Chapter 4

 

Solutions to simple boundary and initial value problems

 

 

 

4.3 Spherically symmetric solution to quasi-static large strain elasticity problems

 

4.3.1 Summary of governing equations of finite elasticity in Cartesian components

 

This section is intended to illustrate the nature of solutions to elasticity problems with large shape changes. 

 

We are given the following information

1.      The geometry of the solid

2.      A constitutive law for the material (i.e. the hyperelastic strain energy potential)

3.      The body force density b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaa aa@34C6@  (per unit mass) (if any)

4.      Prescribed boundary tractions t i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG0bWaaSbaaSqaaiaadMgaaeqaaa aa@34D8@  and/or boundary displacements u i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaa aa@34D9@

 

To simplify the problem we will assume

The solid is stress free in its undeformed configuration;

Temperature changes during deformation are neglected;

The solid is incompressible

 

With these assumptions, we wish to calculate the displacement field u i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaa aa@34D9@ , the left Cauchy-Green deformation tensor B ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGcbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@3595@  and the stress field σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3691@  satisfying the following equations:

 

 Displacement MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E8@ strain relation B ij = F ik F jk F ij = δ ij + u i x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGcbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadAeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGa amOramaaBaaaleaacaWGQbGaam4AaaqabaGccaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiab gUcaRmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaO qaaiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaaaa@618F@        

 Incompressibility condition J=det(F)=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeyypa0Jaciizaiaacwgaca GG0bGaaiikaiaahAeacaGGPaGaeyypa0JaaGymaaaa@3B4F@

 

  Stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E8@ strain relation σ ij =2[ ( U I 1 + I 1 U I 2 ) B ij ( I 1 U I ¯ 1 +2 I 2 U I ¯ 2 ) δ ij 3 U I 2 B ik B kj ]+p δ ij 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaaikdadaWadaqaamaabmaabaWaaSaaaeaacqGH ciITcaWGvbaabaGaeyOaIyRaamysamaaBaaaleaacaaIXaaabeaaaa GccqGHRaWkcaWGjbWaaSbaaSqaaiaaigdaaeqaaOWaaSaaaeaacqGH ciITcaWGvbaabaGaeyOaIyRaamysamaaBaaaleaacaaIYaaabeaaaa aakiaawIcacaGLPaaacaWGcbWaaSbaaSqaaiaadMgacaWGQbaabeaa kiabgkHiTmaabmaabaGaamysamaaBaaaleaacaaIXaaabeaakmaala aabaGaeyOaIyRaamyvaaqaaiabgkGi2kqadMeagaqeamaaBaaaleaa caaIXaaabeaaaaGccqGHRaWkcaaIYaGaamysamaaBaaaleaacaaIYa aabeaakmaalaaabaGaeyOaIyRaamyvaaqaaiabgkGi2kqadMeagaqe amaaBaaaleaacaaIYaaabeaaaaaakiaawIcacaGLPaaadaWcaaqaai abes7aKnaaBaaaleaacaWGPbGaamOAaaqabaaakeaacaaIZaaaaiab gkHiTmaalaaabaGaeyOaIyRaamyvaaqaaiabgkGi2kaadMeadaWgaa WcbaGaaGOmaaqabaaaaOGaamOqamaaBaaaleaacaWGPbGaam4Aaaqa baGccaWGcbWaaSbaaSqaaiaadUgacaWGQbaabeaaaOGaay5waiaaw2 faaiabgUcaRiaadchadaWcaaqaaiabes7aKnaaBaaaleaacaWGPbGa amOAaaqabaaakeaacaaIZaaaaaaa@73F1@

where σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@  is the Cauchy stress tensor,  U( I 1 , I 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacaGGOaGaamysamaaBaaaleaaca aIXaaabeaakiaacYcacaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaaiyk aaaa@36CA@  is the strain energy potential for the elastic solid, p is the hydrostatic part of the stress (which must be determined as part of the solution) and I 1 = B kk I 2 =( I 1 2 B ik B ki )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaamOqamaaBaaaleaacaWGRbGaam4AaaqabaGccaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaadMeadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaqa daqaaiaadMeadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHsislca WGcbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaadkeadaWgaaWcbaGa am4AaiaadMgaaeqaaaGccaGLOaGaayzkaaGaai4laiaaikdaaaa@583E@ .

 Equilibrium Equation σ ij y i +ρ b j =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kabeo8aZnaaBa aaleaacaWGPbGaamOAaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqa aiaadMgaaeqaaaaakiabgUcaRiabeg8aYjaadkgadaWgaaWcbaGaam OAaaqabaGccqGH9aqpcaaIWaaaaa@4207@  

 Traction boundary conditions σ ij n i = t j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaamOBamaaBaaaleaacaWGPbaabeaakiabg2da9iaadsha daWgaaWcbaGaamOAaaqabaaaaa@3BCC@  on parts of the boundary where tractions are known.

