Chapter 4

 

Solutions to simple boundary and initial value problems

 

 

 

In this chapter, we derive exact solutions to several problems involving deformable solids.  The examples have been selected partly because they can easily be solved, partly because they illustrate clearly the role of the various governing equations and boundary conditions in controlling the solution, and partly because the solutions themselves are of some practical interest.

 

 

4.1 Axially and spherically symmetric solutions to quasi-static linear elastic problems

 

 

4.1.1 Summary of governing equations of linear elasticity in Cartesian components

 

At last, we have all the basic equations we need to solve problems involving elastic materials subjected to loading. 

 

Specifically, we are given the following information

1.      Geometry of the solid

2.      Constitutive law for the material (i.e. the linear elastic-stress-strain equations)

3.      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.      Temperature distribution ΔT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcaWGubaaaa@3504@  (if any)

5.      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@

 

In addition, to simplify the problem, we make the following assumptions

1.      All displacements are small.  This means that we can use the infinitesimal strain tensor to characterize deformation; we do not need to distinguish between stress measures, and we do not need to distinguish between deformed and undeformed configurations of the solid when writing equilibrium equations and boundary conditions.

2.      The material is an isotropic, linear elastic solid, with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347D@ , and mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaa@356C@

 

With these assumptions, we need to solve for the displacement field u i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaa aa@34D9@ , the strain field ε ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3675@  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 ε ij = 1 2 ( u i x j + u j x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqa amaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaOqaai abgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOGaey4kaSYaaSaa aeaacqGHciITcaWG1bWaaSbaaSqaaiaadQgaaeqaaaGcbaGaeyOaIy RaamiEamaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaaaaa@49AF@

 Stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E8@ strain relation σ ij = E 1+ν { ε ij + ν 12ν ε kk δ ij } EαΔT 12ν δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaGaaGymaiabgUcaRiab e27aUbaadaGadaqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaaqaba GccqGHRaWkdaWcaaqaaiabe27aUbqaaiaaigdacqGHsislcaaIYaGa eqyVd4gaaiabew7aLnaaBaaaleaacaWGRbGaam4AaaqabaGccqaH0o azdaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGL7bGaayzFaaGaeyOe I0YaaSaaaeaacaWGfbGaeqySdeMaeuiLdqKaamivaaqaaiaaigdacq GHsislcaaIYaGaeqyVd4gaaiabes7aKnaaBaaaleaacaWGPbGaamOA aaqabaaaaa@5D65@

 Equilibrium Equation σ ij x i + ρ 0 b j =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kabeo8aZnaaBa aaleaacaWGPbGaamOAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqa aiaadMgaaeqaaaaakiabgUcaRiabeg8aYnaaBaaaleaacaaIWaaabe aakiaadkgadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaaIWaaaaa@42F6@  (static problems only MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  you need the acceleration terms for dynamic problems)

 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.

 

 

4.1.2 Simplified equations for spherically symmetric linear elasticity problems

 

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).  If the temperature of the sphere is non-uniform, it must also be spherically symmetric (a function of R only).

 

The solution is most conveniently expressed using a spherical-polar coordinate system, illustrated in the figure.  The general procedure for solving problems using spherical and cylindrical coordinates is complicated, and is discussed in detail in Appendix E.  In this section, we summarize the special form of these equations for spherically symmetric problems.

 

As usual, a point in the solid is identified by its spherical-polar co-ordinates (R,θ,ϕ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOuaiaacYcacqaH4oqCca GGSaGaeqy1dyMaaiykaaaa@39D4@ . All vectors and tensors are expressed as components in the basis { e R , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaam OuaaqabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiil aiaahwgadaWgaaWcbaGaeqy1dygabeaaaOGaay5Eaiaaw2haaaaa@3E18@  shown in the figure.  For a spherically symmetric problem

 Position Vector       x=R e R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaeyypa0JaamOuaiaahwgada WgaaWcbaGaamOuaaqabaaaaa@3794@

 Displacement vector u=u(R) e R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0JaamyDaiaacIcaca WGsbGaaiykaiaahwgadaWgaaWcbaGaamOuaaqabaaaaa@39E4@

 Body force vector b= ρ 0 b(R) e R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHIbGaeyypa0JaeqyWdi3aaSbaaS qaaiaaicdaaeqaaOGaamOyaiaacIcacaWGsbGaaiykaiaahwgadaWg aaWcbaGaamOuaaqabaaaaa@3C6E@

 

Here, u( R ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaeWaaeaacaWGsbaacaGLOa Gaayzkaaaaaa@3620@  and b( R ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGIbWaaeWaaeaacaWGsbaacaGLOa Gaayzkaaaaaa@360D@  are scalar functions. The stress and strain tensors (written as components in { e R , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaWGsb aabeaakiaacYcacaWHLbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa aCyzamaaBaaaleaacqaHvpGzaeqaaOGaaiyFaaaa@3DE6@  ) have the form

σ[ σ RR 0 0 0 σ θθ 0 0 0 σ ϕϕ ]ε[ ε RR 0 0 0 ε θθ 0 0 0 ε ϕϕ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCcqGHHjIUdaWadaqaauaabe qadmaaaeaacqaHdpWCdaWgaaWcbaGaamOuaiaadkfaaeqaaaGcbaGa aGimaaqaaiaaicdaaeaacaaIWaaabaGaeq4Wdm3aaSbaaSqaaiabeI 7aXjabeI7aXbqabaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaa cqaHdpWCdaWgaaWcbaGaeqy1dyMaeqy1dygabeaaaaaakiaawUfaca GLDbaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabew7aLj abggMi6oaadmaabaqbaeqabmWaaaqaaiabew7aLnaaBaaaleaacaWG sbGaamOuaaqabaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacq aH1oqzdaWgaaWcbaGaeqiUdeNaeqiUdehabeaaaOqaaiaaicdaaeaa caaIWaaabaGaaGimaaqaaiabew7aLnaaBaaaleaacqaHvpGzcqaHvp GzaeqaaaaaaOGaay5waiaaw2faaaaa@78FB@

and furthermore must satisfy σ θθ = σ ϕϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaaaa@3EB0@   ε θθ = ε ϕϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabew7aLnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaaaa@3E78@ . The tensor components have exactly the same physical interpretation as they did when we used a fixed { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BC5@  basis, except that the subscripts (1,2,3) have been replaced by (R,θ,ϕ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOuaiaacYcacqaH4oqCca GGSaGaeqy1dyMaaiykaaaa@39D3@ .

 

For spherical symmetry, the governing equations of linear elasticity reduce to

 

 Strain Displacement Relations ε RR = du dR ε ϕϕ = ε θθ = u R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0ZaaSaaaeaacaWGKbGaamyDaaqaaiaadsgacaWG sbaaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlabew7aLnaaBaaaleaacqaHvpGzcqaHvpGzaeqaaOGa eyypa0JaeqyTdu2aaSbaaSqaaiabeI7aXjabeI7aXbqabaGccqGH9a qpdaWcaaqaaiaadwhaaeaacaWGsbaaaaaa@5790@

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

σ RR = E ( 1+ν )( 12ν ) { (1ν) ε RR +ν ε θθ +ν ε ϕϕ } EαΔT 12ν σ θθ = σ ϕϕ = E ( 1+ν )( 12ν ) { ε θθ +ν ε RR } EαΔT 12ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaWGsb GaamOuaaqabaGccqGH9aqpdaWcaaqaaiaadweaaeaadaqadaqaaiaa igdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaadaqadaqaaiaaigdacq GHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaaamaacmaabaGaaiik aiaaigdacqGHsislcqaH9oGBcaGGPaGaeqyTdu2aaSbaaSqaaiaadk facaWGsbaabeaakiabgUcaRiabe27aUjabew7aLnaaBaaaleaacqaH 4oqCcqaH4oqCaeqaaOGaey4kaSIaeqyVd4MaeqyTdu2aaSbaaSqaai abew9aMjabew9aMbqabaaakiaawUhacaGL9baacqGHsisldaWcaaqa aiaadweacqaHXoqycqqHuoarcaWGubaabaGaaGymaiabgkHiTiaaik dacqaH9oGBaaaabaGaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7aXbqa baGccqGH9aqpcqaHdpWCdaWgaaWcbaGaeqy1dyMaeqy1dygabeaaki abg2da9maalaaabaGaamyraaqaamaabmaabaGaaGymaiabgUcaRiab e27aUbGaayjkaiaawMcaamaabmaabaGaaGymaiabgkHiTiaaikdacq aH9oGBaiaawIcacaGLPaaaaaWaaiWaaeaacqaH1oqzdaWgaaWcbaGa eqiUdeNaeqiUdehabeaakiabgUcaRiabe27aUjabew7aLnaaBaaale aacaWGsbGaamOuaaqabaaakiaawUhacaGL9baacqGHsisldaWcaaqa aiaadweacqaHXoqycqqHuoarcaWGubaabaGaaGymaiabgkHiTiaaik dacqaH9oGBaaaaaaa@96AD@

 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 WcbaGaamOuaiaadkfaaeqaaaGcbaGaamizaiaadkfaaaGaey4kaSYa aSaaaeaacaaIXaaabaGaamOuaaaadaqadaqaaiaaikdacqaHdpWCda WgaaWcbaGaamOuaiaadkfaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqa aiabeI7aXjabeI7aXbqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaeq y1dyMaeqy1dygabeaaaOGaayjkaiaawMcaaiaaykW7caaMc8Uaey4k aSIaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaamOyamaaBaaaleaaca WGsbaabeaakiabg2da9iaaicdaaaa@58B9@

 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 caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadkfaaeqaaO GaaiikaiaadggacaGGPaGaeyypa0Jaam4zamaaBaaaleaacaWGHbaa beaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caWG1bWaaSbaaSqaaiaadkfaaeqaaOGaaiikaiaadkgacaGGPa Gaeyypa0Jaam4zamaaBaaaleaacaWGIbaabeaaaaa@5A15@

Prescribed Tractions σ RR (a)= t a σ RR (b)= t b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaaiikaiaadggacaGGPaGaeyypa0JaamiDamaaBaaaleaa caWGHbaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaWGsbGaamOuaaqa baGccaGGOaGaamOyaiaacMcacqGH9aqpcaWG0bWaaSbaaSqaaiaadk gaaeqaaaaa@52A2@

 

These results can either be derived as a special case of the general 3D equations of linear elasticity in spherical coordinates, or alternatively can be obtained directly from the formulas in Cartesian components.  Here, we briefly outline the the latter.

1.      Note that we can find the components of { e R , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaam OuaaqabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiil aiaahwgadaWgaaWcbaGaeqy1dygabeaaaOGaay5Eaiaaw2haaaaa@3E18@  in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  basis as follows. First, note that e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaadkfaaeqaaa aa@34B7@  is radial, and can be written in terms of the position vector as x/| x | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaai4lamaaemaabaGaaCiEaa Gaay5bSlaawIa7aaaa@389D@ .  Next, note e ϕ = e 3 × e R /| e 3 × e R | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiabew9aMbqaba GccqGH9aqpcaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaey41aqRaaCyz amaaBaaaleaacaWGsbaabeaakiaac+cadaabdaqaaiaahwgadaWgaa WcbaGaaG4maaqabaGccqGHxdaTcaWHLbWaaSbaaSqaaiaadkfaaeqa aaGccaGLhWUaayjcSdaaaa@4673@  and e θ = e ϕ × e R /| e ϕ × e R | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiabeI7aXbqaba GccqGH9aqpcaWHLbWaaSbaaSqaaiabew9aMbqabaGccqGHxdaTcaWH LbWaaSbaaSqaaiaadkfaaeqaaOGaai4lamaaemaabaGaaCyzamaaBa aaleaacqaHvpGzaeqaaOGaey41aqRaaCyzamaaBaaaleaacaWGsbaa beaaaOGaay5bSlaawIa7aaaa@4877@ .  Using index notation, the components of the basis vectors { e R , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaam OuaaqabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiil aiaahwgadaWgaaWcbaGaeqy1dygabeaaaOGaay5Eaiaaw2haaaaa@3E18@  in { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  are therefore

{ x i R , x 3 x i R 2 δ i3 R 2 x 3 2 , i3j R x j R 2 x 3 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaamaalaaabaGaamiEamaaBa aaleaacaWGPbaabeaaaOqaaiaadkfaaaGaaiilamaalaaabaGaamiE amaaBaaaleaacaaIZaaabeaakiaadIhadaWgaaWcbaGaamyAaaqaba GccqGHsislcaWGsbWaaWbaaSqabeaacaaIYaaaaOGaeqiTdq2aaSba aSqaaiaadMgacaaIZaaabeaaaOqaaiaadkfadaahaaWcbeqaaiaaik daaaGccqGHsislcaWG4bWaa0baaSqaaiaaiodaaeaacaaIYaaaaaaa kiaacYcadaWcaaqaaiabgIGiopaaBaaaleaacaWGPbGaaG4maiaadQ gaaeqaaOGaamOuaiaadIhadaWgaaWcbaGaamOAaaqabaaakeaacaWG sbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamiEamaaDaaaleaaca aIZaaabaGaaGOmaaaaaaaakiaawUhacaGL9baaaaa@55E2@

where R=| x |= x k x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbGaeyypa0ZaaqWaaeaacaWH4b aacaGLhWUaayjcSdGaeyypa0ZaaOaaaeaacaWG4bWaaSbaaSqaaiaa dUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaaaeqaaaaa@3E18@ , δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3416@  is the Kronecker delta and ijk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgIGiopaaBaaaleaacaWGPbGaamOAai aadUgaaeqaaaaa@34E5@  is the permutation symbol.

2.      The components of the (radial) displacement vector in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  basis are u i =u(R) x i /R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaamyDaiaacIcacaWGsbGaaiykaiaadIhadaWgaaWcbaGa amyAaaqabaGccaGGVaGaamOuaaaa@3CBF@ .

3.      To proceed with the algebra, it is helpful to remember that x i / x j = δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadM gaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGc cqGH9aqpcqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaaaa@3F3C@ R/ x j = x j /R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWGsbGaai4laiabgkGi2k aadIhadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaWG4bWaaSbaaSqa aiaadQgaaeqaaOGaai4laiaadkfaaaa@3DF0@  and R 1 / x j = x j / R 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWGsbWaaWbaaSqabeaacq GHsislcaaIXaaaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOA aaqabaGccqGH9aqpcqGHsislcaWG4bWaaSbaaSqaaiaadQgaaeqaaO Gaai4laiaadkfadaahaaWcbeqaaiaaiodaaaaaaa@41A6@

4.      The components of the strain tensor in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  basis therefore follow as

ε ij = 1 2 ( u i x j + u j x i )= du dR x i x j R 2 +u(R)( δ ij R x i x j R 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqa amaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaOqaai abgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOGaey4kaSYaaSaa aeaacqGHciITcaWG1bWaaSbaaSqaaiaadQgaaeqaaaGcbaGaeyOaIy RaamiEamaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaacqGH 9aqpdaWcaaqaaiaadsgacaWG1baabaGaamizaiaadkfaaaWaaSaaae aacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaamiEamaaBaaaleaacaWG QbaabeaaaOqaaiaadkfadaahaaWcbeqaaiaaikdaaaaaaOGaey4kaS IaamyDaiaacIcacaWGsbGaaiykamaabmaabaWaaSaaaeaacqaH0oaz daWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaamOuaaaacqGHsislda WcaaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccaWG4bWaaSbaaSqa aiaadQgaaeqaaaGcbaGaamOuamaaCaaaleqabaGaaG4maaaaaaaaki aawIcacaGLPaaaaaa@65C4@

5.      The strain components ε RR , ε θθ , ε ϕϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOuaiaadk faaeqaaOGaaiilaiabew7aLnaaBaaaleaacqaH4oqCcqaH4oqCaeqa aOGaaiilaiabew7aLnaaBaaaleaacqaHvpGzcqaHvpGzaeqaaaaa@425D@  can then be found as ε RR = e R ε e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0JaaCyzamaaBaaaleaacaWGsbaabeaakiabgwSi xlaahw7acqGHflY1caWHLbWaaSbaaSqaaiaadkfaaeqaaaaa@4118@ , ε θθ = e θ ε e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iaahwgadaWgaaWcbaGaeqiUdehabeaakiab gwSixlaahw7acqGHflY1caWHLbWaaSbaaSqaaiabeI7aXbqabaaaaa@4494@  and ε ϕϕ = e ϕ ε e ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaeqy1dyMaeq y1dygabeaakiabg2da9iaahwgadaWgaaWcbaGaeqy1dygabeaakiab gwSixlaahw7acqGHflY1caWHLbWaaSbaaSqaaiabew9aMbqabaaaaa@44DC@ .  Substituting for the basis vectors and simplifying gives the strain-displacement relations.  For example

ε RR = ε ij x i x j R 2 = du dR x i x i x j x j R 4 +u(R)( δ ij R x i x j R 2 x i x i x j x j R 5 )= du dR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiaadMgacaWGQbaabeaa kmaalaaabaGaamiEamaaBaaaleaacaWGPbaabeaakiaadIhadaWgaa WcbaGaamOAaaqabaaakeaacaWGsbWaaWbaaSqabeaacaaIYaaaaaaa kiabg2da9maalaaabaGaamizaiaadwhaaeaacaWGKbGaamOuaaaada WcaaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccaWG4bWaaSbaaSqa aiaadMgaaeqaaOGaamiEamaaBaaaleaacaWGQbaabeaakiaadIhada WgaaWcbaGaamOAaaqabaaakeaacaWGsbWaaWbaaSqabeaacaaI0aaa aaaakiabgUcaRiaadwhacaGGOaGaamOuaiaacMcadaqadaqaamaala aabaGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaOqaaiaadkfa aaWaaSaaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaamiEamaaBa aaleaacaWGQbaabeaaaOqaaiaadkfadaahaaWcbeqaaiaaikdaaaaa aOGaeyOeI0YaaSaaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaam iEamaaBaaaleaacaWGPbaabeaakiaadIhadaWgaaWcbaGaamOAaaqa baGccaWG4bWaaSbaaSqaaiaadQgaaeqaaaGcbaGaamOuamaaCaaale qabaGaaGynaaaaaaaakiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaa dsgacaWG1baabaGaamizaiaadkfaaaaaaa@70A2@

where we have noted x i x i = R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaO GaamiEamaaBaaaleaacaWGPbaabeaakiabg2da9iaadkfadaahaaWc beqaaiaaikdaaaaaaa@39CE@ . The remaining components are left as an exercise.

