Problems for Chapter 5

 

Analytical Techniques and Solutions for Linear Elastic Solids

 

 

 

5.6.  Solutions to Dynamic Problems

 

 

 

5.6.1.      Consider the Love potentials Ψ i =0ϕ=Asin( p i x i c L t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHOoqwdaWgaaWcbaGaamyAaaqaba GccqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeqy1dyMaey ypa0JaamyqaiGacohacaGGPbGaaiOBaiaacIcacaWGWbWaaSbaaSqa aiaadMgaaeqaaOGaamiEamaaBaaaleaacaWGPbaabeaakiabgkHiTi aadogadaWgaaWcbaGaamitaaqabaGccaWG0bGaaiykaaaa@5991@ , where p i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadMgaaeqaaa aa@34D5@  is a constant unit vector and A is a constant.

5.6.1.1.            Verify that the potentials satisfy the appropriate governing equations

5.6.1.2.            Calculate the stresses and displacements generated from these potentials.

5.6.1.3.            Briefly, interpret the wave motion represented by this solution.

 

 

 

5.6.2.      Consider the Love potentials Ψ i = U i sin( p k x k c s t)ϕ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHOoqwdaWgaaWcbaGaamyAaaqaba GccqGH9aqpcaWGvbWaaSbaaSqaaiaadMgaaeqaaOGaci4CaiaacMga caGGUbGaaiikaiaadchadaWgaaWcbaGaam4AaaqabaGccaWG4bWaaS baaSqaaiaadUgaaeqaaOGaeyOeI0Iaam4yamaaBaaaleaacaWGZbaa beaakiaadshacaGGPaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeqy1dyMa eyypa0JaaGimaaaa@5AF4@ , where p i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadMgaaeqaaa aa@34D5@  is a constant unit vector and U i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGvbWaaSbaaSqaaiaadMgaaeqaaa aa@34BA@  is a constant unit vector.

5.6.2.1.            Find a condition relating U i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGvbWaaSbaaSqaaiaadMgaaeqaaa aa@34BA@  and p i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadMgaaeqaaa aa@34D5@  that must be satisfied for this to be a solution to the governing equations

5.6.2.2.            Calculate the stresses and displacements generated from these potentials.

5.6.2.3.            Briefly, interpret the wave motion represented by this solution.

 

 

 

5.6.3.      Show that Ψ i =0ϕ= 1 R f(tR/ c L ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHOoqwdaWgaaWcbaGaamyAaaqaba GccqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabew 9aMjabg2da9maalaaabaGaaGymaaqaaiaadkfaaaGaamOzaiaacIca caWG0bGaeyOeI0IaamOuaiaac+cacaWGJbWaaSbaaSqaaiaadYeaae qaaOGaaiykaaaa@575B@  satisfy the governing equations for Love potentials.  Find expressions for the corresponding displacement and stress fields.

 

 

 

5.6.4.      Calculate the radial distribution of Von-Mises effective stress surrounding a spherical cavity of radius a, which has pressure p 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaaicdaaeqaaa aa@34A1@  suddenly applied to its surface at time t=0.  Hence, find the location in the solid that is subjected to the largest Von-Mises stress, and the time at which the maximum occurs.  

 

 

 

5.6.5.      Calculate the displacement and stress fields generated by the Love potentials

Ψ i =0ϕ= A R sin{ ω( t(Ra)/ c L )+ξ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHOoqwdaWgaaWcbaGaamyAaaqaba GccqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabew 9aMjabg2da9maalaaabaGaamyqaaqaaiaadkfaaaGaci4CaiaacMga caGGUbWaaiWaaeaacqaHjpWDdaqadaqaaiaadshacqGHsislcaGGOa GaamOuaiabgkHiTiaadggacaGGPaGaai4laiaadogadaWgaaWcbaGa amitaaqabaaakiaawIcacaGLPaaacqGHRaWkcqaH+oaEaiaawUhaca GL9baaaaa@6352@

Calculate the traction acting on the surface at r=a.  Hence, find the Love potential that generates the fields around a spherical cavity with radius a, which is subjected to a harmonic pressure p(t)= p 0 sinωt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbGaaiikaiaadshacaGGPaGaey ypa0JaamiCamaaBaaaleaacaaIWaaabeaakiGacohacaGGPbGaaiOB aiabeM8a3jaadshaaaa@3E96@ .  Plot the amplitude of the surface displacement at r=a (normalized by a) as a function of ωa/ c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHjpWDcaWGHbGaai4laiaadogada WgaaWcbaGaamitaaqabaaaaa@3811@ .