 Displacement boundary conditions u i = d i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaamizamaaBaaaleaacaWGPbaabeaaaaa@37EC@  on parts of the boundary where displacements are known.

 

 

4.3.2 Simplified equations for incompressible spherically symmetric solids

 

A representative spherically symmetric problem is illustrated in the picture.  We consider a hollow, spherical solid, which is subjected to spherically symmetric loading (i.e. internal body forces, as well as tractions or displacements applied to the surface, are independent of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@347C@  and ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzaaa@348E@ , and act in the radial direction only). 

 

The solution is most conveniently expressed using a spherical-polar coordinate system, illustrated in the figure.  For a finite deformation problem, we need a way to characterize the position of material particles in both the undeformed and deformed solid.  To do this, we let (R,Θ,Φ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOuaiaacYcacqqHyoquca GGSaGaeuOPdyKaaiykaaaa@3947@  identify a material particle in the undeformed solid. The coordinates of the same point in the deformed solid is identified by a new set of spherical-polar co-ordinates (r,θ,ϕ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaacYcacqaH4oqCca GGSaGaeqy1dyMaaiykaaaa@39F4@ .  One way to describe the deformation would be to specify each of the deformed coordinates (r,θ,ϕ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaacYcacqaH4oqCca GGSaGaeqy1dyMaaiykaaaa@39F4@  in terms of the reference coordinates (R,Θ,Φ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOuaiaacYcacqqHyoquca GGSaGaeuOPdyKaaiykaaaa@3947@ . For a spherically symmetric deformation, points only move radially, so that

r=f(R)θ=Θϕ=Φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0JaamOzaiaacIcaca WGsbGaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeI7aXjabg2da9i abfI5arjaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeqy1dy Maeyypa0JaaGPaVlabfA6agbaa@6A02@

 

In finite deformation problems vectors and tensors can be expressed as components in a basis { e R , e Θ , e Φ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaam OuaaqabaGccaGGSaGaaCyzamaaBaaaleaacqqHyoquaeqaaOGaaiil aiaahwgadaWgaaWcbaGaeuOPdyeabeaaaOGaay5Eaiaaw2haaaaa@3D8B@  associated with the position of material points in the undeformed solid, or, if more convenient, in a basis { e r , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaam OCaaqabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiil aiaahwgadaWgaaWcbaGaeqy1dygabeaaaOGaay5Eaiaaw2haaaaa@3E38@  associated with material points in the deformed solid.  For spherically symmetric deformations the two bases are identical MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  consequently, we can write

 Position vector in the undeformed solid        x=R e r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaeyypa0JaamOuaiaahwgada WgaaWcbaGaamOCaaqabaaaaa@37B4@

 Position vector in the deformed solid       y=r e r =f(R) e r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH5bGaeyypa0JaamOCaiaahwgada WgaaWcbaGaamOCaaqabaGccqGH9aqpcaWGMbGaaiikaiaadkfacaGG PaGaaCyzamaaBaaaleaacaWGYbaabeaaaaa@3E11@

 Displacement vector u=yx=r e r R e r =(f(R)R) e r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0JaaCyEaiabgkHiTi aahIhacqGH9aqpcaWGYbGaaCyzamaaBaaaleaacaWGYbaabeaakiab gkHiTiaadkfacaWHLbWaaSbaaSqaaiaadkhaaeqaaOGaeyypa0Jaai ikaiaadAgacaGGOaGaamOuaiaacMcacqGHsislcaWGsbGaaiykaiaa hwgadaWgaaWcbaGaamOCaaqabaaaaa@48FF@

 

 

The stress, deformation gradient and deformation tensors tensors (written as components in { e r , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaWGYb aabeaakiaacYcacaWHLbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa aCyzamaaBaaaleaacqaHvpGzaeqaaOGaaiyFaaaa@3E06@  ) have the form

σ[ σ rr 0 0 0 σ θθ 0 0 0 σ ϕϕ ]F[ F rr 0 0 0 F θθ 0 0 0 F ϕϕ ]B[ B rr 0 0 0 B θθ 0 0 0 B ϕϕ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdpGaeyyyIO7aamWaaeaafaqabe WadaaabaGaeq4Wdm3aaSbaaSqaaiaadkhacaWGYbaabeaaaOqaaiaa icdaaeaacaaIWaaabaGaaGimaaqaaiabeo8aZnaaBaaaleaacqaH4o qCcqaH4oqCaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGa eq4Wdm3aaSbaaSqaaiabew9aMjabew9aMbqabaaaaaGccaGLBbGaay zxaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caWHgbGaeyyyIO7aamWaaeaafaqabe WadaaabaGaamOramaaBaaaleaacaWGYbGaamOCaaqabaaakeaacaaI WaaabaGaaGimaaqaaiaaicdaaeaacaWGgbWaaSbaaSqaaiabeI7aXj abeI7aXbqabaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaWG gbWaaSbaaSqaaiabew9aMjabew9aMbqabaaaaaGccaGLBbGaayzxaa GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caWHcbGaeyyyIO7aamWaaeaafaqabeWada aabaGaamOqamaaBaaaleaacaWGYbGaamOCaaqabaaakeaacaaIWaaa baGaaGimaaqaaiaaicdaaeaacaWGcbWaaSbaaSqaaiabeI7aXjabeI 7aXbqabaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaWGcbWa aSbaaSqaaiabew9aMjabew9aMbqabaaaaaGccaGLBbGaayzxaaaaaa@96CD@

