Chapter 4

 

Solutions to simple boundary and initial value problems

 

 

 

4.2 Axially and spherically symmetric solutions to quasi-static elastic-plastic problems

 

In this section, we derive exact solutions to simple boundary value problems involving elastic-perfectly plastic solids.  The solutions are of interest primarily because they illustrate important general features of solids that are loaded beyond yield.  In particular, they illustrate the concepts of

1.      The elastic limit MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  this is the load required to initiate plastic flow in the solid.

2.      The plastic collapse load MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  at this load the displacements in the solid become infinite.

3.      Residual stress - if a solid is loaded beyond the elastic limit and then unloaded, a system of self-equilibrated stress is established in the material.

4.      Shakedown - if an elastic-plastic solid is subjected to cyclic loading, and the maximum load during the cycle exceeds yield, then some plastic deformation must occur in the material during the first load cycle.  However, residual stresses are introduced in the solid, which may prevent plastic flow during subsequent cycles of load.  This process is known as `shakedown’ and the maximum load for which it can occur is known as the `shakedown limit.’  The shakedown limit is often substantially higher than the elastic limit, so the concept of shakedown can often be used to reduce the weight of a design.

5.      Cyclic plasticity - for cyclic loads exceeding the shakedown limit, a region in the solid will be repeatedly plastically deformed.

 

 

4.2.1 Summary of governing equations

 

We are given the following information

1.      Geometry of the solid

2.      Constitutive law for the material (i.e. the elastic-plastic 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, elastic-perfectly plastic solid, with Young’s modulus E Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347D@ , yield stress Y and mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaa@356C@

3.      We will neglect temperature changes.

 

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@

 Incremental stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E8@ strain relation

d ε ij =d ε ij e +d ε ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiabew7aLnaaBaaaleaacaWGPb GaamOAaaqabaGccqGH9aqpcaWGKbGaeqyTdu2aa0baaSqaaiaadMga caWGQbaabaGaamyzaaaakiabgUcaRiaadsgacqaH1oqzdaqhaaWcba GaamyAaiaadQgaaeaacaWGWbaaaaaa@4277@

d ε ij e = 1+ν E ( d σ ij ν 1+ν d σ kk δ ij ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiabew7aLnaaDaaaleaacaWGPb GaamOAaaqaaiaadwgaaaGccqGH9aqpdaWcaaqaaiaaigdacqGHRaWk cqaH9oGBaeaacaWGfbaaamaabmaabaGaamizaiabeo8aZnaaBaaale aacaWGPbGaamOAaaqabaGccqGHsisldaWcaaqaaiabe27aUbqaaiaa igdacqGHRaWkcqaH9oGBaaGaamizaiabeo8aZnaaBaaaleaacaWGRb Gaam4AaaqabaGccqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaaGc caGLOaGaayzkaaaaaa@505E@

d ε ij p ={ 0 3 2 S ij S ij <Y d ε ¯ p 3 2 S ij Y 3 2 S ij S ij =Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH1oqzdaqhaaWcbaGaamyAai aadQgaaeaacaWGWbaaaOGaeyypa0ZaaiqaaeaafaqabeGabaaabaGa aGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8+aaOaaaeaadaWcaaqaaiaaiodaaeaacaaIYaaa aiaadofadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaam4uamaaBaaale aacaWGPbGaamOAaaqabaaabeaakiabgYda8iaadMfaaeaacaWGKbGa fqyTduMbaebadaahaaWcbeqaaiaadchaaaGcdaWcaaqaaiaaiodaae aacaaIYaaaamaalaaabaGaam4uamaaBaaaleaacaWGPbGaamOAaaqa baaakeaacaWGzbaaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8+aaOaaaeaadaWcaaqaaiaaiodaaeaacaaIYaaa aiaadofadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaam4uamaaBaaale aacaWGPbGaamOAaaqabaaabeaakiabg2da9iaadMfaaaaacaGL7baa aaa@8938@

where S ij = σ ij σ kk δ ij /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiab gkHiTiabeo8aZnaaBaaaleaacaWGRbGaam4AaaqabaGccqaH0oazda WgaaWcbaGaamyAaiaadQgaaeqaaOGaai4laiaaiodaaaa@421D@

 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.

