Problems for Chapter 2

 

Governing Equations

 

 

 

2.4.  Work and Energy, the Principle of Virtual Work

 

 

2.4.1.      A solid with volume V is subjected to a distribution of traction t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG0bWaaSbaaSqaaiaadMgaaeqaaa aa@34D9@  on its surface. Assume that the solid is in static equilibrium. By considering a virtual velocity of the form δ v i = A ij y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWG2bWaaSbaaSqaaiaadM gaaeqaaOGaeyypa0JaamyqamaaBaaaleaacaWGPbGaamOAaaqabaGc caWG5bWaaSbaaSqaaiaadQgaaeqaaaaa@3C82@ , where A ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@3595@  is a constant symmetric tensor, use the principle of virtual work to show that the average stress in a solid can be computed from the shape of the solid and the tractions acting on its surface using the expression

1 V V σ ij dV = 1 V S 1 2 ( t i y j + t j y i ) dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaaigdaaeaacaWGwbaaam aapefabaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaadsga caWGwbaaleaacaWGwbaabeqdcqGHRiI8aOGaeyypa0ZaaSaaaeaaca aIXaaabaGaamOvaaaadaWdrbqaamaalaaabaGaaGymaaqaaiaaikda aaWaaeWaaeaacaWG0bWaaSbaaSqaaiaadMgaaeqaaOGaamyEamaaBa aaleaacaWGQbaabeaakiabgUcaRiaadshadaWgaaWcbaGaamOAaaqa baGccaWG5bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaale aacaWGtbaabeqdcqGHRiI8aOGaamizaiaadgeaaaa@512E@

 

 

2.4.2.      The figure shows a cantilever beam that is subjected to surface loading q( x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadghacaGGOaGaamiEamaaBaaaleaaca aIXaaabeaakiaacMcaaaa@34A5@  per unit length.  The state of stress in the beam can be approximated by σ 11 =M( x 1 ) x 2 /I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcaWGnbGaaiikaiaadIhadaWgaaWcbaGaaGymaaqa baGccaGGPaGaamiEamaaBaaaleaacaaIYaaabeaakiaac+cacaWGjb aaaa@3C66@ , where I= A x 2 2 dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeacqGH9aqpdaWdrbqaaiaadIhada qhaaWcbaGaaGOmaaqaaiaaikdaaaGccaWGKbGaamyqaaWcbaGaamyq aaqab0Gaey4kIipaaaa@39A7@  is the area moment of inertia of the beam’s cross section and M( x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eacaGGOaGaamiEamaaBaaaleaaca aIXaaabeaakiaacMcaaaa@3481@  is an arbitrary function (all other stress components are zero).  By considering a virtual velocity field of the form

δ v 1 = w( x 1 ) d x 1 x 2 δ v 2 =w( x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjaadAhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcqGHsisldaWcaaqaaiaadEhacaGGOaGaamiEamaa BaaaleaacaaIXaaabeaakiaacMcaaeaacaWGKbGaamiEamaaBaaale aacaaIXaaabeaaaaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlabes7aKjaadAhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqp caWG3bGaaiikaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGPaaaaa@574E@

where w( x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEhacaGGOaGaamiEamaaBaaaleaaca aIXaaabeaakiaacMcaaaa@34AB@  is an arbitrary function satisfying w=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEhacqGH9aqpcaaIWaaaaa@3324@  at x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaaIWaaaaa@3416@ , show that the beam is in static equilibrium if

0 L M( x 1 ) d 2 w d x 1 2 d x 1 + 0 L q( x 1 )wd x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapehabaGaamytaiaacIcacaWG4bWaaS baaSqaaiaaigdaaeqaaOGaaiykamaalaaabaGaamizamaaCaaaleqa baGaaGOmaaaakiaadEhaaeaacaWGKbGaamiEamaaDaaaleaacaaIXa aabaGaaGOmaaaaaaaabaGaaGimaaqaaiaadYeaa0Gaey4kIipakiaa dsgacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSYaa8qCaeaaca WGXbGaaiikaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGPaGaam4D aiaadsgacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaa WcbaGaaGimaaqaaiaadYeaa0Gaey4kIipaaaa@5063@

By integrating the first integral expression by parts twice, show that the equilibrium equation and boundary conditions for M( x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eacaGGOaGaamiEamaaBaaaleaaca aIXaaabeaakiaacMcaaaa@3481@  are

d 2 M d x 1 2 +q( x 1 )=0M( x 1 )= dM( x 1 ) d x 1 =0 x 1 =L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizamaaCaaaleqabaGaaG Omaaaakiaad2eaaeaacaWGKbGaamiEamaaDaaaleaacaaIXaaabaGa aGOmaaaaaaGccqGHRaWkcaWGXbGaaiikaiaadIhadaWgaaWcbaGaaG ymaaqabaGccaGGPaGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caWGnbGaaiikaiaadIhadaWg aaWcbaGaaGymaaqabaGccaGGPaGaeyypa0ZaaSaaaeaacaWGKbGaam ytaiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiykaaqaaiaa dsgacaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakiabg2da9iaaicdaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caWG4bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0Jaamitaaaa@7B5A@