6.      Finally, to derive the equilibrium equation, note that the stress tensor can be expressed as σ= σ RR e R e R + σ θθ e θ e θ + σ ϕϕ e ϕ e ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdpGaeyypa0Jaeq4Wdm3aaSbaaS qaaiaadkfacaWGsbaabeaakiaahwgadaWgaaWcbaGaamOuaaqabaGc cqGHxkcXcaWHLbWaaSbaaSqaaiaadkfaaeqaaOGaey4kaSIaeq4Wdm 3aaSbaaSqaaiabeI7aXjabeI7aXbqabaGccaWHLbWaaSbaaSqaaiab eI7aXbqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiabeI7aXbqabaGccq GHRaWkcqaHdpWCdaWgaaWcbaGaeqy1dyMaeqy1dygabeaakiaahwga daWgaaWcbaGaeqy1dygabeaakiabgEPielaahwgadaWgaaWcbaGaeq y1dygabeaaaaa@5B07@ .  Substituting for the basis vectors from item(1) above gives

σ ij = σ RR x i x j R 2 + σ θθ i3k R x k R 2 x 3 2 j3n R x n R 2 x 3 2 + σ ϕϕ ( x 3 x i R 2 δ i3 R 2 x 3 2 )( x 3 x j R 2 δ j3 R 2 x 3 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadkfacaWGsbaabeaa kmaalaaabaGaamiEamaaBaaaleaacaWGPbaabeaakiaadIhadaWgaa WcbaGaamOAaaqabaaakeaacaWGsbWaaWbaaSqabeaacaaIYaaaaaaa kiabgUcaRiabeo8aZnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOWaaS aaaeaacqGHiiIZdaWgaaWcbaGaamyAaiaaiodacaWGRbaabeaakiaa dkfacaWG4bWaaSbaaSqaaiaadUgaaeqaaaGcbaGaamOuamaaCaaale qabaGaaGOmaaaakiabgkHiTiaadIhadaqhaaWcbaGaaG4maaqaaiaa ikdaaaaaaOWaaSaaaeaacqGHiiIZdaWgaaWcbaGaamOAaiaaiodaca WGUbaabeaakiaadkfacaWG4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGa amOuamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadIhadaqhaaWcba GaaG4maaqaaiaaikdaaaaaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiab ew9aMjabew9aMbqabaGcdaqadaqaamaalaaabaGaamiEamaaBaaale aacaaIZaaabeaakiaadIhadaWgaaWcbaGaamyAaaqabaGccqGHsisl caWGsbWaaWbaaSqabeaacaaIYaaaaOGaeqiTdq2aaSbaaSqaaiaadM gacaaIZaaabeaaaOqaaiaadkfadaahaaWcbeqaaiaaikdaaaGccqGH sislcaWG4bWaa0baaSqaaiaaiodaaeaacaaIYaaaaaaaaOGaayjkai aawMcaamaabmaabaWaaSaaaeaacaWG4bWaaSbaaSqaaiaaiodaaeqa aOGaamiEamaaBaaaleaacaWGQbaabeaakiabgkHiTiaadkfadaahaa WcbeqaaiaaikdaaaGccqaH0oazdaWgaaWcbaGaamOAaiaaiodaaeqa aaGcbaGaamOuamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadIhada qhaaWcbaGaaG4maaqaaiaaikdaaaaaaaGccaGLOaGaayzkaaaaaa@8A3D@

7.      Substitute the preceding result into the equilibrium equation

σ ij x i + ρ 0 b(R) x j R =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kabeo8aZnaaBa aaleaacaWGPbGaamOAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqa aiaadMgaaeqaaaaakiabgUcaRiabeg8aYnaaBaaaleaacaaIWaaabe aakiaadkgacaGGOaGaamOuaiaacMcadaWcaaqaaiaadIhadaWgaaWc baGaamOAaaqabaaakeaacaWGsbaaaiabg2da9iaaicdaaaa@470B@

and work through a good deal of tedious algebra to see that

( d σ RR dR + 1 R ( 2 σ RR σ θθ σ ϕϕ )+ ρ 0 b(R) ) x j R =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaamaalaaabaGaamizaiabeo 8aZnaaBaaaleaacaWGsbGaamOuaaqabaaakeaacaWGKbGaamOuaaaa cqGHRaWkdaWcaaqaaiaaigdaaeaacaWGsbaaamaabmaabaGaaGOmai abeo8aZnaaBaaaleaacaWGsbGaamOuaaqabaGccqGHsislcqaHdpWC daWgaaWcbaGaeqiUdeNaeqiUdehabeaakiabgkHiTiabeo8aZnaaBa aaleaacqaHvpGzcqaHvpGzaeqaaaGccaGLOaGaayzkaaGaey4kaSIa eqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaamOyaiaacIcacaWGsbGaai ykaaGaayjkaiaawMcaamaalaaabaGaamiEamaaBaaaleaacaWGQbaa beaaaOqaaiaadkfaaaGaeyypa0JaaGimaaaa@5B59@

This result can also be obtained using the virtual work principle (see problems for Sect 2.4 for details)

 

 

 

4.1.3 General solution to the spherically symmetric linear elasticity problem

 

Our goal is to solve the equations given in Section 4.1.2 for the displacement, strain and stress in the sphere.  To do so,

1.      Substitute the strain-displacement relations into the stress-strain law to show that

[ σ RR σ θθ ]= E ( 1+ν )( 12ν ) [ 1ν 2ν ν 1 ][ du dR u R ] EαΔT 12ν [ 1 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqaceaaaeaacqaHdp WCdaWgaaWcbaGaamOuaiaadkfaaeqaaaGcbaGaeq4Wdm3aaSbaaSqa aiabeI7aXjabeI7aXbqabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaaS aaaeaacaWGfbaabaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGL OaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbGaay jkaiaawMcaaaaadaWadaqaauaabeqaciaaaeaacaaIXaGaeyOeI0Ia eqyVd4gabaGaaGOmaiabe27aUbqaaiabe27aUbqaaiaaigdaaaaaca GLBbGaayzxaaWaamWaaeaafaqabeGabaaabaWaaSaaaeaacaWGKbGa amyDaaqaaiaadsgacaWGsbaaaaqaamaalaaabaGaamyDaaqaaiaadk faaaaaaaGaay5waiaaw2faaiabgkHiTmaalaaabaGaamyraiabeg7a Hjabfs5aejaadsfaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbaada WadaqaauaabeqaceaaaeaacaaIXaaabaGaaGymaaaaaiaawUfacaGL Dbaaaaa@6946@

2.      Substitute this expression for the stress into the equilibrium equation and rearrange the result to see that

d 2 u d R 2 + 2 R du dR 2u R 2 = d dR { 1 R 2 d dR ( R 2 u ) }= α( 1+ν ) ( 1ν ) dΔT dR ( 1+ν )( 12ν ) E( 1ν ) ρ 0 b(R) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgadaahaaWcbeqaai aaikdaaaGccaWG1baabaGaamizaiaadkfadaahaaWcbeqaaiaaikda aaaaaOGaey4kaSYaaSaaaeaacaaIYaaabaGaamOuaaaadaWcaaqaai aadsgacaWG1baabaGaamizaiaadkfaaaGaeyOeI0YaaSaaaeaacaaI YaGaamyDaaqaaiaadkfadaahaaWcbeqaaiaaikdaaaaaaOGaeyypa0 ZaaSaaaeaacaWGKbaabaGaamizaiaadkfaaaWaaiWaaeaadaWcaaqa aiaaigdaaeaacaWGsbWaaWbaaSqabeaacaaIYaaaaaaakmaalaaaba GaamizaaqaaiaadsgacaWGsbaaamaabmaabaGaamOuamaaCaaaleqa baGaaGOmaaaakiaadwhaaiaawIcacaGLPaaaaiaawUhacaGL9baacq GH9aqpdaWcaaqaaiabeg7aHnaabmaabaGaaGymaiabgUcaRiabe27a UbGaayjkaiaawMcaaaqaamaabmaabaGaaGymaiabgkHiTiabe27aUb GaayjkaiaawMcaaaaadaWcaaqaaiaadsgacqqHuoarcaWGubaabaGa amizaiaadkfaaaGaeyOeI0YaaSaaaeaadaqadaqaaiaaigdacqGHRa WkcqaH9oGBaiaawIcacaGLPaaadaqadaqaaiaaigdacqGHsislcaaI YaGaeqyVd4gacaGLOaGaayzkaaaabaGaamyramaabmaabaGaaGymai abgkHiTiabe27aUbGaayjkaiaawMcaaaaacqaHbpGCdaWgaaWcbaGa aGimaaqabaGccaWGIbGaaiikaiaadkfacaGGPaaaaa@7B0D@

 

Given the temperature distribution and body force this equation can easily be integrated to calculate the displacement u.  Two arbitrary constants of integration will appear when you do the integral MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  these must be determined from the boundary conditions at the inner and outer surface of the sphere.  Specifically, the constants must be selected so that either the displacement or the radial stress have prescribed values on the inner and outer surface of the sphere.

 

In the following sections, this procedure is used to derive solutions to various boundary value problems of practical interest.

 

 

4.1.4 Pressurized hollow sphere

 

Assume that

 No body forces act on the sphere

 The sphere has uniform temperature

 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 displacement, strain and stress fields in the sphere are

u= 1 2E( b 3 a 3 ) R 2 { 2( p a a 3 p b b 3 )( 12ν ) R 3 +( p a p b )( 1+ν ) b 3 a 3 } e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0ZaaSaaaeaacaaIXa aabaGaaGOmaiaadweadaqadaqaaiaadkgadaahaaWcbeqaaiaaioda aaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaay zkaaGaamOuamaaCaaaleqabaGaaGOmaaaaaaGcdaGadaqaaiaaikda daqadaqaaiaadchadaWgaaWcbaGaamyyaaqabaGccaWGHbWaaWbaaS qabeaacaaIZaaaaOGaeyOeI0IaamiCamaaBaaaleaacaWGIbaabeaa kiaadkgadaahaaWcbeqaaiaaiodaaaaakiaawIcacaGLPaaadaqada qaaiaaigdacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaGaamOu amaaCaaaleqabaGaaG4maaaakiabgUcaRmaabmaabaGaamiCamaaBa aaleaacaWGHbaabeaakiabgkHiTiaadchadaWgaaWcbaGaamOyaaqa baaakiaawIcacaGLPaaadaqadaqaaiaaigdacqGHRaWkcqaH9oGBai aawIcacaGLPaaacaWGIbWaaWbaaSqabeaacaaIZaaaaOGaamyyamaa CaaaleqabaGaaG4maaaaaOGaay5Eaiaaw2haaiaahwgadaWgaaWcba GaamOuaaqabaaaaa@65BA@

ε RR = 1 E( b 3 a 3 ) R 3 { ( p a a 3 p b b 3 )( 12ν ) R 3 ( p a p b )( 1+ν ) b 3 a 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaamyramaabmaabaGa amOyamaaCaaaleqabaGaaG4maaaakiabgkHiTiaadggadaahaaWcbe qaaiaaiodaaaaakiaawIcacaGLPaaacaWGsbWaaWbaaSqabeaacaaI ZaaaaaaakmaacmaabaWaaeWaaeaacaWGWbWaaSbaaSqaaiaadggaae qaaOGaamyyamaaCaaaleqabaGaaG4maaaakiabgkHiTiaadchadaWg aaWcbaGaamOyaaqabaGccaWGIbWaaWbaaSqabeaacaaIZaaaaaGcca GLOaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbGa ayjkaiaawMcaaiaadkfadaahaaWcbeqaaiaaiodaaaGccqGHsislda qadaqaaiaadchadaWgaaWcbaGaamyyaaqabaGccqGHsislcaWGWbWa aSbaaSqaaiaadkgaaeqaaaGccaGLOaGaayzkaaWaaeWaaeaacaaIXa Gaey4kaSIaeqyVd4gacaGLOaGaayzkaaGaamOyamaaCaaaleqabaGa aG4maaaakiaadggadaahaaWcbeqaaiaaiodaaaaakiaawUhacaGL9b aaaaa@64EA@

ε θθ = ε ϕϕ = 1 2E( b 3 a 3 ) R 3 { 2( p a a 3 p b b 3 )( 12ν ) R 3 +( p a p b )( 1+ν ) b 3 a 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabew7aLnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiaadweadaqada qaaiaadkgadaahaaWcbeqaaiaaiodaaaGccqGHsislcaWGHbWaaWba aSqabeaacaaIZaaaaaGccaGLOaGaayzkaaGaamOuamaaCaaaleqaba GaaG4maaaaaaGcdaGadaqaaiaaikdadaqadaqaaiaadchadaWgaaWc baGaamyyaaqabaGccaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0 IaamiCamaaBaaaleaacaWGIbaabeaakiaadkgadaahaaWcbeqaaiaa iodaaaaakiaawIcacaGLPaaadaqadaqaaiaaigdacqGHsislcaaIYa GaeqyVd4gacaGLOaGaayzkaaGaamOuamaaCaaaleqabaGaaG4maaaa kiabgUcaRmaabmaabaGaamiCamaaBaaaleaacaWGHbaabeaakiabgk HiTiaadchadaWgaaWcbaGaamOyaaqabaaakiaawIcacaGLPaaadaqa daqaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaacaWGIbWaaW baaSqabeaacaaIZaaaaOGaamyyamaaCaaaleqabaGaaG4maaaaaOGa ay5Eaiaaw2haaaaa@6E88@

σ RR = ( p a a 3 p b b 3 ) ( b 3 a 3 ) ( p a p b ) b 3 a 3 ( b 3 a 3 ) R 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0ZaaSaaaeaadaqadaqaaiaadchadaWgaaWcbaGa amyyaaqabaGccaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam iCamaaBaaaleaacaWGIbaabeaakiaadkgadaahaaWcbeqaaiaaioda aaaakiaawIcacaGLPaaaaeaadaqadaqaaiaadkgadaahaaWcbeqaai aaiodaaaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIZaaaaaGccaGL OaGaayzkaaaaaiaaykW7caaMc8UaeyOeI0YaaSaaaeaadaqadaqaai aadchadaWgaaWcbaGaamyyaaqabaGccqGHsislcaWGWbWaaSbaaSqa aiaadkgaaeqaaaGccaGLOaGaayzkaaGaamOyamaaCaaaleqabaGaaG 4maaaakiaadggadaahaaWcbeqaaiaaiodaaaaakeaadaqadaqaaiaa dkgadaahaaWcbeqaaiaaiodaaaGccqGHsislcaWGHbWaaWbaaSqabe aacaaIZaaaaaGccaGLOaGaayzkaaGaamOuamaaCaaaleqabaGaaG4m aaaaaaaaaa@5E49@      σ θθ = σ ϕϕ = ( p a a 3 p b b 3 ) ( b 3 a 3 ) + ( p a p b ) b 3 a 3 2( b 3 a 3 ) R 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaadaqadaqaaiaadchadaWgaaWcbaGaam yyaaqabaGccaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamiC amaaBaaaleaacaWGIbaabeaakiaadkgadaahaaWcbeqaaiaaiodaaa aakiaawIcacaGLPaaaaeaadaqadaqaaiaadkgadaahaaWcbeqaaiaa iodaaaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIZaaaaaGccaGLOa GaayzkaaaaaiabgUcaRmaalaaabaWaaeWaaeaacaWGWbWaaSbaaSqa aiaadggaaeqaaOGaeyOeI0IaamiCamaaBaaaleaacaWGIbaabeaaaO GaayjkaiaawMcaaiaadkgadaahaaWcbeqaaiaaiodaaaGccaWGHbWa aWbaaSqabeaacaaIZaaaaaGcbaGaaGOmamaabmaabaGaamOyamaaCa aaleqabaGaaG4maaaakiabgkHiTiaadggadaahaaWcbeqaaiaaioda aaaakiaawIcacaGLPaaacaWGsbWaaWbaaSqabeaacaaIZaaaaaaaaa a@6431@

 

Derivation:  The solution can be found by applying the procedure outlined in Sect 4.1.3.

