5. Conservation Laws for Continua

 

In this section, we generalize Newton’s laws of motion (conservation of linear and angular momentum); mass conservation; and the laws of thermodynamics for a continuum.

 

 

5.1 Mass Conservation

 

The total mass of any subregion within a deformable solid must be conserved.  We can write express this condition as a constraint in several different ways:

 

In integral form:

d dt V 0 ρ(X)dV = d dt V ρ(y,t)dV =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqyWdiNaaiikaiaahIfacaGGPaGaamizaiaadAfa aSqaaiaadAfadaWgaaadbaGaaGimaaqabaaaleqaniabgUIiYdGccq GH9aqpdaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaadaWdrbqaaiab eg8aYjaacIcacaWH5bGaaiilaiaadshacaGGPaGaamizaiaadAfaaS qaaiaadAfaaeqaniabgUIiYdGccqGH9aqpcaqGWaaaaa@4D72@

Or, (using Reynolds transport relation) we can write a local mass conservation equation

V ( ρ t | x=const +ρ v i y i ) dV=0 ρ t | x=const +ρ v i y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaWaaeWaaeaadaabcaqaamaala aabaGaeyOaIyRaeqyWdihabaGaeyOaIyRaamiDaaaaaiaawIa7amaa BaaaleaacaWH4bGaeyypa0Jaam4yaiaad+gacaWGUbGaam4Caiaads haaeqaaOGaey4kaSIaeqyWdi3aaSaaaeaacqGHciITcaWG2bWaaSba aSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGPb aabeaaaaaakiaawIcacaGLPaaaaSqaaiaadAfaaeqaniabgUIiYdGc caWGKbGaamOvaiabg2da9iaaicdacqGHshI3daabcaqaamaalaaaba GaeyOaIyRaeqyWdihabaGaeyOaIyRaamiDaaaaaiaawIa7amaaBaaa leaacaWH4bGaeyypa0Jaam4yaiaad+gacaWGUbGaam4Caiaadshaae qaaOGaey4kaSIaeqyWdi3aaSaaaeaacqGHciITcaWG2bWaaSbaaSqa aiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabe aaaaGccqGH9aqpcaaIWaaaaa@6C8E@

 

Alternatively, in spatial form

 

ρ t | y=const + ρ v i y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaWaaSaaaeaacqGHciITcqaHbp GCaeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaahMhacqGH 9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccqGHRaWkda WcaaqaaiabgkGi2kabeg8aYjaadAhadaWgaaWcbaGaamyAaaqabaaa keaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaakiabg2da9i aaicdaaaa@4A0F@

 

 

 

5.2 Linear momentum balance in terms of Cauchy stress

 

Let σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3433@  denote the Cauchy stress distribution within a deformed solid.  Assume that the solid is subjected to a body force b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkgadaWgaaWcbaGaamyAaaqabaaaaa@3278@ , and let u i , v i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGcca GGSaGaaGPaVlaaykW7caWG2bWaaSbaaSqaaiaadMgaaeqaaaaa@3870@  and a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaamyAaaqabaaaaa@3277@  denote the displacement, velocity and acceleration of a material particle at position y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaaaaa@328F@   in the deformed solid.

 

Newton’s third law of motion (F=ma) can be expressed as

y σ+ρb=ρa     or          σ ij y i +ρ b j =ρ a j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirpaaBaaaleaacaWH5baabeaaki abgwSixlaaho8acqGHRaWkcqaHbpGCcaWHIbGaeyypa0JaeqyWdiNa aCyyaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae4Baiaabkhaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccadaWcaaqaaiabgkGi2kabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaakeaacqGHciITcaaMc8UaamyEamaaBaaaleaacaWGPbaabeaa aaGccqGHRaWkcqaHbpGCcaaMc8UaamOyamaaBaaaleaacaWGQbaabe aakiabg2da9iabeg8aYjaaykW7caWGHbWaaSbaaSqaaiaadQgaaeqa aaaa@5FB6@

Written out in full

σ 11 y 1 + σ 21 y 2 + σ 31 y 3 +ρ b 1 =ρ a 1 σ 12 y 1 + σ 22 y 2 + σ 32 y 3 +ρ b 2 =ρ a 2 σ 13 y 1 + σ 23 y 2 + σ 33 y 3 +ρ b 3 =ρ a 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaalaaabaGaeyOaIyRaeq4Wdm 3aaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiabgkGi2kaadMhadaWg aaWcbaGaaGymaaqabaaaaOGaey4kaSYaaSaaaeaacqGHciITcqaHdp WCdaWgaaWcbaGaaGOmaiaaigdaaeqaaaGcbaGaeyOaIyRaamyEamaa BaaaleaacaaIYaaabeaaaaGccqGHRaWkdaWcaaqaaiabgkGi2kabeo 8aZnaaBaaaleaacaaIZaGaaGymaaqabaaakeaacqGHciITcaWG5bWa aSbaaSqaaiaaiodaaeqaaaaakiabgUcaRiabeg8aYjaadkgadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcqaHbpGCcaWGHbWaaSbaaSqaaiaa igdaaeqaaaGcbaWaaSaaaeaacqGHciITcqaHdpWCdaWgaaWcbaGaaG ymaiaaikdaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaaIXaaa beaaaaGccqGHRaWkdaWcaaqaaiabgkGi2kabeo8aZnaaBaaaleaaca aIYaGaaGOmaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaaikda aeqaaaaakiabgUcaRmaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaSqaai aaiodacaaIYaaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaaG4m aaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaaleaacaaIYaaabe aakiabg2da9iabeg8aYjaadggadaWgaaWcbaGaaGOmaaqabaaakeaa daWcaaqaaiabgkGi2kabeo8aZnaaBaaaleaacaaIXaGaaG4maaqaba aakeaacqGHciITcaWG5bWaaSbaaSqaaiaaigdaaeqaaaaakiabgUca RmaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaaikdacaaIZaaabe aaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaaGOmaaqabaaaaOGaey4k aSYaaSaaaeaacqGHciITcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaae qaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaaIZaaabeaaaaGccqGH RaWkcqaHbpGCcaWGIbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0Jaeq yWdiNaamyyamaaBaaaleaacaaIZaaabeaaaaaa@9D80@

Note that the derivative is taken with respect to position in the actual, deformed solid. For the special (but rather common) case of a solid in static equilibrium in the absence of body forces

σ ij y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadMgacaWGQbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGa amyAaaqabaaaaOGaeyypa0JaaGimaaaa@3AFB@

 

Derivation: Recall that the resultant force acting on an arbitrary volume of material V within a solid is

P i = A T i (n)dA+ V ρ b i dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWdrbqaaiaadsfadaWgaaWcbaGaamyAaaqabaGccaGGOaGa aCOBaiaacMcacaWGKbGaamyqaiabgUcaRaWcbaGaamyqaaqab0Gaey 4kIipakmaapefabaGaeqyWdiNaaGPaVlaadkgadaWgaaWcbaGaamyA aaqabaGccaaMc8UaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIiYd aaaa@4928@

where T(n) is the internal traction acting on the surface A with normal n that bounds V.

 

The linear momentum of the volume V is

Λ i = V ρ v i dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfU5amnaaBaaaleaacaWGPbaabeaaki abg2da9maapefabaGaeqyWdiNaaGPaVlaadAhadaWgaaWcbaGaamyA aaqabaaabaGaamOvaaqab0Gaey4kIipakiaadsgacaWGwbaaaa@3E4E@

where v is the velocity vector of a material particle in the deformed solid. Express T in terms of σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3433@  and set P i =d Λ i /dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWGKbGaeu4MdW0aaSbaaSqaaiaadMgaaeqaaOGaai4laiaa dsgacaWG0baaaa@397D@

A σ ji n j dA+ V ρ b i dV = d dt { V ρ v i dV } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaeq4Wdm3aaSbaaSqaaiaadQ gacaWGPbaabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGccaWGKbGa amyqaiabgUcaRaWcbaGaamyqaaqab0Gaey4kIipakmaapefabaGaeq yWdiNaaGPaVlaadkgadaWgaaWcbaGaamyAaaqabaGccaaMc8Uaamiz aiaadAfaaSqaaiaadAfaaeqaniabgUIiYdGccqGH9aqpdaWcaaqaai aadsgaaeaacaWGKbGaamiDaaaadaGadaqaamaapefabaGaeqyWdiNa aGPaVlaadAhadaWgaaWcbaGaamyAaaqabaaabaGaamOvaaqab0Gaey 4kIipakiaadsgacaWGwbaacaGL7bGaayzFaaaaaa@582E@

Apply the divergence theorem to convert the first integral into a volume integral, and note that the Reynolds transport equation implies that

d dt { V ρ v i dV }= V ρ a i dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaacmaabaWaa8quaeaacqaHbpGCcaaMc8UaamODamaaBaaaleaa caWGPbaabeaaaeaacaWGwbaabeqdcqGHRiI8aOGaamizaiaadAfaai aawUhacaGL9baacqGH9aqpdaWdrbqaaiabeg8aYjaadggadaWgaaWc baGaamyAaaqabaaabaGaamOvaaqab0Gaey4kIipakiaadsgacaWGwb aaaa@4969@

so

V σ ji y j dV+ V ρ b i dV = V ρ a i dV V ( σ ji y j +ρ b i ρ a i ) dV=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaWaaSaaaeaacqGHciITcqaHdp WCdaWgaaWcbaGaamOAaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaa BaaaleaacaWGQbaabeaaaaGccaWGKbGaamOvaiabgUcaRaWcbaGaam Ovaaqab0Gaey4kIipakmaapefabaGaeqyWdiNaaGPaVlaadkgadaWg aaWcbaGaamyAaaqabaGccaaMc8UaamizaiaadAfaaSqaaiaadAfaae qaniabgUIiYdGccqGH9aqpdaWdrbqaaiabeg8aYjaadggadaWgaaWc baGaamyAaaqabaaabaGaamOvaaqab0Gaey4kIipakiaadsgacaWGwb GaeyO0H49aa8quaeaadaqadaqaamaalaaabaGaeyOaIyRaeq4Wdm3a aSbaaSqaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaa WcbaGaamOAaaqabaaaaOGaey4kaSIaeqyWdiNaaGPaVlaadkgadaWg aaWcbaGaamyAaaqabaGccqGHsislcqaHbpGCcaaMc8UaamyyamaaBa aaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaWcbaGaamOvaaqab0Ga ey4kIipakiaaykW7caaMc8UaamizaiaadAfacqGH9aqpcaaIWaaaaa@7791@

Since this must hold for every volume of material within a solid, it follows that

σ ji y j +ρ b i =ρ a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGa amOAaaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaaleaacaWGPb aabeaakiabg2da9iabeg8aYjaadggadaWgaaWcbaGaamyAaaqabaaa aa@42AF@

as stated.

 

We can also write this in spatial form by recalling that

a i = v i y k v k + v i t | y i =const MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamyAaaqabaaa keaacqGHciITcaWG5bWaaSbaaSqaaiaadUgaaeqaaaaakiaadAhada WgaaWcbaGaam4AaaqabaGccqGHRaWkdaabcaqaamaalaaabaGaeyOa IyRaamODamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadshaaa aacaGLiWoadaWgaaWcbaGaamyEamaaBaaameaacaWGPbaabeaaliab g2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabeaaaaa@4D3A@

so that

σ ji y j +ρ b i =ρ( v i y k v k + v i t | y i =const ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGa amOAaaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaaleaacaWGPb aabeaakiabg2da9iabeg8aYnaabmaabaWaaSaaaeaacqGHciITcaWG 2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaale aacaWGRbaabeaaaaGccaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaey4k aSYaaqGaaeaadaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamyAaa qabaaakeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaadMha daWgaaadbaGaamyAaaqabaWccqGH9aqpcaWGJbGaam4Baiaad6gaca WGZbGaamiDaaqabaaakiaawIcacaGLPaaaaaa@5C04@

 

 

 

5.3 Angular momentum balance in terms of Cauchy stress

 

Conservation of angular momentum for a continuum requires that the Cauchy stress satisfy

σ ji = σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGQbGaamyAaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaaaa @390F@

i.e. the stress tensor must be symmetric.

 

Derivation: write down the equation for balance of angular momentum for the region V within the  deformed solid

A y×TdA + V y×ρbdV = d dt { V y×ρvdV } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaCyEaiabgEna0kaahsfaca aMc8UaamizaiaadgeaaSqaaiaadgeaaeqaniabgUIiYdGccqGHRaWk daWdrbqaaiaahMhacqGHxdaTcqaHbpGCcaaMc8UaaCOyaiaadsgaca WGwbaaleaacaWGwbaabeqdcqGHRiI8aOGaeyypa0ZaaSaaaeaacaWG KbaabaGaamizaiaadshaaaWaaiWaaeaadaWdrbqaaiaahMhacqGHxd aTcqaHbpGCcaaMc8UaaCODaiaadsgacaWGwbaaleaacaWGwbaabeqd cqGHRiI8aaGccaGL7bGaayzFaaaaaa@5A3D@

Here, the left hand side is the resultant moment (about the origin) exerted by tractions and body forces acting on a general region within a solid.  The right hand side is the total angular momentum of the solid about the origin.

 

We can write the same expression using index notation

A ijk y j T k dA + V ijk y j b k ρdV = d dt { V ijk y j v k ρdV } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaeyicI48aaSbaaSqaaiaadM gacaWGQbGaam4AaaqabaGccaWG5bWaaSbaaSqaaiaadQgaaeqaaOGa amivamaaBaaaleaacaWGRbaabeaakiaadsgacaWGbbaaleaacaWGbb aabeqdcqGHRiI8aOGaey4kaSYaa8quaeaacqGHiiIZdaWgaaWcbaGa amyAaiaadQgacaWGRbaabeaakiaadMhadaWgaaWcbaGaamOAaaqaba GccaWGIbWaaSbaaSqaaiaadUgaaeqaaOGaeqyWdiNaaGPaVlaadsga caWGwbaaleaacaWGwbaabeqdcqGHRiI8aOGaeyypa0ZaaSaaaeaaca WGKbaabaGaamizaiaadshaaaWaaiWaaeaadaWdrbqaaiabgIGiopaa BaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaamyEamaaBaaaleaaca WGQbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqaHbpGCcaaM c8UaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIiYdaakiaawUhaca GL9baaaaa@66CB@

Express T in terms of σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3443@  and re-write the first integral as a volume integral using the divergence theorem

A ijk y j T k dA = A ijk y j σ mk n m dA = V y m ( ijk y j σ mk )dV = V ijk ( δ jm σ mk + y j σ mk x m )dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaa8quaeaacqGHiiIZdaWgaaWcba GaamyAaiaadQgacaWGRbaabeaakiaadMhadaWgaaWcbaGaamOAaaqa baGccaWGubWaaSbaaSqaaiaadUgaaeqaaOGaamizaiaadgeaaSqaai aadgeaaeqaniabgUIiYdGccqGH9aqpdaWdrbqaaiabgIGiopaaBaaa leaacaWGPbGaamOAaiaadUgaaeqaaOGaamyEamaaBaaaleaacaWGQb aabeaakiabeo8aZnaaBaaaleaacaWGTbGaam4AaaqabaGccaWGUbWa aSbaaSqaaiaad2gaaeqaaOGaamizaiaadgeaaSqaaiaadgeaaeqani abgUIiYdGccqGH9aqpdaWdrbqaamaalaaabaGaeyOaIylabaGaeyOa IyRaamyEamaaBaaaleaacaWGTbaabeaaaaGcdaqadaqaaiabgIGiop aaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaamyEamaaBaaaleaa caWGQbaabeaakiabeo8aZnaaBaaaleaacaWGTbGaam4Aaaqabaaaki aawIcacaGLPaaacaWGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIipa aOqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua eyypa0Zaa8quaeaacqGHiiIZdaWgaaWcbaGaamyAaiaadQgacaWGRb aabeaakmaabmaabaGaeqiTdq2aaSbaaSqaaiaadQgacaWGTbaabeaa kiabeo8aZnaaBaaaleaacaWGTbGaam4AaaqabaGccqGHRaWkcaWG5b WaaSbaaSqaaiaadQgaaeqaaOWaaSaaaeaacqGHciITcqaHdpWCdaWg aaWcbaGaamyBaiaadUgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaale aacaWGTbaabeaaaaaakiaawIcacaGLPaaacaWGKbGaamOvaaWcbaGa amOvaaqab0Gaey4kIipaaaaa@023F@

We may also show that

d dt { V ijk y j v k ρdV }= V ijk y j a k ρdV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaacmaabaWaa8quaeaacqGHiiIZdaWgaaWcbaGaamyAaiaadQga caWGRbaabeaakiaadMhadaWgaaWcbaGaamOAaaqabaGccaWG2bWaaS baaSqaaiaadUgaaeqaaOGaeqyWdiNaaGPaVlaadsgacaWGwbaaleaa caWGwbaabeqdcqGHRiI8aaGccaGL7bGaayzFaaGaeyypa0Zaa8quae aacqGHiiIZdaWgaaWcbaGaamyAaiaadQgacaWGRbaabeaakiaadMha daWgaaWcbaGaamOAaaqabaGccaWGHbWaaSbaaSqaaiaadUgaaeqaaO GaeqyWdiNaaGPaVlaadsgacaWGwbaaleaacaWGwbaabeqdcqGHRiI8 aaaa@586C@

Substitute the last two results into the angular momentum balance equation to see that

