Chapter 4

 

Solutions to simple boundary and initial value problems

 

 

 

 

4.4 Simple dynamic solutions for linear elastic materials

 

In this section we summarize and derive the solutions to various elementary problems in dynamic linear elasticity.

 

4.4.1: Surface subjected to time varying normal pressure

 

An isotropic, linear elastic half space with shear modulus μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBaaa@347C@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@  and mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaa@356C@  occupies the region x 2 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGOmaaqabaGccqGH+aGpcaaIWaaaaa@38B9@  .  The solid is at rest and stress free at time t=0.  For t>0 it is subjected to a uniform pressure p(t) on x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGOmaaqabaGccqGH9aqpcaaIWaaaaa@38B7@  as shown in the picture. 

 

Solution: The displacement and stress fields in the solid (as a function of time and position) are

u 2 ( x 2 ,t)={ c L E (1+ν)(12ν) (1ν) 0 t x 2 / c L p(τ)dτ x 2 <t c L 0 x 2 >t c L σ 22 ={ p(t x 2 / c L ) x 2 <t c L 0 x 2 >t c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamyDam aaBaaaleaacaaIYaaabeaakiaacIcacaWG4bWaaSbaaSqaaiaaikda aeqaaOGaaiilaiaadshacaGGPaGaeyypa0ZaaiqaaeaafaqabeGaba aabaWaaSaaaeaacaWGJbWaaSbaaSqaaiaadYeaaeqaaaGcbaGaamyr aaaadaWcaaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4MaaiykaiaacI cacaaIXaGaeyOeI0IaaGOmaiabe27aUjaacMcaaeaacaGGOaGaaGym aiabgkHiTiabe27aUjaacMcaaaWaa8qCaeaacaWGWbGaaiikaiabes 8a0jaacMcacaWGKbGaeqiXdqNaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyipaWJaamiDaiaado gadaWgaaWcbaGaamitaaqabaaabaGaaGimaaqaaiaadshacqGHsisl caWG4bWaaSbaaWqaaiaaikdaaeqaaSGaai4laiaadogadaWgaaadba GaamitaaqabaaaniabgUIiYdaakeaacaaIWaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaadIhadaWgaaWcbaGaaGOmaaqaba GccqGH+aGpcaWG0bGaam4yamaaBaaaleaacaWGmbaabeaaaaaakiaa wUhaaaqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH9a qpdaGabaqaauaabeqaceaaaeaacqGHsislcaWGWbGaaiikaiaadsha cqGHsislcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaai4laiaadogada WgaaWcbaGaamitaaqabaGccaGGPaGaaGPaVlaaykW7caaMc8UaaGPa VlaadIhadaWgaaWcbaGaaGOmaaqabaGccqGH8aapcaWG0bGaam4yam aaBaaaleaacaWGmbaabeaaaOqaaiaaicdacaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadIhadaWgaaWcbaGaaGOm aaqabaGccqGH+aGpcaWG0bGaam4yamaaBaaaleaacaWGmbaabeaaaa aakiaawUhaaaaaaa@1C6F@

where c L = E(1ν)/ ρ 0 (1+ν)( 12ν ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaqhaa WcbaGaamitaaqaaaaakiabg2da9maakaaabaGaamyraiaacIcacaaI XaGaeyOeI0IaeqyVd4Maaiykaiaac+cacqaHbpGCdaWgaaWcbaGaaG imaaqabaGccaGGOaGaaGymaiabgUcaRiabe27aUjaacMcadaqadaqa aiaaigdacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaaleqaaa aa@4B52@  is the speed of longitudinal wave propagation through the solid.  All other displacement and stress components are zero.  For the particular case of a constant (i.e. time independent) pressure, magnitude σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa aaleaacaaIWaaabeaaaaa@37B1@ , applied to the surface

u 2 ( x 2 ,t)={ (12ν)(1+ν) (1ν) σ 0 E ( c L t x 2 ) x 2 < c L t 0 x 2 > c L t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaGOmaaqabaGccaGGOaGaamiEamaaBaaaleaacaaIYaaabeaa kiaacYcacaWG0bGaaiykaiabg2da9maaceaabaqbaeqabiqaaaqaam aalaaabaGaaiikaiaaigdacqGHsislcaaIYaGaeqyVd4Maaiykaiaa cIcacaaIXaGaey4kaSIaeqyVd4MaaiykaaqaaiaacIcacaaIXaGaey OeI0IaeqyVd4MaaiykaaaadaWcaaqaaiabeo8aZnaaBaaaleaacaaI WaaabeaaaOqaaiaadweaaaGaaiikaiaadogadaWgaaWcbaGaamitaa qabaGccaWG0bGaeyOeI0IaamiEamaaBaaaleaacaaIYaaabeaakiaa cMcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaadIhadaWgaaWcbaGaaGOmaaqabaGccqGH8aapcaWGJbWaaSba aSqaaiaadYeaaeqaaOGaamiDaaqaaiaaicdacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaamiEamaaBaaaleaacaaIYaaabeaa kiabg6da+iaadogadaWgaaWcbaGaamitaaqabaGccaWG0baaaaGaay 5Eaaaaaa@C551@               σ 22 ={ σ 0 x 2 < c L t 0 x 2 > c L t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa aaleaacaaIYaGaaGOmaaqabaGccqGH9aqpdaGabaqaauaabeqaceaa aeaacqGHsislcqaHdpWCdaWgaaWcbaGaaGimaaqabaGccaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadIhadaWg aaWcbaGaaGOmaaqabaGccqGH8aapcaWGJbWaaSbaaSqaaiaadYeaae qaaOGaamiDaaqaaiaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaamiEamaaBaaaleaacaaIYaaabeaakiabg6da+iaadogadaWgaaWc baGaamitaaqabaGccaWG0baaaaGaay5Eaaaaaa@6B21@

Evidently, a stress pulse equal in magnitude to the surface pressure propagates vertically through the half-space with speed c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaWgaa WcbaGaamitaaqabaaaaa@36FB@ .

 

Notice that the velocity of the solid is constant in the region 0< x 2 <t c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdacqGH8aapcaWG4bWaaSbaaSqaai aaikdaaeqaaOGaeyipaWJaamiDaiaadogadaWgaaWcbaGaamitaaqa baaaaa@37F7@ , and the velocity is related to the pressure by

v 2 = c L (12ν)(1+ν) (1ν) σ 0 E = ρ 0 c L σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAhadaWgaa WcbaGaaGOmaaqabaGccqGH9aqpcaWGJbWaaSbaaSqaaiaadYeaaeqa aOWaaSaaaeaacaGGOaGaaGymaiabgkHiTiaaikdacqaH9oGBcaGGPa GaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaaabaGaaiikaiaaigda cqGHsislcqaH9oGBcaGGPaaaamaalaaabaGaeq4Wdm3aaSbaaSqaai aaicdaaeqaaaGcbaGaamyraaaacqGH9aqpdaWcaaqaaiabeg8aYnaa BaaaleaacaaIWaaabeaaaOqaaiaadogadaWgaaWcbaGaamitaaqaba aaaOGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaaa@54C1@

 

Derivation: The solution can be derived as follows. The governing equations are

 The strain-displacement relation  ε ij =( u i / x j + u j / x i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maabmaabaGaeyOaIyRaamyDamaaBaaaleaacaWG Pbaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaO Gaey4kaSIaeyOaIyRaamyDamaaBaaaleaacaWGQbaabeaakiaac+ca cqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaa Gaai4laiaaikdaaaa@48F7@

 The elastic stress-strain equations   σ ij =E{ ε ij +ν ε kk δ ij /(12ν) }/(1+ν) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaWGfbWaaiWaaeaacqaH1oqzdaWgaaWcbaGaamyA aiaadQgaaeqaaOGaey4kaSIaeqyVd4MaeqyTdu2aaSbaaSqaaiaadU gacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGc caGGVaGaaiikaiaaigdacqGHsislcaaIYaGaeqyVd4MaaiykaaGaay 5Eaiaaw2haaiaac+cacaGGOaGaaGymaiabgUcaRiabe27aUjaacMca aaa@5190@

 The linear momentum balance equation  σ ij / x i = ρ 0 2 u j / t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaadMga aeqaaOGaeyypa0JaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaeyOaIy 7aaWbaaSqabeaacaaIYaaaaOGaamyDamaaBaaaleaacaWGQbaabeaa kiaac+cacqGHciITcaWG0bWaaWbaaSqabeaacaaIYaaaaaaa@466E@

Now:

1.      Symmetry considerations indicate that the displacement field must have the form

u 1 = u 3 =0 u 2 =u( x 2 ,t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaWG1bWaaSbaaSqaaiaaiodaaeqa aOGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaamyDamaaBaaaleaacaaIYaaabeaakiabg2da 9iaadwhacaGGOaGaamiEamaaBaaaleaacaaIYaaabeaakiaacYcaca WG0bGaaiykaaaa@50EB@

Substituting this equation into the strain-displacement equations shows that the only nonzero component of strain is ε 22 =u/ x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0JaeyOaIyRaamyDaiaac+cacqGHciITcaWG4bWa aSbaaSqaaiaaikdaaeqaaaaa@3D7F@ .