1.      Note that the governing equation for u (Sect 4.1.3) reduces to

d dR { 1 R 2 d dR ( R 2 u ) }=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgaaeaacaWGKbGaam OuaaaadaGadaqaamaalaaabaGaaGymaaqaaiaadkfadaahaaWcbeqa aiaaikdaaaaaaOWaaSaaaeaacaWGKbaabaGaamizaiaadkfaaaWaae WaaeaacaWGsbWaaWbaaSqabeaacaaIYaaaaOGaamyDaaGaayjkaiaa wMcaaaGaay5Eaiaaw2haaiabg2da9iaaicdaaaa@430B@

2.      Integrating twice gives

u=AR+ B R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bGaeyypa0Jaamyqaiaadkfacq GHRaWkdaWcaaqaaiaadkeaaeaacaWGsbWaaWbaaSqabeaacaaIYaaa aaaaaaa@39DC@

where A and B are constants of integration to be determined.

3.      The radial stress follows by substituting into the stress-displacement formulas

σ RR = E ( 1+ν )( 12ν ) { ( 1ν ) du dR +2ν u R }= E ( 1+ν )( 12ν ) { ( 1+ν )A2( 12ν ) B R 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaWaaeWaaeaacaaIXaGa ey4kaSIaeqyVd4gacaGLOaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0 IaaGOmaiabe27aUbGaayjkaiaawMcaaaaadaGadaqaamaabmaabaGa aGymaiabgkHiTiabe27aUbGaayjkaiaawMcaamaalaaabaGaamizai aadwhaaeaacaWGKbGaamOuaaaacqGHRaWkcaaIYaGaeqyVd42aaSaa aeaacaWG1baabaGaamOuaaaaaiaawUhacaGL9baacqGH9aqpdaWcaa qaaiaadweaaeaadaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaawIca caGLPaaadaqadaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gacaGLOa GaayzkaaaaamaacmaabaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4ga caGLOaGaayzkaaGaamyqaiabgkHiTiaaikdadaqadaqaaiaaigdacq GHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaWaaSaaaeaacaWGcbaa baGaamOuamaaCaaaleqabaGaaG4maaaaaaaakiaawUhacaGL9baaaa a@70FB@

4.      To satisfy the boundary conditions, A and B must be chosen so that σ RR (R=a)= p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaaiikaiaadkfacqGH9aqpcaWGHbGaaiykaiabg2da9iab gkHiTiaadchadaWgaaWcbaGaamyyaaqabaaaaa@3E83@  and σ RR (R=b)= p b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaaiikaiaadkfacqGH9aqpcaWGIbGaaiykaiabg2da9iab gkHiTiaadchadaWgaaWcbaGaamOyaaqabaaaaa@3E85@  (the stress is negative because the pressure is compressive).  This gives two equations for A and B that are easily solved to find

A= ( p b b 3 p a a 3 )( 12ν ) ( a 3 b 3 )E B= ( p b p a )( 1+ν ) b 3 a 3 2( a 3 b 3 )E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbGaeyypa0ZaaSaaaeaadaqada qaaiaadchadaWgaaWcbaGaamOyaaqabaGccaWGIbWaaWbaaSqabeaa caaIZaaaaOGaeyOeI0IaamiCamaaBaaaleaacaWGHbaabeaakiaadg gadaahaaWcbeqaaiaaiodaaaaakiaawIcacaGLPaaadaqadaqaaiaa igdacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaabaWaaeWaae aacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamOyamaaCaaa leqabaGaaG4maaaaaOGaayjkaiaawMcaaiaadweaaaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caWGcbGaeyypa0ZaaSaaaeaada qadaqaaiaadchadaWgaaWcbaGaamOyaaqabaGccqGHsislcaWGWbWa aSbaaSqaaiaadggaaeqaaaGccaGLOaGaayzkaaWaaeWaaeaacaaIXa Gaey4kaSIaeqyVd4gacaGLOaGaayzkaaGaamOyamaaCaaaleqabaGa aG4maaaakiaadggadaahaaWcbeqaaiaaiodaaaaakeaacaaIYaWaae WaaeaacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamOyamaa CaaaleqabaGaaG4maaaaaOGaayjkaiaawMcaaiaadweaaaaaaa@79DE@

5.      Finally, expressions for displacement, strain and stress follow by substituting for A and B in the formula for u in (2), and using the formulas for strain and stress in terms of u in Section 4.1.2.

 

 

4.1.5 Gravitating sphere

 

A planet under its own gravitational attraction may be idealized (rather crudely) as a solid sphere with radius a, with the following loading

 A body force b=(gR/a) e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHIbGaeyypa0JaeyOeI0Iaaiikai aadEgacaaMc8UaamOuaiaac+cacaWGHbGaaiykaiaahwgadaWgaaWc baGaamOuaaqabaaaaa@3DD5@  per unit mass, where g is the acceleration due to gravity at the surface of the sphere

 A uniform temperature distribution

 A traction free surface at R=a

 

The displacement, strain and stress in the sphere follow as

u= ( 12ν ) 10aE( 1ν ) ρ 0 gR{ ( 1+ν ) R 2 ( 3ν ) a 2 } e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0ZaaSaaaeaadaqada qaaiaaigdacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaabaGa aGymaiaaicdacaWGHbGaamyramaabmaabaGaaGymaiabgkHiTiabe2 7aUbGaayjkaiaawMcaaaaacqaHbpGCdaWgaaWcbaGaaGimaaqabaGc caWGNbGaamOuamaacmaabaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4 gacaGLOaGaayzkaaGaamOuamaaCaaaleqabaGaaGOmaaaakiabgkHi TmaabmaabaGaaG4maiabgkHiTiabe27aUbGaayjkaiaawMcaaiaadg gadaahaaWcbeqaaiaaikdaaaaakiaawUhacaGL9baacaWHLbWaaSba aSqaaiaadkfaaeqaaaaa@597B@

ε RR = ( 12ν ) 10aE( 1ν ) ρ 0 g{ 3( 1+ν ) R 2 ( 3ν ) a 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0ZaaSaaaeaadaqadaqaaiaaigdacqGHsislcaaI YaGaeqyVd4gacaGLOaGaayzkaaaabaGaaGymaiaaicdacaWGHbGaam yramaabmaabaGaaGymaiabgkHiTiabe27aUbGaayjkaiaawMcaaaaa cqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGNbWaaiWaaeaacaaIZa WaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzkaaGaamOu amaaCaaaleqabaGaaGOmaaaakiabgkHiTmaabmaabaGaaG4maiabgk HiTiabe27aUbGaayjkaiaawMcaaiaadggadaahaaWcbeqaaiaaikda aaaakiaawUhacaGL9baaaaa@59FD@

ε θθ = ε ϕϕ = ( 12ν ) 10aE( 1ν ) ρ 0 g{ ( 1+ν ) R 2 ( 3ν ) a 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabew7aLnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaadaqadaqaaiaaigdacqGHsislcaaIYa GaeqyVd4gacaGLOaGaayzkaaaabaGaaGymaiaaicdacaWGHbGaamyr amaabmaabaGaaGymaiabgkHiTiabe27aUbGaayjkaiaawMcaaaaacq aHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGNbWaaiWaaeaadaqadaqa aiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaacaWGsbWaaWbaaS qabeaacaaIYaaaaOGaeyOeI0YaaeWaaeaacaaIZaGaeyOeI0IaeqyV d4gacaGLOaGaayzkaaGaamyyamaaCaaaleqabaGaaGOmaaaaaOGaay 5Eaiaaw2haaaaa@6171@

σ RR = ρ 0 g(3ν) 10a( 1ν ) ( R 2 a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0ZaaSaaaeaacqaHbpGCdaWgaaWcbaGaaGimaaqa baGccaWGNbGaaiikaiaaiodacqGHsislcqaH9oGBcaGGPaaabaGaaG ymaiaaicdacaWGHbWaaeWaaeaacaaIXaGaeyOeI0IaeqyVd4gacaGL OaGaayzkaaaaamaabmaabaGaamOuamaaCaaaleqabaGaaGOmaaaaki abgkHiTiaadggadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaa aaa@4D37@             σ θθ = σ ϕϕ = ρ 0 g 10a( 1ν ) { ( 3ν+1 ) R 2 ( 3ν ) a 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba GccaWGNbaabaGaaGymaiaaicdacaWGHbWaaeWaaeaacaaIXaGaeyOe I0IaeqyVd4gacaGLOaGaayzkaaaaamaacmaabaWaaeWaaeaacaaIZa GaeqyVd4Maey4kaSIaaGymaaGaayjkaiaawMcaaiaadkfadaahaaWc beqaaiaaikdaaaGccqGHsisldaqadaqaaiaaiodacqGHsislcqaH9o GBaiaawIcacaGLPaaacaWGHbWaaWbaaSqabeaacaaIYaaaaaGccaGL 7bGaayzFaaaaaa@5BF7@

 

Derivation:

1.      Begin by writing the governing equation for u given in 4.1.3 as

d dR { 1 R 2 d dR ( R 2 u ) }= ( 1+ν )( 12ν ) E( 1ν ) ρ 0 gR a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgaaeaacaWGKbGaam OuaaaadaGadaqaamaalaaabaGaaGymaaqaaiaadkfadaahaaWcbeqa aiaaikdaaaaaaOWaaSaaaeaacaWGKbaabaGaamizaiaadkfaaaWaae WaaeaacaWGsbWaaWbaaSqabeaacaaIYaaaaOGaamyDaaGaayjkaiaa wMcaaaGaay5Eaiaaw2haaiabg2da9maalaaabaWaaeWaaeaacaaIXa Gaey4kaSIaeqyVd4gacaGLOaGaayzkaaWaaeWaaeaacaaIXaGaeyOe I0IaaGOmaiabe27aUbGaayjkaiaawMcaaaqaaiaadweadaqadaqaai aaigdacqGHsislcqaH9oGBaiaawIcacaGLPaaaaaWaaSaaaeaacqaH bpGCdaWgaaWcbaGaaGimaaqabaGccaWGNbGaamOuaaqaaiaadggaaa aaaa@5800@

2.      Integrating

u= ( 1+ν )( 12ν ) E( 1ν ) ρ 0 g R 3 10a +AR+ B R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bGaeyypa0ZaaSaaaeaadaqada qaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaadaqadaqaaiaa igdacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaabaGaamyram aabmaabaGaaGymaiabgkHiTiabe27aUbGaayjkaiaawMcaaaaadaWc aaqaaiabeg8aYnaaBaaaleaacaaIWaaabeaakiaadEgacaWGsbWaaW baaSqabeaacaaIZaaaaaGcbaGaaGymaiaaicdacaWGHbaaaiabgUca RiaadgeacaWGsbGaey4kaSYaaSaaaeaacaWGcbaabaGaamOuamaaCa aaleqabaGaaGOmaaaaaaaaaa@52D6@

where A and B are constants of integration that must be determined from boundary conditions.

3.      The radial stress follows from the formulas in 4.1.3 as

σ RR = E ( 1+ν )( 12ν ) { ( 1ν ) du dR +2ν u R }= ρ 0 g(3ν) R 2 10a(1ν) + E ( 1+ν )( 12ν ) { ( 1+ν )A2( 12ν ) B R 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaWaaeWaaeaacaaIXaGa ey4kaSIaeqyVd4gacaGLOaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0 IaaGOmaiabe27aUbGaayjkaiaawMcaaaaadaGadaqaamaabmaabaGa aGymaiabgkHiTiabe27aUbGaayjkaiaawMcaamaalaaabaGaamizai aadwhaaeaacaWGKbGaamOuaaaacqGHRaWkcaaIYaGaeqyVd42aaSaa aeaacaWG1baabaGaamOuaaaaaiaawUhacaGL9baacqGH9aqpdaWcaa qaaiabeg8aYnaaBaaaleaacaaIWaaabeaakiaadEgacaGGOaGaaG4m aiabgkHiTiabe27aUjaacMcacaWGsbWaaWbaaSqabeaacaaIYaaaaa GcbaGaaGymaiaaicdacaWGHbGaaiikaiaaigdacqGHsislcqaH9oGB caGGPaaaaiabgUcaRmaalaaabaGaamyraaqaamaabmaabaGaaGymai abgUcaRiabe27aUbGaayjkaiaawMcaamaabmaabaGaaGymaiabgkHi TiaaikdacqaH9oGBaiaawIcacaGLPaaaaaWaaiWaaeaadaqadaqaai aaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaacaWGbbGaeyOeI0Ia aGOmamaabmaabaGaaGymaiabgkHiTiaaikdacqaH9oGBaiaawIcaca GLPaaadaWcaaqaaiaadkeaaeaacaWGsbWaaWbaaSqabeaacaaIZaaa aaaaaOGaay5Eaiaaw2haaaaa@8322@

4.      Finally, the constants A and B can be determined as follows: (i) The stress must be finite at R0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbGaeyOKH4QaaGimaaaa@3644@ , which is only possible if B=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGcbGaeyypa0JaaGimaaaa@354D@ .  (ii) The surface of the sphere is traction free, which requires σ RR =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0JaaGimaaaa@382D@  at R=a.  Substituting the latter condition into the formula for stress in (3) and solving for A gives

A= ( 12ν )( 3ν ) ρ 0 ga 10E(1ν) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbGaeyypa0JaeyOeI0YaaSaaae aadaqadaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzk aaWaaeWaaeaacaaIZaGaeyOeI0IaeqyVd4gacaGLOaGaayzkaaGaeq yWdi3aaSbaaSqaaiaaicdaaeqaaOGaam4zaiaadggaaeaacaaIXaGa aGimaiaadweacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcaaaaaaa@4B99@

5.      The final formulas for stress and strain follow by substituting the result of (4) back into (2), and using the formulas in Section 4.1.2.