V ijk ( δ jm σ mk + y j σ mk x m )dV + V ijk y j b k ρdV = V ijk y j a k ρdV V ijk δ jm σ mk = V ijk y j ( σ mk y m +ρ b k ρ a k )dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaa8quaeaacqGHiiIZdaWgaaWcba GaamyAaiaadQgacaWGRbaabeaakmaabmaabaGaeqiTdq2aaSbaaSqa aiaadQgacaWGTbaabeaakiabeo8aZnaaBaaaleaacaWGTbGaam4Aaa qabaGccqGHRaWkcaWG5bWaaSbaaSqaaiaadQgaaeqaaOWaaSaaaeaa cqGHciITcqaHdpWCdaWgaaWcbaGaamyBaiaadUgaaeqaaaGcbaGaey OaIyRaamiEamaaBaaaleaacaWGTbaabeaaaaaakiaawIcacaGLPaaa caWGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabgUcaRmaape fabaGaeyicI48aaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGccaWG 5bWaaSbaaSqaaiaadQgaaeqaaOGaamOyamaaBaaaleaacaWGRbaabe aakiabeg8aYjaaykW7caWGKbGaamOvaaWcbaGaamOvaaqab0Gaey4k Iipakiabg2da9maapefabaGaeyicI48aaSbaaSqaaiaadMgacaWGQb Gaam4AaaqabaGccaWG5bWaaSbaaSqaaiaadQgaaeqaaOGaamyyamaa BaaaleaacaWGRbaabeaakiabeg8aYjaaykW7caWGKbGaamOvaaWcba GaamOvaaqab0Gaey4kIipaaOqaaiabgkDiEpaapefabaGaeyicI48a aSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGccqaH0oazdaWgaaWcba GaamOAaiaad2gaaeqaaOGaeq4Wdm3aaSbaaSqaaiaad2gacaWGRbaa beaakiabg2da9iabgkHiTaWcbaGaamOvaaqab0Gaey4kIipakmaape fabaGaeyicI48aaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGccaWG 5bWaaSbaaSqaaiaadQgaaeqaaOWaaeWaaeaadaWcaaqaaiabgkGi2k abeo8aZnaaBaaaleaacaWGTbGaam4AaaqabaaakeaacqGHciITcaWG 5bWaaSbaaSqaaiaad2gaaeqaaaaakiabgUcaRiabeg8aYjaadkgada WgaaWcbaGaam4AaaqabaGccqGHsislcqaHbpGCcaWGHbWaaSbaaSqa aiaadUgaaeqaaaGccaGLOaGaayzkaaGaamizaiaadAfaaSqaaiaadA faaeqaniabgUIiYdaaaaa@A5B8@

The integral on the right hand side of this expression is zero, because the stresses must satisfy the linear momentum balance equation.  Since this holds for any volume V, we conclude that

ijk δ jm σ mk = ijk σ jk =0 imn ijk σ jk =0 ( δ jm δ kn δ mk δ nj ) σ jk =0 σ mn σ nm =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeyicI48aaSbaaSqaaiaadMgaca WGQbGaam4AaaqabaGccqaH0oazdaWgaaWcbaGaamOAaiaad2gaaeqa aOGaeq4Wdm3aaSbaaSqaaiaad2gacaWGRbaabeaakiabg2da9iabgI GiopaaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaeq4Wdm3aaSba aSqaaiaadQgacaWGRbaabeaakiabg2da9iaaicdaaeaacqGHshI3cq GHiiIZdaWgaaWcbaGaamyAaiaad2gacaWGUbaabeaakiabgIGiopaa BaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaeq4Wdm3aaSbaaSqaai aadQgacaWGRbaabeaakiabg2da9iaaicdaaeaacqGHshI3daqadaqa aiabes7aKnaaBaaaleaacaWGQbGaamyBaaqabaGccqaH0oazdaWgaa WcbaGaam4Aaiaad6gaaeqaaOGaeyOeI0IaeqiTdq2aaSbaaSqaaiaa d2gacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGUbGaamOAaaqaba aakiaawIcacaGLPaaacqaHdpWCdaWgaaWcbaGaamOAaiaadUgaaeqa aOGaeyypa0JaaGimaaqaaiabgkDiElabeo8aZnaaBaaaleaacaWGTb GaamOBaaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaamOBaiaad2ga aeqaaOGaeyypa0JaaGimaaaaaa@7EE1@

which is the result we wanted.

 

 

 

5.4 Equations of motion in terms of other stress measures

 

In terms of nominal and material stress the balance of linear momentum is

S+ ρ 0 b= ρ 0 a S ij x i + ρ 0 b j = ρ 0 a j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgEGirlabgw SixJqabiaa=nfacqGHRaWkcqaHbpGCdaWgaaWcbaGaaGimaaqabaGc caWFIbGaeyypa0JaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaaCyyai aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVpaalaaabaGaam4uamaaBaaaleaacaWGPbGaamOAaaqabaaake aacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaakiabgUcaRiab eg8aYnaaBaaaleaacaaIWaaabeaakiaadkgadaWgaaWcbaGaamOAaa qabaGccqGH9aqpcqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGHbWa aSbaaSqaaiaadQgaaeqaaaaa@6284@

[ Σ F T ]+ ρ 0 b= ρ 0 a ( Σ ik F jk ) x i + ρ 0 b j = ρ 0 a j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgEGirlabgw SixpaadmaabaacceGae83OdmLaeyyXICncbeGaa4NramaaCaaaleqa baGaa4hvaaaaaOGaay5waiaaw2faaiabgUcaRiabeg8aYnaaBaaale aacaaIWaaabeaakiaa+jgacqGH9aqpcqaHbpGCdaWgaaWcbaGaaGim aaqabaGccaWHHbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVpaalaaabaGaeyOaIy7aaeWaaeaa cqqHJoWudaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamOramaaBaaale aacaWGQbGaam4AaaqabaaakiaawIcacaGLPaaaaeaacqGHciITcaWG 4bWaaSbaaSqaaiaadMgaaeqaaaaakiabgUcaRiabeg8aYnaaBaaale aacaaIWaaabeaakiaadkgadaWgaaWcbaGaamOAaaqabaGccqGH9aqp cqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGHbWaaSbaaSqaaiaadQ gaaeqaaaaa@714C@

Note that the derivatives are taken with respect to position in the undeformed solid.

 

The angular momentum balance equation is

FS= [ FS ] T Σ= Σ Τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaykW6caaMcS UaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcSocbeGaa8Nraiab gwSixlaa=nfacqGH9aqpdaWadaqaaiaa=zeacqGHflY1caWFtbaaca GLBbGaayzxaaWaaWbaaSqabeaacaWGubaaaOGaaGPaRlaaykW6caaM cSUaaGPaRlaaykW6caaMcSUaaGPaRJGabiab+n6atjabg2da9iab+n 6atnaaCaaaleqabaGae4hPdqfaaaaa@5D81@

 

 

To derive these results, you can start with the integral form of the linear momentum balance in terms of Cauchy stress

A σ ji n j dA+ V ρ b i dV = d dt { V ρ v i dV } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaeq4Wdm3aaSbaaSqaaiaadQ gacaWGPbaabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGccaWGKbGa amyqaiabgUcaRaWcbaGaamyqaaqab0Gaey4kIipakmaapefabaGaeq yWdiNaaGPaVlaadkgadaWgaaWcbaGaamyAaaqabaGccaaMc8Uaamiz aiaadAfaaSqaaiaadAfaaeqaniabgUIiYdGccqGH9aqpdaWcaaqaai aadsgaaeaacaWGKbGaamiDaaaadaGadaqaamaapefabaGaeqyWdiNa aGPaVlaadAhadaWgaaWcbaGaamyAaaqabaaabaGaamOvaaqab0Gaey 4kIipakiaadsgacaWGwbaacaGL7bGaayzFaaaaaa@582E@

Recall that area elements in the deformed and undeformed solids are related through

dA n i =J F ki 1 n k 0 d A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbGaamOBamaaDaaaleaaca WGPbaabaaaaOGaeyypa0JaamOsaiaadAeadaqhaaWcbaGaam4Aaiaa dMgaaeaacqGHsislcaaIXaaaaOGaamOBamaaDaaaleaacaWGRbaaba GaaGimaaaakiaadsgacaWGbbWaaSbaaSqaaiaaicdaaeqaaaaa@3FF4@

In addition, volume elements are related by dV=Jd V 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamOvaiabg2da9iaadQeaca WGKbGaamOvamaaBaaaleaacaaIWaaabeaaaaa@3909@ .  We can use these results to re-write the integrals as integrals over a volume in the undeformed solid as

A0 σ ji J F kj 1 n k 0 d A 0 + V0 ρ b i Jd V 0 = d dt { V0 ρ v i Jd V 0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaeq4Wdm3aaSbaaSqaaiaadQ gacaWGPbaabeaakiaadQeacaWGgbWaa0baaSqaaiaadUgacaWGQbaa baGaeyOeI0IaaGymaaaakiaad6gadaqhaaWcbaGaam4Aaaqaaiaaic daaaGccaWGKbGaamyqamaaBaaaleaacaaIWaaabeaakiabgUcaRaWc baGaamyqaiaaicdaaeqaniabgUIiYdGcdaWdrbqaaiabeg8aYjaayk W7caWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVlaadQeacaWGKbGa amOvamaaBaaaleaacaaIWaaabeaaaeaacaWGwbGaaGimaaqab0Gaey 4kIipakiabg2da9maalaaabaGaamizaaqaaiaadsgacaWG0baaamaa cmaabaWaa8quaeaacqaHbpGCcaaMc8UaamODamaaBaaaleaacaWGPb aabeaaaeaacaWGwbGaaGimaaqab0Gaey4kIipakiaadQeacaWGKbGa amOvamaaBaaaleaacaaIWaaabeaaaOGaay5Eaiaaw2haaaaa@64C9@

Finally, recall that S ij =J F ik 1 σ kj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaam4uaOWaaSbaaSqaaiaadMgaca WGQbaabeaakiabg2da9iaadQeacaWGgbWaa0baaSqaaiaadMgacaWG RbaabaGaeyOeI0IaaGymaaaakiabeo8aZnaaBaaaleaacaWGRbGaam OAaaqabaaaaa@3EB7@  and that Jρ= ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeqyWdiNaeyypa0JaeqyWdi 3aaSbaaSqaaiaaicdaaeqaaaaa@3901@  to see that

A0 S ki n k 0 d A 0 + V0 ρ 0 b i d V 0 = d dt { V0 ρ 0 v i d V 0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaam4uaiaaxcW7daWgaaWcba Gaam4AaiaadMgaaeqaaOGaamOBamaaDaaaleaacaWGRbaabaGaaGim aaaakiaadsgacaWGbbWaaSbaaSqaaiaaicdaaeqaaOGaey4kaScale aacaWGbbGaaGimaaqab0Gaey4kIipakmaapefabaGaeqyWdiNaaGPa VpaaBaaaleaacaaIWaaabeaakiaadkgadaWgaaWcbaGaamyAaaqaba GccaaMc8UaamizaiaadAfadaWgaaWcbaGaaGimaaqabaaabaGaamOv aiaaicdaaeqaniabgUIiYdGccqGH9aqpdaWcaaqaaiaadsgaaeaaca WGKbGaamiDaaaadaGadaqaamaapefabaGaeqyWdiNaaGPaVpaaBaaa leaacaaIWaaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaaabaGaam OvaiaaicdaaeqaniabgUIiYdGccaWGKbGaamOvamaaBaaaleaacaaI WaaabeaaaOGaay5Eaiaaw2haaaaa@6051@

Apply the divergence theorem to the first term and rearrange

V ( S ji x j + ρ 0 b i ρ 0 d v i dt ) d V 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaWaaeWaaeaadaWcaaqaaiabgk Gi2kaadofadaWgaaWcbaGaamOAaiaadMgaaeqaaaGcbaGaeyOaIyRa amiEamaaBaaaleaacaWGQbaabeaaaaGccqGHRaWkcqaHbpGCdaWgaa WcbaGaaGimaaqabaGccaaMc8UaamOyamaaBaaaleaacaWGPbaabeaa kiabgkHiTiabeg8aYnaaBaaaleaacaaIWaaabeaakiaaykW7daWcaa qaaiaadsgacaWG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaamizaiaa dshaaaaacaGLOaGaayzkaaaaleaacaWGwbaabeqdcqGHRiI8aOGaaG PaVlaaykW7caWGKbGaamOvamaaBaaaleaacaaIWaaabeaakiabg2da 9iaaicdaaaa@55DC@

Once again, since this must hold for any material volume, we conclude that

S ij x i + ρ 0 b j = ρ 0 a j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam 4uamaaBaaaleaacaWGPbGaamOAaaqabaaakeaacqGHciITcaWG4bWa aSbaaSqaaiaadMgaaeqaaaaakiabgUcaRiabeg8aYnaaBaaaleaaca aIWaaabeaakiaadkgadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcqaH bpGCdaWgaaWcbaGaaGimaaqabaGccaWGHbWaaSbaaSqaaiaadQgaae qaaaaa@46DE@

The linear momentum balance equation in terms of material stress follows directly, by substituting into this equation for S ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@35A7@  in terms of Σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHJoWudaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3653@

 

The angular momentum balance equation can be derived simply by substituting into the momentum balance equation in terms of Cauchy stress σ ij = σ ji MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadQgacaWGPbaabeaa aaa@3B6E@

 

 

 

5.5 Work done by Cauchy stresses

 

Consider a solid with mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaa@356C@  in its initial configuration, and density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCaaa@3486@  in the deformed solid. Let σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3443@  denote the Cauchy stress distribution within the solid.  Assume that the solid is subjected to a body force b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkgadaWgaaWcbaGaamyAaaqabaaaaa@3278@  (per unit mass), and let u i , v i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGcca GGSaGaaGPaVlaaykW7caWG2bWaaSbaaSqaaiaadMgaaeqaaaaa@3870@  and a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaamyAaaqabaaaaa@3277@  denote the displacement, velocity and acceleration of a material particle at position y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaaaaa@328F@   in the deformed solid. In addition, let

D ij = 1 2 ( v i y j + v j y i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGebWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaeWaaeaa daWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamyAaaqabaaakeaacq GHciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaaaakiabgUcaRmaalaaa baGaeyOaIyRaamODamaaBaaaleaacaWGQbaabeaaaOqaaiabgkGi2k aadMhadaWgaaWcbaGaamyAaaqabaaaaaGccaGLOaGaayzkaaaaaa@48D6@

denote the stretch rate in the solid.

 

The rate of work done by Cauchy stresses per unit deformed volume is then σ ij D ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaamiramaaBaaaleaacaWGPbGaamOAaaqabaaaaa@396E@ .  This energy is either dissipated as heat or stored as internal energy in the solid, depending on the material behavior.

 

We shall show that the rate of work done by internal forces acting on any sub-volume V bounded by a surface A in the deformed solid can be calculated from

r ˙ = A T i (n) v i dA + V ρ b i v i dV = V σ ij D ij dV + d dt { V 1 2 ρ v i v i dV } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaai aadsfadaqhaaWcbaGaamyAaaqaaiaacIcacaWHUbGaaiykaaaakiaa dAhadaWgaaWcbaGaamyAaaqabaGccaWGKbGaamyqaaWcbaGaamyqaa qab0Gaey4kIipakiabgUcaRmaapefabaGaeqyWdiNaamOyamaaBaaa leaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGccaWGKb GaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabg2da9maapefabaGa eq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaadseadaWgaaWcba GaamyAaiaadQgaaeqaaOGaamizaiaadAfaaSqaaiaadAfaaeqaniab gUIiYdGccqGHRaWkdaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaada GadaqaamaapefabaWaaSaaaeaacaaIXaaabaGaaGOmaaaacqaHbpGC caWG2bWaaSbaaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPb aabeaakiaadsgacaWGwbaaleaacaWGwbaabeqdcqGHRiI8aaGccaGL 7bGaayzFaaaaaa@6AD2@

Here, the two terms on the left hand side represent the rate of work done by tractions and body forces acting on the solid (work done = force x velocity).  The first term on the right-hand side can be interpreted as the work done by Cauchy stresses; the second term is the rate of change of kinetic energy. 

 

Derivation: Substitute for T i (n) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaa0baaSqaaiaadMgaaeaaca GGOaGaaCOBaiaacMcaaaaaaa@370A@  in terms of Cauchy stress to see that

r ˙ = A T i (n) v i dA + V ρ b i v i dV = A n j σ ji v i dA + V ρ b i v i dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaai aadsfadaqhaaWcbaGaamyAaaqaaiaacIcacaWHUbGaaiykaaaakiaa dAhadaWgaaWcbaGaamyAaaqabaGccaWGKbGaamyqaaWcbaGaamyqaa qab0Gaey4kIipakiabgUcaRmaapefabaGaeqyWdiNaamOyamaaBaaa leaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGccaWGKb GaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabg2da9maapefabaGa amOBamaaBaaaleaacaWGQbaabeaakiabeo8aZnaaBaaaleaacaWGQb GaamyAaaqabaGccaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaamizaiaa dgeaaSqaaiaadgeaaeqaniabgUIiYdGccqGHRaWkdaWdrbqaaiabeg 8aYjaadkgadaWgaaWcbaGaamyAaaqabaGccaWG2bWaaSbaaSqaaiaa dMgaaeqaaOGaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIiYdaaaa@6552@

Now, apply the divergence theorem to the first term on the right hand side

r ˙ = V y j ( σ ji v i )dV + V ρ b i v i dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaam aalaaabaGaeyOaIylabaGaeyOaIyRaamyEamaaBaaaleaacaWGQbaa beaaaaGcdaqadaqaaiabeo8aZnaaBaaaleaacaWGQbGaamyAaaqaba GccaWG2bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaamiz aiaadAfaaSqaaiaadAfaaeqaniabgUIiYdGccqGHRaWkdaWdrbqaai abeg8aYjaadkgadaWgaaWcbaGaamyAaaqabaGccaWG2bWaaSbaaSqa aiaadMgaaeqaaOGaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIiYd aaaa@51F1@

Evaluate the derivative and collect together the terms involving body force and stress divergence

r ˙ = V { σ ji v i y j +( σ ji y j +ρ b i ) v i }dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaam aacmaabaGaeq4Wdm3aaSbaaSqaaiaadQgacaWGPbaabeaakmaalaaa baGaeyOaIyRaamODamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2k aadMhadaWgaaWcbaGaamOAaaqabaaaaOGaey4kaSYaaeWaaeaadaWc aaqaaiabgkGi2kabeo8aZnaaBaaaleaacaWGQbGaamyAaaqabaaake aacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaaaakiabgUcaRiab eg8aYjaadkgadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaca WG2bWaaSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzFaaGaamizaiaa dAfaaSqaaiaadAfaaeqaniabgUIiYdaaaa@58E6@

Recall the equation of motion

σ ji y j +ρ b i =ρ a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGa amOAaaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaaleaacaWGPb aabeaakiabg2da9iabeg8aYjaadggadaWgaaWcbaGaamyAaaqabaaa aa@42BF@

and note that since the stress is symmetric σ ij = σ ji MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadQgacaWGPbaabeaa aaa@3B6E@