 

 

 

2.4.3.      The figure shows a plate with a clamped edge that is subjected to a pressure p( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacaGGOaGaamiEamaaBaaaleaaca aIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiyk aaaa@3743@  on its surface.  The state of stress in the plate can be approximated by

σ αβ = M αβ ( x 1 , x 2 ) x 3 /3 h 3 σ 33 = σ 3α =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacqaHXoqycqaHYo GyaeqaaOGaeyypa0JaamytamaaBaaaleaacqaHXoqycqaHYoGyaeqa aOGaaiikaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEam aaBaaaleaacaaIYaaabeaakiaacMcacaWG4bWaaSbaaSqaaiaaioda aeqaaOGaai4laiaaiodacaWGObWaaWbaaSqabeaacaaIZaaaaOGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIZa GaaG4maaqabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaG4maiabeg7a HbqabaGccqGH9aqpcaaIWaaaaa@64B3@

where the subscripts α,β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjaacYcacqaHYoGyaaa@3458@  can have values 1 or 2, and M αβ ( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaeqySdeMaeqOSdi gabeaakiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaa dIhadaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@3A96@  is a tensor valued function.  By considering a virtual velocity of the form

δ v α = w x α x 3 δ v 3 =w( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjaadAhadaWgaaWcbaGaeqySde gabeaakiabg2da9iabgkHiTmaalaaabaGaeyOaIyRaam4Daaqaaiab gkGi2kaadIhadaWgaaWcbaGaeqySdegabeaaaaGccaWG4bWaaSbaaS qaaiaaiodaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH0oazcaWG2bWaaS baaSqaaiaaiodaaeqaaOGaeyypa0Jaam4DaiaacIcacaWG4bWaaSba aSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqaba GccaGGPaaaaa@5BDE@

where w( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEhacaGGOaGaamiEamaaBaaaleaaca aIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiyk aaaa@374A@  is an arbitrary function satisfying w=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEhacqGH9aqpcaaIWaaaaa@3324@  on the edge of the plate, show that the beam is in static equilibrium if

A M αβ ( x 1 ) 2 w x α x β dA+ A p( x 1 , x 2 )wdA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapehabaGaamytamaaBaaaleaacqaHXo qycqaHYoGyaeqaaOGaaiikaiaadIhadaWgaaWcbaGaaGymaaqabaGc caGGPaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWG3b aabaGaeyOaIyRaamiEamaaBaaaleaacqaHXoqyaeqaaOGaeyOaIyRa amiEamaaBaaaleaacqaHYoGyaeqaaaaaaeaacaWGbbaabaaaniabgU IiYdGccaWGKbGaamyqaiabgUcaRmaapehabaGaamiCaiaacIcacaWG 4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaG OmaaqabaGccaGGPaGaam4DaiaadsgacaWGbbGaeyypa0JaaGimaaWc baGaamyqaaqaaaqdcqGHRiI8aaaa@57F8@

By applying the divergence theorem appropriately, show that the governing equation for M αβ ( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaeqySdeMaeqOSdi gabeaakiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaa dIhadaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@3A96@  is

2 M αβ x α x β +p=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaamytamaaBaaaleaacqaHXoqycqaHYoGyaeqaaaGcbaGa eyOaIyRaamiEamaaBaaaleaacqaHXoqyaeqaaOGaeyOaIyRaamiEam aaBaaaleaacqaHYoGyaeqaaaaakiabgUcaRiaadchacqGH9aqpcaaI Waaaaa@4322@

 

 

 

 

2.4.4.      The shell shown in the figure is subjected to a radial body force b=b(R) e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkgacqGH9aqpcaWGIbGaaiikaiaadk facaGGPaGaaCyzamaaBaaaleaacaWGsbaabeaaaaa@3761@ , and a radial pressure p a , p b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaamyyaaqabaGcca GGSaGaamiCamaaBaaaleaacaWGIbaabeaaaaa@3531@  acting on the surfaces at R=a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacqGH9aqpcaWGHbaaaa@332B@  and R=b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacqGH9aqpcaWGIbaaaa@332C@ . The loading induces a spherically symmetric state of stress in the shell, which can be expressed in terms of its components in a spherical-polar coordinate system as σ RR e R e R + σ θθ e θ e θ + σ ϕϕ e ϕ e ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGsbGaamOuaa qabaGccaWHLbWaaSbaaSqaaiaadkfaaeqaaOGaey4LIqSaaCyzamaa BaaaleaacaWGsbaabeaakiabgUcaRiabeo8aZnaaBaaaleaacqaH4o qCcqaH4oqCaeqaaOGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4L IqSaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaSIaeq4Wdm3aaS baaSqaaiabew9aMjabew9aMbqabaGccaWHLbWaaSbaaSqaaiabew9a MbqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiabew9aMbqabaaaaa@5654@ .   By considering a virtual velocity of the form δv=w(R) e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjaahAhacqGH9aqpcaWG3bGaai ikaiaadkfacaGGPaGaaCyzamaaBaaaleaacaWGsbaabeaaaaa@392F@ , show that the stress state is in static equilibrium if