and furthermore must satisfy σ θθ = σ ϕϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaaaa@3EB0@   F rr = F θθ B θθ = B ϕϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadkhacaWGYb aabeaakiabg2da9iaadAeadaWgaaWcbaGaeqiUdeNaeqiUdehabeaa kiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaadkeadaWgaaWcbaGaeqiUdeNaeqiUdehabeaakiabg2da 9iaadkeadaWgaaWcbaGaeqy1dyMaeqy1dygabeaaaaa@52FD@ .

 

For spherical symmetry, the governing equations reduce to

 Strain Displacement Relations F rr = df dR F ϕϕ = F θθ = f(R) R B rr = ( df dR ) 2 B ϕϕ = B θθ = ( f(R) R ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadkhacaWGYb aabeaakiabg2da9maalaaabaGaamizaiaadAgaaeaacaWGKbGaamOu aaaacaaMc8UaaGPaVlaaykW7caaMc8UaamOramaaBaaaleaacqaHvp GzcqaHvpGzaeqaaOGaeyypa0JaamOramaaBaaaleaacqaH4oqCcqaH 4oqCaeqaaOGaeyypa0ZaaSaaaeaacaWGMbGaaiikaiaadkfacaGGPa aabaGaamOuaaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaamOqamaaBaaaleaacaWGYbGaamOCaaqabaGccqGH9aqpda qadaqaamaalaaabaGaamizaiaadAgaaeaacaWGKbGaamOuaaaaaiaa wIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccaaMc8UaaGPaVlaayk W7caWGcbWaaSbaaSqaaiabew9aMjabew9aMbqabaGccqGH9aqpcaWG cbWaaSbaaSqaaiabeI7aXjabeI7aXbqabaGccqGH9aqpdaqadaqaam aalaaabaGaamOzaiaacIcacaWGsbGaaiykaaqaaiaadkfaaaaacaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@7A8C@

 Incompressibility condition ( df dR ) ( f(R) R ) 2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaamaalaaabaGaamizaiaadA gaaeaacaWGKbGaamOuaaaaaiaawIcacaGLPaaadaqadaqaamaalaaa baGaamOzaiaacIcacaWGsbGaaiykaaqaaiaadkfaaaaacaGLOaGaay zkaaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaGymaaaa@4032@

 Stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E8@ Strain relations

σ rr =2[ ( U I 1 + I 1 U I 2 ) B rr I 1 3 U I 1 2 I 2 3 U I ¯ 2 U I 2 B rr 2 ]+p σ θθ = σ ϕϕ =2[ ( U I 1 + I 1 U I 2 ) B θθ I 1 3 U I 1 2 I 2 3 U I ¯ 2 U I 2 B θθ 2 ]+p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacqaHdpWCdaWgaaWcbaGaamOCai aadkhaaeqaaOGaeyypa0JaaGOmamaadmaabaWaaeWaaeaadaWcaaqa aiabgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaigdaae qaaaaakiabgUcaRiaadMeadaWgaaWcbaGaaGymaaqabaGcdaWcaaqa aiabgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaikdaae qaaaaaaOGaayjkaiaawMcaaiaadkeadaWgaaWcbaGaamOCaiaadkha aeqaaOGaeyOeI0YaaSaaaeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaa GcbaGaaG4maaaadaWcaaqaaiabgkGi2kaadwfaaeaacqGHciITcaWG jbWaaSbaaSqaaiaaigdaaeqaaaaakiabgkHiTmaalaaabaGaaGOmai aadMeadaWgaaWcbaGaaGOmaaqabaaakeaacaaIZaaaamaalaaabaGa eyOaIyRaamyvaaqaaiabgkGi2kqadMeagaqeamaaBaaaleaacaaIYa aabeaaaaGccqGHsisldaWcaaqaaiabgkGi2kaadwfaaeaacqGHciIT caWGjbWaaSbaaSqaaiaaikdaaeqaaaaakiaadkeadaqhaaWcbaGaam OCaiaadkhaaeaacaaIYaaaaaGccaGLBbGaayzxaaGaey4kaSIaamiC aaqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyypa0 Jaeq4Wdm3aaSbaaSqaaiabew9aMjabew9aMbqabaGccqGH9aqpcaaI YaWaamWaaeaadaqadaqaamaalaaabaGaeyOaIyRaamyvaaqaaiabgk Gi2kaadMeadaWgaaWcbaGaaGymaaqabaaaaOGaey4kaSIaamysamaa BaaaleaacaaIXaaabeaakmaalaaabaGaeyOaIyRaamyvaaqaaiabgk Gi2kaadMeadaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaGa amOqamaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyOeI0YaaSaaae aacaWGjbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaG4maaaadaWcaaqa aiabgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaigdaae qaaaaakiabgkHiTmaalaaabaGaaGOmaiaadMeadaWgaaWcbaGaaGOm aaqabaaakeaacaaIZaaaamaalaaabaGaeyOaIyRaamyvaaqaaiabgk Gi2kqadMeagaqeamaaBaaaleaacaaIYaaabeaaaaGccqGHsisldaWc aaqaaiabgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaik daaeqaaaaakiaadkeadaqhaaWcbaGaeqiUdeNaeqiUdehabaGaaGOm aaaaaOGaay5waiaaw2faaiabgUcaRiaadchaaaaa@AC3D@