σ ji v i y j = 1 2 ( σ ij + σ ji ) v i y j = 1 2 σ ij ( v i y j + v j y i )= σ ij D ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOAaiaadM gaaeqaaOWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaadMgaaeqa aaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGQbaabeaaaaGccqGH9a qpdaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaabaGaeq4Wdm3aaSba aSqaaiaadMgacaWGQbaabeaakiabgUcaRiabeo8aZnaaBaaaleaaca WGQbGaamyAaaqabaaakiaawIcacaGLPaaadaWcaaqaaiabgkGi2kaa dAhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaSbaaS qaaiaadQgaaeqaaaaakiabg2da9maalaaabaGaaGymaaqaaiaaikda aaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakmaabmaabaWaaS aaaeaacqGHciITcaWG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOa IyRaamyEamaaBaaaleaacaWGQbaabeaaaaGccqGHRaWkdaWcaaqaai abgkGi2kaadAhadaWgaaWcbaGaamOAaaqabaaakeaacqGHciITcaWG 5bWaaSbaaSqaaiaadMgaaeqaaaaaaOGaayjkaiaawMcaaiabg2da9i abeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccaWGebWaaSbaaSqa aiaadMgacaWGQbaabeaaaaa@7034@

to see that

r ˙ = V { σ ij D ij +ρ a i v i }dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaam aacmaabaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaadsea daWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIaeqyWdiNaamyyam aaBaaaleaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaaa kiaawUhacaGL9baacaWGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIi paaaa@4962@

Finally, note that

V ρ a i v i dV = V 0 ρ o d v i dt v i d V 0 = V 0 ρ o 1 2 d dt ( v i v i )d V 0 = d dt ( V 0 1 2 ρ 0 ( v i v i )d V 0 )= d dt ( V 0 1 2 ρ 0 ( v i v i )d V 0 )= d dt V 1 2 ρ( v i v i )dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaapefabaGaeqyWdiNaamyyam aaBaaaleaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGc caWGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabg2da9maape fabaGaeqyWdi3aaSbaaSqaaiaad+gaaeqaaOWaaSaaaeaacaWGKbGa amODamaaBaaaleaacaWGPbaabeaaaOqaaiaadsgacaWG0baaaiaadA hadaWgaaWcbaGaamyAaaqabaGccaWGKbGaamOvamaaBaaaleaacaaI Waaabeaakiabg2da9aWcbaGaamOvamaaBaaameaacaaIWaaabeaaaS qab0Gaey4kIipakmaapefabaGaeqyWdi3aaSbaaSqaaiaad+gaaeqa aOWaaSaaaeaacaaIXaaabaGaaGOmaaaadaWcaaqaaiaadsgaaeaaca WGKbGaamiDaaaadaqadaqaaiaadAhadaWgaaWcbaGaamyAaaqabaGc caWG2bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaamizai aadAfadaWgaaWcbaGaaGimaaqabaaabaGaamOvamaaBaaameaacaaI WaaabeaaaSqab0Gaey4kIipaaOqaaiabg2da9maalaaabaGaamizaa qaaiaadsgacaWG0baaamaabmaabaWaa8quaeaadaWcaaqaaiaaigda aeaacaaIYaaaaiabeg8aYnaaBaaaleaacaaIWaaabeaakmaabmaaba GaamODamaaBaaaleaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyA aaqabaaakiaawIcacaGLPaaacaWGKbGaamOvamaaBaaaleaacaaIWa aabeaaaeaacaWGwbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8 aaGccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaWGKbaabaGaamizai aadshaaaWaaeWaaeaadaWdrbqaamaalaaabaGaaGymaaqaaiaaikda aaGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacaWG2bWaaS baaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPbaabeaaaOGa ayjkaiaawMcaaiaadsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaaqaai aadAfadaWgaaadbaGaaGimaaqabaaaleqaniabgUIiYdaakiaawIca caGLPaaacqGH9aqpdaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaada WdrbqaamaalaaabaGaaGymaaqaaiaaikdaaaGaeqyWdi3aaeWaaeaa caWG2bWaaSbaaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPb aabeaaaOGaayjkaiaawMcaaiaadsgacaWGwbaaleaacaWGwbWaaSba aWqaaaqabaaaleqaniabgUIiYdaaaaa@A2C0@

Finally, substitution leads to

r ˙ = A T i (n) v i dA + V ρ b i v i dV = V σ ij D ij dV + d dt { V 1 2 ρ v i v i dV } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaai aadsfadaqhaaWcbaGaamyAaaqaaiaacIcacaWHUbGaaiykaaaakiaa dAhadaWgaaWcbaGaamyAaaqabaGccaWGKbGaamyqaaWcbaGaamyqaa qab0Gaey4kIipakiabgUcaRmaapefabaGaeqyWdiNaamOyamaaBaaa leaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGccaWGKb GaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabg2da9maapefabaGa eq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaadseadaWgaaWcba GaamyAaiaadQgaaeqaaOGaamizaiaadAfaaSqaaiaadAfaaeqaniab gUIiYdGccqGHRaWkdaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaada GadaqaamaapefabaWaaSaaaeaacaaIXaaabaGaaGOmaaaacqaHbpGC caWG2bWaaSbaaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPb aabeaakiaadsgacaWGwbaaleaacaWGwbaabeqdcqGHRiI8aaGccaGL 7bGaayzFaaaaaa@6AD2@

as required.

 

 

5.6 Rate of mechanical work in terms of other stress measures

 

 The rate of work done per unit undeformed volume by Kirchhoff stress is τ ij D ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHepaDdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaamiramaaBaaaleaacaWGPbGaamOAaaqabaaaaa@3970@

 The rate of work done per unit undeformed volume by Nominal stress is S ij F ˙ ji MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgacaWGQb aabeaakiqadAeagaGaamaaBaaaleaacaWGQbGaamyAaaqabaaaaa@388E@

 The rate of work done per unit undeformed volume by Material stress is Σ ij E ˙ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHJoWudaWgaaWcbaGaamyAaiaadQ gaaeqaaOGabmyrayaacaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@3939@

 

This shows that nominal stress and deformation gradient are work conjugate, as are material stress and Lagrange strain.

 

In addition, the rate of work done on a volume V 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaa aa@3487@  of the undeformed solid can be expressed as

r ˙ = A T i (n) v i dA + V ρ b i v i dV = V 0 τ ij D ij d V 0 + d dt { V 0 1 2 ρ 0 v i v i d V 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaai aadsfadaqhaaWcbaGaamyAaaqaaiaacIcacaWHUbGaaiykaaaakiaa dAhadaWgaaWcbaGaamyAaaqabaGccaWGKbGaamyqaaWcbaGaamyqaa qab0Gaey4kIipakiabgUcaRmaapefabaGaeqyWdiNaamOyamaaBaaa leaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGccaWGKb GaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabg2da9maapefabaGa eqiXdq3aaSbaaSqaaiaadMgacaWGQbaabeaakiaadseadaWgaaWcba GaamyAaiaadQgaaeqaaOGaamizaiaadAfadaWgaaWcbaGaaGimaaqa baaabaGaamOvamaaBaaameaacaaIWaaabeaaaSqab0Gaey4kIipaki abgUcaRmaalaaabaGaamizaaqaaiaadsgacaWG0baaamaacmaabaWa a8quaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiabeg8aYnaaBaaale aacaaIWaaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGccaWG2bWa aSbaaSqaaiaadMgaaeqaaOGaamizaiaadAfadaWgaaWcbaGaaGimaa qabaaabaGaamOvamaaBaaameaacaaIWaaabeaaaSqab0Gaey4kIipa aOGaay5Eaiaaw2haaaaa@6F5E@

r ˙ = A T i (n) v i dA + V ρ b i v i dV = V 0 S ij F ˙ ji d V 0 + d dt { V 0 1 2 ρ 0 v i v i d V 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaai aadsfadaqhaaWcbaGaamyAaaqaaiaacIcacaWHUbGaaiykaaaakiaa dAhadaWgaaWcbaGaamyAaaqabaGccaWGKbGaamyqaaWcbaGaamyqaa qab0Gaey4kIipakiabgUcaRmaapefabaGaeqyWdiNaamOyamaaBaaa leaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGccaWGKb GaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabg2da9maapefabaGa am4uamaaBaaaleaacaWGPbGaamOAaaqabaGcceWGgbGbaiaadaWgaa WcbaGaamOAaiaadMgaaeqaaOGaamizaiaadAfadaWgaaWcbaGaaGim aaqabaaabaGaamOvamaaBaaameaacaaIWaaabeaaaSqab0Gaey4kIi pakiabgUcaRmaalaaabaGaamizaaqaaiaadsgacaWG0baaamaacmaa baWaa8quaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiabeg8aYnaaBa aaleaacaaIWaaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGccaWG 2bWaaSbaaSqaaiaadMgaaeqaaOGaamizaiaadAfadaWgaaWcbaGaaG imaaqabaaabaGaamOvamaaBaaameaacaaIWaaabeaaaSqab0Gaey4k IipaaOGaay5Eaiaaw2haaaaa@6E7C@

r ˙ = A T i (n) v i dA + V ρ b i v i dV = V 0 Σ ij E ˙ ij d V 0 + d dt { V 0 1 2 ρ 0 v i v i d V 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaai aadsfadaqhaaWcbaGaamyAaaqaaiaacIcacaWHUbGaaiykaaaakiaa dAhadaWgaaWcbaGaamyAaaqabaGccaWGKbGaamyqaaWcbaGaamyqaa qab0Gaey4kIipakiabgUcaRmaapefabaGaeqyWdiNaamOyamaaBaaa leaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGccaWGKb GaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabg2da9maapefabaGa eu4Odm1aaSbaaSqaaiaadMgacaWGQbaabeaakiqadweagaGaamaaBa aaleaacaWGPbGaamOAaaqabaGccaWGKbGaamOvamaaBaaaleaacaaI WaaabeaaaeaacaWGwbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGHRi I8aOGaey4kaSYaaSaaaeaacaWGKbaabaGaamizaiaadshaaaWaaiWa aeaadaWdrbqaamaalaaabaGaaGymaaqaaiaaikdaaaGaeqyWdi3aaS baaSqaaiaaicdaaeqaaOGaamODamaaBaaaleaacaWGPbaabeaakiaa dAhadaWgaaWcbaGaamyAaaqabaGccaWGKbGaamOvamaaBaaaleaaca aIWaaabeaaaeaacaWGwbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGH RiI8aaGccaGL7bGaayzFaaaaaa@6F27@

 

Derivations: The proof of the first result (and the stress power of Kirchhoff stress) is straightforward and is left as an exercise.  To show the second result, note that T i (n) dA= n j 0 S ji d A 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaa0baaSqaaiaadMgaaeaaca GGOaGaaCOBaiaacMcaaaGccaWGKbGaamyqaiabg2da9iaad6gadaqh aaWcbaGaamOAaaqaaiaaicdaaaGccaWGtbWaaSbaaSqaaiaadQgaca WGPbaabeaakiaadsgacaWGbbWaaSbaaSqaaiaaicdaaeqaaaaa@421C@  and dV=Jd V 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamOvaiabg2da9iaadQeaca WGKbGaamOvamaaBaaaleaacaaIWaaabeaaaaa@3909@  to re-write the integrals over the undeformed solid; then and apply the divergence theorem to see that

r ˙ = V 0 x j ( S ji v i )d V 0 + V0 ρ b i v i Jd V 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaam aalaaabaGaeyOaIylabaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaa beaaaaGcdaqadaqaaiaadofadaWgaaWcbaGaamOAaiaadMgaaeqaaO GaamODamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaiaadsga caWGwbWaaSbaaSqaaiaaicdaaeqaaaqaaiaadAfadaWgaaadbaGaaG imaaqabaaaleqaniabgUIiYdGccqGHRaWkdaWdrbqaaiabeg8aYjaa dkgadaWgaaWcbaGaamyAaaqabaGccaWG2bWaaSbaaSqaaiaadMgaae qaaOGaamOsaiaadsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaaqaaiaa dAfacaaIWaaabeqdcqGHRiI8aaaa@5536@

Evaluate the derivative, recall that Jρ= ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeqyWdiNaeyypa0JaeqyWdi 3aaSbaaSqaaiaaicdaaeqaaaaa@3901@  and use the equation of motion

S ij x i + ρ 0 b j = ρ 0 d v j dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaykW7daWcaa qaaiaadofadaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOaIyRa amiEamaaBaaaleaacaWGPbaabeaaaaGccqGHRaWkcqaHbpGCdaWgaa WcbaGaaGimaaqabaGccaWGIbWaaSbaaSqaaiaadQgaaeqaaOGaeyyp a0JaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOWaaSaaaeaacaWGKbGaam ODamaaBaaaleaacaWGQbaabeaaaOqaaiaadsgacaWG0baaaaaa@4B73@

to see that

r ˙ = V 0 S ji v i x j d V 0 + V0 ρ 0 d v i dt v i d V 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaai aadofadaWgaaWcbaGaamOAaiaadMgaaeqaaOWaaSaaaeaacqGHciIT caWG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBa aaleaacaWGQbaabeaaaaGccaWGKbGaamOvamaaBaaaleaacaaIWaaa beaaaeaacaWGwbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aO Gaey4kaSYaa8quaeaacqaHbpGCdaWgaaWcbaGaaGimaaqabaGcdaWc aaqaaiaadsgacaWG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaamizai aadshaaaGaamODamaaBaaaleaacaWGPbaabeaakiaadsgacaWGwbWa aSbaaSqaaiaaicdaaeqaaaqaaiaadAfacaaIWaaabeqdcqGHRiI8aa aa@56BD@

Finally, note that v i / x j =( u ˙ i / x j )= F ˙ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWG2bWaaSbaaSqaaiaadM gaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGc cqGH9aqpdaqadaqaaiabgkGi2kqadwhagaGaamaaBaaaleaacaWGPb aabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaGc caGLOaGaayzkaaGaeyypa0JabmOrayaacaWaaSbaaSqaaiaadMgaca WGQbaabeaaaaa@48C0@  and re-write the second integral as a kinetic energy term as before to obtain the required result.

 

The third result follows by straightforward algebraic manipulations MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  note that by definition

S ij F ˙ ji = Σ ik F jk F ˙ ji MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgacaWGQb aabeaakiqadAeagaGaamaaBaaaleaacaWGQbGaamyAaaqabaGccqGH 9aqpcqqHJoWudaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamOramaaDa aaleaacaWGQbGaam4AaaqaaaaakiqadAeagaGaamaaBaaaleaacaWG QbGaamyAaaqabaaaaa@42F4@

Since Σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHJoWudaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3653@  is symmetric it follows that

Σ ik F jk F ˙ ji = 1 2 ( Σ ik + Σ ki ) F jk F ˙ ji = Σ ik 1 2 ( F jk F ˙ ji + F ji F ˙ jk )= Σ ik E ˙ ik MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHJoWudaWgaaWcbaGaamyAaiaadU gaaeqaaOGaamOramaaDaaaleaacaWGQbGaam4AaaqaaaaakiqadAea gaGaamaaBaaaleaacaWGQbGaamyAaaqabaGccqGH9aqpdaWcaaqaai aaigdaaeaacaaIYaaaamaabmaabaGaeu4Odm1aaSbaaSqaaiaadMga caWGRbaabeaakiabgUcaRiabfo6atnaaBaaaleaacaWGRbGaamyAaa qabaaakiaawIcacaGLPaaacaWGgbWaa0baaSqaaiaadQgacaWGRbaa baaaaOGabmOrayaacaWaaSbaaSqaaiaadQgacaWGPbaabeaakiabg2 da9iabfo6atnaaBaaaleaacaWGPbGaam4AaaqabaGcdaWcaaqaaiaa igdaaeaacaaIYaaaamaabmaabaGaamOramaaDaaaleaacaWGQbGaam 4AaaqaaaaakiqadAeagaGaamaaBaaaleaacaWGQbGaamyAaaqabaGc cqGHRaWkcaWGgbWaa0baaSqaaiaadQgacaWGPbaabaaaaOGabmOray aacaWaaSbaaSqaaiaadQgacaWGRbaabeaaaOGaayjkaiaawMcaaiab g2da9iabfo6atnaaBaaaleaacaWGPbGaam4AaaqabaGcceWGfbGbai aadaWgaaWcbaGaamyAaiaadUgaaeqaaaaa@69B1@

 

5.7 Rate of mechanical work for infinitesimal deformations

 

For infintesimal motions all stress measures are equal; and all strain rate measures can be approximated by the infinitesimal strain tensor ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1oaaaa@3407@ .  The rate of work done by stresses per unit volume of either deformed or undeformed solid (the difference is neglected) can be expressed as σ ij ε ˙ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGafqyTduMbaiaadaWgaaWcbaGaamyAaiaadQgaaeqaaaaa @3A55@ , and the work done on a volume V 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaa aa@3487@  of the solid is

r ˙ = A T i (n) v i dA + V ρ b i v i dV = V 0 σ ij ε ˙ ij d V 0 + d dt { V 0 1 2 ρ 0 v i v i d V 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGYbGbaiaacqGH9aqpdaWdrbqaai aadsfadaqhaaWcbaGaamyAaaqaaiaacIcacaWHUbGaaiykaaaakiaa dAhadaWgaaWcbaGaamyAaaqabaGccaWGKbGaamyqaaWcbaGaamyqaa qab0Gaey4kIipakiabgUcaRmaapefabaGaeqyWdiNaamOyamaaBaaa leaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGccaWGKb GaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabg2da9maapefabaGa eq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiqbew7aLzaacaWaaS baaSqaaiaadMgacaWGQbaabeaakiaadsgacaWGwbWaaSbaaSqaaiaa icdaaeqaaaqaaiaadAfadaWgaaadbaGaaGimaaqabaaaleqaniabgU IiYdGccqGHRaWkdaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaadaGa daqaamaapefabaWaaSaaaeaacaaIXaaabaGaaGOmaaaacqaHbpGCda WgaaWcbaGaaGimaaqabaGccaWG2bWaaSbaaSqaaiaadMgaaeqaaOGa amODamaaBaaaleaacaWGPbaabeaakiaadsgacaWGwbWaaSbaaSqaai aaicdaaeqaaaqaaiaadAfadaWgaaadbaGaaGimaaqabaaaleqaniab gUIiYdaakiaawUhacaGL9baaaaa@7043@

 

 

5.8 The principle of Virtual Work

 

The principle of virtual work forms the basis for the finite element method in the mechanics of solids and so will be discussed in detail in this section.