 

 

4.1.6 Sphere with steady state heat flow

 

The deformation and stress in a sphere that is heated on the inside (or outside), and has reached its steady state temperature distribution can be calculated as follows.  Assume that

 No body force acts on the sphere

 The temperature distribution in the sphere is

T= T b b T a a ba + ( T a T b )ab (ba)R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubGaeyypa0ZaaSaaaeaacaWGub WaaSbaaSqaaiaadkgaaeqaaOGaamOyaiabgkHiTiaadsfadaWgaaWc baGaamyyaaqabaGccaWGHbaabaGaamOyaiabgkHiTiaadggaaaGaey 4kaSYaaSaaaeaadaqadaqaaiaadsfadaWgaaWcbaGaamyyaaqabaGc cqGHsislcaWGubWaaSbaaSqaaiaadkgaaeqaaaGccaGLOaGaayzkaa GaamyyaiaadkgaaeaacaGGOaGaamOyaiabgkHiTiaadggacaGGPaGa amOuaaaaaaa@4C1E@

where T a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadggaaeqaaa aa@34B1@  and T b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadkgaaeqaaa aa@34B2@  are the temperatures at the inner and outer surfaces.  The total rate of heat loss from the sphere is Q ˙ =4πk( T a T b )ab/(ba) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGrbGbaiaacqGH9aqpcaaI0aGaeq iWdaNaam4AaiaacIcacaWGubWaaSbaaSqaaiaadggaaeqaaOGaeyOe I0IaamivamaaBaaaleaacaWGIbaabeaakiaacMcacaWGHbGaamOyai aac+cacaGGOaGaamOyaiabgkHiTiaadggacaGGPaaaaa@44DA@ , where k is the thermal conductivity.

 The surfaces at R=a  and R=b are traction free.

The displacement, strain and stress fields in the sphere follow as

u= α ( 1ν ) ( T a T b )a 2( b 3 a 3 ) { (1+ν)b( a 2 +ab+ b 2 )+2(ν a 2 a 2 νabν b 2 )R(1+ν) a 2 b 3 R 2 }+ T b αR MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bGaeyypa0ZaaSaaaeaacqaHXo qyaeaadaqadaqaaiaaigdacqGHsislcqaH9oGBaiaawIcacaGLPaaa aaWaaSaaaeaadaqadaqaaiaadsfadaWgaaWcbaGaamyyaaqabaGccq GHsislcaWGubWaaSbaaSqaaiaadkgaaeqaaaGccaGLOaGaayzkaaGa amyyaaqaaiaaikdacaGGOaGaamOyamaaCaaaleqabaGaaG4maaaaki abgkHiTiaadggadaahaaWcbeqaaiaaiodaaaGccaGGPaaaamaacmaa baGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaGaamOyaiaacIcaca WGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyyaiaadkgacqGH RaWkcaWGIbWaaWbaaSqabeaacaaIYaaaaOGaaiykaiabgUcaRiaaik dacaGGOaGaeqyVd4MaamyyamaaCaaaleqabaGaaGOmaaaakiabgkHi TiaadggadaahaaWcbeqaaiaaikdaaaGccqGHsislcqaH9oGBcaWGHb GaamOyaiabgkHiTiabe27aUjaadkgadaahaaWcbeqaaiaaikdaaaGc caGGPaGaamOuaiabgkHiTiaacIcacaaIXaGaey4kaSIaeqyVd4Maai ykamaalaaabaGaamyyamaaCaaaleqabaGaaGOmaaaakiaadkgadaah aaWcbeqaaiaaiodaaaaakeaacaWGsbWaaWbaaSqabeaacaaIYaaaaa aaaOGaay5Eaiaaw2haaiabgUcaRiaadsfadaWgaaWcbaGaamOyaaqa baGccqaHXoqycaWGsbaaaa@7D5C@

ε RR = α ( 1ν ) ( T a T b )a ( b 3 a 3 ) { (ν a 2 a 2 νabν b 2 )+(1+ν) a 2 b 3 R 3 }+ T b α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0ZaaSaaaeaacqaHXoqyaeaadaqadaqaaiaaigda cqGHsislcqaH9oGBaiaawIcacaGLPaaaaaWaaSaaaeaadaqadaqaai aadsfadaWgaaWcbaGaamyyaaqabaGccqGHsislcaWGubWaaSbaaSqa aiaadkgaaeqaaaGccaGLOaGaayzkaaGaamyyaaqaaiaacIcacaWGIb WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamyyamaaCaaaleqabaGa aG4maaaakiaacMcaaaWaaiWaaeaacaGGOaGaeqyVd4MaamyyamaaCa aaleqabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqaaiaaikda aaGccqGHsislcqaH9oGBcaWGHbGaamOyaiabgkHiTiabe27aUjaadk gadaahaaWcbeqaaiaaikdaaaGccaGGPaGaey4kaSIaaiikaiaaigda cqGHRaWkcqaH9oGBcaGGPaWaaSaaaeaacaWGHbWaaWbaaSqabeaaca aIYaaaaOGaamOyamaaCaaaleqabaGaaG4maaaaaOqaaiaadkfadaah aaWcbeqaaiaaiodaaaaaaaGccaGL7bGaayzFaaGaey4kaSIaamivam aaBaaaleaacaWGIbaabeaakiabeg7aHbaa@6DA9@

ε θθ = ε ϕϕ = α ( 1ν ) ( T a T b )a 2( b 3 a 3 )R { (1+ν)b( a 2 +ab+ b 2 )+2(ν a 2 a 2 νabν b 2 )R(1+ν) a 2 b 3 R 2 }+ T b α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabew7aLnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaacqaHXoqyaeaadaqadaqaaiaaigdacq GHsislcqaH9oGBaiaawIcacaGLPaaaaaWaaSaaaeaadaqadaqaaiaa dsfadaWgaaWcbaGaamyyaaqabaGccqGHsislcaWGubWaaSbaaSqaai aadkgaaeqaaaGccaGLOaGaayzkaaGaamyyaaqaaiaaikdacaGGOaGa amOyamaaCaaaleqabaGaaG4maaaakiabgkHiTiaadggadaahaaWcbe qaaiaaiodaaaGccaGGPaGaamOuaaaadaGadaqaaiaacIcacaaIXaGa ey4kaSIaeqyVd4MaaiykaiaadkgacaGGOaGaamyyamaaCaaaleqaba GaaGOmaaaakiabgUcaRiaadggacaWGIbGaey4kaSIaamOyamaaCaaa leqabaGaaGOmaaaakiaacMcacqGHRaWkcaaIYaGaaiikaiabe27aUj aadggadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGHbWaaWbaaSqa beaacaaIYaaaaOGaeyOeI0IaeqyVd4MaamyyaiaadkgacqGHsislcq aH9oGBcaWGIbWaaWbaaSqabeaacaaIYaaaaOGaaiykaiaadkfacqGH sislcaGGOaGaaGymaiabgUcaRiabe27aUjaacMcadaWcaaqaaiaadg gadaahaaWcbeqaaiaaikdaaaGccaWGIbWaaWbaaSqabeaacaaIZaaa aaGcbaGaamOuamaaCaaaleqabaGaaGOmaaaaaaaakiaawUhacaGL9b aacqGHRaWkcaWGubWaaSbaaSqaaiaadkgaaeqaaOGaeqySdegaaa@881E@

σ RR = Eαν ( 1ν ) ( T a T b )ab ( b 3 a 3 ) (Ra)(Rb)(Ra+Rb+ab) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0ZaaSaaaeaacaWGfbGaeqySdeMaeqyVd4gabaWa aeWaaeaacaaIXaGaeyOeI0IaeqyVd4gacaGLOaGaayzkaaaaamaala aabaWaaeWaaeaacaWGubWaaSbaaSqaaiaadggaaeqaaOGaeyOeI0Ia amivamaaBaaaleaacaWGIbaabeaaaOGaayjkaiaawMcaaiaadggaca WGIbaabaGaaiikaiaadkgadaahaaWcbeqaaiaaiodaaaGccqGHsisl caWGHbWaaWbaaSqabeaacaaIZaaaaOGaaiykaaaacaGGOaGaamOuai abgkHiTiaadggacaGGPaGaaiikaiaadkfacqGHsislcaWGIbGaaiyk aiaacIcacaWGsbGaamyyaiabgUcaRiaadkfacaWGIbGaey4kaSIaam yyaiaadkgacaGGPaaaaa@5F32@

σ θθ = σ ϕϕ = Eα 2( 1ν ) ( T a T b )ab ( b 3 a 3 ) { 2(a+b) a 2 +ab+ b 2 R a 2 b 2 R 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaacaWGfbGaeqySdegabaGaaGOmamaabm aabaGaaGymaiabgkHiTiabe27aUbGaayjkaiaawMcaaaaadaWcaaqa amaabmaabaGaamivamaaBaaaleaacaWGHbaabeaakiabgkHiTiaads fadaWgaaWcbaGaamOyaaqabaaakiaawIcacaGLPaaacaWGHbGaamOy aaqaaiaacIcacaWGIbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam yyamaaCaaaleqabaGaaG4maaaakiaacMcaaaWaaiWaaeaacaaIYaGa aiikaiaadggacqGHRaWkcaWGIbGaaiykaiabgkHiTmaalaaabaGaam yyamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadggacaWGIbGaey4k aSIaamOyamaaCaaaleqabaGaaGOmaaaaaOqaaiaadkfaaaGaeyOeI0 YaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaamOyamaaCaaa leqabaGaaGOmaaaaaOqaaiaadkfadaahaaWcbeqaaiaaiodaaaaaaa GccaGL7bGaayzFaaaaaa@6C9F@

 

Derivation:

1.      The differential equation for u given in 4.1.3 reduces to

d dR { 1 R 2 d dR ( R 2 u ) }= α( 1+ν ) ( 1ν ) ( T a T b )ab (ba) R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgaaeaacaWGKbGaam OuaaaadaGadaqaamaalaaabaGaaGymaaqaaiaadkfadaahaaWcbeqa aiaaikdaaaaaaOWaaSaaaeaacaWGKbaabaGaamizaiaadkfaaaWaae WaaeaacaWGsbWaaWbaaSqabeaacaaIYaaaaOGaamyDaaGaayjkaiaa wMcaaaGaay5Eaiaaw2haaiabg2da9iabgkHiTmaalaaabaGaeqySde 2aaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzkaaaabaWa aeWaaeaacaaIXaGaeyOeI0IaeqyVd4gacaGLOaGaayzkaaaaamaala aabaWaaeWaaeaacaWGubWaaSbaaSqaaiaadggaaeqaaOGaeyOeI0Ia amivamaaBaaaleaacaWGIbaabeaaaOGaayjkaiaawMcaaiaadggaca WGIbaabaGaaiikaiaadkgacqGHsislcaWGHbGaaiykaiaadkfadaah aaWcbeqaaiaaikdaaaaaaaaa@5CC5@

2.      Integrating

u= α( 1+ν ) 2( 1ν ) ( T a T b )ab (ba) +AR+ B R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bGaeyypa0ZaaSaaaeaacqaHXo qydaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaaaeaa caaIYaWaaeWaaeaacaaIXaGaeyOeI0IaeqyVd4gacaGLOaGaayzkaa aaamaalaaabaWaaeWaaeaacaWGubWaaSbaaSqaaiaadggaaeqaaOGa eyOeI0IaamivamaaBaaaleaacaWGIbaabeaaaOGaayjkaiaawMcaai aadggacaWGIbaabaGaaiikaiaadkgacqGHsislcaWGHbGaaiykaaaa cqGHRaWkcaWGbbGaamOuaiabgUcaRmaalaaabaGaamOqaaqaaiaadk fadaahaaWcbeqaaiaaikdaaaaaaaaa@5341@

where A and B are constants of integration.

3.      The radial stress follows from the formulas in 4.1.3 as

σ RR = E ( 1+ν )( 12ν ) { ( 1ν ) du dR +2ν u R } EαΔT 12ν = Eνα ( 12ν )( 1ν ) ( T a T b )ab (ba)R + E ( 1+ν )( 12ν ) { ( 1+ν )A2( 12ν ) B R 3 } Eα 12ν { T b b T a a ba + ( T a T b )ab (ba)R } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaWGsb GaamOuaaqabaGccqGH9aqpdaWcaaqaaiaadweaaeaadaqadaqaaiaa igdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaadaqadaqaaiaaigdacq GHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaaamaacmaabaWaaeWa aeaacaaIXaGaeyOeI0IaeqyVd4gacaGLOaGaayzkaaWaaSaaaeaaca WGKbGaamyDaaqaaiaadsgacaWGsbaaaiabgUcaRiaaikdacqaH9oGB daWcaaqaaiaadwhaaeaacaWGsbaaaaGaay5Eaiaaw2haaiabgkHiTm aalaaabaGaamyraiabeg7aHjabfs5aejaadsfaaeaacaaIXaGaeyOe I0IaaGOmaiabe27aUbaaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeyypa0ZaaSaaaeaa caWGfbGaeqyVd4MaeqySdegabaWaaeWaaeaacaaIXaGaeyOeI0IaaG Omaiabe27aUbGaayjkaiaawMcaamaabmaabaGaaGymaiabgkHiTiab e27aUbGaayjkaiaawMcaaaaadaWcaaqaamaabmaabaGaamivamaaBa aaleaacaWGHbaabeaakiabgkHiTiaadsfadaWgaaWcbaGaamOyaaqa baaakiaawIcacaGLPaaacaWGHbGaamOyaaqaaiaacIcacaWGIbGaey OeI0IaamyyaiaacMcacaWGsbaaaiabgUcaRmaalaaabaGaamyraaqa amaabmaabaGaaGymaiabgUcaRiabe27aUbGaayjkaiaawMcaamaabm aabaGaaGymaiabgkHiTiaaikdacqaH9oGBaiaawIcacaGLPaaaaaWa aiWaaeaadaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPa aacaWGbbGaeyOeI0IaaGOmamaabmaabaGaaGymaiabgkHiTiaaikda cqaH9oGBaiaawIcacaGLPaaadaWcaaqaaiaadkeaaeaacaWGsbWaaW baaSqabeaacaaIZaaaaaaaaOGaay5Eaiaaw2haaaqaaiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlabgkHiTmaalaaabaGaamyraiab eg7aHbqaaiaaigdacqGHsislcaaIYaGaeqyVd4gaamaacmaabaWaaS aaaeaacaWGubWaaSbaaSqaaiaadkgaaeqaaOGaamOyaiabgkHiTiaa dsfadaWgaaWcbaGaamyyaaqabaGccaWGHbaabaGaamOyaiabgkHiTi aadggaaaGaey4kaSYaaSaaaeaadaqadaqaaiaadsfadaWgaaWcbaGa amyyaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaadkgaaeqaaaGcca GLOaGaayzkaaGaamyyaiaadkgaaeaacaGGOaGaamOyaiabgkHiTiaa dggacaGGPaGaamOuaaaaaiaawUhacaGL9baaaaaa@F671@

4.      The boundary conditions require that σ rr =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0JaaGimaaaa@386D@  at r=a and r=b.  Substituting these conditions into the result of step (3) gives two equations for A and B which can be solved to see that

A= (1ν)( T b b 3 T a a 3 )+( T a T b )νab(a+b) ( 1ν )( a 3 b 3 ) B= α( T a T b )( 1+ν ) 2( 1ν ) a 3 b 3 ( b 3 a 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbGaeyypa0ZaaSaaaeaacaGGOa GaaGymaiabgkHiTiabe27aUjaacMcacaGGOaGaamivamaaBaaaleaa caWGIbaabeaakiaadkgadaahaaWcbeqaaiaaiodaaaGccqGHsislca WGubWaaSbaaSqaaiaadggaaeqaaOGaamyyamaaCaaaleqabaGaaG4m aaaakiaacMcacqGHRaWkcaGGOaGaamivamaaBaaaleaacaWGHbaabe aakiabgkHiTiaadsfadaWgaaWcbaGaamOyaaqabaGccaGGPaGaeqyV d4MaamyyaiaadkgacaGGOaGaamyyaiabgUcaRiaadkgacaGGPaaaba WaaeWaaeaacaaIXaGaeyOeI0IaeqyVd4gacaGLOaGaayzkaaWaaeWa aeaacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamOyamaaCa aaleqabaGaaG4maaaaaOGaayjkaiaawMcaaaaacaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaadkeacqGH9aqpdaWcaaqaaiabeg7a HnaabmaabaGaamivamaaBaaaleaacaWGHbaabeaakiabgkHiTiaads fadaWgaaWcbaGaamOyaaqabaaakiaawIcacaGLPaaadaqadaqaaiaa igdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaaaeaacaaIYaWaaeWaae aacaaIXaGaeyOeI0IaeqyVd4gacaGLOaGaayzkaaaaamaalaaabaGa amyyamaaCaaaleqabaGaaG4maaaakiaadkgadaahaaWcbeqaaiaaio daaaaakeaadaqadaqaaiaadkgadaahaaWcbeqaaiaaiodaaaGccqGH sislcaWGHbWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaayzkaaaaaa aa@90DF@

 

 

 

4.1.7 Simplified equations for axially symmetric linear elasticity problems

 

 

 

 

Two examples of axially symmetric problems are illustrated in the picture.  In both cases the solid is a circular cylinder, which is subjected to axially 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 z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG6baaaa@33C5@ , and act in the radial direction only).  If the temperature of the sphere is non-uniform, it must also be axially symmetric (a function of r only).  Finally, the solid can spin with steady angular velocity about the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa aa@349D@  axis.

