2.3 Equations of motion and equilibrium for deformable solids

 

In this section, we generalize Newton’s laws of motion (conservation of linear and angular momentum) to a deformable solid.

 

 

2.3.1 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 one can show (see Appendix D) 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.

 

 

 

2.3.2 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 x 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 IyRaamiEamaaBaaaleaacaWGTbaabeaaaaGcdaqadaqaaiabgIGiop aaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaamyEamaaBaaaleaa caWGQbaabeaakiabeo8aZnaaBaaaleaacaWGTbGaam4Aaaqabaaaki aawIcacaGLPaaacaWGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIipa aOqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua eyypa0Zaa8quaeaacqGHiiIZdaWgaaWcbaGaamyAaiaadQgacaWGRb aabeaakmaabmaabaGaeqiTdq2aaSbaaSqaaiaadQgacaWGTbaabeaa kiabeo8aZnaaBaaaleaacaWGTbGaam4AaaqabaGccqGHRaWkcaWG5b WaaSbaaSqaaiaadQgaaeqaaOWaaSaaaeaacqGHciITcqaHdpWCdaWg aaWcbaGaamyBaiaadUgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaale aacaWGTbaabeaaaaaakiaawIcacaGLPaaacaWGKbGaamOvaaWcbaGa amOvaaqab0Gaey4kIipaaaaa@023E@

We may also show (see Appendix D) 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.

 

 

 

2.3.3 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 (or see Appendix D for a reminder) 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@