 Equilibrium Equations

d σ rr dr + 1 r ( 2 σ rr σ θθ σ ϕϕ )+ ρ 0 b r =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacqaHdpWCdaWgaa WcbaGaamOCaiaadkhaaeqaaaGcbaGaamizaiaadkhaaaGaey4kaSYa aSaaaeaacaaIXaaabaGaamOCaaaadaqadaqaaiaaikdacqaHdpWCda WgaaWcbaGaamOCaiaadkhaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqa aiabeI7aXjabeI7aXbqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaeq y1dyMaeqy1dygabeaaaOGaayjkaiaawMcaaiaaykW7caaMc8Uaey4k aSIaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaamOyamaaBaaaleaaca WGYbaabeaakiabg2da9iaaicdaaaa@5999@

 Boundary Conditions

 

Prescribed Displacements u r (a)= g a u r (b)= g b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadkhaaeqaaO GaaiikaiaadggacaGGPaGaeyypa0Jaam4zamaaBaaaleaacaWGHbaa beaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caWG1bWaaSbaaSqaaiaadkhaaeqaaOGaaiikaiaadkgacaGGPa Gaeyypa0Jaam4zamaaBaaaleaacaWGIbaabeaaaaa@5A55@

Prescribed Tractions σ rr (a)= t a σ rr (b)= t b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaaiikaiaadggacaGGPaGaeyypa0JaamiDamaaBaaaleaa caWGHbaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaWGYbGaamOCaaqa baGccaGGOaGaamOyaiaacMcacqGH9aqpcaWG0bWaaSbaaSqaaiaadk gaaeqaaaaa@5322@

 

 

4.3.3 Pressurized hollow sphere made from an incompressible rubber

 

As an example, consider a pressurized hollow rubber shell, as shown in the picture. Assume that

 Before deformation, the sphere has inner radius A and outer radius B

 After deformation, the sphere has inner radius a and outer radius b

 The solid is made from an incompressible Mooney-Rivlin solid, with strain energy potential

U= μ 1 2 ( I 1 3)+ μ 2 2 ( I 2 3) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyvaiabg2da9maalaaabaGaeqiVd0 2aaSbaaSqaaiaaigdaaeqaaaGcbaGaaGOmaaaacaGGOaGaamysamaa BaaaleaacaaIXaaabeaakiabgkHiTiaaiodacaGGPaGaey4kaSYaaS aaaeaacqaH8oqBdaWgaaWcbaGaaGOmaaqabaaakeaacaaIYaaaaiaa cIcacaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaaG4maiaacM caaaa@43FD@

 No body forces act on the sphere

 The inner surface r=a is subjected to pressure p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaa aa@34CD@

 The outer surface r=b is subjected to pressure p b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadkgaaeqaaa aa@34CE@

 

The deformed radii a,b of the inner and outer surfaces of the spherical shell are related to the pressure by

p a p b μ 1 =2( 1 β 1 α )+ 1 2 ( 1 β 4 1 α 4 ) 2 μ 2 μ 1 ( βα )+ μ 2 μ 1 ( 1 β 2 1 α 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamiCamaaBaaaleaacaWGHb aabeaakiabgkHiTiaadchadaWgaaWcbaGaamOyaaqabaaakeaacqaH 8oqBdaWgaaWcbaGaaGymaaqabaaaaOGaeyypa0JaaGOmamaabmaaba WaaSaaaeaacaaIXaaabaGaeqOSdigaaiabgkHiTmaalaaabaGaaGym aaqaaiabeg7aHbaaaiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaaig daaeaacaaIYaaaamaabmaabaWaaSaaaeaacaaIXaaabaGaeqOSdi2a aWbaaSqabeaacaaI0aaaaaaakiabgkHiTmaalaaabaGaaGymaaqaai abeg7aHnaaCaaaleqabaGaaGinaaaaaaaakiaawIcacaGLPaaacqGH sisldaWcaaqaaiaaikdacqaH8oqBdaWgaaWcbaGaaGOmaaqabaaake aacqaH8oqBdaWgaaWcbaGaaGymaaqabaaaaOWaaeWaaeaacqaHYoGy cqGHsislcqaHXoqyaiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiabeY 7aTnaaBaaaleaacaaIYaaabeaaaOqaaiabeY7aTnaaBaaaleaacaaI XaaabeaaaaGcdaqadaqaamaalaaabaGaaGymaaqaaiabek7aInaaCa aaleqabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaaeaacqaH XoqydaahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGaayzkaaaaaa@690D@