 

The two solids have different shapes.  In the first case, the length of the cylinder is substantially greater than any cross-sectional dimension.  In the second case, the length of the cylinder is much less than its outer radius. 

 

 

The state of stress and strain in the solid depends on the loads applied to the ends of the cylinder. Specifically

 If the cylinder is completely prevented from stretching in the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa aa@349D@  direction a state of plane strain exists in the solid.  This is an exact solution to the 3D equations of elasticity, is valid for a cylinder with any length, and is accurate everywhere in the cylinder.

 If the top and bottom surface of the short plate-like cylinder are free of traction, a state of plane stress exists in the solid.  This is an approximate solution to the 3D equations of elasticity, and is accurate only if the cylinder’s length is much less than its diameter 

 If the top and bottom  ends of the long cylinder are subjected to a prescribed force (or the ends are free of force) a state of generalized plane strain exists in the cylinder.  This is an approximate solution, which is accurate only away from the ends of a long cylinder.  As a rule of thumb, the solution is applicable approximately three cylinder radii away from the ends.

 

The solution is most conveniently expressed using a spherical-polar coordinate system, illustrated in the figure.  A point in the solid is identified by its spherical-polar co-ordinates (r,θ,z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaacYcacqaH4oqCca GGSaGaamOEaiaacMcaaaa@392B@ . All vectors and tensors are expressed as components in the basis { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaam OCaaqabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiil aiaahwgadaWgaaWcbaGaamOEaaqabaaakiaawUhacaGL9baaaaa@3D6F@  shown in the figure.  For an axially symmetric problem

  Position Vector       x=r e r +z e z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaeyypa0JaamOCaiaahwgada WgaaWcbaGaamOCaaqabaGccqGHRaWkcaWG6bGaaCyzamaaBaaaleaa caWG6baabeaaaaa@3BD8@

  Displacement vector u=u(r) e r + ε zz z e z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0JaamyDaiaacIcaca WGYbGaaiykaiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHRaWkcqaH 1oqzdaWgaaWcbaGaamOEaiaadQhaaeqaaOGaamOEaiaahwgadaWgaa WcbaGaamOEaaqabaaaaa@4203@

  Body force vector b= ρ 0 b(r) e r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHIbGaeyypa0JaeqyWdi3aaSbaaS qaaiaaicdaaeqaaOGaamOyaiaacIcacaWGYbGaaiykaiaahwgadaWg aaWcbaGaamOCaaqabaaaaa@3CAE@

  Acceleration vector a= ω 2 r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHHbGaeyypa0JaeyOeI0IaeqyYdC 3aaWbaaSqabeaacaaIYaaaaOGaamOCaaaa@395A@

Here, u( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaeWaaeaacaWGYbaacaGLOa Gaayzkaaaaaa@3640@  and b( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGIbWaaeWaaeaacaWGYbaacaGLOa Gaayzkaaaaaa@362D@  are scalar functions.

 

The stress and strain tensors (written as components in { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaWGYb aabeaakiaacYcacaWHLbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa aCyzamaaBaaaleaacaWG6baabeaakiaac2haaaa@3D3D@  ) have the form

σ[ σ rr 0 0 0 σ θθ 0 0 0 σ zz ]ε[ ε rr 0 0 0 ε θθ 0 0 0 ε zz ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCcqGHHjIUdaWadaqaauaabe qadmaaaeaacqaHdpWCdaWgaaWcbaGaamOCaiaadkhaaeqaaaGcbaGa aGimaaqaaiaaicdaaeaacaaIWaaabaGaeq4Wdm3aaSbaaSqaaiabeI 7aXjabeI7aXbqabaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaa cqaHdpWCdaWgaaWcbaGaamOEaiaadQhaaeqaaaaaaOGaay5waiaaw2 faaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqyTduMaey yyIO7aamWaaeaafaqabeWadaaabaGaeqyTdu2aaSbaaSqaaiaadkha caWGYbaabeaaaOqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiabew 7aLnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaaGcbaGaaGimaaqaaiaa icdaaeaacaaIWaaabaGaeqyTdu2aaSbaaSqaaiaadQhacaWG6baabe aaaaaakiaawUfacaGLDbaaaaa@7657@

 

For axial symmetry, the governing equations of linear elasticity reduce to

 

 Strain Displacement Relations ε rr = du dr ε θθ = u r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaWGKbGaamyDaaqaaiaadsgacaWG YbaaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlabew7aLnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGa eyypa0ZaaSaaaeaacaWG1baabaGaamOCaaaaaaa@519D@

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

[ σ rr σ θθ σ zz ]= E (1+ν)(12ν) [ 1ν ν ν ν 1ν ν ν ν 1ν ][ ε rr ε θθ ε zz ] EαΔT 12ν [ 1 1 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadeaaaeaacqaHdp WCdaWgaaWcbaGaamOCaiaadkhaaeqaaaGcbaGaeq4Wdm3aaSbaaSqa aiabeI7aXjabeI7aXbqabaaakeaacqaHdpWCdaWgaaWcbaGaamOEai aadQhaaeqaaaaaaOGaay5waiaaw2faaiabg2da9maalaaabaGaamyr aaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4MaaiykaiaacIcacaaIXa GaeyOeI0IaaGOmaiabe27aUjaacMcaaaWaamWaaeaafaqabeWadaaa baGaaGymaiabgkHiTiabe27aUbqaaiabe27aUbqaaiabe27aUbqaai abe27aUbqaaiaaigdacqGHsislcqaH9oGBaeaacqaH9oGBaeaacqaH 9oGBaeaacqaH9oGBaeaacaaIXaGaeyOeI0IaeqyVd4gaaaGaay5wai aaw2faamaadmaabaqbaeqabmqaaaqaaiabew7aLnaaBaaaleaacaWG YbGaamOCaaqabaaakeaacqaH1oqzdaWgaaWcbaGaeqiUdeNaeqiUde habeaaaOqaaiabew7aLnaaBaaaleaacaWG6bGaamOEaaqabaaaaaGc caGLBbGaayzxaaGaeyOeI0YaaSaaaeaacaWGfbGaeqySdeMaeuiLdq KaamivaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gaamaadmaabaqb aeqabmqaaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaaaaiaawUfaca GLDbaaaaa@8169@

    where ε zz =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOEaiaadQ haaeqaaOGaeyypa0JaaGimaaaa@3861@  for plane strain, and constant for generalized plane strain.

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

[ σ rr σ θθ ]= E 1 ν 2 [ 1 ν ν 1 ][ ε rr ε θθ ] EαΔT 1ν [ 1 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqaceaaaeaacqaHdp WCdaWgaaWcbaGaamOCaiaadkhaaeqaaaGcbaGaeq4Wdm3aaSbaaSqa aiabeI7aXjabeI7aXbqabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaaS aaaeaacaWGfbaabaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGa aGOmaaaaaaGcdaWadaqaauaabeqaciaaaeaacaaIXaaabaGaeqyVd4 gabaGaeqyVd4gabaGaaGymaaaaaiaawUfacaGLDbaadaWadaqaauaa beqaceaaaeaacqaH1oqzdaWgaaWcbaGaamOCaiaadkhaaeqaaaGcba GaeqyTdu2aaSbaaSqaaiabeI7aXjabeI7aXbqabaaaaaGccaGLBbGa ayzxaaGaeyOeI0YaaSaaaeaacaWGfbGaeqySdeMaeuiLdqKaamivaa qaaiaaigdacqGHsislcqaH9oGBaaWaamWaaeaafaqabeGabaaabaGa aGymaaqaaiaaigdaaaaacaGLBbGaayzxaaaaaa@62B9@

σ zz =0 ε zz = ν E ( σ rr + σ θθ )+αΔT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOEaiaadQ haaeqaaOGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqyTdu2aaS baaSqaaiaadQhacaWG6baabeaakiabg2da9iabgkHiTmaalaaabaGa eqyVd4gabaGaamyraaaadaqadaqaaiabeo8aZnaaBaaaleaacaWGYb GaamOCaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiU dehabeaaaOGaayjkaiaawMcaaiabgUcaRiabeg7aHjabfs5aejaads faaaa@624D@

 Equation of motion

d σ rr dr + 1 r ( σ rr σ θθ )+ ρ 0 b r = ρ 0 ω 2 r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacqaHdpWCdaWgaa WcbaGaamOCaiaadkhaaeqaaaGcbaGaamizaiaadkhaaaGaey4kaSYa aSaaaeaacaaIXaaabaGaamOCaaaadaqadaqaaiabeo8aZnaaBaaale aacaWGYbGaamOCaaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaeqiU deNaeqiUdehabeaaaOGaayjkaiaawMcaaiaaykW7caaMc8Uaey4kaS IaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaamOyamaaBaaaleaacaWG Ybaabeaakiabg2da9iabgkHiTiabeg8aYnaaBaaaleaacaaIWaaabe aakiabeM8a3naaCaaaleqabaGaaGOmaaaakiaadkhaaaa@5901@

 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@

Plane strain solution ε zz =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOEaiaadQ haaeqaaOGaeyypa0JaaGimaaaa@3860@

Generalized plane strain solution, with axial force F z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadQhaaeqaaa aa@34BC@  applied to cylinder:

a b 2πr σ zz dr= F z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdXbqaaiaaikdacqaHapaCcaWGYb Gaeq4Wdm3aaSbaaSqaaiaadQhacaWG6baabeaaaeaacaWGHbaabaGa amOyaaqdcqGHRiI8aOGaamizaiaadkhacqGH9aqpcaWGgbWaaSbaaS qaaiaadQhaaeqaaaaa@4334@

 

These results can either be derived as a special case of the general 3D equations of linear elasticity in spherical coordinates, or alternatively can be obtained directly from the formulas in Cartesian components.  Here, we briefly outline the the latter.

1.      Note that we can find the components of { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaam OCaaqabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiil aiaahwgadaWgaaWcbaGaamOEaaqabaaakiaawUhacaGL9baaaaa@3D6F@  in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  basis as follows. First, note that e r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaadkhaaeqaaa aa@34D7@  is radial MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  a radial unit vector can be written in terms of the position vector as x/| x | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaai4lamaaemaabaGaaCiEaa Gaay5bSlaawIa7aaaa@389D@ .  Next, note e θ = e 3 × e r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiabeI7aXbqaba GccqGH9aqpcaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaey41aqRaaCyz amaaBaaaleaacaWGYbaabeaaaaa@3CAF@  and e z = e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaadQhaaeqaaO Gaeyypa0JaaCyzamaaBaaaleaacaaIZaaabeaaaaa@37C6@ .  Using index notation, the components of the basis vectors { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaam OCaaqabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiil aiaahwgadaWgaaWcbaGaamOEaaqabaaakiaawUhacaGL9baaaaa@3D6F@  in { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  are therefore

{ x α r , i3α x α r , δ i3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaamaalaaabaGaamiEamaaBa aaleaacqaHXoqyaeqaaaGcbaGaamOCaaaacaGGSaGaeyicI48aaSba aSqaaiaadMgacaaIZaGaeqySdegabeaakmaalaaabaGaamiEamaaBa aaleaacqaHXoqyaeqaaaGcbaGaamOCaaaacaGGSaGaeqiTdq2aaSba aSqaaiaadMgacaaIZaaabeaaaOGaay5Eaiaaw2haaaaa@4693@

where r= x α x α = x 1 2 + x 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0ZaaOaaaeaacaWG4b WaaSbaaSqaaiabeg7aHbqabaGccaWG4bWaaSbaaSqaaiabeg7aHbqa baaabeaakiabg2da9maakaaabaGaamiEamaaDaaaleaacaaIXaaaba GaaGOmaaaakiabgUcaRiaadIhadaqhaaWcbaGaaGOmaaqaaiaaikda aaaabeaaaaa@41BC@ , and we use the convention that Greek subscripts range from 1 to 2.

2.      The components of the (radial) displacement vector in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  basis are u α =u(r) x α /r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiabeg7aHbqaba GccqGH9aqpcaWG1bGaaiikaiaadkhacaGGPaGaamiEamaaBaaaleaa cqaHXoqyaeqaaOGaai4laiaadkhaaaa@3E61@ .

3.      To proceed with the algebra, it is helpful to remember that x α / x α = δ αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWG4bWaaSbaaSqaaiabeg 7aHbqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacqaHXoqyaeqa aOGaeyypa0JaeqiTdq2aaSbaaSqaaiabeg7aHjabek7aIbqabaaaaa@4200@ r/ x α = x α /r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWGYbGaai4laiabgkGi2k aadIhadaWgaaWcbaGaeqySdegabeaakiabg2da9iaadIhadaWgaaWc baGaeqySdegabeaakiaac+cacaWGYbaaaa@3F90@  and r 1 / x α = x α / r 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWGYbWaaWbaaSqabeaacq GHsislcaaIXaaaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaeqyS degabeaakiabg2da9iabgkHiTiaadIhadaWgaaWcbaGaeqySdegabe aakiaac+cacaWGYbWaaWbaaSqabeaacaaIZaaaaaaa@4346@

4.      The components of the strain tensor in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  basis therefore follow as

ε αβ = 1 2 ( u α x β + u β x α )= du dr x α x β r 2 +u(r)( δ αβ r x α x β r 3 ) ε zz = ε 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaeqySdeMaeq OSdigabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaeWa aeaadaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaeqySdegabeaaaO qaaiabgkGi2kaadIhadaWgaaWcbaGaeqOSdigabeaaaaGccqGHRaWk daWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaeqOSdigabeaaaOqaai abgkGi2kaadIhadaWgaaWcbaGaeqySdegabeaaaaaakiaawIcacaGL PaaacqGH9aqpdaWcaaqaaiaadsgacaWG1baabaGaamizaiaadkhaaa WaaSaaaeaacaWG4bWaaSbaaSqaaiabeg7aHbqabaGccaWG4bWaaSba aSqaaiabek7aIbqabaaakeaacaWGYbWaaWbaaSqabeaacaaIYaaaaa aakiabgUcaRiaadwhacaGGOaGaamOCaiaacMcadaqadaqaamaalaaa baGaeqiTdq2aaSbaaSqaaiabeg7aHjabek7aIbqabaaakeaacaWGYb aaaiabgkHiTmaalaaabaGaamiEamaaBaaaleaacqaHXoqyaeqaaOGa amiEamaaBaaaleaacqaHYoGyaeqaaaGcbaGaamOCamaaCaaaleqaba GaaG4maaaaaaaakiaawIcacaGLPaaacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH1oqzdaWgaaWcba GaamOEaiaadQhaaeqaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiaaioda caaIZaaabeaaaaa@84C7@

5.      The strain components ε rr , ε θθ , ε zz MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOCaiaadk haaeqaaOGaaiilaiabew7aLnaaBaaaleaacqaH4oqCcqaH4oqCaeqa aOGaaiilaiabew7aLnaaBaaaleaacaWG6bGaamOEaaqabaaaaa@410B@  can then be found as ε rr = e r ε e r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0JaaCyzamaaBaaaleaacaWGYbaabeaakiabgwSi xlaahw7acqGHflY1caWHLbWaaSbaaSqaaiaadkhaaeqaaaaa@4198@ , ε θθ = e θ ε e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iaahwgadaWgaaWcbaGaeqiUdehabeaakiab gwSixlaahw7acqGHflY1caWHLbWaaSbaaSqaaiabeI7aXbqabaaaaa@4494@  and ε zz = e z ε e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOEaiaadQ haaeqaaOGaeyypa0JaaCyzamaaBaaaleaacaWG6baabeaakiabgwSi xlaahw7acqGHflY1caWHLbWaaSbaaSqaaiaadQhaaeqaaaaa@41B8@ .  Substituting for the basis vectors and simplifying gives the strain-displacement relations.  For example

ε rr = ε αβ x α x β r 2 = du dr x α x α x β x β r 4 +u(r)( δ αβ r x α x β r 2 x α x α x β x β r 5 )= du dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiabeg7aHjabek7aIbqa baGcdaWcaaqaaiaadIhadaWgaaWcbaGaeqySdegabeaakiaadIhada WgaaWcbaGaeqOSdigabeaaaOqaaiaadkhadaahaaWcbeqaaiaaikda aaaaaOGaeyypa0ZaaSaaaeaacaWGKbGaamyDaaqaaiaadsgacaWGYb aaamaalaaabaGaamiEamaaBaaaleaacqaHXoqyaeqaaOGaamiEamaa BaaaleaacqaHXoqyaeqaaOGaamiEamaaBaaaleaacqaHYoGyaeqaaO GaamiEamaaBaaaleaacqaHYoGyaeqaaaGcbaGaamOCamaaCaaaleqa baGaaGinaaaaaaGccqGHRaWkcaWG1bGaaiikaiaadkhacaGGPaWaae WaaeaadaWcaaqaaiabes7aKnaaBaaaleaacqaHXoqycqaHYoGyaeqa aaGcbaGaamOCaaaadaWcaaqaaiaadIhadaWgaaWcbaGaeqySdegabe aakiaadIhadaWgaaWcbaGaeqOSdigabeaaaOqaaiaadkhadaahaaWc beqaaiaaikdaaaaaaOGaeyOeI0YaaSaaaeaacaWG4bWaaSbaaSqaai abeg7aHbqabaGccaWG4bWaaSbaaSqaaiabeg7aHbqabaGccaWG4bWa aSbaaSqaaiabek7aIbqabaGccaWG4bWaaSbaaSqaaiabek7aIbqaba aakeaacaWGYbWaaWbaaSqabeaacaaI1aaaaaaaaOGaayjkaiaawMca aiabg2da9maalaaabaGaamizaiaadwhaaeaacaWGKbGaamOCaaaaaa a@7CFA@

where we have noted x α x α = r 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiabeg7aHbqaba GccaWG4bWaaSbaaSqaaiabeg7aHbqabaGccqGH9aqpcaWGYbWaaWba aSqabeaacaaIYaaaaaaa@3B50@ . The remaining components are left as an exercise.