2.      The stress-strain law then shows that

σ 22 = E(1ν) (1+ν)(12ν) u x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0ZaaSaaaeaacaWGfbGaaiikaiaaigdacqGHsisl cqaH9oGBcaGGPaaabaGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPa GaaiikaiaaigdacqGHsislcaaIYaGaeqyVd4MaaiykaaaadaWcaaqa aiabgkGi2kaadwhaaeaacqGHciITcaWG4bWaaSbaaSqaaiaaikdaae qaaaaaaaa@4CAE@

In addition, the shear stresses are all zero (because the shear strains are zero), and while σ 11 , σ 22 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaa aa@3A4C@  are nonzero, they are independent of x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaa aa@34AA@  and x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaa aa@34AC@ .

3.      The only nonzero linear momentum balance equation is therefore

σ 22 / x 2 = ρ 0 2 u/ t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqaHdpWCdaWgaaWcbaGaaG OmaiaaikdaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaaGOm aaqabaGccqGH9aqpcqaHbpGCdaWgaaWcbaGaaGimaaqabaGccqGHci ITdaahaaWcbeqaaiaaikdaaaGccaWG1bGaai4laiabgkGi2kaadsha daahaaWcbeqaaiaaikdaaaaaaa@46A9@

Substituting for stress from (2) yields

2 u x 2 2 = 1 c L 2 2 u t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaey OaIy7aaWbaaSqabeaacaaIYaaaaOGaamyDaaqaaiabgkGi2kaadIha daqhaaWcbaGaaGOmaaqaaiaaikdaaaaaaOGaeyypa0ZaaSaaaeaaca aIXaaabaGaam4yamaaDaaaleaacaWGmbaabaGaaGOmaaaaaaGcdaWc aaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadwhaaeaacqGHci ITcaWG0bWaaWbaaSqabeaacaaIYaaaaaaaaaa@47A5@

where

c L 2 = E(1ν) ρ 0 (1+ν)( 12ν ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaqhaa WcbaGaamitaaqaaiaaikdaaaGccqGH9aqpdaWcaaqaaiaadweacaGG OaGaaGymaiabgkHiTiabe27aUjaacMcaaeaacqaHbpGCdaWgaaWcba GaaGimaaqabaGccaGGOaGaaGymaiabgUcaRiabe27aUjaacMcadaqa daqaaiaaigdacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaaaa aa@4B50@

4.      This is a 1-D wave equation with general solution

u( x 2 ,t)=f(t x 2 / c L )+g(t+ x 2 / c L ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacaGGOa GaamiEamaaBaaaleaacaaIYaaabeaakiaacYcacaWG0bGaaiykaiab g2da9iaadAgacaGGOaGaamiDaiabgkHiTiaadIhadaWgaaWcbaGaaG OmaaqabaGccaGGVaGaam4yamaaBaaaleaacaWGmbaabeaakiaacMca cqGHRaWkcaWGNbGaaiikaiaadshacqGHRaWkcaWG4bWaaSbaaSqaai aaikdaaeqaaOGaai4laiaadogadaWgaaWcbaGaamitaaqabaGccaGG Paaaaa@4E45@

where f and g are two functions that must be chosen to satisfy boundary and initial conditions.

5.      The initial conditions are

u( x 2 ,0)=f( x 2 / c L )+g( x 2 / c L )=0 u t = f ( x 2 / c L )+ g ( x 2 / c L )=0 } x 2 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaciaabaqbae qabiqaaaqaaiaadwhacaGGOaGaamiEamaaBaaaleaacaaIYaaabeaa kiaacYcacaaIWaGaaiykaiabg2da9iaadAgacaGGOaGaeyOeI0Iaam iEamaaBaaaleaacaaIYaaabeaakiaac+cacaWGJbWaaSbaaSqaaiaa dYeaaeqaaOGaaiykaiabgUcaRiaadEgacaGGOaGaamiEamaaBaaale aacaaIYaaabeaakiaac+cacaWGJbWaaSbaaSqaaiaadYeaaeqaaOGa aiykaiabg2da9iaaicdaaeaadaWcaaqaaiabgkGi2kaadwhaaeaacq GHciITcaWG0baaaiabg2da9iqadAgagaqbaiaacIcacqGHsislcaWG 4bWaaSbaaSqaaiaaikdaaeqaaOGaai4laiaadogadaWgaaWcbaGaam itaaqabaGccaGGPaGaey4kaSIabm4zayaafaGaaiikaiaadIhadaWg aaWcbaGaaGOmaaqabaGccaGGVaGaam4yamaaBaaaleaacaWGmbaabe aakiaacMcacqGH9aqpcaaIWaaaaaGaayzFaaGaamiEamaaBaaaleaa caaIYaaabeaakiabgwMiZkaaicdaaaa@69B2@

where the prime denotes differentiation with respect to its argument.  Solving these equations (differentiate the first equation and then solve for f',g' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbGaai4jaiaacYcacaWGNbGaai 4jaaaa@36A3@  and integrate) shows that

f( x 2 / c L )=g( x 2 / c L )=A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaGGOa GaeyOeI0IaamiEamaaBaaaleaacaaIYaaabeaakiaac+cacaWGJbWa aSbaaSqaaiaadYeaaeqaaOGaaiykaiabg2da9iabgkHiTiaadEgaca GGOaGaamiEamaaBaaaleaacaaIYaaabeaakiaac+cacaWGJbWaaSba aSqaaiaadYeaaeqaaOGaaiykaiabg2da9iaadgeaaaa@475D@

where A is some constant.

6.      Observe that t+ x 2 / c L 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshacqGHRa WkcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaai4laiaadogadaWgaaWc baGaamitaaqabaGccqGHLjYScaaIWaaaaa@3DF4@  for t>0, so that g(t+ x 2 / c L )=A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbGaaiikaiaadshacqGHRaWkca WG4bWaaSbaaSqaaiaaikdaaeqaaOGaai4laiaadogadaWgaaWcbaGa amitaaqabaGccaGGPaGaeyypa0JaeyOeI0Iaamyqaaaa@3E30@ .  Substituting this result back into the solution in (4) gives u( x 2 ,t)=f(t x 2 / c L )A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacaGGOa GaamiEamaaBaaaleaacaaIYaaabeaakiaacYcacaWG0bGaaiykaiab g2da9iaadAgacaGGOaGaamiDaiabgkHiTiaadIhadaWgaaWcbaGaaG OmaaqabaGccaGGVaGaam4yamaaBaaaleaacaWGmbaabeaakiaacMca cqGHsislcaWGbbaaaa@4667@ .