 

 

 

5.6.6.      Calculate the distribution of kinetic and potential energy near the surface of a half-space that contains a Rayleigh wave with displacement amplitude U 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGvbWaaSbaaSqaaiaaicdaaeqaaa aa@3486@  and wave number k.  Take Poisson’s ratio ν=0.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBcqGH9aqpcaaIWaGaaiOlai aaiodaaaa@37AD@ .  Calculate the total energy per unit area of the wave (find the total energy in one wavelength, then divide by the wavelength).  Estimate the energy per unit area in Rayleigh waves associated with earthquakes.

 

 

 

5.6.7.      The figure shows a surface-acoustic-wave device that is intended to act as a narrow band-pass filter.  A piezoelectric substrate has two transducers attached to its surface MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  one acts as an “input” transducer and the other as “output.”  The transducers are electrodes: a charge can be applied to the input transducer; or detected on the output.  Applying a charge to the input transducer induces a strain on the surface of the substrate: at an appropriate frequency, this will excite a Rayleigh wave in the solid.  The wave propagates to the “output” electrodes, and the resulting deformation of the substrate induces a charge that can be detected.   If the electrodes have spacing d, calculate the frequency at which the surface will be excited.   Estimate the spacing required for a 1GHz filter made from AlN with Young’s modulus 345 GPa and Poisson’s ratio 0.3

 

 

 

5.6.8.      The figure shows a thin elastic strip, which is bonded to rigid solids on both its surfaces. The strip has shear modulus μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTbaa@321E@  and wave speed c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4Caaqabaaaaa@3274@ , and acts as a wave-guide.  The goal of this problem is to calculate the displacement field associated with transverse wave propagation down the strip.

5.6.8.1.            Assume that the displacement has the form

u 3 =f( x 2 )exp(ik( x 1 ct)) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaWGMbGaaiikaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGG PaGaciyzaiaacIhacaGGWbGaaiikaiabgkHiTiaadMgacaWGRbGaai ikaiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGJbGaamiD aiaacMcacaGGPaaaaa@44A2@

By substituting into the Cauchy-Navier equation, show that

d 2 d x 2 2 f( x 2 )+ k 2 ( c 2 c s 2 1 )f( x 2 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizamaaCaaaleqabaGaaG OmaaaaaOqaaiaadsgacaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaa aaaakiaadAgacaGGOaGaamiEamaaBaaaleaacaaIYaaabeaakiaacM cacqGHRaWkcaWGRbWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaadaWc aaqaaiaadogadaahaaWcbeqaaiaaikdaaaaakeaacaWGJbWaa0baaS qaaiaadohaaeaacaaIYaaaaaaakiabgkHiTiaaigdaaiaawIcacaGL PaaacaWGMbGaaiikaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGPa Gaeyypa0JaaGimaaaa@4AC3@

Hence, write down the general solution for f( x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacaGGOaGaamiEamaaBaaaleaaca aIYaaabeaakiaacMcaaaa@349B@

5.6.8.2.            Show that the boundary conditions admit solutions of the form

f( x 2 )={ Asin(nπ x 2 /H) Bcos((π/2+nπ) x 2 /H) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacaGGOaGaamiEamaaBaaaleaaca aIYaaabeaakiaacMcacqGH9aqpdaGabaqaauaabeqaceaaaeaacaWG bbGaci4CaiaacMgacaGGUbGaaiikaiaad6gacqaHapaCcaWG4bWaaS baaSqaaiaaikdaaeqaaOGaai4laiaadIeacaGGPaaabaGaamOqaiGa cogacaGGVbGaai4CaiaacIcacaGGOaGaeqiWdaNaai4laiaaikdacq GHRaWkcaWGUbGaeqiWdaNaaiykaiaadIhadaWgaaWcbaGaaGOmaaqa baGccaGGVaGaamisaiaacMcaaaaacaGL7baaaaa@5257@