Suppose that a deformable solid is subjected to loading that induces a displacement field u(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaaiikaiaahIhacaGGPaaaaa@361E@ , and a velocity field v(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaaiikaiaahIhacaGGPaaaaa@361F@ .  The loading consists of a prescribed displacement on part of the boundary (denoted by S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdaaeqaaa aa@3485@  ), together with a traction t (which may be zero in places) applied to the rest of the boundary (denoted by S 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaikdaaeqaaa aa@3486@  ).  The loading induces a Cauchy stress σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@ .  The stress field satisfies the angular momentum balance equation σ ij = σ ji MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadQgacaWGPbaabeaa aaa@3B6E@ .

 

The principle of virtual work is a different way of re-writing partial differential equation for linear moment balance

σ ji y j +ρ b i =ρ d v i dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGa amOAaaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaaleaacaWGPb aabeaakiabg2da9iabeg8aYnaalaaabaGaamizaiaadAhadaWgaaWc baGaamyAaaqabaaakeaacaWGKbGaamiDaaaaaaa@45B9@

in an equivalent integral form, which is much better suited for computer solution.

 

To express the principle, we define a kinematically admissible virtual velocity field δv(y) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaaCODaiaacIcacaWH5bGaai ykaaaa@35DC@ , satisfying  δv=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaaCODaiabg2da9iaaicdaaa a@3541@  on S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4uamaaBaaaleaacaaIXaaabeaaaa a@329C@ .  You can visualize this field as a small change in the velocity of the solid, if you like, but it is really just an arbitrary differentiable vector field.  The term `kinematically admissible’ is just a complicated way of saying that the field is continuous, differentiable, and satisfies δv=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaaCODaiabg2da9iaaicdaaa a@3541@  on S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4uamaaBaaaleaacaaIXaaabeaaaa a@329C@  - that is to say, if you perturb the velocity by δv(y) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaaCODaiaacIcacaWH5bGaai ykaaaa@35DC@ , the boundary conditions on displacement are still satisfied.

 

In addition, we define an associated virtual velocity gradient, and virtual stretch rate as

δ L ij = δ v i y j δ D ij = 1 2 ( δ v i y j + δ v j y i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGmbWaaSbaaSqaaiaadM gacaWGQbaabeaakiabg2da9maalaaabaGaeyOaIyRaeqiTdqMaamOD amaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcba GaamOAaaqabaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7cqaH0oazcaWGebWaaSbaaSqaaiaadMga caWGQbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaae WaaeaadaWcaaqaaiabgkGi2kabes7aKjaadAhadaWgaaWcbaGaamyA aaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaaaaki abgUcaRmaalaaabaGaeyOaIyRaeqiTdqMaamODamaaBaaaleaacaWG QbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaa GccaGLOaGaayzkaaaaaa@7652@

 

The principal of virtual work may be stated in two ways.

 

First version of the principle of virtual work

 

The first is not very interesting, but we will state it anyway.  Suppose that the Cauchy stress satisfies:

1.      The boundary condition n i σ ij = t j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO Gaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iaadsha daWgaaWcbaGaamOAaaqabaaaaa@3BCD@  on S 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaikdaaeqaaa aa@3486@

2.      The linear momentum balance equation

σ ji y j +ρ b i =ρ d v i dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGa amOAaaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaaleaacaWGPb aabeaakiabg2da9iabeg8aYnaalaaabaGaamizaiaadAhadaWgaaWc baGaamyAaaqabaaakeaacaWGKbGaamiDaaaaaaa@45B9@

Then the virtual work equation

V σ ij δ D ij dV+ V ρ d v i dt δ v i dV V ρ b i δ v i dV S 2 t i δ v i dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaKaaahaacqaHdpWCkmaaBaaale aacaWGPbGaamOAaaqabaGccqaH0oazcaWGebWaaSbaaSqaaiaadMga caWGQbaabeaajaaWcaaMc8UaamizaiaadAfacqGHRaWkkmaapefaja aWbaGaeqyWdiNcdaWcaaqaaiaadsgacaWG2bWaaSbaaSqaaiaadMga aeqaaaGcbaGaamizaiaadshaaaqcaaSaeqiTdqMaamODaOWaaSbaaS qaaiaadMgaaeqaaaqcbaCaaiaadAfaaeqajmaWcqGHRiI8aOGaamiz aiaadAfajaaWcqGHsislkmaapefajaaWbaGaeqyWdiNaamOyaOWaaS baaSqaaiaadMgaaeqaaKaaalabes7aKjaadAhakmaaBaaaleaacaWG PbaabeaajaaWcaWGKbGaamOvaiabgkHiTOWaa8quaKaaahaacaWG0b GcdaWgaaWcbaGaamyAaaqabaqcaaSaeqiTdqMaamODaOWaaSbaaSqa aiaadMgaaeqaaaqcbaCaaiaadofalmaaBaaajiaWbaGaaGOmaaqaba aajeaWbeqcdaSaey4kIipaaKqaahaacaWGwbaabeqcdaSaey4kIipa aSqaaiaadAfaaeqaniabgUIiYdGccaWGKbGaamyqaiabg2da9iaaic daaaa@79B4@

is satisfied for all virtual velocity fields.

 

Proof:  Observe that since the Cauchy stress is symmetric

σ ij δ D ij = 1 2 σ ij ( δ v i y j + δ v j y i )= 1 2 ( σ ji δ v i y j + σ ij δ v j y i )= σ ji δ v i y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabes7aKjaadseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa eyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacqaHdpWCdaWgaaWcba GaamyAaiaadQgaaeqaaOWaaeWaaeaadaWcaaqaaiabgkGi2kabes7a KjaadAhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaS baaSqaaiaadQgaaeqaaaaakiabgUcaRmaalaaabaGaeyOaIyRaeqiT dqMaamODamaaBaaaleaacaWGQbaabeaaaOqaaiabgkGi2kaadMhada WgaaWcbaGaamyAaaqabaaaaaGccaGLOaGaayzkaaGaeyypa0ZaaSaa aeaacaaIXaaabaGaaGOmaaaadaqadaqaaiabeo8aZnaaBaaaleaaca WGQbGaamyAaaqabaGcdaWcaaqaaiabgkGi2kabes7aKjaadAhadaWg aaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadQ gaaeqaaaaakiabgUcaRiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqa baGcdaWcaaqaaiabgkGi2kabes7aKjaadAhadaWgaaWcbaGaamOAaa qabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaaaOGa ayjkaiaawMcaaiabg2da9iabeo8aZnaaBaaaleaacaWGQbGaamyAaa qabaGcdaWcaaqaaiabgkGi2kabes7aKjaadAhadaWgaaWcbaGaamyA aaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaaaaaa a@7F39@

Next, note that

σ ji v i y j = y j ( σ ji δ v i ) σ ji y j δ v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadQgacaWGPb aabeaakmaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGPbaabeaa aOqaaiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaaaOGaeyypa0 ZaaSaaaeaacqGHciITaeaacqGHciITcaWG5bWaaSbaaSqaaiaadQga aeqaaaaakmaabmaabaGaeq4Wdm3aaSbaaSqaaiaadQgacaWGPbaabe aakiabes7aKjaadAhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGL PaaacqGHsisldaWcaaqaaiabgkGi2kabeo8aZnaaBaaaleaacaWGQb GaamyAaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqa aaaakiabes7aKjaadAhadaWgaaWcbaGaamyAaaqabaaaaa@5877@

Finally, substituting the latter identity into the virtual work equation, applying the divergence theorem, using the linear momentum balance equation and boundary conditions on σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaaC4Wdaaa@32F5@  and δv(y) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaaCODaiaacIcacaWH5bGaai ykaaaa@35DC@  we obtain the required result.

 

Second version of the principle of virtual work

 

The converse of this statement is much more interesting and useful.  Suppose that σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3444@  satisfies the virtual work equation

V σ ij δ D ij dV+ V ρ d v i dt δ v i dV V ρ b i δ v i dV S 2 t i δ v i dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaKaaahaacqaHdpWCkmaaBaaale aacaWGPbGaamOAaaqabaGccqaH0oazcaWGebWaaSbaaSqaaiaadMga caWGQbaabeaajaaWcaaMc8UaamizaiaadAfacqGHRaWkkmaapefaja aWbaGaeqyWdiNcdaWcaaqaaiaadsgacaWG2bWaaSbaaSqaaiaadMga aeqaaaGcbaGaamizaiaadshaaaqcaaSaeqiTdqMaamODaOWaaSbaaS qaaiaadMgaaeqaaaqcbaCaaiaadAfaaeqajmaWcqGHRiI8aOGaamiz aiaadAfajaaWcqGHsislkmaapefajaaWbaGaeqyWdiNaamOyaOWaaS baaSqaaiaadMgaaeqaaKaaalabes7aKjaadAhakmaaBaaaleaacaWG PbaabeaajaaWcaWGKbGaamOvaiabgkHiTOWaa8quaKaaahaacaWG0b GcdaWgaaWcbaGaamyAaaqabaqcaaSaeqiTdqMaamODaOWaaSbaaSqa aiaadMgaaeqaaaqcbaCaaiaadofalmaaBaaajiaWbaGaaGOmaaqaba aajeaWbeqcdaSaey4kIipaaKqaahaacaWGwbaabeqcdaSaey4kIipa aSqaaiaadAfaaeqaniabgUIiYdGccaWGKbGaamyqaiabg2da9iaaic daaaa@79B4@

for all virtual velocity fields δv(y) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaaCODaiaacIcacaWH5bGaai ykaaaa@35DC@ .  Then the stress field must satisfy

3.      The boundary condition n i σ ij = t j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO Gaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iaadsha daWgaaWcbaGaamOAaaqabaaaaa@3BCD@  on S 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaikdaaeqaaa aa@3486@

4.      The linear momentum balance equation

σ ji y j +ρ b i =ρ d v i dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGa amOAaaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaaleaacaWGPb aabeaakiabg2da9iabeg8aYnaalaaabaGaamizaiaadAhadaWgaaWc baGaamyAaaqabaaakeaacaWGKbGaamiDaaaaaaa@45B9@

The significance of this result is that it gives us an alternative way to solve for a stress field that satisfies the linear momentum balance equation, which avoids having to differentiate the stress.  It is not easy to differentiate functions accurately in the computer, but it is easy to integrate them.  The virtual work statement is the starting point for any finite element solution involving deformable solids.

 

 

 

 

Proof: Follow the same preliminary steps as before, i..e.

σ ij δ D ij = 1 2 σ ij ( δ v i y j + δ v j y i )= 1 2 ( σ ji δ v i y j + σ ij δ v j y i )= σ ji δ v i y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabes7aKjaadseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa eyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacqaHdpWCdaWgaaWcba GaamyAaiaadQgaaeqaaOWaaeWaaeaadaWcaaqaaiabgkGi2kabes7a KjaadAhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaS baaSqaaiaadQgaaeqaaaaakiabgUcaRmaalaaabaGaeyOaIyRaeqiT dqMaamODamaaBaaaleaacaWGQbaabeaaaOqaaiabgkGi2kaadMhada WgaaWcbaGaamyAaaqabaaaaaGccaGLOaGaayzkaaGaeyypa0ZaaSaa aeaacaaIXaaabaGaaGOmaaaadaqadaqaaiabeo8aZnaaBaaaleaaca WGQbGaamyAaaqabaGcdaWcaaqaaiabgkGi2kabes7aKjaadAhadaWg aaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadQ gaaeqaaaaakiabgUcaRiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqa baGcdaWcaaqaaiabgkGi2kabes7aKjaadAhadaWgaaWcbaGaamOAaa qabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaaaOGa ayjkaiaawMcaaiabg2da9iabeo8aZnaaBaaaleaacaWGQbGaamyAaa qabaGcdaWcaaqaaiabgkGi2kabes7aKjaadAhadaWgaaWcbaGaamyA aaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaaaaaa a@7F39@

σ ji v i y j = y j ( σ ji δ v i ) σ ji y j δ v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadQgacaWGPb aabeaakmaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGPbaabeaa aOqaaiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaaaOGaeyypa0 ZaaSaaaeaacqGHciITaeaacqGHciITcaWG5bWaaSbaaSqaaiaadQga aeqaaaaakmaabmaabaGaeq4Wdm3aaSbaaSqaaiaadQgacaWGPbaabe aakiabes7aKjaadAhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGL PaaacqGHsisldaWcaaqaaiabgkGi2kabeo8aZnaaBaaaleaacaWGQb GaamyAaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqa aaaakiabes7aKjaadAhadaWgaaWcbaGaamyAaaqabaaaaa@5867@

and substitute into the virtual work equation

V { y j ( σ ji δ v i ) σ ji y j δ v i }dV+ V ρ d v i dt δ v i dV V ρ b i δ v i dV S 2 t i δ v i dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaKaaahaakmaacmaabaWaaSaaae aacqGHciITaeaacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaaaa kmaabmaabaGaeq4Wdm3aaSbaaSqaaiaadQgacaWGPbaabeaakiabes 7aKjaadAhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaacqGH sisldaWcaaqaaiabgkGi2kabeo8aZnaaBaaaleaacaWGQbGaamyAaa qabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaaaakiab es7aKjaadAhadaWgaaWcbaGaamyAaaqabaGccaaMc8oacaGL7bGaay zFaaqcaaSaamizaiaadAfacqGHRaWkkmaapefajaaWbaGaeqyWdiNc daWcaaqaaiaadsgacaWG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaam izaiaadshaaaqcaaSaeqiTdqMaamODaOWaaSbaaSqaaiaadMgaaeqa aaqcbaCaaiaadAfaaeqajmaWcqGHRiI8aOGaamizaiaadAfajaaWcq GHsislkmaapefajaaWbaGaeqyWdiNaamOyaOWaaSbaaSqaaiaadMga aeqaaKaaalabes7aKjaadAhakmaaBaaaleaacaWGPbaabeaajaaWca WGKbGaamOvaiabgkHiTOWaa8quaKaaahaacaWG0bGcdaWgaaWcbaGa amyAaaqabaqcaaSaeqiTdqMaamODaOWaaSbaaSqaaiaadMgaaeqaaa qcbaCaaiaadofalmaaBaaajiaWbaGaaGOmaaqabaaajeaWbeqcdaSa ey4kIipaaKqaahaacaWGwbaabeqcdaSaey4kIipaaSqaaiaadAfaae qaniabgUIiYdGccaWGKbGaamyqaiabg2da9iaaicdaaaa@8F40@

Apply the divergence theorem to the first term in the first integral, and recall that δv=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaaCODaiabg2da9iaaicdaaa a@3531@  on S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4uamaaBaaaleaacaaIXaaabeaaaa a@328C@ , we see that

V { σ ji y j +ρ b i ρ d v i dt }δ v i dV + S 2 ( σ ji n j t i )δ v i dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOeI0Yaa8quaKaaahaakmaacmaaba WaaSaaaeaacqGHciITcqaHdpWCdaWgaaWcbaGaamOAaiaadMgaaeqa aaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGQbaabeaaaaGccqGHRa WkcqaHbpGCcaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaeqyW di3aaSaaaeaacaWGKbGaamODamaaBaaaleaacaWGPbaabeaaaOqaai aadsgacaWG0baaaiaaykW7aiaawUhacaGL9baajaaWcqaH0oazcaWG 2bGcdaWgaaqcbaCaaiaadMgaaeqaaKaaalaadsgacaWGwbaaleaaca WGwbaabeqdcqGHRiI8aOGaey4kaSYaa8quaeaadaqadaqaaiabeo8a ZnaaBaaaleaacaWGQbGaamyAaaqabaGccaWGUbWaaSbaaSqaaiaadQ gaaeqaaOGaeyOeI0IaamiDamaaBaaaleaacaWGPbaabeaaaOGaayjk aiaawMcaaiabes7aKjaadAhadaWgaaWcbaGaamyAaaqabaaabaGaam 4uamaaBaaameaacaaIYaaabeaaaSqab0Gaey4kIipakiaadsgacaWG bbGaeyypa0JaaGimaaaa@6D1D@

Since this must hold for all virtual velocity fields we could choose

δ v i =f(y){ σ ji y j +ρ b i ρ d v i dt } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaamODamaaBaaaleaacaWGPb aabeaakiabg2da9iaadAgacaGGOaGaaCyEaiaacMcadaGadaqaamaa laaabaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaadQgacaWGPbaabeaaaO qaaiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaaaOGaey4kaSIa eqyWdiNaamOyamaaBaaaleaacaWGPbaabeaakiabgkHiTiabeg8aYn aalaaabaGaamizaiaadAhadaWgaaWcbaGaamyAaaqabaaakeaacaWG KbGaamiDaaaacaaMc8oacaGL7bGaayzFaaaaaa@51C4@

where f(y)=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOzaiaacIcacaWH5bGaaiykaiabg2 da9iaaicdaaaa@35D3@  is an arbitrary function that is positive everywhere inside the solid, but is equal to zero on S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4uaaaa@31B5@ .  For this choice, the virtual work equation reduces to

V f( y ){ σ ji y j +ρ b i ρ d v i dt }{ σ ki y k +ρ b i ρ d v i dt }dV =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOeI0Yaa8quaKaaahaakiaadAgada qadaqaaiaahMhaaiaawIcacaGLPaaadaGadaqaamaalaaabaGaeyOa IyRaeq4Wdm3aaSbaaSqaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2k aadMhadaWgaaWcbaGaamOAaaqabaaaaOGaey4kaSIaeqyWdiNaamOy amaaBaaaleaacaWGPbaabeaakiabgkHiTiabeg8aYnaalaaabaGaam izaiaadAhadaWgaaWcbaGaamyAaaqabaaakeaacaWGKbGaamiDaaaa caaMc8oacaGL7bGaayzFaaWaaiWaaeaadaWcaaqaaiabgkGi2kabeo 8aZnaaBaaaleaacaWGRbGaamyAaaqabaaakeaacqGHciITcaWG5bWa aSbaaSqaaiaadUgaaeqaaaaakiabgUcaRiabeg8aYjaadkgadaWgaa WcbaGaamyAaaqabaGccqGHsislcqaHbpGCdaWcaaqaaiaadsgacaWG 2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaamizaiaadshaaaGaaGPaVd Gaay5Eaiaaw2haaKaaalaadsgacaWGwbaaleaacaWGwbaabeqdcqGH RiI8aOGaeyypa0JaaGimaaaa@6F4D@

and since the integrand is positive everywhere the only way the equation can be satisfied is if

σ ji y j +ρ b i =ρ d v i dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGa amOAaaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaaleaacaWGPb aabeaakiabg2da9iabeg8aYnaalaaabaGaamizaiaadAhadaWgaaWc baGaamyAaaqabaaakeaacaWGKbGaamiDaaaaaaa@45B9@

Given this, we can next choose a virtual velocity field that satisfies

δ v i =( σ ji n j t i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiTdqMaamODamaaBaaaleaacaWGPb aabeaakiabg2da9maabmaabaGaeq4Wdm3aaSbaaSqaaiaadQgacaWG Pbaabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGccqGHsislcaWG0b WaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaaaa@401A@

on S 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGOmaaqabaaaaa@3238@ .  For this choice (and noting that the volume integral is zero) the virtual work equation reduces to

+ S 2 ( σ ji n j t i )( σ ki n k t i ) dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaey4kaSYaa8quaeaadaqadaqaaiabeo 8aZnaaBaaaleaacaWGQbGaamyAaaqabaGccaWGUbWaaSbaaSqaaiaa dQgaaeqaaOGaeyOeI0IaamiDamaaBaaaleaacaWGPbaabeaaaOGaay jkaiaawMcaamaabmaabaGaeq4Wdm3aaSbaaSqaaiaadUgacaWGPbaa beaakiaad6gadaWgaaWcbaGaam4AaaqabaGccqGHsislcaWG0bWaaS baaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaaleaacaWGtbWaaSba aWqaaiaaikdaaeqaaaWcbeqdcqGHRiI8aOGaamizaiaadgeacqGH9a qpcaaIWaaaaa@4E44@

Again, the integrand is positive everywhere (it is a perfect square) and so can vanish only if

σ ji n j = t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadQgacaWGPb aabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaWG0bWa aSbaaSqaaiaadMgaaeqaaaaa@39D6@

as stated.