where α=a/A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjabg2da9iaadggacaGGVaGaam yqaaaa@356C@ , β=b/B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjabg2da9iaadkgacaGGVaGaam Oqaaaa@3570@ , and α,β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjaacYcacqaHYoGyaaa@3458@  are related by

B 3 A 3 = 1 α 3 1 β 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamOqamaaCaaaleqabaGaaG 4maaaaaOqaaiaadgeadaahaaWcbeqaaiaaiodaaaaaaOGaeyypa0Za aSaaaeaacaaIXaGaeyOeI0IaeqySde2aaWbaaSqabeaacaaIZaaaaa GcbaGaaGymaiabgkHiTiabek7aInaaCaaaleqabaGaaG4maaaaaaaa aa@3D71@

Provided the pressure is not too large (see below), the preceding two equations can be solved for α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3207@  and β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIbaa@3209@  given the pressure and properties of the shell (for graphing purposes, it is better to assume a value for α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3207@ , calculate the corresponding β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIbaa@3209@ , and then determine the pressure).

 

The position  r of a material particle after deformation is related to its position R before deformation by

r A = ( R 3 A 3 + α 3 1 ) 1/3 R A = ( r 3 A 3 α 3 +1 ) 1/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadkhaaeaacaWGbbaaai abg2da9maabmaabaWaaSaaaeaacaWGsbWaaWbaaSqabeaacaaIZaaa aaGcbaGaamyqamaaCaaaleqabaGaaG4maaaaaaGccqGHRaWkcqaHXo qydaahaaWcbeqaaiaaiodaaaGccqGHsislcaaIXaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaIXaGaai4laiaaiodaaaGccaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7daWcaaqaaiaadkfaaeaaca WGbbaaaiabg2da9maabmaabaWaaSaaaeaacaWGYbWaaWbaaSqabeaa caaIZaaaaaGcbaGaamyqamaaCaaaleqabaGaaG4maaaaaaGccqGHsi slcqaHXoqydaahaaWcbeqaaiaaiodaaaGccqGHRaWkcaaIXaaacaGL OaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4laiaaiodaaaaaaa@68C1@

The deformation tensor distribution in the sphere is

B rr = (R/r) 4 B θθ = B ϕϕ = (r/R) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGcbWaaSbaaSqaaiaadkhacaWGYb aabeaakiabg2da9iaacIcacaWGsbGaai4laiaadkhacaGGPaWaaWba aSqabeaacaaI0aaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaamOqamaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyypa0Ja amOqamaaBaaaleaacqaHvpGzcqaHvpGzaeqaaOGaeyypa0Jaaiikai aadkhacaGGVaGaamOuaiaacMcadaahaaWcbeqaaiaaikdaaaaaaa@548D@

The Cauchy stress in the sphere is

σ rr = μ 1 ( 2R r + R 4 2 r 4 ) μ 2 ( 2r R R 2 r 2 )+C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcqaH8oqBdaWgaaWcbaGaaGymaaqabaGcdaqadaqa amaalaaabaGaaGOmaiaadkfaaeaacaWGYbaaaiabgUcaRmaalaaaba GaamOuamaaCaaaleqabaGaaGinaaaaaOqaaiaaikdacaWGYbWaaWba aSqabeaacaaI0aaaaaaaaOGaayjkaiaawMcaaiabgkHiTiabeY7aTn aaBaaaleaacaaIYaaabeaakmaabmaabaWaaSaaaeaacaaIYaGaamOC aaqaaiaadkfaaaGaeyOeI0YaaSaaaeaacaWGsbWaaWbaaSqabeaaca aIYaaaaaGcbaGaamOCamaaCaaaleqabaGaaGOmaaaaaaaakiaawIca caGLPaaacqGHRaWkcaWGdbaaaa@4F97@

