Theory of Seismic waves I. Elasticity. Theory of elasticity Seismic waves are stress (mechanical)...

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.

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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

///


• 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)


• 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)


• 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


• 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


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)


• 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

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Theory of Seismic waves I. Elasticity. Theory of elasticity Seismic waves are stress (mechanical)...

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