Introduction to elastic wave equation. Salam Alnabulsi University of Calgary Department of Mathematics and Statistics October 15,2012

Transcription

1 Introdcton to elastc wave eqaton Salam Alnabls Unversty of Calgary Department of Mathematcs and Statstcs October 15,01

2 Otlne Motvaton Elastc wave eqaton Eqaton of moton, Defntons and The lnear Stress- Stran relatonshp The Sesmc Wave Eqaton n Isotropc Meda Sesmc wave eqaton n homogeneos meda Acostc wave eqaton Short Smmary

3 Motvaton Elastc wave eqaton has been wdely sed to descrbe wave propagaton n an elastc medm, sch as sesmc waves n Earth and ltrasonc waves n hman body. Sesmc waves are waves of energy that travel throgh the earth, and are a reslt of an earthqake, exploson, or a volcano.

4 Elastc wave eqaton The standard form for sesmc elastc wave eqaton n homogeneos meda s : ρ ( ). ρ :s the densty :s the dsplacement, : Lame parameters

5 Eqaton of Moton We wll depend on Newton s second law F=ma m : mass ρdx dx 1 a : acceleraton t The total force from stress feld: dx 3 F F F body F F body f x dx dx 1 1 dx dx dx 3 dx 3 dx dx 1 dx 3

6 Eqaton of Moton Combnng these nformaton together we get the Momentm eqaton (Eqaton of Moton) t f where s thedensty, s and s thestresstensor. the dsplacement,

7 Defntons Stress : A measre of the nternal forces actng wthn a deformable body. (The force actng on a sold to deform t) The stress at any pont n an obect, assmed to behave as a contnm, s completely defned by nne component stresses: three orthogonal normal stresses and sx orthogonal shear stresses.

8 Ths can be expressed as a second-order tensor known as the Cachy stress tensor.

9 Defntons Stran : A local measre of relatve change n the dsplacement feld, that s, the spatal gradents n the dsplacement feld. And t related to deformaton, or change n shape, of a materal rather than any change n poston. e 1 ( )

10 Some possble strans for two- dmensonal element

11 The lnear Stress-Stran Relatonshp Stress and Stran are lnked n elastc meda by Stress - Stran or constttve relatonshp. The most general lnear relatonshp between Stress and Stran s : where, C kl e kl C Stffness(or Elastc coeffcent) C kl s termed the elastc tensor.

12 The lnear Stress-Stran Relatonshp The elastc tensor C kl, s forth-order wth 81 components ( 1,,k,l 3 ). Becase of the symmetry of the stress and stran tensors and the thermodynamc consderatons, only 1 of these components are ndependent. The 1 components are necessary to specfy the stress-stran relatonshp for the most general form of an elastc sold.

13 The lnear Stress-Stran Relatonshp The materal s sotropc f the propertes of the sold are the same n all drectons. The materal s ansotropc f the propertes of the meda vary wth drecton.

14 The lnear Stress-Stran Relatonshp If we assme sotropy, the nmber of the ndependent parameters s redced to two : C ( where and are called the Lame parameters δ kl 0 for : A measre of the resstance of the materal xy e xy kl 1for, δ l : Has no smple physcal explanaton. k k l ) to shearng

15 The lnear Stress-Stran Relatonshp The stress-stran eqaton for an sotropc meda : [ kl ( l k k l )] e kl e kk e

16 The lnear Stress-Stran Relatonshp The lnear sotropc stress-stran relatonshp e kk e The stran tensor s defned as : 1 e ( ) (1) () Sbstttng for () n (1) we obtan : k k ( ) (3)

17 The Sesmc Wave Eqaton n Isotropc Meda Sbstttng (3) n the homogeneos eqaton of moton : ] ) ( [ k k t k k k k ) ( k k k k ) (

18 The Sesmc Wave Eqaton n Isotropc Meda Defnng we can wrte ths n vector form as t T ρ (. ).[ ( ) ] ( ). se the vector dentty weobtan : ρ (. ).[ ( ) T. ] ( ).

19 The Sesmc Wave Eqaton n Isotropc Meda Ths s one form of the sesmc wave eqaton ρ (. ).[ ( ) The frst two terms on the (r.h.s) nvolve gradent n the Lame parameters and are nonzero whenever the materal s nhomogeneos (.e. : contans velocty gradent) Incldng these factors makes the eqatons very complcated and dffclt to solve effcently. T ] ( ).

20 The Sesmc Wave Eqaton n Isotropc Meda If velocty s only a fncton of depth, then the materal can be modeled as a seres of homogeneos layers. Wthn each layer, there are no gradents n the Lames parameters and so these terms go to zero. The standard form for sesmc wave eqaton n homogeneos meda s : ρ ( ). Note : Here we neglected the gravty and velocty gradent terms and has assmed a lnear, sotropc Earth model

21 Sesmc Wave Eqaton n homogeneos meda If, and are constants,the waveeqaton s smplfed. as: thes- wherethe P - wave velocty wave velocty

22

23 Acostc Wave Eqaton If the Lame parameter µ = 0 (.e. No shearng ) then we get : 1 c t where c the speedof propagaton In ths case, the Elastc wave eqaton s redced to an acostc wave eqaton.

24 Short Smmary We ntrodced defntons of Stress and Stran and the relatonshp between them. We depend on Newton s nd law to get the eqaton of moton and from t we Derve the general form of Elastc wave eqaton. We smplfy t to the standard form by modelng the materal as seres of homogeneos layers. We dscssed two types of waves P-waves(Compressonal) S-waves(Shear) Fnally, f we assme no shearng then we redced t to an acostc wave eqaton.