σ θθ = μ 1 ( 2R r R 4 2 r 4 + r 2 R 2 ) μ 2 ( 2r R r 4 R 4 )+C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4o qCaeqaaOGaeyypa0JaeqiVd02aaSbaaSqaaiaaigdaaeqaaOWaaeWa aeaadaWcaaqaaiaaikdacaWGsbaabaGaamOCaaaacqGHsisldaWcaa qaaiaadkfadaahaaWcbeqaaiaaisdaaaaakeaacaaIYaGaamOCamaa CaaaleqabaGaaGinaaaaaaGccqGHRaWkdaWcaaqaaiaadkhadaahaa WcbeqaaiaaikdaaaaakeaacaWGsbWaaWbaaSqabeaacaaIYaaaaaaa aOGaayjkaiaawMcaaiabgkHiTiabeY7aTnaaBaaaleaacaaIYaaabe aakmaabmaabaWaaSaaaeaacaaIYaGaamOCaaqaaiaadkfaaaGaeyOe I0YaaSaaaeaacaWGYbWaaWbaaSqabeaacaaI0aaaaaGcbaGaamOuam aaCaaaleqabaGaaGinaaaaaaaakiaawIcacaGLPaaacqGHRaWkcaWG dbaaaa@55CA@

C= μ 1 2 ( 2 α + 1 2 α 4 + 2 β + 1 2 β 4 )+ μ 2 2 ( 2α 1 α 2 +2β 1 β 2 ) p a + p b 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeacqGH9aqpcqGHsisldaWcaaqaai abeY7aTnaaBaaaleaacaaIXaaabeaaaOqaaiaaikdaaaWaaeWaaeaa daWcaaqaaiaaikdaaeaacqaHXoqyaaGaey4kaSYaaSaaaeaacaaIXa aabaGaaGOmaiabeg7aHnaaCaaaleqabaGaaGinaaaaaaGccqGHRaWk daWcaaqaaiaaikdaaeaacqaHYoGyaaGaey4kaSYaaSaaaeaacaaIXa aabaGaaGOmaiabek7aInaaCaaaleqabaGaaGinaaaaaaaakiaawIca caGLPaaacqGHRaWkdaWcaaqaaiabeY7aTnaaBaaaleaacaaIYaaabe aaaOqaaiaaikdaaaWaaeWaaeaacaaIYaGaeqySdeMaeyOeI0YaaSaa aeaacaaIXaaabaGaeqySde2aaWbaaSqabeaacaaIYaaaaaaakiabgU caRiaaikdacqaHYoGycqGHsisldaWcaaqaaiaaigdaaeaacqaHYoGy daahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaeyOeI0YaaS aaaeaacaWGWbWaaSbaaSqaaiaadggaaeqaaOGaey4kaSIaamiCamaa BaaaleaacaWGIbaabeaaaOqaaiaaikdaaaaaaa@62A2@

 

The variation of the internal radius of the spherical shell with applied pressure is plotted in the figure, for μ 2 / μ 1 =0.04 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaaIYaaabeaaki aac+cacqaH8oqBdaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIWaGa aiOlaiaaicdacaaI0aaaaa@3A54@  (a representative value for a typical rubber).  For comparison, the linear elastic solution (obtained by setting E=3( μ 1 + μ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacqGH9aqpcaaIZaGaaiikaiabeY 7aTnaaBaaaleaacaaIXaaabeaakiabgUcaRiabeY7aTnaaBaaaleaa caaIYaaabeaakiaacMcacaaMc8oaaa@3C0A@  and ν=1/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUjabg2da9iaaigdacaGGVaGaaG Omaaaa@3550@  in the formulas given in section 4.1.4) is also shown.  Note that:

1.      The small strain solution is accurate for u/A<0.05 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhacaGGVaGaamyqaiabgYda8iaaic dacaGGUaGaaGimaiaaiwdaaaa@36C4@

2.      The relationship between pressure and displacement is nonlinear in the large deformation regime.

3.      As the internal radius of the sphere increases, the pressure reaches a maximum, and thereafter decreases (this will be familiar behavior to anyone who has inflated a balloon).  This is  because the wall thickness of the shell decreases as the sphere expands.

 

The stress distribution for various displacements in the shell is plotted in the figures below, for p b =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaamOyaaqabaGccq GH9aqpcaaIWaaaaa@343A@ , μ 2 / μ 1 =0.04 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaaIYaaabeaaki aac+cacqaH8oqBdaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIWaGa aiOlaiaaicdacaaI0aaaaa@3A54@  and B/A=3.  The radial stress remains close to the linear elastic solution even in the large deformation regime.  The hoop stress distribution is significantly altered as the deformation increases, however.

