that k, m, and r are constants, we have
x
uk
x
uxx
x
xxx
)3
4(
)2()2(
2
2)2(
t
xu
x
u
xx
Equation of Motion
• The final form of the last equation can be written in the form;
• This equation equates force per unit volume to mass per unit volume times acceleration.
• The equation means that Pressure is given by the average of the normal stress components the may cause a change in volume per unit volume.
2
2
)2(2
2
x
u
t
xu x
Equation of Motion
• For an applied pressure P producing a volume change V of a volume V, substituting the k is the modulus of incompressibility (bulk modulus) in the last equation, we will find:
2
2)3
4(
2
2
x
uk
t
xu x
2
22
2
2
x
upV
t
xu x
)
3
4(k
pV
Ask students to get the wave equation for Shear wave
Giving P wave equation
Equation of Motionin
Three dimensions
Equation of Motion (3D)
• The total force acting on the parallelepiped in the x-direction is given by
• Making use of the Newton’s second law of motion Mass x Acceleration = resulting force
• Taking u as the displacement in the x-direction, we will have
• The equation can be written in the following form
• Where is density34
dz dy dx zyx
dz dy dxt
u xzxyxxx )(2
2
dz dy dx zyx
dz dy dxz
dz dy dxy
dz dy dxx
F xzxyxxxzxyxxx )(
zyxt
u xzxyxxx
2
2
(1)
Separation of equations of motion in vector form
• From Hook’s law, the generalized relationship between stress, strain and displacement is given by
• Substituting stress components in the equation of motion (1), we have for the x-direction
• Note that for a homogeneous isotropic solid, moduli and are constant with respect to x, y and z that do not vary with position. Thus
35
ijijij 2j
iijij
x
u
2
z
u
x
u
zy
u
x
u
yx
u
xt
u xzxyxx )()2(2
2
2
2
2
2
2
2
2
2
z
u
y
u
x
u
z
u
y
u
x
u
xxt
u xxxxyxx
Separation of equations of motion
• As we know that represent the divergence
operator. If it is applied to: – a vector it produces a scalar– a tensor it produces a vector
• It gives the change in volume per unit volume associated with the deformation ( = V/V ). It expresses the local rate of expansion of the vector field.
36
uz
u
y
u
x
u zyx
z
u
y
u
x
u
u
u
u
zyxu zyx
z
y
x
///
Separation of equations of motion
• The gradient operator is a vector containing three partial derivatives. • When applied to
– a scalar, it produces a vector, – a vector, it produces a tensor.
• The gradient vector of a scalar quantity defines the direction in which it increases fastest; the magnitude equals the rate of change in that direction.
• Thus, the equation of motion can be written as
37
zuzuzu
yuyuyu
xuxuxu
uuu
z
y
x
u
zyx
zyx
zyx
zyx
///
///
///
/
/
/
2
2
2
2
2
2
2
2
z
u
y
u
x
u
xt
u xxxx
• Thus, the equation of motion in the x-direction can be written as
38
x
x uxt
u 2
2
2
(2)
Separation of equations of motion
• Similar to x-direction, the equations of motion in y- and z- directions are given respectively as
• From equations 2, 3, and 4, we have
39
y
y uyt
u2
2
2
zz u
zt
u 2
2
2
(3)
(4)
u zyx
t
u 2
2
2
(5)
Separation of equations of motion
• Then, equation (5) can be written in the form
• From the vector analysis, we have
• Also, the displacement u can be represented in terms of scalar and vector potentials, via Helmholtz’ theorem, then
• Then equation (6) can be written as
40
u u t
u 2
2
2
(6)
u uu 2
u
t
2
2
Vectors analysis: 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!
z
u
y
u
x
u zyx
u
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!
y
u
x
ux
u
z
uz
u
y
u
uuuzyx
xy
zx
yz
zyx
kji
u
The curl of a vector field represents the rotational part of that field (e.g. shear waves in a seismic wavefield)
Vectors analysis
• Background of mathematics
42
Separation of equations of motion
• From the vector analysis, the characteristics of potentials in terms of divergence and curl give that ()=0 and ()=0.
• The last equation can be summarized in the following form
• Using potentials, we can break up the wave equation into two equations
43
t
2
2
2
22
2
222
t
t
2
22
2
222
t
t
Separation of equations of motion
44
0 t
2
222
0
2
22
t
2
2
2 2
t
2
2
2
t
2
Scalar wave equation(divergence)
Vector wave equation(curl)
Separation of equations of motion
• The scalar potential that satisfies the scalar wave equation gives divergence of a displacement that associated with a change in volume (u 0). – This solution produces P waves– No shear motion is associated (u = 0)
• In the vector potential that satisfies the vector wave equation, the displacement is curl (rotation, u 0) that no associated with a change in volume change occurs (u = 0). – This solution produces shear motions generating S- waves of probably two
independent polarizations– No P- wave is associated
45