7.        Next, use the boundary condition σ 22 =p(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0JaeyOeI0IaamiCaiaacIcacaWG0bGaaiykaaaa @3B71@  at x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaaGimaaaa@3675@  to see that

σ 22 = E(1ν) (1+ν)(12ν) u x 2 =p(t) E(1ν) (1+ν)(12ν) ( 1 c L )f'(t)=p(t) f(t x 2 / c L )= c L E (1+ν)(12ν) (1ν) 0 t x 2 / c L p(τ)dτ +B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeq4Wdm 3aaSbaaSqaaiaaikdacaaIYaaabeaakiabg2da9maalaaabaGaamyr aiaacIcacaaIXaGaeyOeI0IaeqyVd4MaaiykaaqaaiaacIcacaaIXa Gaey4kaSIaeqyVd4MaaiykaiaacIcacaaIXaGaeyOeI0IaaGOmaiab e27aUjaacMcaaaWaaSaaaeaacqGHciITcaWG1baabaGaeyOaIyRaam iEamaaBaaaleaacaaIYaaabeaaaaGccqGH9aqpcqGHsislcaWGWbGa aiikaiaadshacaGGPaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabgkDiElaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7daWcaaqaaiaadweacaGGOa GaaGymaiabgkHiTiabe27aUjaacMcaaeaacaGGOaGaaGymaiabgUca Riabe27aUjaacMcacaGGOaGaaGymaiabgkHiTiaaikdacqaH9oGBca GGPaaaamaabmaabaWaaSaaaeaacqGHsislcaaIXaaabaGaam4yamaa BaaaleaacaWGmbaabeaaaaaakiaawIcacaGLPaaacaWGMbGaai4jai aacIcacaWG0bGaaiykaiabg2da9iabgkHiTiaadchacaGGOaGaamiD aiaacMcaaeaacqGHshI3caWGMbGaaiikaiaadshacqGHsislcaWG4b WaaSbaaSqaaiaaikdaaeqaaOGaai4laiaadogadaWgaaWcbaGaamit aaqabaGccaGGPaGaeyypa0ZaaSaaaeaacaWGJbWaaSbaaSqaaiaadY eaaeqaaaGcbaGaamyraaaadaWcaaqaaiaacIcacaaIXaGaey4kaSIa eqyVd4MaaiykaiaacIcacaaIXaGaeyOeI0IaaGOmaiabe27aUjaacM caaeaacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcaaaWaa8qCaeaa caWGWbGaaiikaiabes8a0jaacMcacaWGKbGaeqiXdqhaleaacaaIWa aabaGaamiDaiabgkHiTiaadIhadaWgaaadbaGaaGOmaaqabaWccaGG VaGaam4yamaaBaaameaacaWGmbaabeaaa0Gaey4kIipakiabgUcaRi aadkeaaaaa@BED5@

where B is a constant of integration.

8.      Finally, B can be determined by setting t=0 in the result of (7) and recalling from step (5) that f( x 2 / c L )=A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaGGOa GaeyOeI0IaamiEamaaBaaaleaacaaIYaaabeaakiaac+cacaWGJbWa aSbaaSqaaiaadYeaaeqaaOGaaiykaiabg2da9iaadgeaaaa@3E94@ .  This shows that B=-A and so

u 2 ( x 2 ,t)= c L E (1+ν)(12ν) (1ν) 0 t x 2 / c L p(τ)dτ σ 22 =p(t x 2 / c L ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FgYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamyDam aaBaaaleaacaaIYaaabeaakiaacIcacaWG4bWaaSbaaSqaaiaaikda aeqaaOGaaiilaiaadshacaGGPaGaeyypa0ZaaSaaaeaacaWGJbWaaS baaSqaaiaadYeaaeqaaaGcbaGaamyraaaadaWcaaqaaiaacIcacaaI XaGaey4kaSIaeqyVd4MaaiykaiaacIcacaaIXaGaeyOeI0IaaGOmai abe27aUjaacMcaaeaacaGGOaGaaGymaiabgkHiTiabe27aUjaacMca aaWaa8qCaeaacaWGWbGaaiikaiabes8a0jaacMcacaWGKbGaeqiXdq haleaacaaIWaaabaGaamiDaiabgkHiTiaadIhadaWgaaadbaGaaGOm aaqabaWccaGGVaGaam4yamaaBaaameaacaWGmbaabeaaa0Gaey4kIi paaOqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH9aqp cqGHsislcaWGWbGaaiikaiaadshacqGHsislcaWG4bWaaSbaaSqaai aaikdaaeqaaOGaai4laiaadogadaWgaaWcbaGaamitaaqabaGccaGG Paaaaaa@6D6E@

as stated.

 

 

4.4.2: Surface subjected to time varying shear traction

 

An isotropic, linear elastic half space with shear modulus μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBaaa@348C@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@348E@  and mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaa@357C@  occupies the region x 2 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGOmaaqabaGccqGH+aGpcaaIWaaaaa@38C7@  .  The solid is at rest and stress free at time t=0.  For t>0 it is subjected to a uniform anti-plane shear traction p(t) on x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGOmaaqabaGccqGH9aqpcaaIWaaaaa@38C5@ .  Calculate the displacement, stress and strain fields in the solid.

 

It is straightforward to show that in this case

u 3 ( x 2 ,t)= 2(1+ν) c s E 0 t x 2 / c s p(τ)dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaG4maaqabaGccaGGOaGaamiEamaaBaaaleaacaaIYaaabeaa kiaacYcacaWG0bGaaiykaiabg2da9maalaaabaGaaGOmaiaacIcaca aIXaGaey4kaSIaeqyVd4MaaiykaiaadogadaWgaaWcbaGaam4Caaqa baaakeaacaWGfbaaamaapehabaGaamiCaiaacIcacqaHepaDcaGGPa Gaamizaiabes8a0bWcbaGaaGimaaqaaiaadshacqGHsislcaWG4bWa aSbaaWqaaiaaikdaaeqaaSGaai4laiaadogadaWgaaadbaGaam4Caa qabaaaniabgUIiYdaaaa@55C1@

σ 32 =p(t x 2 / c s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa aaleaacaaIZaGaaGOmaaqabaGccqGH9aqpcqGHsislcaWGWbGaaiik aiaadshacqGHsislcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaai4lai aadogadaWgaaWcbaGaam4CaaqabaGccaGGPaaaaa@4359@

where c s 2 = E 2(1+ν)ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaqhaa WcbaGaam4CaaqaaiaaikdaaaGccqGH9aqpdaWcaaqaaiaadweaaeaa caaIYaGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaGaeqyWdihaaa aa@40E5@  is the speed of shear waves propagating through the solid.  The details are left as an exercise.

 

 

4.4.3: 1-D Bar subjected to end loading

 

This solution is a cheat, because it doesn’t satisfy the full 3D equations of elasticity, but it turns out to be quite accurate.

 

A long thin rod occupying the region x 1 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGymaaqabaGccqGH+aGpcaaIWaaaaa@38C6@  is made from a homogeneous, isotropic, linear elastic material with Young’s modulus E and mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg8aYnaaBa aaleaacaaIWaaabeaaaaa@37BC@ .  At time t<0 it is at rest and free of stress.  At time t=0 it is subjected to a pressure p(t) at one end.  Calculate the displacement and stress fields in the solid.

 

We cheat by modeling this as a 1-D problem.  We assume that σ 11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa aaleaacaaIXaGaaGymaaqabaaaaa@387B@  is the only nonzero stress component, in which case the constitutive law and balance of linear momentum require that

σ 11 =E u 1 x 1 σ 11 x 1 = ρ 0 2 u 1 t 2 2 u 1 x 1 2 = 1 c B 2 2 u 1 t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeq4Wdm 3aaSbaaSqaaiaaigdacaaIXaaabeaakiabg2da9iaadweadaWcaaqa aiabgkGi2kaadwhadaWgaaWcbaGaaGymaaqabaaakeaacqGHciITca WG4bWaaSbaaSqaaiaaigdaaeqaaaaakiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVpaalaaabaGaeyOaIyRaeq4Wdm3aaSba aSqaaiaaigdacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcba GaaGymaaqabaaaaOGaeyypa0JaeqyWdi3aaSbaaSqaaiaaicdaaeqa aOWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWG1bWaaS baaSqaaiaaigdaaeqaaaGcbaGaeyOaIyRaamiDamaaCaaaleqabaGa aGOmaaaaaaaakeaacqGHshI3daWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadwhadaWgaaWcbaGaaGymaaqabaaakeaacqGHciIT caWG4bWaaSbaaSqaaiaaigdaaeqaaOWaaWbaaSqabeaacaaIYaaaaa aakiabg2da9maalaaabaGaaGymaaqaaiaadogadaqhaaWcbaGaamOq aaqaaiaaikdaaaaaaOWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaik daaaGccaWG1bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOaIyRaamiD amaaCaaaleqabaGaaGOmaaaaaaaaaaa@82C4@

where c B 2 =E/ ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaqhaa WcbaGaamOqaaqaaiaaikdaaaGccqGH9aqpcaWGfbGaai4laiabeg8a YnaaBaaaleaacaaIWaaabeaaaaa@3CE3@  is the wave speed. This equation is exact for ν=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabe27aUjabg2 da9iaaicdaaaa@388E@  but cannot be correct in general, since transverse motion is neglected.  In practice waves are repeatedly reflected off the sides of the bar, which behaves as a wave-guide (see Sect 5.6.5 for more discussion of wave-guides).

