Introduction to elastic wave equation

Salam AlnabulsiUniversity of Calgary

Department of Mathematics and Statistics October 15,2012

Outline

• Motivation

• Elastic wave equation

– Equation of motion, Definitions and The linear Stress-Strain relationship

• The Seismic Wave Equation in Isotropic Media

• Seismic wave equation in homogeneous media

• Acoustic wave equation

• Short Summary

Motivation

• Elastic wave equation has been widely used to describe wave propagation in an elastic medium, such as seismic waves in Earth and ultrasonic waves in human body.

• Seismic waves are waves of energy that travel through the earth, and are a result of an earthquake, explosion, or a volcano.

Elastic wave equation

• The standard form for seismic elastic wave equation in homogeneous media is :

parameters Lame :,

ntdisplaceme theis :

density theis :

.)2(

u

ρ

uuuρ

Equation of Motion

• We will depend on Newton’s second law F=ma

• The total force from stress field:

on accelerati:

mass:m

2

2

321

t

ua

dxdxρdx

body

ii FFF

321321

321

dxdxdxdxdxdxF

dxdxdxfF

ijjxi

i

body

i

j

ij

Equation of Motion

• Combining these information together we get the Momentum equation (Equation of Motion)

fiijj

t

u

2

2

tensor.stress theis and

nt,displaceme theisu density, theis where•

Definitions

• Stress : A measure of the internal forces acting within a deformable body.

(The force acting on a solid to deform it)

The stress at any point in an object, assumed to behave as a continuum, is completely defined by nine component stresses: three orthogonal normal stresses and six orthogonal shear stresses.

This can be expressed as a second-order tensor known as the Cauchy stress tensor.

333231

232221

131211

Definitions

• Strain : A local measure of relative change in the displacement field, that is , the spatial gradients in the displacement field. And it related to deformation, or change in shape, of a material rather than any change in position.

)(2

1ijjiij

uue

Some possible strains for two- dimensional element

The linear Stress-Strain Relationship

• Stress and Strain are linked in elastic media by Stress -Strain or constitutive relationship.

• The most general linear relationship between Stress and Strain is :

where,

• Cijkl is termed the elastic tensor.

klijklij eC

)tcoefficien Elasticor(StiffnessC

The linear Stress-Strain Relationship

• The elastic tensor Cijkl , is forth-order with 81 components ( 1 ≤ i,j,k,l ≤ 3 ).

• Because of the symmetry of the stress and strain tensors and the thermodynamic considerations, only 21 of these components are independent.

• The 21 components are necessary to specify the stress-strain relationship for the most general form of an elastic solid.

The linear Stress-Strain Relationship

• The material is isotropic if the properties of the solid are the same in all directions.

• The material is anisotropic if the properties of the media vary with direction.

The linear Stress-Strain Relationship

• If we assume isotropy , the number of the independent parameters is reduced to two :

n.explanatio physical simple no Has :

shearing tomaterial theof resistance theof measureA :

for 0 , for 1

thecalled are and where

)(

2

xy

xy

e

ijij

jlikjkilklijijkl

jiδjiδ

etersLame param

C

The linear Stress-Strain Relationship

• The stress-strain equation for an isotropic media :

ijkkij

kljlikjkilklijij

ee

e

2

)]([

The linear Stress-Strain Relationship

• The linear isotropic stress-strain relationship

• The strain tensor is defined as :

• Substituting for (2) in (1) we obtain :

(1) 2 ijkkijij ee

(2) )(2

1ijjiij

uue

(3) )( ijjikkijij uuu

The Seismic Wave Equation in Isotropic Media

• Substituting (3) in the homogeneous equation of motion :

] )([2

2

ijjikkijj uuut

u

ijjjijijjijkkikki uuuuuu )(

ijjjjikkiijjijkki uuuuuu )(

The Seismic Wave Equation in Isotropic Media

uuuuuuρ

t

uu

T 2

2

2

.)(])(.[).(

as form in vector thiscan write we Defining

u.uuity ctor identuse the ve 2

uuuuuuρ T .)2(])(.[).(:obtain we

The Seismic Wave Equation in Isotropic Media

• This is one form of the seismic wave equation

• The first two terms on the (r.h.s) involve gradient in the Lame parameters and are non-zero whenever the material is inhomogeneous (i.e. : contains velocity gradient)

• Including these factors makes the equations very complicated and difficult to solve efficiently.

uuuuuuρ T .)2(])(.[).(

The Seismic Wave Equation in Isotropic Media

• If velocity is only a function of depth , then the material can be modeled as a series of homogeneous layers.

• Within each layer , there are no gradients in the Lames parameters and so these terms go to zero.

• The standard form for seismic wave equation in homogeneous media is :

• Note : Here we neglected the gravity and velocity gradient terms and has assumed a linear , isotropic Earth model

uuuρ .)2(

Seismic Wave Equation in homogeneous media

2

22

22

velocity wave-S the

velocity wave-P thewhere

.

: as simplified

isequation wave theconstants, are and , If

uuu

Acoustic Wave Equation

• If the Lame parameter µ = 0 (i.e. No shearing ) then we get :

• In this case, the Elastic wave equation is reduced to an acoustic wave equation.

npropagatio of speed the where 22

2

2

12

c

tc

Short Summary

• We introduced definitions of Stress and Strain and the relationship between them.

• We depend on Newton’s 2nd law to get the equation of motion and from it we Derive the general form of Elastic wave equation .

• We simplify it to the standard form by modeling the material as series of homogeneous layers.

• We discussed two types of waves– P-waves(Compressional)– S-waves(Shear)

• Finally, if we assume no shearing then we reduced it to an acoustic wave equation .