6.      Finally, to derive the equilibrium equation, note that the stress tensor can be expressed as σ= σ rr e r e r + σ θθ e θ e θ + σ zz e z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdpGaeyypa0Jaeq4Wdm3aaSbaaS qaaiaadkhacaWGYbaabeaakiaahwgadaWgaaWcbaGaamOCaaqabaGc cqGHxkcXcaWHLbWaaSbaaSqaaiaadkhaaeqaaOGaey4kaSIaeq4Wdm 3aaSbaaSqaaiabeI7aXjabeI7aXbqabaGccaWHLbWaaSbaaSqaaiab eI7aXbqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiabeI7aXbqabaGccq GHRaWkcqaHdpWCdaWgaaWcbaGaamOEaiaadQhaaeqaaOGaaCyzamaa BaaaleaacaWG6baabeaakiabgEPielaahwgadaWgaaWcbaGaamOEaa qabaaaaa@5863@ .  Substituting for the basis vectors from item(1) above gives

σ αβ = σ rr x α x β r 2 + σ θθ i3γ x γ r β3κ x κ r σ zz = σ 33 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqySdeMaeq OSdigabeaakiabg2da9iabeo8aZnaaBaaaleaacaWGYbGaamOCaaqa baGcdaWcaaqaaiaadIhadaWgaaWcbaGaeqySdegabeaakiaadIhada WgaaWcbaGaeqOSdigabeaaaOqaaiaadkhadaahaaWcbeqaaiaaikda aaaaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7aXbqaba GccqGHiiIZdaWgaaWcbaGaamyAaiaaiodacqaHZoWzaeqaaOWaaSaa aeaacaWG4bWaaSbaaSqaaiabeo7aNbqabaaakeaacaWGYbaaaiabgI GiopaaBaaaleaacqaHYoGycaaIZaGaeqOUdSgabeaakmaalaaabaGa amiEamaaBaaaleaacqaH6oWAaeqaaaGcbaGaamOCaaaacaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8Uaeq4Wdm3aaSbaaSqaaiaadQhacaWG6baabeaakiabg2da9iabeo 8aZnaaBaaaleaacaaIZaGaaG4maaqabaaaaa@754A@

7.      Substitute the preceding result into the equilibrium equation

σ ij x i + ρ 0 b(r) x j r =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kabeo8aZnaaBa aaleaacaWGPbGaamOAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqa aiaadMgaaeqaaaaakiabgUcaRiabeg8aYnaaBaaaleaacaaIWaaabe aakiaadkgacaGGOaGaamOCaiaacMcadaWcaaqaaiaadIhadaWgaaWc baGaamOAaaqabaaakeaacaWGYbaaaiabg2da9iaaicdaaaa@474B@

and crank through a good deal of tedious algebra to see that

( d σ rr dr + 1 r ( σ rr σ θθ )+ ρ 0 b(r) ) x α r =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaamaalaaabaGaamizaiabeo 8aZnaaBaaaleaacaWGYbGaamOCaaqabaaakeaacaWGKbGaamOCaaaa cqGHRaWkdaWcaaqaaiaaigdaaeaacaWGYbaaamaabmaabaGaeq4Wdm 3aaSbaaSqaaiaadkhacaWGYbaabeaakiabgkHiTiabeo8aZnaaBaaa leaacqaH4oqCcqaH4oqCaeqaaaGccaGLOaGaayzkaaGaey4kaSIaeq yWdi3aaSbaaSqaaiaaicdaaeqaaOGaamOyaiaacIcacaWGYbGaaiyk aaGaayjkaiaawMcaamaalaaabaGaamiEamaaBaaaleaacqaHXoqyae qaaaGcbaGaamOCaaaacqGH9aqpcaaIWaaaaa@55D7@

This result can also be obtained using the virtual work principle (see problems for Sect 2.4 for details)

 

 

4.1.8 General solution to the axisymmetric boundary value problem

 

Our goal is to solve the equations given in Section 4.1.2 for the displacement, strain and stress in the sphere.  To do so,

1.      Substitute the strain-displacement relations into the stress-strain law to show that, for generalized plane strain

[ σ rr σ θθ σ zz ]= E (1+ν)(12ν) [ 1ν ν ν ν 1ν ν ν ν 1ν ][ du dr u r ε zz ] EαΔT 12ν [ 1 1 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadeaaaeaacqaHdp WCdaWgaaWcbaGaamOCaiaadkhaaeqaaaGcbaGaeq4Wdm3aaSbaaSqa aiabeI7aXjabeI7aXbqabaaakeaacqaHdpWCdaWgaaWcbaGaamOEai aadQhaaeqaaaaaaOGaay5waiaaw2faaiabg2da9maalaaabaGaamyr aaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4MaaiykaiaacIcacaaIXa GaeyOeI0IaaGOmaiabe27aUjaacMcaaaWaamWaaeaafaqabeWadaaa baGaaGymaiabgkHiTiabe27aUbqaaiabe27aUbqaaiabe27aUbqaai abe27aUbqaaiaaigdacqGHsislcqaH9oGBaeaacqaH9oGBaeaacqaH 9oGBaeaacqaH9oGBaeaacaaIXaGaeyOeI0IaeqyVd4gaaaGaay5wai aaw2faamaadmaabaqbaeqabmqaaaqaamaalaaabaGaamizaiaadwha aeaacaWGKbGaamOCaaaaaeaadaWcaaqaaiaadwhaaeaacaWGYbaaaa qaaiabew7aLnaaBaaaleaacaWG6bGaamOEaaqabaaaaaGccaGLBbGa ayzxaaGaeyOeI0YaaSaaaeaacaWGfbGaeqySdeMaeuiLdqKaamivaa qaaiaaigdacqGHsislcaaIYaGaeqyVd4gaamaadmaabaqbaeqabmqa aaqaaiaaigdaaeaacaaIXaaabaGaaGymaaaaaiaawUfacaGLDbaaaa a@7E29@

where ε zz MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOEaiaadQ haaeqaaaaa@3697@  is constant.  The equivalent expression for plane stress is

[ σ rr σ θθ ]= E 1 ν 2 [ 1 ν ν 1 ][ du dr u r ] EαΔT 1ν [ 1 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqaceaaaeaacqaHdp WCdaWgaaWcbaGaamOCaiaadkhaaeqaaaGcbaGaeq4Wdm3aaSbaaSqa aiabeI7aXjabeI7aXbqabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaaS aaaeaacaWGfbaabaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGa aGOmaaaaaaGcdaWadaqaauaabeqaciaaaeaacaaIXaaabaGaeqyVd4 gabaGaeqyVd4gabaGaaGymaaaaaiaawUfacaGLDbaadaWadaqaauaa beqaceaaaeaadaWcaaqaaiaadsgacaWG1baabaGaamizaiaadkhaaa aabaWaaSaaaeaacaWG1baabaGaamOCaaaaaaaacaGLBbGaayzxaaGa eyOeI0YaaSaaaeaacaWGfbGaeqySdeMaeuiLdqKaamivaaqaaiaaig dacqGHsislcqaH9oGBaaWaamWaaeaafaqabeGabaaabaGaaGymaaqa aiaaigdaaaaacaGLBbGaayzxaaaaaa@5F79@

2.      Substitute these expressions for the stress into the equilibrium equation and rearrange the result to see that, for generalized plane strain

2 u r 2 + 1 r u r u r 2 = r { 1 r r ( ru ) }= α( 1+ν ) ( 1ν ) ΔT r ( 1+ν )( 12ν ) E(1ν) ρ 0 (b+ ω 2 r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadwhaaeaacqGHciITcaWGYbWaaWbaaSqabeaacaaI YaaaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiaadkhaaaWaaSaaae aacqGHciITcaWG1baabaGaeyOaIyRaamOCaaaacqGHsisldaWcaaqa aiaadwhaaeaacaWGYbWaaWbaaSqabeaacaaIYaaaaaaakiabg2da9m aalaaabaGaeyOaIylabaGaeyOaIyRaamOCaaaadaGadaqaamaalaaa baGaaGymaaqaaiaadkhaaaWaaSaaaeaacqGHciITaeaacqGHciITca WGYbaaamaabmaabaGaamOCaiaadwhaaiaawIcacaGLPaaaaiaawUha caGL9baacqGH9aqpdaWcaaqaaiabeg7aHnaabmaabaGaaGymaiabgU caRiabe27aUbGaayjkaiaawMcaaaqaamaabmaabaGaaGymaiabgkHi Tiabe27aUbGaayjkaiaawMcaaaaadaWcaaqaaiabgkGi2kabfs5aej aadsfaaeaacqGHciITcaWGYbaaaiabgkHiTmaalaaabaWaaeWaaeaa caaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzkaaWaaeWaaeaacaaIXa GaeyOeI0IaaGOmaiabe27aUbGaayjkaiaawMcaaaqaaiaadweacaGG OaGaaGymaiabgkHiTiabe27aUjaacMcaaaGaeqyWdi3aaSbaaSqaai aaicdaaeqaaOGaaiikaiaadkgacqGHRaWkcqaHjpWDdaahaaWcbeqa aiaaikdaaaGccaWGYbGaaiykaaaa@81FE@

while for plane stress

2 u r 2 + 1 r u r u r 2 = r { 1 r r ( ru ) }=α( 1+ν ) ΔT r ( 1 ν 2 ) E ρ 0 (b+ ω 2 r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadwhaaeaacqGHciITcaWGYbWaaWbaaSqabeaacaaI YaaaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiaadkhaaaWaaSaaae aacqGHciITcaWG1baabaGaeyOaIyRaamOCaaaacqGHsisldaWcaaqa aiaadwhaaeaacaWGYbWaaWbaaSqabeaacaaIYaaaaaaakiabg2da9m aalaaabaGaeyOaIylabaGaeyOaIyRaamOCaaaadaGadaqaamaalaaa baGaaGymaaqaaiaadkhaaaWaaSaaaeaacqGHciITaeaacqGHciITca WGYbaaamaabmaabaGaamOCaiaadwhaaiaawIcacaGLPaaaaiaawUha caGL9baacqGH9aqpcqaHXoqydaqadaqaaiaaigdacqGHRaWkcqaH9o GBaiaawIcacaGLPaaadaWcaaqaaiabgkGi2kabfs5aejaadsfaaeaa cqGHciITcaWGYbaaaiabgkHiTmaalaaabaWaaeWaaeaacaaIXaGaey OeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaa baGaamyraaaacqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaGGOaGaam OyaiabgUcaRiabeM8a3naaCaaaleqabaGaaGOmaaaakiaadkhacaGG Paaaaa@73A5@

 

Given the temperature distribution and body force these equations can be integrated to calculate the displacement u.  Two arbitrary constants of integration will appear when you do the integral MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  these must be determined from the boundary conditions at the inner and outer surface of the sphere.  Specifically, the constants must be selected so that either the displacement or the radial stress have prescribed values on the inner and outer surface of the cylinder.  Finally, for the generalized plane strain solution, the axial strain ε zz MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOEaiaadQ haaeqaaaaa@3697@  must be determined, using the equation for the axial force acting on the ends of the cylinder.

 

In the following sections, this procedure is used to derive solutions to various boundary value problems of practical interest.

 

 

4.1.9 Long (generalized plane strain) cylinder subjected to internal and external pressure.

 

We consider a long hollow cylinder with internal radius a and external radius b as shown in the figure. 

Assume that

 No body forces act on the cylinder

 The cylinder has zero angular velocity

 The sphere has uniform temperature

 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@

 For the plane strain solution, the cylinder does not stretch parallel to its axis.  For the generalized plane strain solution, the ends of the cylinder are subjected to an axial force F z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadQhaaeqaaa aa@34BC@  as shown.  In particular, for a closed ended cylinder the axial force exerted by the pressure inside the cylinder acting on the closed ends is F z =π( p a a 2 p b b 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadQhaaeqaaO Gaeyypa0JaeqiWda3aaeWaaeaacaWGWbWaaSbaaSqaaiaadggaaeqa aOGaamyyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadchadaWgaa WcbaGaamOyaaqabaGccaWGIbWaaWbaaSqabeaacaaIYaaaaaGccaGL OaGaayzkaaaaaa@41D5@

 

The displacement, strain and stress fields in the cylinder are

u= ( 1+ν ) a 2 b 2 E( b 2 a 2 ) { ( p a p b ) r +( 12ν ) ( p a a 2 p b b 2 ) a 2 b 2 r } e r ν ε zz r e r + ε zz z e z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0ZaaSaaaeaadaqada qaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaacaWGHbWaaWba aSqabeaacaaIYaaaaOGaamOyamaaCaaaleqabaGaaGOmaaaaaOqaai aadweadaqadaqaaiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHsisl caWGHbWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaaamaacm aabaWaaSaaaeaadaqadaqaaiaadchadaWgaaWcbaGaamyyaaqabaGc cqGHsislcaWGWbWaaSbaaSqaaiaadkgaaeqaaaGccaGLOaGaayzkaa aabaGaamOCaaaacqGHRaWkdaqadaqaaiaaigdacqGHsislcaaIYaGa eqyVd4gacaGLOaGaayzkaaWaaSaaaeaadaqadaqaaiaadchadaWgaa WcbaGaamyyaaqabaGccaWGHbWaaWbaaSqabeaacaaIYaaaaOGaeyOe I0IaamiCamaaBaaaleaacaWGIbaabeaakiaadkgadaahaaWcbeqaai aaikdaaaaakiaawIcacaGLPaaaaeaacaWGHbWaaWbaaSqabeaacaaI YaaaaOGaamOyamaaCaaaleqabaGaaGOmaaaaaaGccaWGYbaacaGL7b GaayzFaaGaaCyzamaaBaaaleaacaWGYbaabeaakiabgkHiTiabe27a Ujabew7aLnaaBaaaleaacaWG6bGaamOEaaqabaGccaWGYbGaaCyzam aaBaaaleaacaWGYbaabeaakiabgUcaRiabew7aLnaaBaaaleaacaWG 6bGaamOEaaqabaGccaWG6bGaaCyzamaaBaaaleaacaWG6baabeaaaa a@773D@