 

 It is straightforward to solve the equation to see that

u 1 ( x 2 ,t)= c B E 0 t x 1 / c B p(τ)dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaGymaaqabaGccaGGOaGaamiEamaaBaaaleaacaaIYaaabeaa kiaacYcacaWG0bGaaiykaiabg2da9maalaaabaGaam4yamaaBaaale aacaWGcbaabeaaaOqaaiaadweaaaWaa8qCaeaacaWGWbGaaiikaiab es8a0jaacMcacaWGKbGaeqiXdqhaleaacaaIWaaabaGaamiDaiabgk HiTiaadIhadaWgaaadbaGaaGymaaqabaWccaGGVaGaam4yamaaBaaa meaacaWGcbaabeaaa0Gaey4kIipaaaa@5002@

σ 11 =p(t x 1 / c B ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa aaleaacaaIXaGaaGymaaqabaGccqGH9aqpcqGHsislcaWGWbGaaiik aiaadshacqGHsislcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaai4lai aadogadaWgaaWcbaGaamOqaaqabaGccaGGPaaaaa@4334@

 

 

4.4.4 Plane waves in an infinite solid

 

A plane wave that travels in direction p at speed c has a displacement field of the form

u i = a i f(ct x k p k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaamyAaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaadMgaaeqa aOGaamOzaiaacIcacaWGJbGaamiDaiabgkHiTiaadIhadaWgaaWcba Gaam4AaaqabaGccaWGWbWaaSbaaSqaaiaadUgaaeqaaOGaaiykaaaa @4396@

where p is a unit vector.  Again, to visualize this motion, consider the special case

u={ 0ct< x k p k u i = a i (ct x k p k )/cct x k p k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwhacqGH9a qpdaGabaqaauaabeqaceaaaeaacaaIWaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caWGJbGaamiDaiabgYda8iaadIhadaWg aaWcbaGaam4AaaqabaGccaWGWbWaaSbaaSqaaiaadUgaaeqaaaGcba GaamyDamaaBaaaleaacaWGPbaabeaakiabg2da9iaadggadaWgaaWc baGaamyAaaqabaGccaGGOaGaam4yaiaadshacqGHsislcaWG4bWaaS baaSqaaiaadUgaaeqaaOGaamiCamaaBaaaleaacaWGRbaabeaakiaa cMcacaGGVaGaam4yaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaadogacaWG0bGaeyyzImRaamiEamaaBaaaleaacaWGRbaabeaa kiaadchadaWgaaWcbaGaam4AaaqabaaaaaGccaGL7baaaaa@A8FA@

In this solution, the wave has a planar front, with normal vector p.  The wave travels in direction p at speed c.  Ahead of the front, the solid is at rest.  Behind it, the solid has velocity a.  For ap=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHfl Y1caWHWbGaeyypa0JaaGimaaaa@3B03@  the particle velocity is perpendicular to the wave velocity.  For a=αp MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGH9a qpcqaHXoqycaWHWbaaaa@399E@  the particle velocity is parallel to the wave velocity.  These two cases are like the shear and longitudinal waves discussed in the preceding sections.

 

We seek plane wave solutions of the Cauchy-Navier equation of motion

C ijkl 2 u k x j x l = ρ 0 2 u i t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaWgaa WcbaGaamyAaiaadQgacaWGRbGaamiBaaqabaGcdaWcaaqaaiabgkGi 2oaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaam4Aaaqaba aakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaOGaeyOaIyRa amiEamaaBaaaleaacaWGSbaabeaaaaGccqGH9aqpcqaHbpGCdaWgaa WcbaGaaGimaaqabaGcdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOm aaaakiaadwhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG0b WaaWbaaSqabeaacaaIYaaaaaaaaaa@50E4@

Substituting a plane wave solution for u we see that

A ik a k f(ct x j p j )=ρ a i c 2 f(ct x j p j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaamyAaiaadUgaaeqaaOGaamyyamaaBaaaleaacaWGRbaabeaa kiaadAgacaGGOaGaam4yaiaadshacqGHsislcaWG4bWaaSbaaSqaai aadQgaaeqaaOGaamiCamaaBaaaleaacaWGQbaabeaakiaacMcacqGH 9aqpcqaHbpGCcaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaam4yamaaCa aaleqabaGaaGOmaaaakiaadAgacaGGOaGaam4yaiaadshacqGHsisl caWG4bWaaSbaaSqaaiaadQgaaeqaaOGaamiCamaaBaaaleaacaWGQb aabeaakiaacMcaaaa@5335@

where

A jk = C ijkl p i p l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaamOAaiaadUgaaeqaaOGaeyypa0Jaam4qamaaBaaaleaacaWG PbGaamOAaiaadUgacaWGSbaabeaakiaadchadaWgaaWcbaGaamyAaa qabaGccaWGWbWaaSbaaSqaaiaadYgaaeqaaaaa@41D0@

is a symmetric, positive definite tensor known as the `Acoustic Tensor.’  Plane wave solutions to the Cauchy-Navier equation must therefore satisfy

( A ik ρ 0 c 2 δ ik ) a k =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGbb WaaSbaaSqaaiaadMgacaWGRbaabeaakiabgkHiTiabeg8aYnaaBaaa leaacaaIWaaabeaakiaadogadaahaaWcbeqaaiaaikdaaaGccqaH0o azdaWgaaWcbaGaamyAaiaadUgaaeqaaOGaaiykaiaadggadaWgaaWc baGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@4638@

This requires

det( A ik ρ 0 c 2 δ ik )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacsgacaGGLb GaaiiDaiaacIcacaWGbbWaaSbaaSqaaiaadMgacaWGRbaabeaakiab gkHiTiabeg8aYnaaBaaaleaacaaIWaaabeaakiaadogadaahaaWcbe qaaiaaikdaaaGccqaH0oazdaWgaaWcbaGaamyAaiaadUgaaeqaaOGa aiykaiabg2da9iaaicdaaaa@46F7@

Evidently for any wave propagation direction, there are three wave speeds, and three corresponding displacement directions, which follow from the eigenvalues and eigenvectors of A ij / ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaai4laiabeg8aYnaaBaaaleaacaaI Waaabeaaaaa@3B3A@   For the special case of an isotropic solid

C ijkl =μ( δ il δ jk + δ ik δ jl )+ 2μν 12ν δ ij δ kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaWgaa WcbaGaamyAaiaadQgacaWGRbGaamiBaaqabaGccqGH9aqpcqaH8oqB daqadaqaaiabes7aKnaaBaaaleaacaWGPbGaamiBaaqabaGccqaH0o azdaWgaaWcbaGaamOAaiaadUgaaeqaaOGaey4kaSIaeqiTdq2aaSba aSqaaiaadMgacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGQbGaam iBaaqabaaakiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaaikdacqaH 8oqBcqaH9oGBaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbaacqaH0o azdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeqiTdq2aaSbaaSqaaiaa dUgacaWGSbaabeaaaaa@5E75@

where μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTbaa@322E@  is the shear modulus and ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3230@  is the Poisson’s ratio of the solid.  The acoustic tensor follows as