 

 

5.9 The Virtual Work equation in terms of other stress measures.

 

It is often convenient to implement the virtual work equation in a finite element code using different stress measures. 

 

To do so, we define

1.      The actual deformation gradient in the solid F ij = δ ij + u i x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiab gUcaRmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaO qaaiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaaaa@3FF8@

2.      The virtual rate of change of deformation gradient  δ F ˙ ij = δ v i y k F kj = δ v i x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjqadAeagaGaamaaBaaaleaaca WGPbGaamOAaaqabaGccqGH9aqpdaWcaaqaaiabgkGi2kabes7aKjaa dAhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaSbaaS qaaiaadUgaaeqaaaaakiaadAeadaWgaaWcbaGaam4AaiaadQgaaeqa aOGaeyypa0ZaaSaaaeaacqGHciITcqaH0oazcaWG2bWaaSbaaSqaai aadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaa aaaaaa@4B5C@

3.      The virtual rate of change of Lagrange strain δ E ˙ ij = 1 2 ( F ki δ F ˙ kj +δ F ˙ ki F kj ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjqadweagaGaamaaBaaaleaaca WGPbGaamOAaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaa amaabmaabaGaamOramaaBaaaleaacaWGRbGaamyAaaqabaGccqaH0o azceWGgbGbaiaadaWgaaWcbaGaam4AaiaadQgaaeqaaOGaey4kaSIa eqiTdqMabmOrayaacaWaaSbaaSqaaiaadUgacaWGPbaabeaakiaadA eadaWgaaWcbaGaam4AaiaadQgaaeqaaaGccaGLOaGaayzkaaaaaa@48C5@

In addition, we define (in the usual way)

1.      Kirchhoff stress  τ ij =J σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaaGPaVlabes8a0PWaaSbaaSqaai aadMgacaWGQbaabeaakiabg2da9iaadQeacqaHdpWCdaWgaaWcbaGa amyAaiaadQgaaeqaaaaa@3CA6@

2.      Nominal (First Piola-Kirchhoff) stress   S ij =J F ik 1 σ kj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaam4uaOWaaSbaaSqaaiaadMgaca WGQbaabeaakiabg2da9iaadQeacaWGgbWaa0baaSqaaiaadMgacaWG RbaabaGaeyOeI0IaaGymaaaakiabeo8aZnaaBaaaleaacaWGRbGaam OAaaqabaaaaa@3EB7@

3.      Material (Second Piola-Kirchhoff) stress   Σ ij =J F ik 1 σ kl F jl 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaGPaVlabfo6atnaaBaaaleaacaWGPb GaamOAaaqabaGccqGH9aqpcaWGkbGaamOramaaDaaaleaacaWGPbGa am4AaaqaaiabgkHiTiaaigdaaaGccqaHdpWCdaWgaaWcbaGaam4Aai aadYgaaeqaaOGaamOramaaDaaaleaacaWGQbGaamiBaaqaaiabgkHi Tiaaigdaaaaaaa@44A7@

 

In terms of these quantities, the virtual work equation may be expressed as

V0 τ ij δ D ij d V 0 + V 0 ρ 0 d v i dt δ v i d V 0 V0 ρ 0 b i δ v i d V 0 S 2 t i δ v i dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaKaaahaacqaHepaDkmaaBaaale aacaWGPbGaamOAaaqabaGccqaH0oazcaWGebWaaSbaaSqaaiaadMga caWGQbaabeaajaaWcaaMc8UaamizaiaadAfakmaaBaaaleaacaaIWa aabeaajaaWcqGHRaWkkmaapefajaaWbaGaeqyWdiNcdaWgaaWcbaGa aGimaaqabaGcdaWcaaqaaiaadsgacaWG2bWaaSbaaSqaaiaadMgaae qaaaGcbaGaamizaiaadshaaaqcaaSaeqiTdqMaamODaOWaaSbaaSqa aiaadMgaaeqaaaqcbaCaaiaadAfalmaaBaaameaacaaIWaaabeaaaK qaahqajmaWcqGHRiI8aOGaamizaiaadAfadaWgaaWcbaGaaGimaaqa baqcaaSaeyOeI0IcdaWdrbqcaaCaaiabeg8aYPWaaSbaaSqaaiaaic daaeqaaKaaalaadkgakmaaBaaaleaacaWGPbaabeaajaaWcqaH0oaz caWG2bGcdaWgaaWcbaGaamyAaaqabaqcaaSaamizaiaadAfakmaaBa aaleaacaaIWaaabeaajaaWcqGHsislkmaapefajaaWbaGaamiDaOWa aSbaaSqaaiaadMgaaeqaaKaaalabes7aKjaadAhakmaaBaaaleaaca WGPbaabeaaaKqaahaacaWGtbWcdaWgaaqccaCaaiaaikdaaeqaaaqc baCabKWaalabgUIiYdaajeaWbaGaamOvaiaaicdaaeqajmaWcqGHRi I8aaWcbaGaamOvaiaaicdaaeqaniabgUIiYdGccaWGKbGaamyqaiab g2da9iaaicdaaaa@83E7@

V0 S ij δ F ˙ ji d V 0 + V 0 ρ 0 d v i dt δ v i d V 0 V0 ρ 0 b i δ v i d V 0 S 2 t i δ v i dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaKaaahaacaWGtbGcdaWgaaWcba GaamyAaiaadQgaaeqaaOGaeqiTdqMabmOrayaacaWaaSbaaSqaaiaa dQgacaWGPbaabeaajaaWcaaMc8UaamizaiaadAfakmaaBaaaleaaca aIWaaabeaajaaWcqGHRaWkkmaapefajaaWbaGaeqyWdiNcdaWgaaWc baGaaGimaaqabaGcdaWcaaqaaiaadsgacaWG2bWaaSbaaSqaaiaadM gaaeqaaaGcbaGaamizaiaadshaaaqcaaSaeqiTdqMaamODaOWaaSba aSqaaiaadMgaaeqaaaqcbaCaaiaadAfalmaaBaaameaacaaIWaaabe aaaKqaahqajmaWcqGHRiI8aOGaamizaiaadAfadaWgaaWcbaGaaGim aaqabaqcaaSaeyOeI0IcdaWdrbqcaaCaaiabeg8aYPWaaSbaaSqaai aaicdaaeqaaKaaalaadkgakmaaBaaaleaacaWGPbaabeaajaaWcqaH 0oazcaWG2bGcdaWgaaWcbaGaamyAaaqabaqcaaSaamizaiaadAfakm aaBaaaleaacaaIWaaabeaajaaWcqGHsislkmaapefajaaWbaGaamiD aOWaaSbaaSqaaiaadMgaaeqaaKaaalabes7aKjaadAhakmaaBaaale aacaWGPbaabeaaaKqaahaacaWGtbWcdaWgaaqccaCaaiaaikdaaeqa aaqcbaCabKWaalabgUIiYdaajeaWbaGaamOvaiaaicdaaeqajmaWcq GHRiI8aaWcbaGaamOvaiaaicdaaeqaniabgUIiYdGccaWGKbGaamyq aiabg2da9iaaicdaaaa@8305@

V0 Σ ij δ E ˙ ij d V 0 + V 0 ρ 0 d v i dt δ v i d V 0 V0 ρ 0 b i δ v i d V 0 S 2 t i δ v i dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaKaaahaacqqHJoWukmaaBaaale aacaWGPbGaamOAaaqabaGccqaH0oazceWGfbGbaiaadaWgaaWcbaGa amyAaiaadQgaaeqaaKaaalaaykW7caWGKbGaamOvaOWaaSbaaSqaai aaicdaaeqaaKaaalabgUcaROWaa8quaKaaahaacqaHbpGCkmaaBaaa leaacaaIWaaabeaakmaalaaabaGaamizaiaadAhadaWgaaWcbaGaam yAaaqabaaakeaacaWGKbGaamiDaaaajaaWcqaH0oazcaWG2bGcdaWg aaWcbaGaamyAaaqabaaajeaWbaGaamOvaSWaaSbaaWqaaiaaicdaae qaaaqcbaCabKWaalabgUIiYdGccaWGKbGaamOvamaaBaaaleaacaaI WaaabeaajaaWcqGHsislkmaapefajaaWbaGaeqyWdiNcdaWgaaWcba GaaGimaaqabaqcaaSaamOyaOWaaSbaaSqaaiaadMgaaeqaaKaaalab es7aKjaadAhakmaaBaaaleaacaWGPbaabeaajaaWcaWGKbGaamOvaO WaaSbaaSqaaiaaicdaaeqaaKaaalabgkHiTOWaa8quaKaaahaacaWG 0bGcdaWgaaWcbaGaamyAaaqabaqcaaSaeqiTdqMaamODaOWaaSbaaS qaaiaadMgaaeqaaaqcbaCaaiaadofalmaaBaaajiaWbaGaaGOmaaqa baaajeaWbeqcdaSaey4kIipaaKqaahaacaWGwbGaaGimaaqabKWaal abgUIiYdaaleaacaWGwbGaaGimaaqab0Gaey4kIipakiaadsgacaWG bbGaeyypa0JaaGimaaaa@83B0@

Note that all the volume integrals are now taken over the undeformed solid MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  this is convenient for computer applications, because the shape of the undeformed solid is known.  The area integral is evaluated over the deformed solid, unfortunately.  It can be expressed as an equivalent integral over the undeformed solid, but the result is messy and will be deferred until we actually need to do it.

 

5.10 The Virtual Work equation for infinitesimal deformations.

 

For infintesimal motions, the Cauchy, Nominal, and Material stress tensors are equal; and the virtual stretch rate can be replaced by the virtual infinitesimal strain rate

δ ε ˙ ij = 1 2 ( δ v i x j + δ v j x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcuaH1oqzgaGaamaaBaaale aacaWGPbGaamOAaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaI YaaaamaabmaabaWaaSaaaeaacqGHciITcqaH0oazcaWG2bWaaSbaaS qaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaa beaaaaGccqGHRaWkdaWcaaqaaiabgkGi2kabes7aKjaadAhadaWgaa WcbaGaamOAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadMga aeqaaaaaaOGaayjkaiaawMcaaaaa@4EAA@

There is no need to distinguish between the volume or surface area of the deformed and undeformed solid.  The virtual work equation can thus be expressed as

V0 σ ij δ ε ˙ ij d V 0 + V 0 ρ 0 d v i dt δ v i d V 0 V0 ρ 0 b i δ v i d V 0 S 2 t i δ v i d A 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaKaaahaacqaHdpWCkmaaBaaale aacaWGPbGaamOAaaqabaGccqaH0oazcuaH1oqzgaGaamaaBaaaleaa caWGPbGaamOAaaqabaqcaaSaaGPaVlaadsgacaWGwbGcdaWgaaWcba GaaGimaaqabaqcaaSaey4kaSIcdaWdrbqcaaCaaiabeg8aYPWaaSba aSqaaiaaicdaaeqaaOWaaSaaaeaacaWGKbGaamODamaaBaaaleaaca WGPbaabeaaaOqaaiaadsgacaWG0baaaKaaalabes7aKjaadAhakmaa BaaaleaacaWGPbaabeaaaKqaahaacaWGwbWcdaWgaaadbaGaaGimaa qabaaajeaWbeqcdaSaey4kIipakiaadsgacaWGwbWaaSbaaSqaaiaa icdaaeqaaKaaalabgkHiTOWaa8quaKaaahaacqaHbpGCkmaaBaaale aacaaIWaaabeaajaaWcaWGIbGcdaWgaaWcbaGaamyAaaqabaqcaaSa eqiTdqMaamODaOWaaSbaaSqaaiaadMgaaeqaaKaaalaadsgacaWGwb GcdaWgaaWcbaGaaGimaaqabaqcaaSaeyOeI0IcdaWdrbqcaaCaaiaa dshakmaaBaaaleaacaWGPbaabeaajaaWcqaH0oazcaWG2bGcdaWgaa WcbaGaamyAaaqabaaajeaWbaGaam4uaSWaaSbaaKGaahaacaaIYaaa beaaaKqaahqajmaWcqGHRiI8aaqcbaCaaiaadAfacaaIWaaabeqcda Saey4kIipaaSqaaiaadAfacaaIWaaabeqdcqGHRiI8aOGaamizaiaa dgeadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIWaaaaa@85BC@

for all kinematically admissible velocity fields.

 

 

As a special case, this expression can be applied to a quasi-static state with v i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaaIWaaaaa@3447@ . Then, for a stress state σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@  satisfying the static equilibrium equation σ ij /d x i + ρ 0 b j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaGGVaGaamizaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGH RaWkcqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGIbWaaSbaaSqaai aadQgaaeqaaOGaeyypa0JaaGimaaaa@3F59@  and boundary conditions σ ij n j = t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaWGUbWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaamiDamaa BaaaleaacaWGPbaabeaaaaa@396F@  on S 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGOmaaqabaaaaa@3228@ , the virtual work equation reduces to

V0 σ ij δ ε ij d V 0 = V0 ρ 0 b i δ u i d V 0 + S 2 t i δ u i dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaKaaahaacqaHdpWCkmaaBaaale aacaWGPbGaamOAaaqabaGccqaH0oazcqaH1oqzdaWgaaWcbaGaamyA aiaadQgaaeqaaKaaalaaykW7caWGKbGaamOvaOWaaSbaaSqaaiaaic daaeqaaaqaaiaadAfacaaIWaaabeqdcqGHRiI8aOGaeyypa0Zaa8qu aeaacqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGIbWaaSbaaSqaai aadMgaaeqaaOGaeqiTdqMaamyDamaaBaaaleaacaWGPbaabeaakiaa dsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSYaa8quaeaaca WG0bWaaSbaaSqaaiaadMgaaeqaaOGaeqiTdqMaamyDamaaBaaaleaa caWGPbaabeaakiaadsgacaWGbbaaleaacaWGtbWaaSbaaWqaaiaaik daaeqaaaWcbeqdcqGHRiI8aaWcbaGaamOvaiaaicdaaeqaniabgUIi Ydaaaa@6051@

In which δ u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjaadwhadaWgaaWcbaGaamyAaa qabaaaaa@3421@  are kinematically admissible displacements components (δ u i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacqaH0oazcaWG1bWaaSbaaSqaai aadMgaaeqaaOGaeyypa0JaaGimaaaa@3697@  on S2) and δ ε ij =( δ u i / x j +δ u j / x i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcqaH1oqzdaWgaaWcbaGaam yAaiaadQgaaeqaaOGaeyypa0ZaaeWaaeaacqGHciITcqaH0oazcaWG 1bWaaSbaaSqaaiaadMgaaeqaaOGaai4laiaadIhadaWgaaWcbaGaam OAaaqabaGccqGHRaWkcqGHciITcqaH0oazcaWG1bWaaSbaaSqaaiaa dQgaaeqaaOGaai4laiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawI cacaGLPaaacaGGVaGaaGOmaaaa@4D01@ .

 

Conversely, if  the stress state σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@  satisfies V0 σ ij δ ε ij d V 0 = V0 ρ 0 b i δ u i d V 0 + S 2 t i δ u i dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaa8quaKaaahaacqaHdpWCkmaaBaaale aacaWGPbGaamOAaaqabaGccqaH0oazcqaH1oqzdaWgaaWcbaGaamyA aiaadQgaaeqaaKaaalaaykW7caWGKbGaamOvaOWaaSbaaSqaaiaaic daaeqaaaqaaiaadAfacaaIWaaabeqdcqGHRiI8aOGaeyypa0Zaa8qu aeaacqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGIbWaaSbaaSqaai aadMgaaeqaaOGaeqiTdqMaamyDamaaBaaaleaacaWGPbaabeaakiaa dsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSYaa8quaeaaca WG0bWaaSbaaSqaaiaadMgaaeqaaOGaeqiTdqMaamyDamaaBaaaleaa caWGPbaabeaakiaadsgacaWGbbaaleaacaWGtbWaaSbaaWqaaiaaik daaeqaaaWcbeqdcqGHRiI8aaWcbaGaamOvaiaaicdaaeqaniabgUIi Ydaaaa@6051@  for every set of kinematically admissible virtual displacements, then the stress state σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@  satisfies the static equilibrium equation σ ij /d x i + ρ 0 b j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaGGVaGaamizaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGH RaWkcqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGIbWaaSbaaSqaai aadQgaaeqaaOGaeyypa0JaaGimaaaa@3F59@  and boundary conditions σ ij n j = t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaWGUbWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaamiDamaa BaaaleaacaWGPbaabeaaaaa@396F@  on S 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGOmaaqabaaaaa@3228@ .