ε rr = ( 1+ν ) a 2 b 2 E( b 2 a 2 ) { ( p a p b ) r 2 +( 12ν ) ( p a a 2 p b b 2 ) a 2 b 2 }ν ε zz ε θθ = ( 1+ν ) a 2 b 2 E( b 2 a 2 ) { ( p a p b ) r 2 +( 12ν ) ( p a a 2 p b b 2 ) a 2 b 2 }ν ε zz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabew7aLnaaBaaaleaacaWGYb GaamOCaaqabaGccqGH9aqpdaWcaaqaamaabmaabaGaaGymaiabgUca Riabe27aUbGaayjkaiaawMcaaiaadggadaahaaWcbeqaaiaaikdaaa GccaWGIbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamyramaabmaabaGa amOyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbe qaaiaaikdaaaaakiaawIcacaGLPaaaaaWaaiWaaeaacqGHsisldaWc aaqaamaabmaabaGaamiCamaaBaaaleaacaWGHbaabeaakiabgkHiTi aadchadaWgaaWcbaGaamOyaaqabaaakiaawIcacaGLPaaaaeaacaWG YbWaaWbaaSqabeaacaaIYaaaaaaakiabgUcaRmaabmaabaGaaGymai abgkHiTiaaikdacqaH9oGBaiaawIcacaGLPaaadaWcaaqaamaabmaa baGaamiCamaaBaaaleaacaWGHbaabeaakiaadggadaahaaWcbeqaai aaikdaaaGccqGHsislcaWGWbWaaSbaaSqaaiaadkgaaeqaaOGaamOy amaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaqaaiaadggada ahaaWcbeqaaiaaikdaaaGccaWGIbWaaWbaaSqabeaacaaIYaaaaaaa aOGaay5Eaiaaw2haaiabgkHiTiabe27aUjabew7aLnaaBaaaleaaca WG6bGaamOEaaqabaaakeaacqaH1oqzdaWgaaWcbaGaeqiUdeNaeqiU dehabeaakiabg2da9maalaaabaWaaeWaaeaacaaIXaGaey4kaSIaeq yVd4gacaGLOaGaayzkaaGaamyyamaaCaaaleqabaGaaGOmaaaakiaa dkgadaahaaWcbeqaaiaaikdaaaaakeaacaWGfbWaaeWaaeaacaWGIb WaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyyamaaCaaaleqabaGa aGOmaaaaaOGaayjkaiaawMcaaaaadaGadaqaamaalaaabaWaaeWaae aacaWGWbWaaSbaaSqaaiaadggaaeqaaOGaeyOeI0IaamiCamaaBaaa leaacaWGIbaabeaaaOGaayjkaiaawMcaaaqaaiaadkhadaahaaWcbe qaaiaaikdaaaaaaOGaey4kaSYaaeWaaeaacaaIXaGaeyOeI0IaaGOm aiabe27aUbGaayjkaiaawMcaamaalaaabaWaaeWaaeaacaWGWbWaaS baaSqaaiaadggaaeqaaOGaamyyamaaCaaaleqabaGaaGOmaaaakiab gkHiTiaadchadaWgaaWcbaGaamOyaaqabaGccaWGIbWaaWbaaSqabe aacaaIYaaaaaGccaGLOaGaayzkaaaabaGaamyyamaaCaaaleqabaGa aGOmaaaakiaadkgadaahaaWcbeqaaiaaikdaaaaaaaGccaGL7bGaay zFaaGaeyOeI0IaeqyVd4MaeqyTdu2aaSbaaSqaaiaadQhacaWG6baa beaaaaaa@A9AB@

σ rr ={ ( p a a 2 p b b 2 ) b 2 a 2 a 2 b 2 ( b 2 a 2 ) r 2 ( p a p b ) } σ θθ ={ ( p a a 2 p b b 2 ) b 2 a 2 + a 2 b 2 ( b 2 a 2 ) r 2 ( p a p b ) } σ zz =2ν ( p a a 2 p b b 2 ) b 2 a 2 +E ε zz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaWGYb GaamOCaaqabaGccqGH9aqpdaGadaqaamaalaaabaWaaeWaaeaacaWG WbWaaSbaaSqaaiaadggaaeqaaOGaamyyamaaCaaaleqabaGaaGOmaa aakiabgkHiTiaadchadaWgaaWcbaGaamOyaaqabaGccaWGIbWaaWba aSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaabaGaamOyamaaCaaale qabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqaaiaaikdaaaaa aOGaeyOeI0YaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaam OyamaaCaaaleqabaGaaGOmaaaaaOqaamaabmaabaGaamOyamaaCaaa leqabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqaaiaaikdaaa aakiaawIcacaGLPaaacaWGYbWaaWbaaSqabeaacaaIYaaaaaaakmaa bmaabaGaamiCamaaBaaaleaacaWGHbaabeaakiabgkHiTiaadchada WgaaWcbaGaamOyaaqabaaakiaawIcacaGLPaaaaiaawUhacaGL9baa aeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaakiabg2da9m aacmaabaWaaSaaaeaadaqadaqaaiaadchadaWgaaWcbaGaamyyaaqa baGccaWGHbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamiCamaaBa aaleaacaWGIbaabeaakiaadkgadaahaaWcbeqaaiaaikdaaaaakiaa wIcacaGLPaaaaeaacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0 IaamyyamaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkdaWcaaqaaiaa dggadaahaaWcbeqaaiaaikdaaaGccaWGIbWaaWbaaSqabeaacaaIYa aaaaGcbaWaaeWaaeaacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaeyOe I0IaamyyamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiaadk hadaahaaWcbeqaaiaaikdaaaaaaOWaaeWaaeaacaWGWbWaaSbaaSqa aiaadggaaeqaaOGaeyOeI0IaamiCamaaBaaaleaacaWGIbaabeaaaO GaayjkaiaawMcaaaGaay5Eaiaaw2haaaqaaiabeo8aZnaaBaaaleaa caWG6bGaamOEaaqabaGccqGH9aqpcaaIYaGaeqyVd42aaSaaaeaada qadaqaaiaadchadaWgaaWcbaGaamyyaaqabaGccaWGHbWaaWbaaSqa beaacaaIYaaaaOGaeyOeI0IaamiCamaaBaaaleaacaWGIbaabeaaki aadkgadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaaaeaacaWG IbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyyamaaCaaaleqaba GaaGOmaaaaaaGccqGHRaWkcaWGfbGaeqyTdu2aaSbaaSqaaiaadQha caWG6baabeaaaaaa@A316@

where ε zz =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOEaiaadQ haaeqaaOGaeyypa0JaaGimaaaa@3861@  for plane strain, while

ε zz = F z πE( b 2 a 2 ) 2ν E ( p a a 2 p b b 2 ) ( b 2 a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOEaiaadQ haaeqaaOGaeyypa0ZaaSaaaeaacaWGgbWaaSbaaSqaaiaadQhaaeqa aaGcbaGaeqiWdaNaamyraiaacIcacaWGIbWaaWbaaSqabeaacaaIYa aaaOGaeyOeI0IaamyyamaaCaaaleqabaGaaGOmaaaakiaacMcaaaGa eyOeI0YaaSaaaeaacaaIYaGaeqyVd4gabaGaamyraaaadaWcaaqaam aabmaabaGaamiCamaaBaaaleaacaWGHbaabeaakiaadggadaahaaWc beqaaiaaikdaaaGccqGHsislcaWGWbWaaSbaaSqaaiaadkgaaeqaaO GaamOyamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaqaaiaa cIcacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyyamaaCa aaleqabaGaaGOmaaaakiaacMcaaaaaaa@56C7@

for generalized plane strain.

 

Derivation: These results can be derived as follows.  The governing equation reduces to

2 u r 2 + 1 r u r u r 2 = r { 1 r r ( ru ) }=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadwhaaeaacqGHciITcaWGYbWaaWbaaSqabeaacaaI YaaaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiaadkhaaaWaaSaaae aacqGHciITcaWG1baabaGaeyOaIyRaamOCaaaacqGHsisldaWcaaqa aiaadwhaaeaacaWGYbWaaWbaaSqabeaacaaIYaaaaaaakiabg2da9m aalaaabaGaeyOaIylabaGaeyOaIyRaamOCaaaadaGadaqaamaalaaa baGaaGymaaqaaiaadkhaaaWaaSaaaeaacqGHciITaeaacqGHciITca WGYbaaamaabmaabaGaamOCaiaadwhaaiaawIcacaGLPaaaaiaawUha caGL9baacqGH9aqpcaaIWaaaaa@56A4@

The equation can be integrated to see that

u=Ar+ B r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bGaeyypa0Jaamyqaiaadkhacq GHRaWkdaWcaaqaaiaadkeaaeaacaWGYbaaaaaa@3932@

The radial stress follows as

σ rr = E ( 1+ν )( 12ν ) { (1ν) u r +ν u r }= E ( 1+ν )( 12ν ) { A(12ν) B r 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaWaaeWaaeaacaaIXaGa ey4kaSIaeqyVd4gacaGLOaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0 IaaGOmaiabe27aUbGaayjkaiaawMcaaaaadaGadaqaaiaacIcacaaI XaGaeyOeI0IaeqyVd4MaaiykamaalaaabaGaeyOaIyRaamyDaaqaai abgkGi2kaadkhaaaGaey4kaSIaeqyVd42aaSaaaeaacaWG1baabaGa amOCaaaaaiaawUhacaGL9baacqGH9aqpdaWcaaqaaiaadweaaeaada qadaqaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaadaqadaqa aiaaigdacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaaamaacm aabaGaamyqaiabgkHiTiaacIcacaaIXaGaeyOeI0IaaGOmaiabe27a UjaacMcadaWcaaqaaiaadkeaaeaacaWGYbWaaWbaaSqabeaacaaIYa aaaaaaaOGaay5Eaiaaw2haaaaa@6BDE@

The boundary conditions are σ rr (r=a)= p a σ rr (r=b)= p b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaaiikaiaadkhacqGH9aqpcaWGHbGaaiykaiabg2da9iab gkHiTiaadchadaWgaaWcbaGaamyyaaqabaGccaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaamOCaiaadkha aeqaaOGaaiikaiaadkhacqGH9aqpcaWGIbGaaiykaiabg2da9iabgk HiTiaadchadaWgaaWcbaGaamOyaaqabaaaaa@544D@  (the stresses are negative because the pressure is compressive).  This yields two equations for A and B that area easily solved to see that

A= ( 1+ν )( 12ν ) E ( p a a 2 p b b 2 ) b 2 a 2 B= ( 1+ν ) E a 2 b 2 b 2 a 2 ( p a p b ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbGaeyypa0ZaaSaaaeaadaqada qaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaadaqadaqaaiaa igdacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaabaGaamyraa aadaWcaaqaamaabmaabaGaamiCamaaBaaaleaacaWGHbaabeaakiaa dggadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGWbWaaSbaaSqaai aadkgaaeqaaOGaamOyamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaa wMcaaaqaaiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGHb WaaWbaaSqabeaacaaIYaaaaaaakiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamOqai abg2da9maalaaabaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGL OaGaayzkaaaabaGaamyraaaadaWcaaqaaiaadggadaahaaWcbeqaai aaikdaaaGccaWGIbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOyamaa CaaaleqabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqaaiaaik daaaaaaOWaaeWaaeaacaWGWbWaaSbaaSqaaiaadggaaeqaaOGaeyOe I0IaamiCamaaBaaaleaacaWGIbaabeaaaOGaayjkaiaawMcaaaaa@7664@

The remaining results follow by elementary algebraic manipulations.

 

 

 

4.1.10 Spinning circular plate

 

We consider a thin solid plate with radius a that spins with angular speed ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHjpWDaaa@3493@  about its axis. Assume that

 No body forces act on the disk

 The disk has constant angular velocity

 The disk has uniform temperature

 The outer surface r=a and the top and bottom faces of the disk are free of traction.

 The disk is sufficiently thin to ensure a state of plane stress in the disk.

u=(1ν) ρ 0 ω 2 8E { ( 3+ν ) a 2 r( 1+ν ) r 3 } e r +z ε zz e z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0Jaaiikaiaaigdacq GHsislcqaH9oGBcaGGPaWaaSaaaeaacqaHbpGCdaWgaaWcbaGaaGim aaqabaGccqaHjpWDdaahaaWcbeqaaiaaikdaaaaakeaacaaI4aGaam yraaaadaGadaqaamaabmaabaGaaG4maiabgUcaRiabe27aUbGaayjk aiaawMcaaiaadggadaahaaWcbeqaaiaaikdaaaGccaWGYbGaeyOeI0 YaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzkaaGaamOC amaaCaaaleqabaGaaG4maaaaaOGaay5Eaiaaw2haaiaahwgadaWgaa WcbaGaamOCaaqabaGccqGHRaWkcaWG6bGaeqyTdu2aaSbaaSqaaiaa dQhacaWG6baabeaakiaahwgadaWgaaWcbaGaamOEaaqabaaaaa@5C15@

ε rr =(1ν) ρ 0 ω 2 8E { ( 3+ν ) a 2 3( 1+ν ) r 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0JaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaWa aSaaaeaacqaHbpGCdaWgaaWcbaGaaGimaaqabaGccqaHjpWDdaahaa WcbeqaaiaaikdaaaaakeaacaaI4aGaamyraaaadaGadaqaamaabmaa baGaaG4maiabgUcaRiabe27aUbGaayjkaiaawMcaaiaadggadaahaa WcbeqaaiaaikdaaaGccqGHsislcaaIZaWaaeWaaeaacaaIXaGaey4k aSIaeqyVd4gacaGLOaGaayzkaaGaamOCamaaCaaaleqabaGaaGOmaa aaaOGaay5Eaiaaw2haaaaa@54B8@            ε θθ =(1ν) ρ 0 ω 2 8E { ( 3+ν ) a 2 ( 1+ν ) r 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiyk amaalaaabaGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaeqyYdC3aaW baaSqabeaacaaIYaaaaaGcbaGaaGioaiaadweaaaWaaiWaaeaadaqa daqaaiaaiodacqGHRaWkcqaH9oGBaiaawIcacaGLPaaacaWGHbWaaW baaSqabeaacaaIYaaaaOGaeyOeI0YaaeWaaeaacaaIXaGaey4kaSIa eqyVd4gacaGLOaGaayzkaaGaamOCamaaCaaaleqabaGaaGOmaaaaaO Gaay5Eaiaaw2haaaaa@5578@

ε zz =ν ρ 0 ω 2 8E { 2( 3+ν ) a 2 (3ν+2) r 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOEaiaadQ haaeqaaOGaeyypa0JaeyOeI0IaeqyVd42aaSaaaeaacqaHbpGCdaWg aaWcbaGaaGimaaqabaGccqaHjpWDdaahaaWcbeqaaiaaikdaaaaake aacaaI4aGaamyraaaadaGadaqaaiaaikdadaqadaqaaiaaiodacqGH RaWkcqaH9oGBaiaawIcacaGLPaaacaWGHbWaaWbaaSqabeaacaaIYa aaaOGaeyOeI0IaaiikaiaaiodacqaH9oGBcqGHRaWkcaaIYaGaaiyk aiaadkhadaahaaWcbeqaaiaaikdaaaaakiaawUhacaGL9baaaaa@5341@

σ rr =( 3+ν ) ρ 0 ω 2 8 { a 2 r 2 } σ θθ = ρ 0 ω 2 8 { (3+ν) a 2 (3ν+1) r 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaWGYb GaamOCaaqabaGccqGH9aqpdaqadaqaaiaaiodacqGHRaWkcqaH9oGB aiaawIcacaGLPaaadaWcaaqaaiabeg8aYnaaBaaaleaacaaIWaaabe aakiabeM8a3naaCaaaleqabaGaaGOmaaaaaOqaaiaaiIdaaaWaaiWa aeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamOCamaaCa aaleqabaGaaGOmaaaaaOGaay5Eaiaaw2haaaqaaiabeo8aZnaaBaaa leaacqaH4oqCcqaH4oqCaeqaaOGaeyypa0ZaaSaaaeaacqaHbpGCda WgaaWcbaGaaGimaaqabaGccqaHjpWDdaahaaWcbeqaaiaaikdaaaaa keaacaaI4aaaamaacmaabaGaaiikaiaaiodacqGHRaWkcqaH9oGBca GGPaGaamyyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaacIcacaaI ZaGaeqyVd4Maey4kaSIaaGymaiaacMcacaWGYbWaaWbaaSqabeaaca aIYaaaaaGccaGL7bGaayzFaaaaaaa@6766@

 

 