 

 

4.2.2 Simplified equations for spherically symmetric 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 with spherical and cylindrical coordinates is complicated, and is discussed in detail in Chapter 10 in the context of modeling deformation in shells.  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 caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaacYcacqaH4oqCca GGSaGaeqy1dyMaaiykaaaa@39F4@ . All vectors and tensors are expressed as components in the Cartesian basis { e r , e θ , e ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaam OCaaqabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiil aiaahwgadaWgaaWcbaGaeqy1dygabeaaaOGaay5Eaiaaw2haaaaa@3E38@  shown in the figure.  For a spherically symmetric problem we have that

 Position Vector       x=r e r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaeyypa0JaamOCaiaahwgada WgaaWcbaGaamOCaaqabaaaaa@37D4@

 Displacement vector u=u(r) e r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0JaamyDaiaacIcaca WGYbGaaiykaiaahwgadaWgaaWcbaGaamOCaaqabaaaaa@3A24@

 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@

 

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 ϕ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaWGYb aabeaakiaacYcacaWHLbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa aCyzamaaBaaaleaacqaHvpGzaeqaaOGaaiyFaaaa@3E06@  ) have the form

σ[ σ rr 0 0 0 σ θθ 0 0 0 σ ϕϕ ]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 qadmaaaeaacqaHdpWCdaWgaaWcbaGaamOCaiaadkhaaeqaaaGcbaGa aGimaaqaaiaaicdaaeaacaaIWaaabaGaeq4Wdm3aaSbaaSqaaiabeI 7aXjabeI7aXbqabaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaa cqaHdpWCdaWgaaWcbaGaeqy1dyMaeqy1dygabeaaaaaakiaawUfaca GLDbaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabew7aLj abggMi6oaadmaabaqbaeqabmWaaaqaaiabew7aLnaaBaaaleaacaWG YbGaamOCaaqabaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacq aH1oqzdaWgaaWcbaGaeqiUdeNaeqiUdehabeaaaOqaaiaaicdaaeaa caaIWaaabaGaaGimaaqaaiabew7aLnaaBaaaleaacqaHvpGzcqaHvp GzaeqaaaaaaOGaay5waiaaw2faaaaa@797B@

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,z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaacYcacqaH4oqCca GGSaGaamOEaiaacMcaaaa@392A@ .

 

For spherical symmetry, the governing equations reduce to

 

 Strain Displacement Relations ε rr = du dr ε ϕϕ = ε θθ = u r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaWGKbGaamyDaaqaaiaadsgacaWG YbaaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlabew7aLnaaBaaaleaacqaHvpGzcqaHvpGzaeqaaOGa eyypa0JaeqyTdu2aaSbaaSqaaiabeI7aXjabeI7aXbqabaGccqGH9a qpdaWcaaqaaiaadwhaaeaacaWGYbaaaaaa@5820@

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

In elastic region(s)

σ rr = E ( 1+ν )( 12ν ) { (1ν) ε rr +ν ε θθ +ν ε ϕϕ } σ θθ = σ ϕϕ = E ( 1+ν )( 12ν ) { ε θθ +ν ε rr } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaWGYb GaamOCaaqabaGccqGH9aqpdaWcaaqaaiaadweaaeaadaqadaqaaiaa igdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaadaqadaqaaiaaigdacq GHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaaamaacmaabaGaaiik aiaaigdacqGHsislcqaH9oGBcaGGPaGaeqyTdu2aaSbaaSqaaiaadk hacaWGYbaabeaakiabgUcaRiabe27aUjabew7aLnaaBaaaleaacqaH 4oqCcqaH4oqCaeqaaOGaey4kaSIaeqyVd4MaeqyTdu2aaSbaaSqaai abew9aMjabew9aMbqabaaakiaawUhacaGL9baaaeaacqaHdpWCdaWg aaWcbaGaeqiUdeNaeqiUdehabeaakiabg2da9iabeo8aZnaaBaaale aacqaHvpGzcqaHvpGzaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaWa aeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzkaaWaaeWaae aacaaIXaGaeyOeI0IaaGOmaiabe27aUbGaayjkaiaawMcaaaaadaGa daqaaiabew7aLnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaey4kaS IaeqyVd4MaeqyTdu2aaSbaaSqaaiaadkhacaWGYbaabeaaaOGaay5E aiaaw2haaaaaaa@83EB@

| σ θθ σ rr |<Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaemaabaGaeq4Wdm3aaSbaaSqaaiabeI 7aXjabeI7aXbqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaamOCaiaa dkhaaeqaaaGccaGLhWUaayjcSdGaeyipaWJaamywaaaa@3FB5@