A ik =μ p l p l δ ik + μ 12ν p i p k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaamyAaiaadUgaaeqaaOGaeyypa0JaeqiVd0MaamiCamaaBaaa leaacaWGSbaabeaakiaadchadaWgaaWcbaGaamiBaaqabaGccqaH0o azdaWgaaWcbaGaamyAaiaadUgaaeqaaOGaey4kaSYaaSaaaeaacqaH 8oqBaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbaacaWGWbWaaSbaaS qaaiaadMgaaeqaaOGaamiCamaaBaaaleaacaWGRbaabeaaaaa@4D7D@

so that

( μ ρ 0 c 2 ) a k + μ 12ν p i a i p k =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaeq iVd0MaeyOeI0IaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaam4yamaa CaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiaadggadaWgaaWcba Gaam4AaaqabaGccqGHRaWkdaWcaaqaaiabeY7aTbqaaiaaigdacqGH sislcaaIYaGaeqyVd4gaaiaadchadaWgaaWcbaGaamyAaaqabaGcca WGHbWaaSbaaSqaaiaadMgaaeqaaOGaamiCamaaBaaaleaacaWGRbaa beaakiabg2da9iaaicdaaaa@4E8D@

By inspection, there are two eigenvectors that satisfy this equation

1.      a i p i =0 c 2 = c 2 2 = ρ 0 /μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaaqabaGccaWGWbWaaSbaaSqaaiaadMgaaeqaaOGaeyyp a0JaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyO0H4TaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadogadaah aaWcbeqaaiaaikdaaaGccqGH9aqpcaWGJbWaa0baaSqaaiaaikdaae aacaaIYaaaaOGaeyypa0JaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGa ai4laiabeY7aTbaa@5B63@                          (Shear wave,  or S-wave)

2.      a i =η p i c 2 = c L 2 =2μ(1ν)/ ρ 0 (12ν) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaaqabaGccqGH9aqpcqaH3oaAcaWGWbWaaSbaaSqaaiaa dMgaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHshI3ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGJbWaaWbaaSqa beaacaaIYaaaaOGaeyypa0Jaam4yamaaDaaaleaacaWGmbaabaGaaG Omaaaakiabg2da9iaaikdacqaH8oqBcaGGOaGaaGymaiabgkHiTiab e27aUjaacMcacaGGVaGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaai ikaiaaigdacqGHsislcaaIYaGaeqyVd4Maaiykaaaa@65C9@   (Longitudinal, or P-wave)

 

The two wave speeds are evidently those we found in our 1-D calculation earlier.  So there are two types of plane wave in an isotropic solid.  The S-wave travels at speed c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaWgaa WcbaGaam4Caaqabaaaaa@3714@ , and material  particles are displaced perpendicular to the direction of motion of the wave.  The P-wave travels at speed c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaWgaa WcbaGaamitaaqabaaaaa@36FB@ , and material particles are displaced parallel to the direction of motion of the wave.

 

 

 

4.4.5: Summary of Wave Speeds in isotropic elastic solids.

 

It is worth summarizing the three wave speeds calculated in the preceding sections.  Recall that

                 c L 2 = E(1ν) ρ 0 (1+ν)(12ν) = 2μ(1ν) ρ 0 (12ν) c s 2 = E 2(1+ν) ρ 0 = μ ρ 0 c B 2 = E ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaqhaa WcbaGaamitaaqaaiaaikdaaaGccqGH9aqpdaWcaaqaaiaadweacaGG OaGaaGymaiabgkHiTiabe27aUjaacMcaaeaacqaHbpGCdaWgaaWcba GaaGimaaqabaGccaGGOaGaaGymaiabgUcaRiabe27aUjaacMcacaGG OaGaaGymaiabgkHiTiaaikdacqaH9oGBcaGGPaaaaiabg2da9maala aabaGaaGOmaiabeY7aTjaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiyk aaqaaiabeg8aYnaaBaaaleaacaaIWaaabeaakiaacIcacaaIXaGaey OeI0IaaGOmaiabe27aUjaacMcaaaGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caWGJbWaa0baaSqaaiaadohaaeaacaaIYaaaaOGa eyypa0ZaaSaaaeaacaWGfbaabaGaaGOmaiaacIcacaaIXaGaey4kaS IaeqyVd4Maaiykaiabeg8aYnaaBaaaleaacaaIWaaabeaaaaGccqGH 9aqpdaWcaaqaaiabeY7aTbqaaiabeg8aYnaaBaaaleaacaaIWaaabe aaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaadogadaqhaaWcbaGaamOqaaqaai aaikdaaaGccqGH9aqpdaWcaaqaaiaadweaaeaacqaHbpGCdaWgaaWc baGaaGimaaqabaaaaaaa@9B89@

It is straightforward to show that, for all positive definite materials (those with positive definite strain energy density MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  a thermodynamic constraint) c L > c S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaWgaa WcbaGaamitaaqabaGccqGH+aGpcaWGJbWaaSbaaSqaaiaadofaaeqa aaaa@39F9@ .  For most real materials c L > c B > c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaWgaa WcbaGaamitaaqabaGccqGH+aGpcaWGJbWaaSbaaSqaaiaadkeaaeqa aOGaeyOpa4Jaam4yamaaBaaaleaacaWGZbaabeaaaaa@3D06@ .

 

There are also special kinds of waves (called Rayleigh and Stoneley waves) that travel near the surface of a solid, or near the interface between two dissimilar solids, respectively.  These waves have their own speeds.  Rayleigh waves are discussed in more detail in Section 5.5.3.

 

 

4.4.6: Reflection of waves traveling normal to a free surface

 

Suppose that a longitudinal wave with stress state

u 1 ( x 1 ,t)= c L E (1+ν)(12ν) (1ν) 0 t x 2 / c L f(τ)dτ u 2 = u 3 =0 σ 11 =f(t x 1 / c L ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamyDam aaBaaaleaacaaIXaaabeaakiaacIcacaWG4bWaaSbaaSqaaiaaigda aeqaaOGaaiilaiaadshacaGGPaGaeyypa0JaeyOeI0YaaSaaaeaaca WGJbWaaSbaaSqaaiaadYeaaeqaaaGcbaGaamyraaaadaWcaaqaaiaa cIcacaaIXaGaey4kaSIaeqyVd4MaaiykaiaacIcacaaIXaGaeyOeI0 IaaGOmaiabe27aUjaacMcaaeaacaGGOaGaaGymaiabgkHiTiabe27a UjaacMcaaaWaa8qCaeaacaWGMbGaaiikaiabes8a0jaacMcacaWGKb GaeqiXdqhaleaacaaIWaaabaGaamiDaiabgkHiTiaadIhadaWgaaad baGaaGOmaaqabaWccaGGVaGaam4yamaaBaaameaacaWGmbaabeaaa0 Gaey4kIipaaOqaaiaadwhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqp caWG1bWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGimaaqaaiabeo 8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGH9aqpcaWGMbGaaiik aiaadshacqGHsislcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaai4lai aadogadaWgaaWcbaGaamitaaqabaGccaGGPaaaaaa@73D5@

is incident on a free surface at x 1 =a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaWGHbaaaa@38F0@ .  Calculate the state of stress in the solid as a function of time, accounting for the stress free surface.

 

To visualize the wave, imagine that it is a front, such as would be generated by applying a constant uniform pressure at x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaaIWaaaaa@38C4@  at time t=0.  The material ahead of the front is at rest, and stress free, while behind the front material has a constant stress and velocity. 