 

 

5.10 The first and second laws of thermodynamics for continua 

 

 

Consider a sub-region V of a deformed solid with surface A.   Define:

* The heat flux vector q , which is defined so that dQ=qndA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGrbGaeyypa0JaaCyCaiabgw Sixlaah6gacaWGKbGaamyqaaaa@3917@  is the heat flux crossing an internal surface with area dA and normal n in the deformed solid;

* The heat supply q , defined so that dQ= qdV is the heat supplied from an external source into a volume element dV in the deformed solid;

* The net heat flux into the solid Q= V qdV A qndA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgfacqGH9aqpdaWdrbqaaiaadghaca WGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabgkHiTmaapefa baGaaCyCaiabgwSixlaah6gacaWGKbGaamyqaaWcbaGaamyqaaqab0 Gaey4kIipaaaa@4214@

*  The net rate of mechanical work done on the solid W= V bv dV+ A nσ vdA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEfacqGH9aqpdaWdrbqaaiaahkgacq GHflY1caWH2baaleaacaWGwbaabeqdcqGHRiI8aOGaamizaiaadAfa cqGHRaWkdaWdrbqaaiaah6gacqGHflY1caWHdpaaleaacaWGbbaabe qdcqGHRiI8aOGaeyyXICTaaCODaiaadsgacaWGbbaaaa@48F5@

*  The total kinetic energy KE= V 1 2 ρ( v i v i )dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeacaWGfbGaeyypa0Zaa8quaeaada WcaaqaaiaaigdaaeaacaaIYaaaaiabeg8aYjaacIcacaWG2bWaaSba aSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPbaabeaakiaacM cacaWGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIipaaaa@40CE@

*  The total internal energy Ε= V ρεdV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfw5afjabg2da9maapefabaGaeqyWdi NaeqyTduMaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIiYdaaaa@3B26@  where ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLbaa@320F@  is the specific internal energy (internal energy per unit mass)

*  The total entropy S= V ρsdV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofacqGH9aqpdaWdrbqaaiabeg8aYj aadohacaWGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIipaaaa@39E7@ , where s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohaaaa@3160@  is the specific entropy (entropy per unit mass)

*  The temperature of the solid θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@321E@ .

*  The net external entropy supplied to the volume dΗ dt = A q θ ndA + V q θ dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiabfE5aibqaaiaads gacaWG0baaaiabg2da9maapefabaGaeyOeI0YaaSaaaeaacaWHXbaa baGaeqiUdehaaiabgwSixlaah6gacaWGKbGaamyqaaWcbaGaamyqaa qab0Gaey4kIipakiabgUcaRmaapefabaWaaSaaaeaacaWGXbaabaGa eqiUdehaaiaadsgacaWGwbaaleaacaWGwbaabeqdcqGHRiI8aaaa@49F4@

*  The specific free energy ψ=εθs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI8a5jabg2da9iabew7aLjabgkHiTi abeI7aXjaadohaaaa@387E@

*  The total free energy Ψ= V ρψdV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfI6azjabg2da9maapefabaGaeqyWdi NaeqiYdKNaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIiYdaaaa@3B74@

 

The first law of thermodynamics then requires that

d dt (Ε+KE)=Q+W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaaiaacIcacqqHvoqrcqGHRaWkcaWGlbGaamyraiaacMcacqGH9aqp caWGrbGaey4kaSIaam4vaaaa@3C1A@

for any volume V.  

 

This condition can also be expressed as

ρ ε t | x=const = σ ij D ij q i y i +q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaeiaabaWaaSaaaeaacqGHci ITcqaH1oqzaeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaa hIhacqGH9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccq GH9aqpcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamiramaa BaaaleaacaWGPbGaamOAaaqabaGccqGHsisldaWcaaqaaiabgkGi2k aadghadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaSba aSqaaiaadMgaaeqaaaaakiabgUcaRiaadghaaaa@51CC@

 

To see this,

  1. recall that

W= V b i v i dV+ A σ ij n i v j dA= V σ ij D ij dV + d dt { V 1 2 ρ v i v i dV }= V σ ij D ij dV + d dt (KE) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEfacqGH9aqpdaWdrbqaaiaadkgada WgaaWcbaGaamyAaaqabaGccaWG2bWaaSbaaSqaaiaadMgaaeqaaaqa aiaadAfaaeqaniabgUIiYdGccaWGKbGaamOvaiabgUcaRmaapefaba Gaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaad6gadaWgaaWc baGaamyAaaqabaGccaWG2bWaaSbaaSqaaiaadQgaaeqaaaqaaiaadg eaaeqaniabgUIiYdGccaWGKbGaamyqaiabg2da9maapefabaGaeq4W dm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaadseadaWgaaWcbaGaam yAaiaadQgaaeqaaOGaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIi YdGccqGHRaWkdaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaadaGada qaamaapefabaWaaSaaaeaacaaIXaaabaGaaGOmaaaacqaHbpGCcaWG 2bWaaSbaaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPbaabe aakiaadsgacaWGwbaaleaacaWGwbaabeqdcqGHRiI8aaGccaGL7bGa ayzFaaGaeyypa0Zaa8quaeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaamiramaaBaaaleaacaWGPbGaamOAaaqabaGccaWGKbGa amOvaaWcbaGaamOvaaqab0Gaey4kIipakiabgUcaRmaalaaabaGaam izaaqaaiaadsgacaWG0baaaiaacIcacaWGlbGaamyraiaacMcaaaa@7B60@

  1. the divergence theorem gives

Q= V qdV A qndA = V ( q q i y i )dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgfacqGH9aqpdaWdrbqaaiaadghaca WGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIipakiabgkHiTmaapefa baGaaCyCaiabgwSixlaah6gacaWGKbGaamyqaaWcbaGaamyqaaqab0 Gaey4kIipakiabg2da9maapefabaWaaeWaaeaacaWGXbGaeyOeI0Ya aSaaaeaacqGHciITcaWGXbWaaSbaaSqaaiaadMgaaeqaaaGcbaGaey OaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaa caWGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIipaaaa@5291@

  1. Therefore

d dt ( V ρεdV +KE )= V ( q q i y i )dV + V σ ij D ij dV + d dt (KE) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaabmaabaWaa8quaeaacqaHbpGCcqaH1oqzcaWGKbGaamOvaaWc baGaamOvaaqab0Gaey4kIipakiabgUcaRiaadUeacaWGfbaacaGLOa GaayzkaaGaeyypa0Zaa8quaeaadaqadaqaaiaadghacqGHsisldaWc aaqaaiabgkGi2kaadghadaWgaaWcbaGaamyAaaqabaaakeaacqGHci ITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaaaOGaayjkaiaawMcaaiaa dsgacaWGwbaaleaacaWGwbaabeqdcqGHRiI8aOGaey4kaSYaa8quae aacqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamiramaaBaaa leaacaWGPbGaamOAaaqabaGccaWGKbGaamOvaaWcbaGaamOvaaqab0 Gaey4kIipakiabgUcaRmaalaaabaGaamizaaqaaiaadsgacaWG0baa aiaacIcacaWGlbGaamyraiaacMcaaaa@6356@

  1. Note also that

d dt V ρεdV = d dt V 0 ρ 0 εdV = V 0 ρ 0 dε dt dV = V ρ dε dt dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqyWdiNaeqyTduMaamizaiaadAfaaSqaaiaadAfa aeqaniabgUIiYdGccqGH9aqpdaWcaaqaaiaadsgaaeaacaWGKbGaam iDaaaadaWdrbqaaiabeg8aYnaaBaaaleaacaaIWaaabeaakiabew7a LjaadsgacaWGwbaaleaacaWGwbWaaSbaaWqaaiaaicdaaeqaaaWcbe qdcqGHRiI8aOGaeyypa0Zaa8quaeaacqaHbpGCdaWgaaWcbaGaaGim aaqabaGcdaWcaaqaaiaadsgacqaH1oqzaeaacaWGKbGaamiDaaaaca WGKbGaamOvaaWcbaGaamOvamaaBaaameaacaaIWaaabeaaaSqab0Ga ey4kIipakiabg2da9maapefabaGaeqyWdi3aaSaaaeaacaWGKbGaeq yTdugabaGaamizaiaadshaaaGaamizaiaadAfaaSqaaiaadAfaaeqa niabgUIiYdaaaa@6408@

where ρ 0 =Jρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaki abg2da9iaadQeacqaHbpGCaaa@36AD@  is the mass density per unit reference volume.  Finally

V ρ dε dt dV = V ( q q i y i )dV + V σ ij D ij dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaeqyWdi3aaSaaaeaacaWGKb GaeqyTdugabaGaamizaiaadshaaaGaamizaiaadAfaaSqaaiaadAfa aeqaniabgUIiYdGccqGH9aqpdaWdrbqaamaabmaabaGaamyCaiabgk HiTmaalaaabaGaeyOaIyRaamyCamaaBaaaleaacaWGPbaabeaaaOqa aiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaaGccaGLOaGaay zkaaGaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIiYdGccqGHRaWk daWdrbqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccaWGeb WaaSbaaSqaaiaadMgacaWGQbaabeaakiaadsgacaWGwbaaleaacaWG wbaabeqdcqGHRiI8aaaa@5897@

This must hold for all V, giving the required result.

 

 

The second law of thermodynamics specifies that the net entropy production within V must be non-negative, i.e.

dS dt dΗ dt 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadofaaeaacaWGKb GaamiDaaaacqGHsisldaWcaaqaaiaadsgacqqHxoasaeaacaWGKbGa amiDaaaacqGHLjYScaaIWaaaaa@3BD0@

This can also be expressed as

ρ s t + ( q i /θ) y i q θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaalaaabaGaeyOaIyRaam4Caa qaaiabgkGi2kaadshaaaGaey4kaSYaaSaaaeaacqGHciITcaGGOaGa amyCamaaBaaaleaacaWGPbaabeaakiaac+cacqaH4oqCcaGGPaaaba GaeyOaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGccqGHsisldaWc aaqaaiaadghaaeaacqaH4oqCaaGaeyyzImRaaGimaaaa@48DA@

(this condition is know as the Clausius-Duhem inequality).

 

To see this, simply substitute the definitions and use the divergence theorem.

 

 

The first and second laws can be combined to yield the free energy imbalance

 

σ ij D ij 1 θ q i θ y i ρ( ψ t +s θ t )0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTmaa laaabaGaaGymaaqaaiabeI7aXbaacaWGXbWaaSbaaSqaaiaadMgaae qaaOWaaSaaaeaacqGHciITcqaH4oqCaeaacqGHciITcaWG5bWaaSba aSqaaiaadMgaaeqaaaaakiabgkHiTiabeg8aYnaabmaabaWaaSaaae aacqGHciITcqaHipqEaeaacqGHciITcaWG0baaaiabgUcaRiaadoha daWcaaqaaiabgkGi2kabeI7aXbqaaiabgkGi2kaadshaaaaacaGLOa GaayzkaaGaeyyzImRaaGimaaaa@5714@

 

This can also be expressed as

W d(KE) dt dΨ dt V ( ρs θ t + 1 θ q i θ y i )dV 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEfacqGHsisldaWcaaqaaiaadsgaca GGOaGaam4saiaadweacaGGPaaabaGaamizaiaadshaaaGaeyOeI0Ya aSaaaeaacaWGKbGaeuiQdKfabaGaamizaiaadshaaaGaeyOeI0Yaa8 quaeaadaqadaqaaiabeg8aYjaadohadaWcaaqaaiabgkGi2kabeI7a XbqaaiabgkGi2kaadshaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaeq iUdehaaiaadghadaWgaaWcbaGaamyAaaqabaGcdaWcaaqaaiabgkGi 2kabeI7aXbqaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaa GccaGLOaGaayzkaaGaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIi YdGccqGHLjYScaaIWaaaaa@5BB3@

 

To see the first result,

  1. note that

ρ s t + ( q i /θ) y i q θ =ρ s t + 1 θ q i y i q θ 1 θ 2 q i θ y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaalaaabaGaeyOaIyRaam4Caa qaaiabgkGi2kaadshaaaGaey4kaSYaaSaaaeaacqGHciITcaGGOaGa amyCamaaBaaaleaacaWGPbaabeaakiaac+cacqaH4oqCcaGGPaaaba GaeyOaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGccqGHsisldaWc aaqaaiaadghaaeaacqaH4oqCaaGaeyypa0JaeqyWdi3aaSaaaeaacq GHciITcaWGZbaabaGaeyOaIyRaamiDaaaacqGHRaWkdaWcaaqaaiaa igdaaeaacqaH4oqCaaWaaSaaaeaacqGHciITcaWGXbWaaSbaaSqaai aadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabeaa aaGccqGHsisldaWcaaqaaiaadghaaeaacqaH4oqCaaGaeyOeI0YaaS aaaeaacaaIXaaabaGaeqiUde3aaWbaaSqabeaacaaIYaaaaaaakiaa dghadaWgaaWcbaGaamyAaaqabaGcdaWcaaqaaiabgkGi2kabeI7aXb qaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaaaa@6936@

  1. Use the first law to see that

ρ s t + 1 θ q i y i q θ 1 θ 2 q i θ y i =ρ s t + 1 θ ( ρ ε t | x=const + σ ij D ij ) 1 θ 2 q i θ y i 0 σ ij D ij ρ t (ψ+θs)+θρ s t 1 θ q i θ y i 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeqyWdi3aaSaaaeaacqGHciITca WGZbaabaGaeyOaIyRaamiDaaaacqGHRaWkdaWcaaqaaiaaigdaaeaa cqaH4oqCaaWaaSaaaeaacqGHciITcaWGXbWaaSbaaSqaaiaadMgaae qaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGccqGH sisldaWcaaqaaiaadghaaeaacqaH4oqCaaGaeyOeI0YaaSaaaeaaca aIXaaabaGaeqiUde3aaWbaaSqabeaacaaIYaaaaaaakiaadghadaWg aaWcbaGaamyAaaqabaGcdaWcaaqaaiabgkGi2kabeI7aXbqaaiabgk Gi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaOGaeyypa0JaeqyWdi3a aSaaaeaacqGHciITcaWGZbaabaGaeyOaIyRaamiDaaaacqGHRaWkda WcaaqaaiaaigdaaeaacqaH4oqCaaWaaeWaaeaacqGHsislcqaHbpGC daabcaqaamaalaaabaGaeyOaIyRaeqyTdugabaGaeyOaIyRaamiDaa aaaiaawIa7amaaBaaaleaacaWH4bGaeyypa0Jaam4yaiaad+gacaWG UbGaam4CaiaadshaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaakiaadseadaWgaaWcbaGaamyAaiaadQgaaeqaaaGc caGLOaGaayzkaaGaeyOeI0YaaSaaaeaacaaIXaaabaGaeqiUde3aaW baaSqabeaacaaIYaaaaaaakiaadghadaWgaaWcbaGaamyAaaqabaGc daWcaaqaaiabgkGi2kabeI7aXbqaaiabgkGi2kaadMhadaWgaaWcba GaamyAaaqabaaaaOGaeyyzImRaaGimaaqaaiabgkDiElabeo8aZnaa BaaaleaacaWGPbGaamOAaaqabaGccaWGebWaaSbaaSqaaiaadMgaca WGQbaabeaakiabgkHiTiabeg8aYnaalaaabaGaeyOaIylabaGaeyOa IyRaamiDaaaacaGGOaGaeqiYdKNaey4kaSIaeqiUdeNaam4CaiaacM cacqGHRaWkcqaH4oqCcqaHbpGCdaWcaaqaaiabgkGi2kaadohaaeaa cqGHciITcaWG0baaaiabgkHiTmaalaaabaGaaGymaaqaaiabeI7aXb aacaWGXbWaaSbaaSqaaiaadMgaaeqaaOWaaSaaaeaacqGHciITcqaH 4oqCaeaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaakiabgw MiZkaaicdaaaaa@B4F6@

where we have noted that temperature is always positive.  This yields the solution.

 

The second result follows by integrating the local form and using the stress-power work expression.

 

 

 

 

 

 

 

5.11 Conservation laws for a control volume

 

To model solids, it is usually convenient to write the equations of motion for a volume that moves with the solid.   When modeling fluids, it is often preferable to consider a fixed spatial volume, through which the fluid moves with time.   To this end,

* We consider a fixed region in space R, bounded by a surface B.  

* Material flows through the region with velocity field v(y,t).

* The solid has mass density per unit deformed volume (in the spatial configuration) ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3228@ ; and is subjected to a body force b per unit mass. 

* A heat flux q flows through the solid; while an external source injects heat flux Q per unit deformed volume.

