Chapter 2

 

Governing Equations

 

 

 

2.4 Work done by stresses; Principle of Virtual Work

 

In this section, we derive formulas that enable you to calculate the work done by stresses acting on a solid.  In addition, we prove the principle of virtual work MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  which is an alternative way of expressing the equations of motion and equilibrium derived in Section 2.3.  The principle of virtual work is the starting point for finite element analysis, and so is a particularly important result.

 

 

2.4.1 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.

 

 

2.4.2 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@

 

 

2.4.3 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@

 

 

 

2.4.4 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.

 

 

 

2.4.5 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.

 

 

 

2.4.6 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@