            

  

 

Derivation

1.      Integrate the incompressibility condition from the inner radius of the sphere to some arbitrary point R

f(A) f(R) [ f(R) ] 2 df= A R R 2 dR f (R) 3 f (A) 3 = R 3 A 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdXbqaamaadmaabaGaamOzaiaacI cacaWGsbGaaiykaaGaay5waiaaw2faamaaCaaaleqabaGaaGOmaaaa kiaadsgacaWGMbGaeyypa0Zaa8qCaeaacaWGsbWaaWbaaSqabeaaca aIYaaaaOGaamizaiaadkfaaSqaaiaadgeaaeaacaWGsbaaniabgUIi YdaaleaacaWGMbGaaiikaiaadgeacaGGPaaabaGaamOzaiaacIcaca WGsbGaaiykaaqdcqGHRiI8aOGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaeyO0H4TaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaamOzaiaacIcacaWGsbGaaiykamaaCaaaleqabaGaaG4maaaa kiabgkHiTiaadAgacaGGOaGaamyqaiaacMcadaahaaWcbeqaaiaaio daaaGccqGH9aqpcaWGsbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Ia amyqamaaCaaaleqabaGaaG4maaaaaaa@6F11@

2.      Note that f(R)=r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbGaaiikaiaadkfacaGGPaGaey ypa0JaamOCaaaa@37DE@  by definition, and f(A)=a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbGaaiikaiaadgeacaGGPaGaey ypa0Jaamyyaaaa@37BC@  since the point at R=A moves to r=a after deformation.  This gives the relationship between the position r of a point in the deformed solid and its position R before deformation

r=f(R)= R 3 + a 3 A 3 3 R= f 1 (r)= r 3 + A 3 a 3 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0JaamOzaiaacIcaca WGsbGaaiykaiabg2da9iaaykW7daGcbaqaaiaadkfadaahaaWcbeqa aiaaiodaaaGccqGHRaWkcaWGHbWaaWbaaSqabeaacaaIZaaaaOGaey OeI0IaamyqamaaCaaaleqabaGaaG4maaaaaeaacaaIZaaaaOGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaadkfacqGH9aqpcaWGMbWaaWbaaSqabeaacqGHsi slcaaIXaaaaOGaaiikaiaadkhacaGGPaGaeyypa0JaaGPaVpaakeaa baGaamOCamaaCaaaleqabaGaaG4maaaakiabgUcaRiaadgeadaahaa WcbeqaaiaaiodaaaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIZaaa aaqaaiaaiodaaaaaaa@7154@

3.      The components of the Cauchy-Green tensor follow as B rr = (R/r) 4 B θθ = B ϕϕ = (r/R) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGcbWaaSbaaSqaaiaadkhacaWGYb aabeaakiabg2da9iaacIcacaWGsbGaai4laiaadkhacaGGPaWaaWba aSqabeaacaaI0aaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaamOqamaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyypa0Ja amOqamaaBaaaleaacqaHvpGzcqaHvpGzaeqaaOGaeyypa0Jaaiikai aadkhacaGGVaGaamOuaiaacMcadaahaaWcbeqaaiaaikdaaaaaaa@548D@

4.      The stresses follow from the stress-strain equation as

σ rr = 2 3 ( μ 1 + μ 2 B θθ )( B rr B θθ )+p σ θθ = σ ϕϕ = 1 3 ( μ 1 + μ 2 B θθ )( B θθ B rr )+p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpdaWcaaqaaiaaikdaaeaacaaIZaaaamaabmaabaGa eqiVd02aaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeqiVd02aaSbaaS qaaiaaikdaaeqaaOGaamOqamaaBaaaleaacqaH4oqCcqaH4oqCaeqa aaGccaGLOaGaayzkaaWaaeWaaeaacaWGcbWaaSbaaSqaaiaadkhaca WGYbaabeaakiabgkHiTiaadkeadaWgaaWcbaGaeqiUdeNaeqiUdeha beaaaOGaayjkaiaawMcaaiabgUcaRiaadchacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWg aaWcbaGaeqiUdeNaeqiUdehabeaakiabg2da9iabeo8aZnaaBaaale aacqaHvpGzcqaHvpGzaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGa aG4maaaadaqadaqaaiabeY7aTnaaBaaaleaacaaIXaaabeaakiabgU caRiabeY7aTnaaBaaaleaacaaIYaaabeaakiaadkeadaWgaaWcbaGa eqiUdeNaeqiUdehabeaaaOGaayjkaiaawMcaamaabmaabaGaamOqam aaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyOeI0IaamOqamaaBaaa leaacaWGYbGaamOCaaqabaaakiaawIcacaGLPaaacqGHRaWkcaWGWb aaaa@82CB@

5.      Substituting these stresses into the equilibrium equation leads to the following differential equation for σ rr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaaaaa@3445@

d σ rr dr + 2 r ( μ 1 + μ 2 B θθ )( B rr B θθ )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiabeo8aZnaaBaaale aacaWGYbGaamOCaaqabaaakeaacaWGKbGaamOCaaaacaaMc8UaaGPa VlabgUcaRmaalaaabaGaaGOmaaqaaiaadkhaaaGaaGPaVpaabmaaba GaeqiVd02aaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeqiVd02aaSba aSqaaiaaikdaaeqaaOGaamOqamaaBaaaleaacqaH4oqCcqaH4oqCae qaaaGccaGLOaGaayzkaaWaaeWaaeaacaWGcbWaaSbaaSqaaiaadkha caWGYbaabeaakiabgkHiTiaadkeadaWgaaWcbaGaeqiUdeNaeqiUde habeaaaOGaayjkaiaawMcaaiabg2da9iaaicdaaaa@561B@