 

At time t=a/ c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshacqGH9a qpcaWGHbGaai4laiaadogadaWgaaWcbaGaamitaaqabaaaaa@3A93@  the front would reach the free surface and be reflected.  Let the horizontal stress associated with the reflected wave be

σ 11 =g(t+ x 1 / c L ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa aaleaacaaIXaGaaGymaaqabaGccqGH9aqpcaWGNbGaaiikaiaadsha cqGHRaWkcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaai4laiaadogada WgaaWcbaGaamitaaqabaGccaGGPaaaaa@422D@

(we need a + in the argument because the wave travels to the left and has negative velocity). For the stress to vanish at the free surface, we must have

f(ta/ c L )+g(t+a/ c L )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaGGOa GaamiDaiabgkHiTiaadggacaGGVaGaam4yamaaBaaaleaacaWGmbaa beaakiaacMcacqGHRaWkcaWGNbGaaiikaiaadshacqGHRaWkcaWGHb Gaai4laiaadogadaWgaaWcbaGaamitaaqabaGccaGGPaGaeyypa0Ja aGimaaaa@4704@

so,

g(t+ x 1 / c L )=f(ta/ c L +( x 1 a)/ c L ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgacaGGOa GaamiDaiabgUcaRiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGVaGa am4yamaaBaaaleaacaWGmbaabeaakiaacMcacqGH9aqpcqGHsislca WGMbGaaiikaiaadshacqGHsislcaWGHbGaai4laiaadogadaWgaaWc baGaamitaaqabaGccqGHRaWkcaGGOaGaamiEamaaBaaaleaacaaIXa aabeaakiabgkHiTiaadggacaGGPaGaai4laiaadogadaWgaaWcbaGa amitaaqabaGccaGGPaaaaa@4FFB@

and the full solution consists of both incident and reflected waves

u 1 ( x 1 ,t)= c L E (1+ν)(12ν) 1ν { 0 t x 2 / c L f(τ)dτ + 0 ta/ c L +( x 2 a)/ c L f(τ)dτ } u 2 = u 3 =0 σ 11 =f(t x 1 / c L )f(ta/ c L +( x 1 a)/ c L ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamyDam aaBaaaleaacaaIXaaabeaakiaacIcacaWG4bWaaSbaaSqaaiaaigda aeqaaOGaaiilaiaadshacaGGPaGaeyypa0JaeyOeI0YaaSaaaeaaca WGJbWaaSbaaSqaaiaadYeaaeqaaaGcbaGaamyraaaadaWcaaqaaiaa cIcacaaIXaGaey4kaSIaeqyVd4MaaiykaiaacIcacaaIXaGaeyOeI0 IaaGOmaiabe27aUjaacMcaaeaacaaIXaGaeyOeI0IaeqyVd4gaamaa cmaabaWaa8qCaeaacaWGMbGaaiikaiabes8a0jaacMcacaWGKbGaeq iXdqhaleaacaaIWaaabaGaamiDaiabgkHiTiaadIhadaWgaaadbaGa aGOmaaqabaWccaGGVaGaam4yamaaBaaameaacaWGmbaabeaaa0Gaey 4kIipakiabgUcaRmaapehabaGaamOzaiaacIcacqaHepaDcaGGPaGa amizaiabes8a0bWcbaGaaGimaaqaaiaadshacqGHsislcaWGHbGaai 4laiaadogadaWgaaadbaGaamitaaqabaWccqGHRaWkcaGGOaGaamiE amaaBaaameaacaaIYaaabeaaliabgkHiTiaadggacaGGPaGaai4lai aadogadaWgaaadbaGaamitaaqabaaaniabgUIiYdaakiaawUhacaGL 9baaaeaacaWG1bWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaamyDam aaBaaaleaacaaIZaaabeaakiabg2da9iaaicdaaeaacqaHdpWCdaWg aaWcbaGaaGymaiaaigdaaeqaaOGaeyypa0JaamOzaiaacIcacaWG0b GaeyOeI0IaamiEamaaBaaaleaacaaIXaaabeaakiaac+cacaWGJbWa aSbaaSqaaiaadYeaaeqaaOGaaiykaiabgkHiTiaadAgacaGGOaGaam iDaiabgkHiTiaadggacaGGVaGaam4yamaaBaaaleaacaWGmbaabeaa kiabgUcaRiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0 IaamyyaiaacMcacaGGVaGaam4yamaaBaaaleaacaWGmbaabeaakiaa cMcaaaaa@9EB8@

 

 

        

 

As a specific example, consider a plane, constant-stress wave that is incident on a free surface. The histories of stress and velocity in the solid are illustrated in the figures above. In this case:

1.      Behind the incident stress wave, the stress is constant, with magnitude σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaaa a@3331@ .   The velocity of the solid is constant, and related to the stress by v 1 =ρ σ 0 / c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAhadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcqGHsislcqaHbpGCcqaHdpWCdaWg aaWcbaGaaGimaaqabaGccaGGVaGaam4yamaaBaaaleaacaWGmbaabe aaaaa@3FF2@

2.      At time t=a/ c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshacqGH9aqpcaWGHbGaai4laiaado gadaWgaaWcbaGaamitaaqabaaaaa@3605@  the stress wave reaches the free surface.  At this time an equal and opposite stress pulse σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiabeo8aZnaaBaaaleaacaaIWa aabeaaaaa@341E@  is reflected from the free surface, and propagates away from the surface.

3.      Behind the reflected wave, the solid is stress free, and, the solid has constant velocity  v 1 =2ρ σ 0 / c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAhadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcqGHsislcaaIYaGaeqyWdiNaeq4W dm3aaSbaaSqaaiaaicdaaeqaaOGaai4laiaadogadaWgaaWcbaGaam itaaqabaaaaa@40AE@

 

 

 

4.4.7: Reflection and Transmission of waves normal to an interface

 

The material on the left has mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg8aYnaaBa aaleaacaaIWaaabeaaaaa@37BC@  and elastic properties that give a longitudinal wave speed c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaWgaa WcbaGaamitaaqabaaaaa@36FB@ .  The corresponding properties for the material on the right are ρ B , c L B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg8aYnaaBa aaleaacaWGcbaabeaakiaacYcacaWGJbWaa0baaSqaaiaadYeaaeaa caWGcbaaaaaa@3B30@ . Suppose that a longitudinal wave with displacement and stress state

u 1 ( x 1 ,t)= c L E (1+ν)(12ν) (1ν) 0 t x 2 / c L f(τ)dτ u 2 = u 3 =0 σ 11 =f(t x 1 / c L ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamyDam aaBaaaleaacaaIXaaabeaakiaacIcacaWG4bWaaSbaaSqaaiaaigda aeqaaOGaaiilaiaadshacaGGPaGaeyypa0JaeyOeI0YaaSaaaeaaca WGJbWaaSbaaSqaaiaadYeaaeqaaaGcbaGaamyraaaadaWcaaqaaiaa cIcacaaIXaGaey4kaSIaeqyVd4MaaiykaiaacIcacaaIXaGaeyOeI0 IaaGOmaiabe27aUjaacMcaaeaacaGGOaGaaGymaiabgkHiTiabe27a UjaacMcaaaWaa8qCaeaacaWGMbGaaiikaiabes8a0jaacMcacaWGKb GaeqiXdqhaleaacaaIWaaabaGaamiDaiabgkHiTiaadIhadaWgaaad baGaaGOmaaqabaWccaGGVaGaam4yamaaBaaameaacaWGmbaabeaaa0 Gaey4kIipaaOqaaiaadwhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqp caWG1bWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGimaaqaaiabeo 8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGH9aqpcaWGMbGaaiik aiaadshacqGHsislcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaai4lai aadogadaWgaaWcbaGaamitaaqabaGccaGGPaaaaaa@73D5@

is incident on a bi-material interface at x 1 =a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaWGHbaaaa@38F0@ .  Calculate the state of stress in the solid as a function of time, accounting for the interface.