 

The conservation laws can be expressed in terms of integrals over the fixed spatial region (which does not move with the solid) as follows

 

Mass Conservation: d dt R ρdV + B ρvndA =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqyWdiNaamizaiaadAfaaSqaaiaadkfaaeqaniab gUIiYdGccqGHRaWkdaWdrbqaaiabeg8aYjaahAhacqGHflY1caWHUb GaamizaiaadgeaaSqaaiaadkeaaeqaniabgUIiYdGccqGH9aqpcaaI Waaaaa@475E@

Linear Momentum Balance B nσdA + R ρbdV = d dt R ρv dV+ B (ρv)vndA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaCOBaiabgwSixlaaho8aca WGKbGaamyqaaWcbaGaamOqaaqab0Gaey4kIipakiabgUcaRmaapefa baGaeqyWdiNaaCOyaiaadsgacaWGwbaaleaacaWGsbaabeqdcqGHRi I8aOGaeyypa0ZaaSaaaeaacaWGKbaabaGaamizaiaadshaaaWaa8qu aeaacqaHbpGCcaWH2baaleaacaWGsbaabeqdcqGHRiI8aOGaamizai aadAfacqGHRaWkdaWdrbqaaiaacIcacqaHbpGCcaWH2bGaaiykaiaa hAhacqGHflY1caWHUbGaamizaiaadgeaaSqaaiaadkeaaeqaniabgU IiYdaaaa@5BC7@

Angular Momentum Balance B y×(nσ)dA + R y×(ρb)dA = d dt R y×ρvdV + B (y×ρv)vndA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaCyEaiabgEna0kaacIcaca WHUbGaeyyXICTaaC4WdiaacMcacaWGKbGaamyqaaWcbaGaamOqaaqa b0Gaey4kIipakiabgUcaRmaapefabaGaaCyEaiabgEna0kaacIcacq aHbpGCcaWHIbGaaiykaiaadsgacaWGbbaaleaacaWGsbaabeqdcqGH RiI8aOGaeyypa0ZaaSaaaeaacaWGKbaabaGaamizaiaadshaaaWaa8 quaeaacaWH5bGaey41aqRaeqyWdiNaaCODaiaadsgacaWGwbaaleaa caWGsbaabeqdcqGHRiI8aOGaey4kaSYaa8quaeaacaGGOaGaaCyEai abgEna0kabeg8aYjaahAhacaGGPaGaaCODaiabgwSixlaah6gacaWG KbGaamyqaaWcbaGaamOqaaqab0Gaey4kIipaaaa@6AC8@

Mechanical Power Balance B (nσ)vdA + R ρbv dV= R σ:D dV+ d dt R 1 2 ρ(vv)dV + B 1 2 ρ(vv)vndA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaiikaiaah6gacqGHflY1ca WHdpGaaiykaiabgwSixlaahAhacaWGKbGaamyqaaWcbaGaamOqaaqa b0Gaey4kIipakiabgUcaRmaapefabaGaeqyWdiNaaCOyaiabgwSixl aahAhaaSqaaiaadkfaaeqaniabgUIiYdGccaWGKbGaamOvaiabg2da 9maapefabaGaaC4WdiaacQdacaWHebaaleaacaWGsbaabeqdcqGHRi I8aOGaamizaiaadAfacqGHRaWkdaWcaaqaaiaadsgaaeaacaWGKbGa amiDaaaadaWdrbqaamaalaaabaGaaGymaaqaaiaaikdaaaGaeqyWdi NaaiikaiaahAhacqGHflY1caWH2bGaaiykaiaadsgacaWGwbaaleaa caWGsbaabeqdcqGHRiI8aOGaey4kaSYaa8quaeaadaWcaaqaaiaaig daaeaacaaIYaaaaiabeg8aYjaacIcacaWH2bGaeyyXICTaaCODaiaa cMcacaWH2bGaeyyXICTaaCOBaiaadsgacaWGbbaaleaacaWGcbaabe qdcqGHRiI8aaaa@7756@

First law of thermodynamics

B (nσ)v dA+ R ρbv dV B qndA + V qdV = d dt R ρ( ε+ 1 2 vv )dV + B ρ( ε+ 1 2 vv )vn dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaiikaiaah6gacqGHflY1ca WHdpGaaiykaiabgwSixlaahAhaaSqaaiaadkeaaeqaniabgUIiYdGc caWGKbGaamyqaiabgUcaRmaapefabaGaeqyWdiNaaCOyaiabgwSixl aahAhaaSqaaiaadkfaaeqaniabgUIiYdGccaWGKbGaamOvaiabgkHi TmaapefabaGaaCyCaiabgwSixlaah6gacaWGKbGaamyqaaWcbaGaam Oqaaqab0Gaey4kIipakiabgUcaRmaapefabaGaamyCaiaadsgacaWG wbaaleaacaWGwbaabeqdcqGHRiI8aOGaeyypa0ZaaSaaaeaacaWGKb aabaGaamizaiaadshaaaWaa8quaeaacqaHbpGCdaqadaqaaiabew7a LjabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGaaCODaiabgwSixl aahAhaaiaawIcacaGLPaaacaWGKbGaamOvaaWcbaGaamOuaaqab0Ga ey4kIipakiabgUcaRmaapefabaGaeqyWdi3aaeWaaeaacqaH1oqzcq GHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaiaahAhacqGHflY1caWH 2baacaGLOaGaayzkaaGaaCODaiabgwSixlaah6gaaSqaaiaadkeaae qaniabgUIiYdGccaWGKbGaamyqaaaa@84E4@

Second law of thermodynamics

d dt R ρsdV + B ρs(vn)dA + B qn θ dA R q θ dV 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqyWdiNaam4CaiaadsgacaWGwbaaleaacaWGsbaa beqdcqGHRiI8aOGaey4kaSYaa8quaeaacqaHbpGCcaWGZbGaaiikai aahAhacqGHflY1caWHUbGaaiykaiaadsgacaWGbbaaleaacaWGcbaa beqdcqGHRiI8aOGaey4kaSYaa8quaeaadaWcaaqaaiaahghacqGHfl Y1caWHUbaabaGaeqiUdehaaiaadsgacaWGbbaaleaacaWGcbaabeqd cqGHRiI8aOGaeyOeI0Yaa8quaeaadaWcaaqaaiaadghaaeaacqaH4o qCaaGaamizaiaadAfaaSqaaiaadkfaaeqaniabgUIiYdGccqGHLjYS caaIWaaaaa@5FAC@

 

 

These results can all be derived from the conservation laws for a material volume, using a similar approach.  Consider the mass conservation equation as a special case.   Start with the local form

ρ t | y=const + ρ v i y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaWaaSaaaeaacqGHciITcqaHbp GCaeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaahMhacqGH 9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccqGHRaWkda WcaaqaaiabgkGi2kabeg8aYjaadAhadaWgaaWcbaGaamyAaaqabaaa keaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaakiabg2da9i aaicdaaaa@4A0F@

Integrate over a fixed spatial volume

R ρ t | y=const dV + R ρ v i y i dV =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaWaaqGaaeaadaWcaaqaaiabgk Gi2kabeg8aYbqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGa aCyEaiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabeaaki aadsgacaWGwbaaleaacaWGsbaabeqdcqGHRiI8aOGaey4kaSYaa8qu aeaadaWcaaqaaiabgkGi2kabeg8aYjaadAhadaWgaaWcbaGaamyAaa qabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaakiaa dsgacaWGwbaaleaacaWGsbaabeqdcqGHRiI8aOGaeyypa0JaaGimaa aa@53ED@

Note the R is independent of time, and use the divergence theorem

d dt R ρ(y,t)dV + B ρ v i n i dA =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqyWdiNaaiikaiaahMhacaGGSaGaamiDaiaacMca caWGKbGaamOvaaWcbaGaamOuaaqab0Gaey4kIipakiabgUcaRmaape fabaGaeqyWdiNaamODamaaBaaaleaacaWGPbaabeaakiaad6gadaWg aaWcbaGaamyAaaqabaGccaWGKbGaamyqaaWcbaGaamOqaaqab0Gaey 4kIipakiabg2da9iaaicdaaaa@4B57@

As a second example, for the linear momentum balance equation, start with the local form

σ ji y j +ρ b i =ρ( v i y k v k + v i t | y i =const ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGa amOAaaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaaleaacaWGPb aabeaakiabg2da9iabeg8aYnaabmaabaWaaSaaaeaacqGHciITcaWG 2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaale aacaWGRbaabeaaaaGccaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaey4k aSYaaqGaaeaadaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamyAaa qabaaakeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaadMha daWgaaadbaGaamyAaaqabaWccqGH9aqpcaWGJbGaam4Baiaad6gaca WGZbGaamiDaaqabaaakiaawIcacaGLPaaaaaa@5C04@

Note that, using mass conservation

y k (ρ v i v k )=ρ v k v i y k + (ρ v k ) y k v i =ρ v k v i y k ρ t v i ρ v k v i y k = y k (ρ v i v k )+ ρ t v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacqGHciITaeaacqGHci ITcaWG5bWaaSbaaSqaaiaadUgaaeqaaaaakiaacIcacqaHbpGCcaWG 2bWaaSbaaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGRbaabe aakiaacMcacqGH9aqpcqaHbpGCcaWG2bWaaSbaaSqaaiaadUgaaeqa aOWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaadMgaaeqaaaGcba GaeyOaIyRaamyEamaaBaaaleaacaWGRbaabeaaaaGccqGHRaWkdaWc aaqaaiabgkGi2kaacIcacqaHbpGCcaWG2bWaaSbaaSqaaiaadUgaae qaaOGaaiykaaqaaiabgkGi2kaadMhadaWgaaWcbaGaam4Aaaqabaaa aOGaamODamaaBaaaleaacaWGPbaabeaakiabg2da9iabeg8aYjaadA hadaWgaaWcbaGaam4AaaqabaGcdaWcaaqaaiabgkGi2kaadAhadaWg aaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadU gaaeqaaaaakiabgkHiTmaalaaabaGaeyOaIyRaeqyWdihabaGaeyOa IyRaamiDaaaacaWG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyO0H4 TaeqyWdiNaamODamaaBaaaleaacaWGRbaabeaakmaalaaabaGaeyOa IyRaamODamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadMhada WgaaWcbaGaam4AaaqabaaaaOGaeyypa0ZaaSaaaeaacqGHciITaeaa cqGHciITcaWG5bWaaSbaaSqaaiaadUgaaeqaaaaakiaacIcacqaHbp GCcaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWG RbaabeaakiaacMcacqGHRaWkdaWcaaqaaiabgkGi2kabeg8aYbqaai abgkGi2kaadshaaaGaamODamaaBaaaleaacaWGPbaabeaaaaaa@8DF8@

Therefore

σ ji y j +ρ b i = y k (ρ v i v k )+ v i ρ t | y i =const +ρ v i t | y i =const = y k (ρ v i v k )+ (ρ v i ) t | y i =const MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGa amOAaaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaaleaacaWGPb aabeaakiabg2da9maalaaabaGaeyOaIylabaGaeyOaIyRaamyEamaa BaaaleaacaWGRbaabeaaaaGccaGGOaGaeqyWdiNaamODamaaBaaale aacaWGPbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccaGGPaGa ey4kaSYaaqGaaeaacaWG2bWaaSbaaSqaaiaadMgaaeqaaOWaaSaaae aacqGHciITcqaHbpGCaeaacqGHciITcaWG0baaaaGaayjcSdWaaSba aSqaaiaadMhadaWgaaadbaGaamyAaaqabaWccqGH9aqpcaWGJbGaam 4Baiaad6gacaWGZbGaamiDaaqabaGccqGHRaWkcqaHbpGCdaabcaqa amaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGPbaabeaaaOqaai abgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaamyEamaaBaaameaa caWGPbaabeaaliabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0b aabeaakiabg2da9maalaaabaGaeyOaIylabaGaeyOaIyRaamyEamaa BaaaleaacaWGRbaabeaaaaGccaGGOaGaeqyWdiNaamODamaaBaaale aacaWGPbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccaGGPaGa ey4kaSYaaqGaaeaadaWcaaqaaiabgkGi2kaacIcacqaHbpGCcaWG2b WaaSbaaSqaaiaadMgaaeqaaOGaaiykaaqaaiabgkGi2kaadshaaaaa caGLiWoadaWgaaWcbaGaamyEamaaBaaameaacaWGPbaabeaaliabg2 da9iaadogacaWGVbGaamOBaiaadohacaWG0baabeaaaaa@90DA@

Integrating this expression over the fixed control volume and using the divergence theorem gives the stated answer.

 

A similar approach can be used to obtain the remaining results MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  details are left as an exercise.

 

 

 

5.12 Transformation of kinematic and kinetic variables under changes of reference frame

Since physical laws must be constructed so as to be independent of the choice of reference frame, the behavior of kinematic and kinetic variables, the field equations, and constitutive equations under a change of reference frame is of interest.  The concept of a reference frame, and the various relations involved in changing reference frames, are both rather obscure concepts.   There are several reasons for this:

1.      One source of confusion arises because Newtonian mechanics relies on the concept of an inertial frame, and Newton’s law F=ma only holds in this frame.   The statement ‘the laws of physics are independent of reference frame’ does not mean that F=ma in all reference frames MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it means that all observers must describe Newton’s laws with respect to the same inertial frame, and must do so in a consistent manner.  Frame indifference is not the theory of relativity…

2.      A second source of confusion stems from the use of a reference configuration to quantify shape changes of a solid.   We nearly always use the undeformed solid as reference, which gives the impression that the reference configuration, like the deformed configuration, is associated with the inertial frame.   In fact, the reference configuration is completely arbitrary, and even though all observers might choose the same initial configuration of a solid as reference, they will all assume that the reference configuration is fixed.   The reference configuration is not attached to the inertial frame.  Moreover, since the reference configuration is arbitrary, two observers could choose different reference configurations, and still devise equations that describe the same physical process.  Of course the exact form of the governing equations will change with the choice of reference configuration.  There are no restrictions governing transformation of reference configuration between observers, beyond the fact that two reference configurations must be related by an invertible 1:1 mapping.

 

To make the concept of a change in reference frame in classical continuum mechanics precise, we first introduce the intertial frame.  As in all preceding discussions, we assume that the inertial frame is a three-dimensional Euclidean space, and let y(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahMhacaGGOaGaamiDaiaacMcaaaa@33BC@  denote a point in the inertial frame.  We then define Newtonian measures of velocity and acceleration vectors in the usual way as

v= y t a= 2 y t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpdaWcaaqaaiabgkGi2k aahMhaaeaacqGHciITcaWG0baaaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaCyyai abg2da9maalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaaCyE aaqaaiabgkGi2kaadshadaahaaWcbeqaaiaaikdaaaaaaaaa@50E0@

 

This inertial frame could be viewed by an observer who rotates and translates with respect to the inertial frame.  To this observer, all physical quantities associated with the inertial frame would appear to be translated and rotated in the opposite sense.  We describe this apparent transformation of space with respect to the observer as a rigid rotation and translation MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  thus, the position vector of a point seen by the observer is related to position in the inertial frame by

y * = y 0 * (t)+Q(t)(y y 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahMhadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWH5bWaa0baaSqaaiaaicdaaeaacaGGQaaaaOGaaiikaiaa dshacaGGPaGaey4kaSIaaCyuaiaacIcacaWG0bGaaiykaiaacIcaca WH5bGaeyOeI0IaaCyEamaaBaaaleaacaaIWaaabeaakiaacMcaaaa@4190@

where y 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahMhadaWgaaWcbaGaaGimaaqabaaaaa@3250@  is an arbitrary fixed point in the inertial frame, y 0 * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahMhadaqhaaWcbaGaaGimaaqaaiaacQ caaaaaaa@32FF@  is an arbitrary vector, and  Q(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahgfacaGGOaGaamiDaiaacMcaaaa@3394@  is a proper orthogonal tensor, representing a rigid rotation.  It is convenient to introduce the spin associated with the relative rotation of the inertial frame and the observer’s frame

Ω= dQ dt Q T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM6acqGH9aqpdaWcaaqaaiaadsgaca WHrbaabaGaamizaiaadshaaaGaaCyuamaaCaaaleqabaGaamivaaaa aaa@3838@

 

We will denote quantities in the observer’s reference frame with a star superscript; -for example mass density, body force, Cauchy stress ρ * , b * , σ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaCaaaleqabaGaaiOkaaaaki aacYcacaWHIbWaaWbaaSqabeaacaGGQaaaaOGaaiilaiaaho8adaah aaWcbeqaaiaacQcaaaaaaa@3867@  ; those without superscripts will be assumed to be defined in the inertial frame.

 

All observable physical quantities must transform in particular ways under a change of observer.  Specifically:

* Scalar quantities, such as density or temperature are invariant MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  they have the same value in all reference frames.

* Quantities such as body force, a line element in the deformed solid; the normal to a surface; the velocity vector, the acceleration vector, and so on, are physical vectors defined in the inertial reference frame.   They can be regarded as connecting two points in the inertial frame, and must transform with the line connecting these two points under a change of reference frame.  Thus, a normal vector to a deformed surface, body force, velocity, acceleration vectors must transform as

b * =Qb n * =Qn v * =Qv a * =Qa MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkgadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWHrbGaaCOyaiaaykW7caaMc8UaaGPaVlaah6gadaahaaWc beqaaiaacQcaaaGccqGH9aqpcaWHrbGaaCOBaiaaykW7caaMc8UaaG PaVlaaykW7caWH2bWaaWbaaSqabeaacaGGQaaaaOGaeyypa0JaaCyu aiaahAhacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWHHb WaaWbaaSqabeaacaGGQaaaaOGaeyypa0JaaCyuaiaahggacaaMc8Ua aGPaVdaa@5A37@

Vectors that transform in this way are said to be frame indifferent, or objective.   Note (1) frame indifference does not mean that vectors are invariant MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  quite the opposite, in fact MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it means that all observers must describe the same physical quantity; (2) vector quantities we make frequent use of in solid mechanics need not necessarily be frame indifferent.   For example, the normal to the reference configuration of a solid would not be frame indifferent; nor would a material fiber within the reference configuration.

* Tensor quantities that map a frame indifferent vector onto another frame independent vector are similarly said to be frame indifferent, or objective. Examples include the stretch rate tensor (which specifies the relative velocity of two ends of an infinitesimal material fiber in the spatial configuration); or Cauchy stress (which maps the normal to a surface in the spatial configuration to the physical traction vector.  A frame indifferent tensor must transform as

σ * =Qσ Q T D * =QD Q T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaho8adaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWHrbGaaC4WdiaahgfadaahaaWcbeqaaiaadsfaaaGccaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaahseadaahaaWcbeqaaiaacQcaaaGccqGH9aqp caWHrbGaaCiraiaahgfadaahaaWcbeqaaiaadsfaaaaaaa@4EED@

Again, not all tensors are frame indifferent.  The deformation gradient; the spin tensor; the Lagrange strain tensor are not frame indifferent.