6.      After substituting for B rr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkeadaWgaaWcbaGaamOCaiaadkhaae qaaaaa@3349@  and B θθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkeadaWgaaWcbaGaeqiUdeNaeqiUde habeaaaaa@34C7@ , and expressing R in terms of r, this equation can be integrated and simplified to see that

σ rr = μ 1 R(4 r 3 + R 3 ) 2 r 4 μ 2 (2 r 3 R 3 ) r 2 R +C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcqaH8oqBdaWgaaWcbaGaaGymaaqabaGcdaWcaaqa aiaadkfacaGGOaGaaGinaiaadkhadaahaaWcbeqaaiaaiodaaaGccq GHRaWkcaWGsbWaaWbaaSqabeaacaaIZaaaaOGaaiykaaqaaiaaikda caWGYbWaaWbaaSqabeaacaaI0aaaaaaakiabgkHiTiabeY7aTnaaBa aaleaacaaIYaaabeaakmaalaaabaGaaiikaiaaikdacaWGYbWaaWba aSqabeaacaaIZaaaaOGaeyOeI0IaamOuamaaCaaaleqabaGaaG4maa aakiaacMcaaeaacaWGYbWaaWbaaSqabeaacaaIYaaaaOGaamOuaaaa cqGHRaWkcaWGdbaaaa@5102@

7.      The boundary conditions require that σ rr = p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcqGHsislcaWGWbWaaSbaaSqaaiaadggaaeqaaaaa @3849@  on (r=a,R=A), while σ rr = p b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcqGHsislcaWGWbWaaSbaaSqaaiaadkgaaeqaaaaa @384A@  on (r=b,R=B), which requires

p a = μ 1 ( 2 α + 1 2 α 4 ) μ 2 ( 2α 1 α 2 )+C p b = μ 1 ( 2 β + 1 2 β 4 ) μ 2 ( 2β 1 β 2 )+C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeyOeI0IaamiCamaaBaaaleaaca WGHbaabeaakiabg2da9iabeY7aTnaaBaaaleaacaaIXaaabeaakmaa bmaabaWaaSaaaeaacaaIYaaabaGaeqySdegaaiabgUcaRmaalaaaba GaaGymaaqaaiaaikdacqaHXoqydaahaaWcbeqaaiaaisdaaaaaaaGc caGLOaGaayzkaaGaeyOeI0IaeqiVd02aaSbaaSqaaiaaikdaaeqaaO WaaeWaaeaacaaIYaGaeqySdeMaeyOeI0YaaSaaaeaacaaIXaaabaGa eqySde2aaWbaaSqabeaacaaIYaaaaaaaaOGaayjkaiaawMcaaiabgU caRiaadoeaaeaacqGHsislcaWGWbWaaSbaaSqaaiaadkgaaeqaaOGa eyypa0JaeqiVd02aaSbaaSqaaiaaigdaaeqaaOWaaeWaaeaadaWcaa qaaiaaikdaaeaacqaHYoGyaaGaey4kaSYaaSaaaeaacaaIXaaabaGa aGOmaiabek7aInaaCaaaleqabaGaaGinaaaaaaaakiaawIcacaGLPa aacqGHsislcqaH8oqBdaWgaaWcbaGaaGOmaaqabaGcdaqadaqaaiaa ikdacqaHYoGycqGHsisldaWcaaqaaiaaigdaaeaacqaHYoGydaahaa WcbeqaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaey4kaSIaam4qaaaa aa@6A8A@

where α=a/A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjabg2da9iaadggacaGGVaGaam yqaaaa@356C@  and β=b/B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjabg2da9iaadkgacaGGVaGaam Oqaaaa@3570@ .  The expression that relates α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3207@  and β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIbaa@3209@  to the pressure follows by subtracting the first equation from the second.   Adding the two equations gives the expression for C.

8.      Finally, the hoop stress follows by noting that, from (4) σ θθ σ rr =( μ 1 + μ 2 B θθ )( B θθ B rr ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4o qCaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadkhacaWGYbaabeaa kiabg2da9maabmaabaGaeqiVd02aaSbaaSqaaiaaigdaaeqaaOGaey 4kaSIaeqiVd02aaSbaaSqaaiaaikdaaeqaaOGaamOqamaaBaaaleaa cqaH4oqCcqaH4oqCaeqaaaGccaGLOaGaayzkaaWaaeWaaeaacaWGcb WaaSbaaSqaaiabeI7aXjabeI7aXbqabaGccqGHsislcaWGcbWaaSba aSqaaiaadkhacaWGYbaabeaaaOGaayjkaiaawMcaaaaa@5194@