 

As before, waves will be reflected at the bi-material interface.  This time, however, some of the energy will be reflected, while some will be transmitted into the adjacent solid.  Guided by the solution to the preceding problem, we assume that the stress associated with the reflected and transmitted waves have the form

σ 11 =g(ta/ c L +( x 1 a)/ c L ) σ 11 =h(ta/ c L ( x 1 a)/ c L B ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeq4Wdm 3aaSbaaSqaaiaaigdacaaIXaaabeaakiabg2da9iaadEgacaGGOaGa amiDaiabgkHiTiaadggacaGGVaGaam4yamaaBaaaleaacaWGmbaabe aakiabgUcaRiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyOe I0IaamyyaiaacMcacaGGVaGaam4yamaaBaaaleaacaWGmbaabeaaki aacMcaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyyp a0JaamiAaiaacIcacaWG0bGaeyOeI0Iaamyyaiaac+cacaWGJbWaaS baaSqaaiaadYeaaeqaaOGaeyOeI0IaaiikaiaadIhadaWgaaWcbaGa aGymaaqabaGccqGHsislcaWGHbGaaiykaiaac+cacaWGJbWaa0baaS qaaiaadYeaaeaacaWGcbaaaOGaaiykaaaaaa@5F6F@

The functions g and h must be chosen to satisfy stress and displacement continuity at the interface.  Stress continuity requires that

f(ta/ c L )+g(ta/ c L )=h(ta/ c L ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaGGOa GaamiDaiabgkHiTiaadggacaGGVaGaam4yamaaBaaaleaacaWGmbaa beaakiaacMcacqGHRaWkcaWGNbGaaiikaiaadshacqGHsislcaWGHb Gaai4laiaadogadaWgaaWcbaGaamitaaqabaGccaGGPaGaeyypa0Ja amiAaiaacIcacaWG0bGaeyOeI0Iaamyyaiaac+cacaWGJbWaaSbaaS qaaiaadYeaaeqaaOGaaiykaaaa@4E09@                    (1)

To satisfy displacement continuity, we make the acceleration continuous

ρ 0 2 u 1 t 2 = σ 11 x 1 f'(ta/ c L ) ρ 0 c L + g'(ta/ c L ) ρ 0 c L = h'(ta/ c L B ) ρ B c L B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqyWdi 3aaSbaaSqaaiaaicdaaeqaaOWaaSaaaeaacqGHciITdaahaaWcbeqa aiaaikdaaaGccaWG1bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOaIy RaamiDamaaCaaaleqabaGaaGOmaaaaaaGccqGH9aqpdaWcaaqaaiab gkGi2kabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaaakeaacqGHci ITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaaaOqaaiabgkDiElabgkHi TmaalaaabaGaamOzaiaacEcacaGGOaGaamiDaiabgkHiTiaadggaca GGVaGaam4yamaaBaaaleaacaWGmbaabeaakiaacMcaaeaacqaHbpGC daWgaaWcbaGaaGimaaqabaGccaWGJbWaaSbaaSqaaiaadYeaaeqaaa aakiabgUcaRmaalaaabaGaam4zaiaacEcacaGGOaGaamiDaiabgkHi TiaadggacaGGVaGaam4yamaaBaaaleaacaWGmbaabeaakiaacMcaae aacqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGJbWaaSbaaSqaaiaa dYeaaeqaaaaakiabg2da9iabgkHiTmaalaaabaGaamiAaiaacEcaca GGOaGaamiDaiabgkHiTiaadggacaGGVaGaam4yamaaDaaaleaacaWG mbaabaGaamOqaaaakiaacMcaaeaacqaHbpGCdaWgaaWcbaGaamOqaa qabaGccaWGJbWaa0baaSqaaiaadYeaaeaacaWGcbaaaaaaaaaa@777D@

which may be integrated to give

f(ta/ c L ) ρ 0 c L + g(ta/ c L ) ρ 0 c L = h(ta/ c L B ) ρ B c L B +C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTmaala aabaGaamOzaiaacIcacaWG0bGaeyOeI0Iaamyyaiaac+cacaWGJbWa aSbaaSqaaiaadYeaaeqaaOGaaiykaaqaaiabeg8aYnaaBaaaleaaca aIWaaabeaakiaadogadaWgaaWcbaGaamitaaqabaaaaOGaey4kaSYa aSaaaeaacaWGNbGaaiikaiaadshacqGHsislcaWGHbGaai4laiaado gadaWgaaWcbaGaamitaaqabaGccaGGPaaabaGaeqyWdi3aaSbaaSqa aiaaicdaaeqaaOGaam4yamaaBaaaleaacaWGmbaabeaaaaGccqGH9a qpcqGHsisldaWcaaqaaiaadIgacaGGOaGaamiDaiabgkHiTiaadgga caGGVaGaam4yamaaDaaaleaacaWGmbaabaGaamOqaaaakiaacMcaae aacqaHbpGCdaWgaaWcbaGaamOqaaqabaGccaWGJbWaa0baaSqaaiaa dYeaaeaacaWGcbaaaaaakiabgUcaRiaadoeaaaa@6137@                (2)

where C is a constant of integration.  Setting t=0 shows that C must vanish, since f=g=h=0 at t=0.   The two conditions (1) and (2) may now be solved for g and h to see that

Reflected wave     σ 11 (r) = β r f(ta/ c L +( x 1 a)/ c L ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaDa aaleaacaaIXaGaaGymaaqaaiaacIcacaWGYbGaaiykaaaakiabg2da 9iabek7aInaaBaaaleaacaWGYbaabeaakiaadAgacaGGOaGaamiDai abgkHiTiaadggacaGGVaGaam4yamaaBaaaleaacaWGmbaabeaakiab gUcaRiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Iaam yyaiaacMcacaGGVaGaam4yamaaBaaaleaacaWGmbaabeaakiaacMca aaa@4EEC@

Transmitted wave σ 11 (t) = β t f(ta/ c L ( x 1 a)/ c L B ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaDa aaleaacaaIXaGaaGymaaqaaiaacIcacaWG0bGaaiykaaaakiabg2da 9iabek7aInaaBaaaleaacaWG0baabeaakiaadAgacaGGOaGaamiDai abgkHiTiaadggacaGGVaGaam4yamaaBaaaleaacaWGmbaabeaakiab gkHiTiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Iaam yyaiaacMcacaGGVaGaam4yamaaDaaaleaacaWGmbaabaGaamOqaaaa kiaacMcaaaa@4FD3@

 

Where the coefficients of reflection and transmission are given by

β r = ρ B c L B ρ 0 c L ρ B c L B + ρ 0 c L β t = 2 ρ B c L B ρ B c L B + ρ 0 c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaWGYbaabeaakiabg2da9maalaaabaGaeqyWdi3aaSbaaSqa aiaadkeaaeqaaOGaam4yamaaDaaaleaacaWGmbaabaGaamOqaaaaki abgkHiTiabeg8aYnaaBaaaleaacaaIWaaabeaakiaadogadaWgaaWc baGaamitaaqabaaakeaacqaHbpGCdaWgaaWcbaGaamOqaaqabaGcca WGJbWaa0baaSqaaiaadYeaaeaacaWGcbaaaOGaey4kaSIaeqyWdi3a aSbaaSqaaiaaicdaaeqaaOGaam4yamaaBaaaleaacaWGmbaabeaaaa GccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7cqaHYoGydaWgaaWcbaGaamiDaa qabaGccqGH9aqpdaWcaaqaaiaaikdacqaHbpGCdaWgaaWcbaGaamOq aaqabaGccaWGJbWaa0baaSqaaiaadYeaaeaacaWGcbaaaaGcbaGaeq yWdi3aaSbaaSqaaiaadkeaaeqaaOGaam4yamaaDaaaleaacaWGmbaa baGaamOqaaaakiabgUcaRiabeg8aYnaaBaaaleaacaaIWaaabeaaki aadogadaWgaaWcbaGaamitaaqabaaaaaaa@7666@

Results for a shear wave approaching the interface follow immediately from the preceding calculation, by simply setting c L = c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaWgaa WcbaGaamitaaqabaGccqGH9aqpcaWGJbWaaSbaaSqaaiaadohaaeqa aaaa@3A19@ .

 

 

 

4.4.8: Simple example involving plane wave propagation: the plate impact experiment

 

A plate impact experiment is used to measure the plastic properties of materials at high rates of strain.  In typical experiment, a large, elastic flyer plate is fired (e.g. by a gas-gun) at a stationary target plate.   The specimen is a thin film of material, which is usually deposited on the surface of the flyer plate.   When the flyer plate impacts the target, plane pressure and shear waves begin to propagate through both plates.  The experiment is designed so that the target and flier plates remain elastic, while the thin film specimen deforms plastically.   A laser interferometer is used to monitor the velocity of the back face of the target plate: these measurements enable the history of stress and strain in the film to be reconstructed.