 

Frame indifference can also be looked at as follows.   Let { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3968@  be an inertial basis.   Under a change of observer, the basis vectors transform as { e 1 * , e 2 * , e 3 * }={Q e 1 ,Q e 2 ,Q e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaa0baaSqaaiaaigdaae aacaGGQaaaaOGaaiilaiaahwgadaqhaaWcbaGaaGOmaaqaaiaacQca aaGccaGGSaGaaCyzamaaDaaaleaacaaIZaaabaGaaiOkaaaakiaac2 hacqGH9aqpcaGG7bGaaCyuaiaahwgadaWgaaWcbaGaaGymaaqabaGc caGGSaGaaCyuaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaC yuaiaahwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@4809@  (relative to the observer, they appear to rotate with the observed frame.  It is important to note that { e 1 * , e 2 * , e 3 * } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaa0baaSqaaiaaigdaae aacaGGQaaaaOGaaiilaiaahwgadaqhaaWcbaGaaGOmaaqaaiaacQca aaGccaGGSaGaaCyzamaaDaaaleaacaaIZaaabaGaaiOkaaaakiaac2 haaaa@3B75@  are time dependent).    Now, we can compute components in either basis

b i * = e i * b * =Q e i Qb= e i Q T Qb= e i b= b i σ ij * = e i * σ * e j * =Q e i Qσ Q T Q e j = σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamOyamaaDaaaleaacaWGPbaaba GaaiOkaaaakiabg2da9iaahwgadaqhaaWcbaGaamyAaaqaaiaacQca aaGccqGHflY1caWHIbWaaWbaaSqabeaacaGGQaaaaOGaeyypa0JaaC yuaiaahwgadaWgaaWcbaGaamyAaaqabaGccqGHflY1caWHrbGaaCOy aiabg2da9iaahwgadaWgaaWcbaGaamyAaaqabaGccqGHflY1caWHrb WaaWbaaSqabeaacaWGubaaaOGaaCyuaiaaykW7caWHIbGaeyypa0Ja aCyzamaaBaaaleaacaWGPbaabeaakiabgwSixlaahkgacqGH9aqpca WGIbWaaSbaaSqaaiaadMgaaeqaaaGcbaGaaGPaVlabeo8aZnaaDaaa leaacaWGPbGaamOAaaqaaiaacQcaaaGccqGH9aqpcaWHLbWaa0baaS qaaiaadMgaaeaacaGGQaaaaOGaeyyXICTaaC4WdmaaCaaaleqabaGa aiOkaaaakiabgwSixlaahwgadaqhaaWcbaGaamOAaaqaaiaacQcaaa GccqGH9aqpcaWHrbGaaCyzamaaBaaaleaacaWGPbaabeaakiabgwSi xlaahgfacaWHdpGaaCyuamaaCaaaleqabaGaamivaaaakiaahgfaca WHLbWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqa aiaadMgacaWGQbaabeaaaaaa@7C74@

The components of a vector with respect to a basis that rotates with an inertial frame is independent of the observer.  This is one interpretation of what we mean by a physical process being independent of the observer.

 

We now examine how several kinematic and kinetic variables commonly used in continuum mechanics transform under a change of observer.   To describe deformations, a reference configuration must be selected.  A material particle in the reference configuration is identified by a time independent vector X in reference space.   The choice of reference space is arbitrary; and there is no reason why different observers will necessarily adopt the same reference space.   Discussions are greatly simplified, however, if we choose to assume that all observers use the same space for the reference configuration (behavior under a change of reference configuration can be treated separately).

 

        

* The deformation mapping transforms as y * (X,t)= y 0 * (t)+Q(t)( y(X,t) y 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahMhadaahaaWcbeqaaiaacQcaaaGcca GGOaGaaCiwaiaacYcacaWG0bGaaiykaiabg2da9iaahMhadaqhaaWc baGaaGimaaqaaiaacQcaaaGccaGGOaGaamiDaiaacMcacqGHRaWkca WHrbGaaiikaiaadshacaGGPaWaaeWaaeaacaWH5bGaaiikaiaahIfa caGGSaGaamiDaiaacMcacqGHsislcaWH5bWaaSbaaSqaaiaaicdaae qaaaGccaGLOaGaayzkaaaaaa@4986@

* The deformation gradient transforms as F * = y * X =Q y X =QF MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAeadaahaaWcbeqaaiaacQcaaaGccq GH9aqpdaWcaaqaaiabgkGi2kaahMhadaahaaWcbeqaaiaacQcaaaaa keaacqGHciITcaWHybaaaiabg2da9iaahgfadaWcaaqaaiabgkGi2k aahMhaaeaacqGHciITcaWHybaaaiabg2da9iaahgfacaWHgbaaaa@4214@

* The right Cauchy Green strain  Lagrange strain, the right stretch tensor are invariant

C * = F *T F * = F T Q T QF=C E * =E U * =U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahoeadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWHgbWaaWbaaSqabeaacaGGQaGaamivaaaakiaahAeadaah aaWcbeqaaiaacQcaaaGccqGH9aqpcaWHgbWaaWbaaSqabeaacaWGub aaaOGaaCyuamaaCaaaleqabaGaamivaaaakiaahgfacaWHgbGaeyyp a0JaaC4qaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaCyramaaCaaaleqabaGaaiOkaaaakiabg2da9iaahwea caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaCyvam aaCaaaleqabaGaaiOkaaaakiabg2da9iaahwfaaaa@5DFD@

* The left Cauchy Green strain, Eulerian strain, left stretch tensor are frame indifferent

B * = F * F *T =QF F T Q T =QC Q T V * =QV Q T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkeadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWHgbWaaWbaaSqabeaacaGGQaaaaOGaaCOramaaCaaaleqa baGaaiOkaiaadsfaaaGccqGH9aqpcaWHrbGaaCOraiaahAeadaahaa WcbeqaaiaadsfaaaGccaWHrbWaaWbaaSqabeaacaWGubaaaOGaeyyp a0JaaCyuaiaahoeacaWHrbWaaWbaaSqabeaacaWGubaaaOGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaahAfadaahaaWcbeqaaiaacQcaaaGccqGH9aqpcaWHrbGaaC OvaiaahgfadaahaaWcbeqaaiaadsfaaaaaaa@583E@

* The velocity gradient and spin tensor transform as

L * = F ˙ * F *1 =( Q ˙ F+Q F ˙ ) F 1 Q T =QL Q T +Ω W * =( L * L *T )/2=QW Q T +Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaCitamaaCaaaleqabaGaaiOkaa aakiabg2da9iqahAeagaGaamaaCaaaleqabaGaaiOkaaaakiaahAea daahaaWcbeqaaiaacQcacqGHsislcaaIXaaaaOGaeyypa0ZaaeWaae aaceWHrbGbaiaacaWHgbGaey4kaSIaaCyuaiqahAeagaGaaaGaayjk aiaawMcaaiaahAeadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaWHrb WaaWbaaSqabeaacaWGubaaaOGaeyypa0JaaCyuaiaahYeacaWHrbWa aWbaaSqabeaacaWGubaaaOGaey4kaSIaaCyQdaqaaiaahEfadaahaa WcbeqaaiaacQcaaaGccqGH9aqpcaGGOaGaaCitamaaCaaaleqabaGa aiOkaaaakiabgkHiTiaahYeadaahaaWcbeqaaiaacQcacaWGubaaaO Gaaiykaiaac+cacaaIYaGaeyypa0JaaCyuaiaahEfacaWHrbWaaWba aSqabeaacaWGubaaaOGaey4kaSIaaCyQdaaaaa@5BF9@

* The velocity and acceleration vectors transform as

v * =Qv=Q dy dt =Q d dt Q T ( y * y 0 * (t))= d y * dt d y 0 * dt Ω( y * y 0 * (t)) a * =Qa=Q d 2 y d t 2 =Q d 2 d t 2 Q T ( y * y 0 * (t))= d 2 y * d t 2 d 2 y 0 * d t 2 +( Ω 2 dΩ dt )( y * y 0 * (t))2Ω( d y * dt d y 0 * (t) dt ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaCODamaaCaaaleqabaGaaiOkaa aakiabg2da9iaahgfacaWH2bGaeyypa0JaaCyuamaalaaabaGaamiz aiaahMhaaeaacaWGKbGaamiDaaaacqGH9aqpcaWHrbWaaSaaaeaaca WGKbaabaGaamizaiaadshaaaGaaCyuamaaCaaaleqabaGaamivaaaa kiaacIcacaWH5bWaaWbaaSqabeaacaGGQaaaaOGaeyOeI0IaaCyEam aaDaaaleaacaaIWaaabaGaaiOkaaaakiaacIcacaWG0bGaaiykaiaa cMcacqGH9aqpdaWcaaqaaiaadsgacaWH5bWaaWbaaSqabeaacaGGQa aaaaGcbaGaamizaiaadshaaaGaeyOeI0YaaSaaaeaacaWGKbGaaCyE amaaDaaaleaacaaIWaaabaGaaiOkaaaaaOqaaiaadsgacaWG0baaai abgkHiTiaahM6acaGGOaGaaCyEamaaCaaaleqabaGaaiOkaaaakiab gkHiTiaahMhadaqhaaWcbaGaaGimaaqaaiaacQcaaaGccaGGOaGaam iDaiaacMcacaGGPaaabaGaaCyyamaaCaaaleqabaGaaiOkaaaakiab g2da9iaahgfacaWHHbGaeyypa0JaaCyuamaalaaabaGaamizamaaCa aaleqabaGaaGOmaaaakiaahMhaaeaacaWGKbGaamiDamaaCaaaleqa baGaaGOmaaaaaaGccqGH9aqpcaWHrbWaaSaaaeaacaWGKbWaaWbaaS qabeaacaaIYaaaaaGcbaGaamizaiaadshadaahaaWcbeqaaiaaikda aaaaaOGaaCyuamaaCaaaleqabaGaamivaaaakiaacIcacaWH5bWaaW baaSqabeaacaGGQaaaaOGaeyOeI0IaaCyEamaaDaaaleaacaaIWaaa baGaaiOkaaaakiaacIcacaWG0bGaaiykaiaacMcacqGH9aqpdaWcaa qaaiaadsgadaahaaWcbeqaaiaaikdaaaGccaWH5bWaaWbaaSqabeaa caGGQaaaaaGcbaGaamizaiaadshadaahaaWcbeqaaiaaikdaaaaaaO GaeyOeI0YaaSaaaeaacaWGKbWaaWbaaSqabeaacaaIYaaaaOGaaCyE amaaDaaaleaacaaIWaaabaGaaiOkaaaaaOqaaiaadsgacaWG0bWaaW baaSqabeaacaaIYaaaaaaakiabgUcaRmaabmaabaGaaCyQdmaaCaaa leqabaGaaGOmaaaakiabgkHiTmaalaaabaGaamizaiaahM6aaeaaca WGKbGaamiDaaaaaiaawIcacaGLPaaacaGGOaGaaCyEamaaCaaaleqa baGaaiOkaaaakiabgkHiTiaahMhadaqhaaWcbaGaaGimaaqaaiaacQ caaaGccaGGOaGaamiDaiaacMcacaGGPaGaeyOeI0IaaGOmaiaahM6a caGGOaWaaSaaaeaacaWGKbGaaCyEamaaCaaaleqabaGaaiOkaaaaaO qaaiaadsgacaWG0baaaiabgkHiTmaalaaabaGaamizaiaahMhadaqh aaWcbaGaaGimaaqaaiaacQcaaaGccaGGOaGaamiDaiaacMcaaeaaca WGKbGaamiDaaaacaGGPaaaaaa@B46F@

(the additional terms in the acceleration can be interpreted as the centripetal and coriolis accelerations)

* The Cauchy stress is frame indifferent σ * =Qσ Q T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaho8adaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWHrbGaaC4WdiaahgfadaahaaWcbeqaaiaadsfaaaaaaa@37AB@  (you can see this from the formal definition, or use the fact that the virtual power must be invariant under a frame change)

* The material stress is frame invariant Σ * =Σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaho6adaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWHJoaaaa@34B1@

* The nominal stress transforms as S * =J (QF) 1 Qσ Q T =J F 1 σ Q T =S Q T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaC4uamaaCaaaleqabaGaaiOkaaaaki abg2da9iaadQeacaGGOaGaaCyuaiaahAeacaGGPaWaaWbaaSqabeaa cqGHsislcaaIXaaaaOGaeyyXICTaaCyuaKaaalaaho8akiaahgfada ahaaWcbeqaaiaadsfaaaGccqGH9aqpcaWGkbGaaCOramaaCaaaleqa baGaeyOeI0IaaGymaaaakiabgwSixNaaalaaho8akiaahgfadaahaa WcbeqaaiaadsfaaaGccqGH9aqpcaWHtbGaaCyuamaaCaaaleqabaGa amivaaaaaaa@4F10@  (note that this transformation rule will differ if the nominal stress is defined as the transpose of the measure used here…)

 

By this time you are probably asking yourself why anyone could possibly care about all this.   This is a fair question MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  issues of frame indifference arise rather rarely in practice. They do come up, however, when we define constitutive equations for a material MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  which must relate deformation measures to internal force measures.  A new constitutive equation is a new physical law MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  and it is important to make sure that the new law behaves correctly under a change of observer.  Most modern constitutive equations try to describe the underlying microscopic processes that govern its response, and if this is done properly, the law will be frame indifferent.   But some constitutive laws are just curve-fits MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  some mathematical relationship between a deformation measure and a force measure MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  and not all possible relationships will transform correctly.  

 

Problems arise most commonly in trying to develop rate forms of constitutive equations, which are intended to relate some measure of strain rate to stress rate.   This is because, even if a vector or tensor is itself frame indifferent, its time derivative is generally not.  For example, although position vector satisfies y * =Qy MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahMhadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWHrbGaaCyEaaaa@3531@  and is frame indifferent, this does not mean that d y * dt =Q dy dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaahMhadaahaaWcbe qaaiaacQcaaaaakeaacaWGKbGaamiDaaaacqGH9aqpcaWHrbWaaSaa aeaacaWGKbGaaCyEaaqaaiaadsgacaWG0baaaaaa@3AE7@ .  Similarly, for the rate of change of Cauchy stress is not frame indifferent, because

d σ * dt = dQ dt σ Q T +Q dσ dt Q T +Qσ dQ dt T =Q dσ dt Q T +Ω σ * σ * ΩQ dσ dt Q T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaaho8adaahaaWcbe qaaiaacQcaaaaakeaacaWGKbGaamiDaaaacqGH9aqpdaWcaaqaaiaa dsgacaWHrbaabaGaamizaiaadshaaaGaaC4WdiaahgfadaahaaWcbe qaaiaadsfaaaGccqGHRaWkcaWHrbWaaSaaaeaacaWGKbGaaC4Wdaqa aiaadsgacaWG0baaaiaahgfadaahaaWcbeqaaiaadsfaaaGccqGHRa WkcaWHrbGaaC4WdmaalaaabaGaamizaiaahgfaaeaacaWGKbGaamiD aaaadaahaaWcbeqaaiaadsfaaaGccqGH9aqpcaWHrbWaaSaaaeaaca WGKbGaaC4WdaqaaiaadsgacaWG0baaaiaahgfadaahaaWcbeqaaiaa dsfaaaGccqGHRaWkcaWHPoGaaC4WdmaaCaaaleqabaGaaiOkaaaaki abgkHiTiaaho8adaahaaWcbeqaaiaacQcaaaGccaWHPoGaeyiyIKRa aCyuamaalaaabaGaamizaiaaho8aaeaacaWGKbGaamiDaaaacaWHrb WaaWbaaSqabeaacaWGubaaaaaa@664B@

In fact, only quantities that are invariant under a change of observer can be differentiated safely with respect to time MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  their time derivatives remain invariant.

 

This means that if constitutive equations are expressed in rate form MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  for example something that looks at first glance like the rate form of an elastic constitutive equation

dσ dt = C ˜ D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaaho8aaeaacaWGKb GaamiDaaaacqGH9aqpceWHdbGbaGaacaWHebaaaa@3740@

(here C is a fourth order constant tensor) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the constitutive equation will not be frame indifferent.  

 

There are various fixes for this MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the constitutive law can be written in terms of invariant quantities (eg by relating the rate of change of material stress to Lagrange strain rate); they can be derived from physical principles, in which case frame indifferent measures usually emerge naturally from the treatment; or frame indifferent measures of time derivatives can be specially constructed.

 

As a specific example, one way to construct a frame indifferent stress rate is to use the rate of change of stress components with respect to a basis that rotates with the solid (this is what an observer rotating with the material would actually see).  This sounds easy MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  we just choose some basis vectors { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHTbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaah2gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyB amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3980@  with each basis vector parallel to a particular material fiber.  But this doesn’t quite work, because of course the basis vectors won’t generally remain orthogonal under an arbitrary deformation.   So rather than attach { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHTbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaah2gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyB amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3980@  to particular material fibers, we simply suppose that they rotate with the average angular velocity of all material fibers passing through a particular point.  This means that

d m i dt =W m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaah2gadaWgaaWcba GaamyAaaqabaaakeaacaWGKbGaamiDaaaacqGH9aqpcaWHxbGaaCyB amaaBaaaleaacaWGPbaabeaaaaa@3953@

where W is the spin tensor.  Now, the time derivative of stress can be written as

dσ dt = d dt ( σ ij m i m j )= d σ ij dt m i m j + σ ij W m i m j + σ ij m i W m j = σ +WσσW MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaaho8aaeaacaWGKb GaamiDaaaacqGH9aqpdaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaa caGGOaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaah2gada WgaaWcbaGaamyAaaqabaGccqGHxkcXcaWHTbWaaSbaaSqaaiaadQga aeqaaOGaaiykaiabg2da9maalaaabaGaamizaiabeo8aZnaaBaaale aacaWGPbGaamOAaaqabaaakeaacaWGKbGaamiDaaaacaWHTbWaaSba aSqaaiaadMgaaeqaaOGaey4LIqSaaCyBamaaBaaaleaacaWGQbaabe aakiabgUcaRiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccaWH xbGaaCyBamaaBaaaleaacaWGPbaabeaakiabgEPielaah2gadaWgaa WcbaGaamOAaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaamyAaiaa dQgaaeqaaOGaaCyBamaaBaaaleaacaWGPbaabeaakiabgEPielaahE facaWHTbWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0ZaaCbiaeaacaWH dpaaleqabaGaey4bIenaaOGaey4kaSIaaC4vaiaaho8acqGHsislca WHdpGaaC4vaaaa@73DC@

Here, the first term can be interpreted as the stress rate seen by an observer rotating with the embedded basis; the second and third are the rates of change of stress arising from the rotation of the material.  The first term is of particular interest, and is called the Jaumann stress rate.  It is defined as

σ = dσ dt Wσ+σW MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaxacabaGaaC4WdaWcbeqaaiabgEGird aakiabg2da9maalaaabaGaamizaiaaho8aaeaacaWGKbGaamiDaaaa cqGHsislcaWHxbGaaC4WdiabgUcaRiaaho8acaWHxbaaaa@3EED@

It is easy to show that σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaxacabaGaaC4WdaWcbeqaaiabgEGird aaaaa@3386@  is frame indifferent.  Many constitutive equations assume that material stretch rate is proportional to this special stress rate.  For example, we could write

σ = C ˜ D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaxacabaGaaC4WdaWcbeqaaiabgEGird aakiabg2da9iqahoeagaacaiaahseaaaa@363E@

Provided that C ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqahoeagaacaaaa@3143@  is a frame indifferent fourth-order tensor, this constitutive equation would be frame indifferent.

 

This raises another question, of course.   What does it mean for a fourth-order tensor to be frame indifferent?   Hopefully you can answer this question for yourself!