where n is an integer, so that

u 3 = U 0 { cos 2 (πn/2)sin(nπ x 2 /2H)+ sin 2 (πn/2)cos(nπ x 2 /2H) }exp(ik( x 1 ct)) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaWGvbWaaSbaaSqaaiaaicdaaeqaaOWaaiWaaeaaciGGJbGa ai4BaiaacohadaahaaWcbeqaaiaaikdaaaGccaGGOaGaeqiWdaNaam OBaiaac+cacaaIYaGaaiykaiGacohacaGGPbGaaiOBaiaacIcacaWG UbGaeqiWdaNaamiEamaaBaaaleaacaaIYaaabeaakiaac+cacaaIYa GaamisaiaacMcacqGHRaWkciGGZbGaaiyAaiaac6gadaahaaWcbeqa aiaaikdaaaGccaGGOaGaeqiWdaNaamOBaiaac+cacaaIYaGaaiykai GacogacaGGVbGaai4CaiaacIcacaWGUbGaeqiWdaNaamiEamaaBaaa leaacaaIYaaabeaakiaac+cacaaIYaGaamisaiaacMcaaiaawUhaca GL9baaciGGLbGaaiiEaiaacchacaGGOaGaamyAaiaadUgacaGGOaGa amiEamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadogacaWG0bGaai ykaiaacMcaaaa@6CF3@

 

5.6.8.3.            Find an expression for the phase velocity of the wave, and plot the phase velocity as a function of kH/n

5.6.8.4.            Calculate the dispersion relation for the wave and hence deduce an expression for the group velocity.  Plot the group velocity as a function of kH/n.

 

 

 

5.6.9.      Consider the Love wave described in Section 5.6.4. 

5.6.9.1.            Consider first a system with  μ= μ f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTjabg2da9iabeY7aTnaaBaaale aacaWGMbaabeaaaaa@35F1@ , c s / c sf =2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4CaaqabaGcca GGVaGaam4yamaaBaaaleaacaWGZbGaamOzaaqabaGccqGH9aqpcaaI Yaaaaa@37F4@ , as discussed at the end of 5.6.4. Find an expression for the group velocity of the wave, and plot a graph showing the group velocity (normalized by shear wave speed in the layer) as a function of  kH

5.6.9.2.            Consider a system with μ= μ f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTjabg2da9iabeY7aTnaaBaaale aacaWGMbaabeaaaaa@35F1@ , c s / c sf =1/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4CaaqabaGcca GGVaGaam4yamaaBaaaleaacaWGZbGaamOzaaqabaGccqGH9aqpcaaI XaGaai4laiaaikdaaaa@3962@ .  Plot graphs showing both the phase velocity and the group velocity in the layer as a function of kH.

 

 

 

 

 

5.6.10.  In this problem, you will investigate the energy associated with wave propagation down a simple wave guide.  Consider an isotropic, linear elastic strip, with thickness 2H, shear modulus μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTbaa@321E@  and wave speed c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4Caaqabaaaaa@3274@  as indicated in the figure.  The solution for a wave propagating in the e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa aa@349B@  direction, with particle velocity u= u 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0JaamyDamaaBaaale aacaaIZaaabeaakiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@388E@  is given in Section 5.6.6 of the text. 

5.6.10.1.        The flux of energy associated with wave propagation along a wave-guide can be computed from the work done by the tractions acting on an internal material surface.  The work done per cycle is given by