 

A full analysis of the plate impact experiment will not be attempted here MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  instead, we illustrate the general procedure for modeling plane wave propagation in the plate impact experiment using a simple example. Suppose that

 Two elastic plates with Young’s modulus E, Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@  and density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3228@  are caused to collide, as shown in the picture.

 As a representative example, we suppose that the target has thickness h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIgaaaa@3155@ , while the projectile has thickness 2h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacaWGObaaaa@3211@ . The thickness of both flyer and target are assumed to be much smaller than any other relevant dimension (so wave reflection off lateral boundaries can be neglected).

 For simplicity, we assume that the faces of flyer and target are perpendicular to the direction of motion.  This means that the particle velocity in both flyer and target remains perpendicular to their surfaces throughout.

 Just prior to impact, the projectile has a uniform velocity v 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaaGimaaqabaaaaa@3249@ , while the target is stationary.

 At impact, plane pressure waves are initiated at the impact surface and propagate (in opposite directions) through both target and projectile.  Our objective is to calculate the history of stress and velocitity in both plates.

 

The resulting stress and motion in the plate is most conveniently displayed on “(x-t) diagrams” as shown in the figures on the next page.  The graphs can be used to deduce the velocity and stress in both flyer and target at any position x and time t in both plates.  The solution consists of triangular regions (of time and position) of constant velocity and stress, separated by lines with slope equal to the longitudinal wave speed c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaamitaaqabaaaaa@324D@  in the two plates (these lines are called “characteristics”).  Note that the stress and velocity have constant discontinuities across each characteristic.

      

The figures illustrate the following sequence of events:

1.      Just after impact, plane pressure waves propagate in opposite directions through the flyer and target.  Behind the traveling wave fronts, both plates have velocity v 1 = v 0 /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWG2bWaaSbaaSqaaiaaicdaaeqaaOGaai4laiaaikdaaaa@36B4@  and are subjected to a stress state σ 11 = σ 22 = σ 33 = σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGa eyypa0Jaeq4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaakiabg2da9i abgkHiTiabeo8aZnaaBaaaleaacaaIWaaabeaaaaa@4163@ , where σ 0 = v 0 c L /2ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaki abg2da9iaadAhadaWgaaWcbaGaaGimaaqabaGccaWGJbWaaSbaaSqa aiaadYeaaeqaaOGaai4laiaaikdacqaHbpGCaaa@3B2A@ .

2.      At time t=h/ c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshacqGH9aqpcaWGObGaai4laiaado gadaWgaaWcbaGaamitaaqabaaaaa@35EC@  the wave propagating in the target plate reaches the free surfaces on the back side of the target.  The wave is reflected from the free surface.  Behind this reflected wave, the target is stress free, and has velocity v 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaaGimaaqabaaaaa@3249@ .  The target thereafter continues to travel at constant speed and remains free of stress indefinitely.

3.      At time t=2h/ c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshacqGH9aqpcaaIYaGaamiAaiaac+ cacaWGJbWaaSbaaSqaaiaadYeaaeqaaaaa@36A8@  there are two simultaneous events: (i) the plane wave in the flyer is reflected off the back surface MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  behind the reflected wave the flyer is stress free and has zero velocity; (ii) the reflected wave in the target reaches the interface.  Since the interface is in compression, and the stress merely drops to zero behind the reflected wave, it passes freely through the interface without reflection.

4.      At time t=3h/ c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshacqGH9aqpcaaIZaGaamiAaiaac+ cacaWGJbWaaSbaaSqaaiaadYeaaeqaaaaa@36A9@  the two reflected waves in the flyer meet at the mid-point of the flyer. Thereafter, the region between the two reflected waves in the flyer becomes tensile.  In addition, the flyer plate has speed v 0 /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaaGimaaqabaGcca GGVaGaaGOmaaaa@33C2@  between the two wavefronts.

5.       At time t=4h/ c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshacqGH9aqpcaaI0aGaamiAaiaac+ cacaWGJbWaaSbaaSqaaiaadYeaaeqaaaaa@36AA@  the reflected wave from the back surface of the flyer reaches the interface.  The stress is tensile behind this wave front, and since the interface between flyer and target cannot support tension in behaves like a free surface, and the wave is reflected off the interface back into the flyer.  At the same time, the reflected wave from the target reaches the back face of the flyer and is reflected for a second time.

6.      Thereafter, the target continues to propagate with constant velocity v 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaaGimaaqabaaaaa@3269@ , while the flyer contains two plane waves that are repeatedly reflected from its two surfaces.  These waves effectively cause the flyer to vibrate, while traveling with average speed v 0 /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaaGimaaqabaGcca GGVaGaaGOmaaaa@33E2@ .

 

Derivation: The solution can be constructed using the simple 1-D solutions given in 4.4.1 and 4.4.6.  For example, to find the stress and velocity associated with the waves generated by the initial impact:

1.      At the moment of impact, both flyer and target are subjected to a sudden pressure. Wave motion in both solids can  be analyzed using the solution given in 4.4.1.

2.      Let Δ v f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadAhadaWgaaWcbaGaamOzaa qabaaaaa@3400@ Δ v t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadAhadaWgaaWcbaGaamiDaa qabaaaaa@340E@  denote the change in velocity of the flyer and target, respectively, as a result of impact.

3.      Let σ 11 = σ f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaamOzaaqabaaaaa@37D7@  and σ 11 = σ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaamiDaaqabaaaaa@37E5@  denote the horizontal stress component behind the wavefronts in the target and flyer just after impact.

4.      From Section 4.4.1 we know that the velocity change and stress are related by

Δ v f =ρ σ f / c L Δ v t =ρ σ t / c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadAhadaWgaaWcbaGaamOzaa qabaGccqGH9aqpcqGHsislcqaHbpGCcqaHdpWCdaWgaaWcbaGaamOz aaqabaGccaGGVaGaam4yamaaBaaaleaacaWGmbaabeaakiaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab fs5aejaadAhadaWgaaWcbaGaamiDaaqabaGccqGH9aqpcqGHsislcq aHbpGCcqaHdpWCdaWgaaWcbaGaamiDaaqabaGccaGGVaGaam4yamaa BaaaleaacaWGmbaabeaaaaa@57F3@

5.      The target and flyer must have the same velocity at the impact surface.  Therefore v 0 Δ v f =Δ v t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaaGimaaqabaGccq GHsislcqqHuoarcaWG2bWaaSbaaSqaaiaadAgaaeqaaOGaeyypa0Ja euiLdqKaamODamaaBaaaleaacaWG0baabeaaaaa@3B6E@

6.      The horizontal stress must be equal in both solids at the impact surface.  Therefore σ f = σ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGMbaabeaaki abg2da9iabeo8aZnaaBaaaleaacaWG0baabeaaaaa@375A@ .

7.      The four equations in steps 4-6 can be solved to yield Δ v f =Δ v t = v 0 /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadAhadaWgaaWcbaGaamOzaa qabaGccqGH9aqpcqqHuoarcaWG2bWaaSbaaSqaaiaadshaaeqaaOGa eyypa0JaamODamaaBaaaleaacaaIWaaabeaakiaac+cacaaIYaaaaa@3D00@ , σ f = σ t = σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGMbaabeaaki abg2da9iabeo8aZnaaBaaaleaacaWG0baabeaakiabg2da9iabgkHi Tiabeo8aZnaaBaaaleaacaaIWaaabeaaaaa@3C00@ , with σ 0 = v 0 c L /2ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaki abg2da9iaadAhadaWgaaWcbaGaaGimaaqabaGccaWGJbWaaSbaaSqa aiaadYeaaeqaaOGaai4laiaaikdacqaHbpGCaaa@3B2A@ .

The changes in stress and velocity that occur at each reflection can then be deduced using the results at the end of Section 4.4.6.  Alternatively the (x-t) diagrams can be constructed directly, by first drawing all the characteristic lines, and then deducing the velocity and stress in each sector of the diagram by noting that (i) the change in stress and velocity across each line must be constant; (ii) the overall momentum of the solid must be conserved, and (iii) the total energy of the solid must be conserved.