a b { σ RR dw dR +( σ θθ + σ ϕϕ ) w R }4π R 2 dR a b b(R)w(R)4π R 2 dR 4π a 2 p a w(a)+4π b 2 p b w(b)=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaa8qCaeaadaGadaqaaiabeo8aZn aaBaaaleaacaWGsbGaamOuaaqabaGcdaWcaaqaaiaadsgacaWG3baa baGaamizaiaadkfaaaGaey4kaSYaaeWaaeaacqaHdpWCdaWgaaWcba GaeqiUdeNaeqiUdehabeaakiabgUcaRiabeo8aZnaaBaaaleaacqaH vpGzcqaHvpGzaeqaaaGccaGLOaGaayzkaaWaaSaaaeaacaWG3baaba GaamOuaaaaaiaawUhacaGL9baacaaI0aGaeqiWdaNaamOuamaaCaaa leqabaGaaGOmaaaakiaadsgacaWGsbaaleaacaWGHbaabaGaamOyaa qdcqGHRiI8aOGaeyOeI0Yaa8qCaeaacaWGIbGaaiikaiaadkfacaGG PaGaam4DaiaacIcacaWGsbGaaiykaiaaisdacqaHapaCcaWGsbWaaW baaSqabeaacaaIYaaaaOGaamizaiaadkfaaSqaaiaadggaaeaacaWG IbaaniabgUIiYdaakeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyOeI0IaaGinaiabec 8aWjaadggadaahaaWcbeqaaiaaikdaaaGccaWGWbWaaSbaaSqaaiaa dggaaeqaaOGaam4DaiaacIcacaWGHbGaaiykaiabgUcaRiaaisdacq aHapaCcaWGIbWaaWbaaSqabeaacaaIYaaaaOGaamiCamaaBaaaleaa caWGIbaabeaakiaadEhacaGGOaGaamOyaiaacMcacqGH9aqpcaaIWa aaaaa@E407@

for all w(R).  Hence, show that the stress state must satisfy

d σ RR dR + 1 R ( 2 σ RR σ θθ σ ϕϕ )+b=0 σ RR = p a ( R=a ) σ RR = p b ( R=b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiabeo8aZnaaBaaale aacaWGsbGaamOuaaqabaaakeaacaWGKbGaamOuaaaacqGHRaWkdaWc aaqaaiaaigdaaeaacaWGsbaaamaabmaabaGaaGOmaiabeo8aZnaaBa aaleaacaWGsbGaamOuaaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGa eqiUdeNaeqiUdehabeaakiabgkHiTiabeo8aZnaaBaaaleaacqaHvp GzcqaHvpGzaeqaaaGccaGLOaGaayzkaaGaey4kaSIaamOyaiabg2da 9iaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSba aSqaaiaadkfacaWGsbaabeaakiabg2da9iabgkHiTiaadchadaWgaa WcbaGaamyyaaqabaGccaaMc8UaaGPaVlaaykW7daqadaqaaiaadkfa cqGH9aqpcaWGHbaacaGLOaGaayzkaaGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaamOu aiaadkfaaeqaaOGaeyypa0JaeyOeI0IaamiCamaaBaaaleaacaWGIb aabeaakiaaykW7caaMc8UaaGPaVpaabmaabaGaamOuaiabg2da9iaa dkgaaiaawIcacaGLPaaacaaMc8oaaa@92AD@

 

 

 

 

 

2.4.5.      In this problem, we consider the internal forces in the polymer specimen described in Problem 2.1.29 and 2.3.5 (you will need to solve 2.1.29 and 2.3.5 before you can attempt this one). Suppose that the specimen is homogeneous, has mass density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCaaa@337F@  in the reference configuration, and may be idealized as a viscous fluid, in which the Kirchhoff stress is related to stretch rate by

τ=μD+pI MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHepGaeyypa0JaeqiVd0MaaCirai abgUcaRiaadchacaWHjbaaaa@3941@

where p is an indeterminate hydrostatic pressure and μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBaaa@3375@  is the viscosity.

 

2.4.5.1.            Calculate the rate of external work done by the torque acting on the rotating exterior cyclinder

2.4.5.2.            Calculate the rate of internal dissipation in the solid as a function of r.

2.4.5.3.            Show that the total internal dissipation is equal to the external work done on the specimen.