Derivation: To derive these results, recall that the governing equation is

2 u r 2 + 1 r u r u r 2 = r { 1 r r ( ru ) }= ( 1 ν 2 ) E ρ 0 ω 2 r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadwhaaeaacqGHciITcaWGYbWaaWbaaSqabeaacaaI YaaaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiaadkhaaaWaaSaaae aacqGHciITcaWG1baabaGaeyOaIyRaamOCaaaacqGHsisldaWcaaqa aiaadwhaaeaacaWGYbWaaWbaaSqabeaacaaIYaaaaaaakiabg2da9m aalaaabaGaeyOaIylabaGaeyOaIyRaamOCaaaadaGadaqaamaalaaa baGaaGymaaqaaiaadkhaaaWaaSaaaeaacqGHciITaeaacqGHciITca WGYbaaamaabmaabaGaamOCaiaadwhaaiaawIcacaGLPaaaaiaawUha caGL9baacqGH9aqpcqGHsisldaWcaaqaamaabmaabaGaaGymaiabgk HiTiabe27aUnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaqa aiaadweaaaGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaeqyYdC3aaW baaSqabeaacaaIYaaaaOGaamOCaaaa@63F4@

The equation can be integrated to see that

u=Ar+ B r ( 1 ν 2 ) 8E ρ 0 ω 2 r 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bGaeyypa0Jaamyqaiaadkhacq GHRaWkdaWcaaqaaiaadkeaaeaacaWGYbaaaiabgkHiTmaalaaabaWa aeWaaeaacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaa GccaGLOaGaayzkaaaabaGaaGioaiaadweaaaGaeqyWdi3aaSbaaSqa aiaaicdaaeqaaOGaeqyYdC3aaWbaaSqabeaacaaIYaaaaOGaamOCam aaCaaaleqabaGaaG4maaaaaaa@48E8@

The radial stress follows as

σ rr = E 1 ν 2 ( du dr +ν u r )= E 1 ν 2 { ( 1+ν )A( 1ν ) B r 2 ( 1 ν 2 ) ρ 0 ω 2 8E ( 3+ν ) r 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaGaaGymaiabgkHiTiab e27aUnaaCaaaleqabaGaaGOmaaaaaaGcdaqadaqaamaalaaabaGaam izaiaadwhaaeaacaWGKbGaamOCaaaacqGHRaWkcqaH9oGBdaWcaaqa aiaadwhaaeaacaWGYbaaaaGaayjkaiaawMcaaiabg2da9maalaaaba GaamyraaqaaiaaigdacqGHsislcqaH9oGBdaahaaWcbeqaaiaaikda aaaaaOWaaiWaaeaadaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaawI cacaGLPaaacaWGbbGaeyOeI0YaaeWaaeaacaaIXaGaeyOeI0IaeqyV d4gacaGLOaGaayzkaaWaaSaaaeaacaWGcbaabaGaamOCamaaCaaale qabaGaaGOmaaaaaaGccqGHsisldaWcaaqaamaabmaabaGaaGymaiab gkHiTiabe27aUnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaai abeg8aYnaaBaaaleaacaaIWaaabeaakiabeM8a3naaCaaaleqabaGa aGOmaaaaaOqaaiaaiIdacaWGfbaaamaabmaabaGaaG4maiabgUcaRi abe27aUbGaayjkaiaawMcaaiaadkhadaahaaWcbeqaaiaaikdaaaaa kiaawUhacaGL9baaaaa@7215@

The radial stress must be bounded at r=0, which is only possible if B=0.  In addition, the radial stress must be zero at r=a, which requires that

A= ρ 0 ω 2 8E ( 3+ν ) (1+ν) a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbGaeyypa0ZaaSaaaeaacqaHbp GCdaWgaaWcbaGaaGimaaqabaGccqaHjpWDdaahaaWcbeqaaiaaikda aaaakeaacaaI4aGaamyraaaadaWcaaqaamaabmaabaGaaG4maiabgU caRiabe27aUbGaayjkaiaawMcaaaqaaiaacIcacaaIXaGaey4kaSIa eqyVd4MaaiykaaaacaWGHbWaaWbaaSqabeaacaaIYaaaaaaa@470B@

The remaining results follow by straightforward algebra.

 

 

4.1.11 Stresses induced by an interference fit between two cylinders

 

Interference fits are often used to secure a bushing or a bearing housing to a shaft.  In this problem we calculate the stress induced by such an interference fit.

 

Consider a hollow cylindrical bushing, with outer radius b and inner radius a.  Suppose that a solid shaft with radius a+Δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbGaey4kaSIaeuiLdqeaaa@35F4@ , with Δ/a<<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcaGGVaGaamyyaiabgYda8i abgYda8iaaigdaaaa@3888@  is inserted into the cylinder as shown.  (In practice, this is done by heating the cylinder or cooling the shaft until they fit, and then letting the system return to thermal equilibrium)

 No body forces act on the solids

 The angular velocity is zero

 The cylinders have uniform temperature

 The shaft slides freely inside the bushing

 The ends of the cylinder are free of force.

 Both the shaft and cylinder have the same Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@

 The cylinder and shaft are sufficiently long to ensure that a state of generalized plane strain can be developed in each solid.

 

The displacements, strains and stresses in the solid shaft  (r<a) are

u= ( 1+ν )( 12ν )Δ( b 2 a 2 ) 2a b 2 r e r 2 ν 2 Δ( b 2 a 2 ) 2a b 2 r e r +2ν Δ( b 2 a 2 ) 2a b 2 z e z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0JaeyOeI0YaaSaaae aadaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaadaqa daqaaiaaigdacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaGaeu iLdqKaaiikaiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG HbWaaWbaaSqabeaacaaIYaaaaOGaaiykaaqaaiaaikdacaWGHbGaam OyamaaCaaaleqabaGaaGOmaaaaaaGccaWGYbGaaCyzamaaBaaaleaa caWGYbaabeaakiabgkHiTiaaikdacqaH9oGBdaahaaWcbeqaaiaaik daaaGcdaWcaaqaaiabfs5aejaacIcacaWGIbWaaWbaaSqabeaacaaI YaaaaOGaeyOeI0IaamyyamaaCaaaleqabaGaaGOmaaaakiaacMcaae aacaaIYaGaamyyaiaadkgadaahaaWcbeqaaiaaikdaaaaaaOGaamOC aiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHRaWkcaaIYaGaeqyVd4 2aaSaaaeaacqqHuoarcaGGOaGaamOyamaaCaaaleqabaGaaGOmaaaa kiabgkHiTiaadggadaahaaWcbeqaaiaaikdaaaGccaGGPaaabaGaaG OmaiaadggacaWGIbWaaWbaaSqabeaacaaIYaaaaaaakiaadQhacaWH LbWaaSbaaSqaaiaadQhaaeqaaaaa@71E0@

ε rr = ε θθ = ( 1+ν )( 12ν )Δ( b 2 a 2 ) 2a b 2 2 ν 2 Δ( b 2 a 2 ) 2a b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiabeI7aXjabeI7aXbqa baGccqGH9aqpcqGHsisldaWcaaqaamaabmaabaGaaGymaiabgUcaRi abe27aUbGaayjkaiaawMcaamaabmaabaGaaGymaiabgkHiTiaaikda cqaH9oGBaiaawIcacaGLPaaacqqHuoarcaGGOaGaamOyamaaCaaale qabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqaaiaaikdaaaGc caGGPaaabaGaaGOmaiaadggacaWGIbWaaWbaaSqabeaacaaIYaaaaa aakiabgkHiTiaaikdacqaH9oGBdaahaaWcbeqaaiaaikdaaaGcdaWc aaqaaiabfs5aejaacIcacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaey OeI0IaamyyamaaCaaaleqabaGaaGOmaaaakiaacMcaaeaacaaIYaGa amyyaiaadkgadaahaaWcbeqaaiaaikdaaaaaaaaa@6375@

σ rr = σ θθ = EΔ( b 2 a 2 ) 2a b 2 σ zz =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7aXbqa baGccqGH9aqpcqGHsisldaWcaaqaaiaadweacqqHuoarcaGGOaGaam OyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqa aiaaikdaaaGccaGGPaaabaGaaGOmaiaadggacaWGIbWaaWbaaSqabe aacaaIYaaaaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaadQhacaWG6baa beaakiabg2da9iaaicdaaaa@6AB1@

In the hollow cylinder, they are

u= ( 1+ν )a r Δ 2 { 1+( 12ν ) r 2 b 2 } e r ν 2 Δa b 2 r e r +2 ν 2 Δa b 2 z e z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0ZaaSaaaeaadaqada qaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaacaWGHbaabaGa amOCaaaadaWcaaqaaiabfs5aebqaaiaaikdaaaWaaiWaaeaacaaIXa Gaey4kaSYaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbGaayjk aiaawMcaamaalaaabaGaamOCamaaCaaaleqabaGaaGOmaaaaaOqaai aadkgadaahaaWcbeqaaiaaikdaaaaaaaGccaGL7bGaayzFaaGaaCyz amaaBaaaleaacaWGYbaabeaakiabgkHiTiabe27aUnaaCaaaleqaba GaaGOmaaaakmaalaaabaGaeuiLdqKaamyyaaqaaiaadkgadaahaaWc beqaaiaaikdaaaaaaOGaamOCaiaahwgadaWgaaWcbaGaamOCaaqaba GccqGHRaWkcaaIYaGaeqyVd42aaWbaaSqabeaacaaIYaaaaOWaaSaa aeaacqqHuoarcaWGHbaabaGaamOyamaaCaaaleqabaGaaGOmaaaaaa GccaWG6bGaaCyzamaaBaaaleaacaWG6baabeaaaaa@639F@

ε rr = ( 1+ν )a r 2 Δ 2 { 1+( 12ν ) r 2 b 2 } ν 2 Δa b 2 ε θθ = ( 1+ν )a r 2 Δ 2 { 1+( 12ν ) r 2 b 2 } ν 2 Δa b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabew7aLnaaBaaaleaacaWGYb GaamOCaaqabaGccqGH9aqpdaWcaaqaamaabmaabaGaaGymaiabgUca Riabe27aUbGaayjkaiaawMcaaiaadggaaeaacaWGYbWaaWbaaSqabe aacaaIYaaaaaaakmaalaaabaGaeuiLdqeabaGaaGOmaaaadaGadaqa aiabgkHiTiaaigdacqGHRaWkdaqadaqaaiaaigdacqGHsislcaaIYa GaeqyVd4gacaGLOaGaayzkaaWaaSaaaeaacaWGYbWaaWbaaSqabeaa caaIYaaaaaGcbaGaamOyamaaCaaaleqabaGaaGOmaaaaaaaakiaawU hacaGL9baacqGHsislcqaH9oGBdaahaaWcbeqaaiaaikdaaaGcdaWc aaqaaiabfs5aejaadggaaeaacaWGIbWaaWbaaSqabeaacaaIYaaaaa aaaOqaaiabew7aLnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyyp a0ZaaSaaaeaadaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaawIcaca GLPaaacaWGHbaabaGaamOCamaaCaaaleqabaGaaGOmaaaaaaGcdaWc aaqaaiabfs5aebqaaiaaikdaaaWaaiWaaeaacaaIXaGaey4kaSYaae WaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbGaayjkaiaawMcaamaa laaabaGaamOCamaaCaaaleqabaGaaGOmaaaaaOqaaiaadkgadaahaa WcbeqaaiaaikdaaaaaaaGccaGL7bGaayzFaaGaeyOeI0IaeqyVd42a aWbaaSqabeaacaaIYaaaaOWaaSaaaeaacqqHuoarcaWGHbaabaGaam OyamaaCaaaleqabaGaaGOmaaaaaaaaaaa@7CD9@

σ rr = EΔa 2 b 2 { 1 b 2 r 2 } σ θθ = EΔa 2 b 2 { 1+ b 2 r 2 } σ zz =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaWGfbGaeuiLdqKaamyyaaqaaiaa ikdacaWGIbWaaWbaaSqabeaacaaIYaaaaaaakmaacmaabaGaaGymai abgkHiTmaalaaabaGaamOyamaaCaaaleqabaGaaGOmaaaaaOqaaiaa dkhadaahaaWcbeqaaiaaikdaaaaaaaGccaGL7bGaayzFaaGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlabeo8aZnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyypa0 ZaaSaaaeaacaWGfbGaeuiLdqKaamyyaaqaaiaaikdacaWGIbWaaWba aSqabeaacaaIYaaaaaaakmaacmaabaGaaGymaiabgUcaRmaalaaaba GaamOyamaaCaaaleqabaGaaGOmaaaaaOqaaiaadkhadaahaaWcbeqa aiaaikdaaaaaaaGccaGL7bGaayzFaaGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8Uaeq4Wdm3aaSbaaSqaaiaadQhacaWG6baabeaakiabg2da9iaaic daaaa@808D@

 

Derivation: These results can be derived using the solution to a pressurized cylinder given in Section 4.1.9. After the shaft is inserted into the tube, a pressure p acts to compress the shaft, and the same pressure pushes outwards to expand the cylinder.  Suppose that this pressure induces a radial displacement u s (r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaWbaaSqabeaacaWGZbaaaO GaaiikaiaadkhacaGGPaaaaa@373F@  in the solid cylinder, and a radial displacement u c (r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaWbaaSqabeaacaWGJbaaaO GaaiikaiaadkhacaGGPaaaaa@372F@  in the hollow tube.  To accommodate the interference, the displacements must satisfy

u c (r=a) u s (r=a)=Δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaWbaaSqabeaacaWGJbaaaO GaaiikaiaadkhacqGH9aqpcaWGHbGaaiykaiabgkHiTiaadwhadaah aaWcbeqaaiaadohaaaGccaGGOaGaamOCaiabg2da9iaadggacaGGPa Gaeyypa0JaeuiLdqeaaa@42D9@

Evaluating the relevant displacements using the formulas in 4.1.9 gives

u s (r=a)= ( 12ν )( 1+ν ) E pa 2 ν 2 E pa u c (r=a)= ( 1+ν ) a 2 b 2 E( b 2 a 2 ) { p a +( 12ν ) pa b 2 }+ 2 ν 2 p a 3 E( b 2 a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadwhadaahaaWcbeqaaiaado haaaGccaGGOaGaamOCaiabg2da9iaadggacaGGPaGaeyypa0JaeyOe I0YaaSaaaeaadaqadaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gaca GLOaGaayzkaaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGa ayzkaaaabaGaamyraaaacaWGWbGaamyyaiabgkHiTmaalaaabaGaaG Omaiabe27aUnaaCaaaleqabaGaaGOmaaaaaOqaaiaadweaaaGaamiC aiaadggaaeaacaWG1bWaaWbaaSqabeaacaWGJbaaaOGaaiikaiaadk hacqGH9aqpcaWGHbGaaiykaiabg2da9maalaaabaWaaeWaaeaacaaI XaGaey4kaSIaeqyVd4gacaGLOaGaayzkaaGaamyyamaaCaaaleqaba GaaGOmaaaakiaadkgadaahaaWcbeqaaiaaikdaaaaakeaacaWGfbWa aeWaaeaacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaamyyam aaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaaadaGadaqaamaa laaabaGaamiCaaqaaiaadggaaaGaey4kaSYaaeWaaeaacaaIXaGaey OeI0IaaGOmaiabe27aUbGaayjkaiaawMcaamaalaaabaGaamiCaiaa dggaaeaacaWGIbWaaWbaaSqabeaacaaIYaaaaaaaaOGaay5Eaiaaw2 haaiabgUcaRmaalaaabaGaaGOmaiabe27aUnaaCaaaleqabaGaaGOm aaaakiaadchacaWGHbWaaWbaaSqabeaacaaIZaaaaaGcbaGaamyrai aacIcacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaamyyamaa CaaaleqabaGaaGOmaaaakiaacMcaaaaaaaa@82A8@

Here, we have assumed that the axial force acting on both the shaft and the tube must vanish separately, since they slide freely relative to one another.  Solving these two equations for p shows that

p= EΔ( b 2 a 2 ) 2a b 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbGaeyypa0ZaaSaaaeaacaWGfb GaeuiLdqKaaiikaiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHsisl caWGHbWaaWbaaSqabeaacaaIYaaaaOGaaiykaaqaaiaaikdacaWGHb GaamOyamaaCaaaleqabaGaaGOmaaaaaaaaaa@406C@

This pressure can then be substituted back into the formulas in 4.1.9 to evaluate the stresses.