P = 1 T 0 T H H σ ij n j d u i dt d x 2 dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaadaqaaiaadcfaaiaawMYicaGLQm cacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGubaaamaapehabaWaa8qC aeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamOBamaaBa aaleaacaWGQbaabeaakmaalaaabaGaamizaiaadwhadaWgaaWcbaGa amyAaaqabaaakeaacaWGKbGaamiDaaaacaWGKbGaamiEamaaBaaale aacaaIYaaabeaakiaadsgacaWG0baaleaacqGHsislcaWGibaabaGa amisaaqdcqGHRiI8aaWcbaGaaGimaaqaaiaadsfaa0Gaey4kIipaaa a@50A2@

where T is the period of oscillation and n j = δ j1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadQgaaeqaaO Gaeyypa0JaeyOeI0IaeqiTdq2aaSbaaSqaaiaadQgacaaIXaaabeaa aaa@3A4C@  is a unit vector normal to an internal plane perpendicular to the direction of wave propagation. Calculate P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaadaqaaiaadcfaaiaawMYicaGLQm caaaa@356B@  for the nth wave propagation mode.

5.6.10.2.        The average kinetic energy of a generic cross-section of the wave-guide can be calculated from

K = 1 T 0 T H H ρ 2 d u i dt d u i dt d x 2 dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaadaqaaiaadUeaaiaawMYicaGLQm cacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGubaaamaapehabaWaa8qC aeaadaWcaaqaaiabeg8aYbqaaiaaikdaaaWaaSaaaeaacaWGKbGaam yDamaaBaaaleaacaWGPbaabeaaaOqaaiaadsgacaWG0baaamaalaaa baGaamizaiaadwhadaWgaaWcbaGaamyAaaqabaaakeaacaWGKbGaam iDaaaacaWGKbGaamiEamaaBaaaleaacaaIYaaabeaakiaadsgacaWG 0baaleaacqGHsislcaWGibaabaGaamisaaqdcqGHRiI8aaWcbaGaaG imaaqaaiaadsfaa0Gaey4kIipaaaa@5234@

Find K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaadaqaaiaadUeaaiaawMYicaGLQm caaaa@3566@  for nth wave propagation mode.

5.6.10.3.        The average potential energy of a generic cross-section of the wave-guide can be calculated from

Φ = 1 T 0 T H H 1 2 σ ij ε ij d x 2 dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaadaqaaiabfA6agbGaayzkJiaawQ Yiaiabg2da9maalaaabaGaaGymaaqaaiaadsfaaaWaa8qCaeaadaWd XbqaamaalaaabaGaaGymaaqaaiaaikdaaaGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaakiabew7aLnaaBaaaleaacaWGPbGaamOAaaqa baGccaWGKbGaamiEamaaBaaaleaacaaIYaaabeaakiaadsgacaWG0b aaleaacqGHsislcaWGibaabaGaamisaaqdcqGHRiI8aaWcbaGaaGim aaqaaiaadsfaa0Gaey4kIipaaaa@4F77@

Find Φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaadaqaaiabfA6agbGaayzkJiaawQ Yiaaaa@3610@  for nth wave propagation mode.  Check that K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaadaqaaiaadUeaaiaawMYicaGLQm caaaa@3566@  = Φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaadaqaaiabfA6agbGaayzkJiaawQ Yiaaaa@3610@

5.6.10.4.        The speed of energy flux down the wave-guide is defined as c e = P /( K + Φ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaadwgaaeqaaO Gaeyypa0ZaaaWaaeaacaWGqbaacaGLPmIaayPkJaGaai4lamaabmaa baWaaaWaaeaacaWGlbaacaGLPmIaayPkJaGaey4kaSYaaaWaaeaacq qHMoGraiaawMYicaGLQmcaaiaawIcacaGLPaaaaaa@4181@ .  Find c e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaadwgaaeqaaa aa@34C4@  for the nth propagation mode, and compare the solution with the expression for the group velocity of the wave

c g = dω dk = c s kH (nπ/2) 2 + k 2 H 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4zaaqabaGccq GH9aqpdaWcaaqaaiaadsgacqaHjpWDaeaacaWGKbGaam4AaaaacqGH 9aqpdaWcaaqaaiaadogadaWgaaWcbaGaam4CaaqabaGccaWGRbGaam isaaqaamaakaaabaGaaiikaiaad6gacqaHapaCcaGGVaGaaGOmaiaa cMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGRbWaaWbaaSqabe aacaaIYaaaaOGaamisamaaCaaaleqabaGaaGOmaaaaaeqaaaaaaaa@47F6@