In plastic region(s)

Yield criterion:   | σ θθ σ rr |=Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaemaabaGaeq4Wdm3aaSbaaSqaaiabeI 7aXjabeI7aXbqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaamOCaiaa dkhaaeqaaaGccaGLhWUaayjcSdGaeyypa0Jaamywaaaa@3FA7@

Strain partition: d ε rr =d ε rr p +d ε rr e d ε ϕϕ =d ε ϕϕ p +d ε ϕϕ e d ε θθ =d ε θθ p +d ε θθ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH1oqzdaWgaaWcbaGaamOCai aadkhaaeqaaOGaeyypa0Jaamizaiabew7aLnaaDaaaleaacaWGYbGa amOCaaqaaiaadchaaaGccqGHRaWkcaWGKbGaeqyTdu2aa0baaSqaai aadkhacaWGYbaabaGaamyzaaaakiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaadsgacqaH1oqzdaWgaaWcbaGaeqy1dyMaeqy1dy gabeaakiabg2da9iaadsgacqaH1oqzdaqhaaWcbaGaeqy1dyMaeqy1 dygabaGaamiCaaaakiabgUcaRiaadsgacqaH1oqzdaqhaaWcbaGaeq y1dyMaeqy1dygabaGaamyzaaaakiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaadsgacqaH1oqzdaWgaaWcbaGaeqiUdeNaeqiUde habeaakiabg2da9iaadsgacqaH1oqzdaqhaaWcbaGaeqiUdeNaeqiU dehabaGaamiCaaaakiabgUcaRiaadsgacqaH1oqzdaqhaaWcbaGaeq iUdeNaeqiUdehabaGaamyzaaaaaaa@81F1@

Elastic strain:    d ε rr e =d σ rr /Eν(d σ θθ +d σ ϕϕ )/E d ε θθ e =d ε ϕϕ e =d σ θθ (1ν)/Eνd σ rr /E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamizaiabew7aLnaaDaaaleaaca WGYbGaamOCaaqaaiaadwgaaaGccqGH9aqpcaWGKbGaeq4Wdm3aaSba aSqaaiaadkhacaWGYbaabeaakiaac+cacaWGfbGaeyOeI0IaeqyVd4 MaaiikaiaadsgacqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaa kiabgUcaRiaadsgacqaHdpWCdaWgaaWcbaGaeqy1dyMaeqy1dygabe aakiaacMcacaGGVaGaamyraaqaaiaadsgacqaH1oqzdaqhaaWcbaGa eqiUdeNaeqiUdehabaGaamyzaaaakiabg2da9iaadsgacqaH1oqzda qhaaWcbaGaeqy1dyMaeqy1dygabaGaamyzaaaakiabg2da9iaadsga cqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaakiaacIcacaaIXa GaeyOeI0IaeqyVd4Maaiykaiaac+cacaWGfbGaeyOeI0IaeqyVd4Ma amizaiabeo8aZnaaBaaaleaacaWGYbGaamOCaaqabaGccaGGVaGaam yraaaaaa@765D@

Flow rule:          d ε rr p =d ε ¯ p ( σ rr σ θθ )/Y d ε θθ p =d ε ϕϕ p =d ε ¯ p ( σ θθ σ rr )/(2Y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamizaiabew7aLnaaDaaaleaaca WGYbGaamOCaaqaaiaadchaaaGccqGH9aqpcaWGKbGafqyTduMbaeba daahaaWcbeqaaiaadchaaaGccaGGOaGaeq4Wdm3aaSbaaSqaaiaadk hacaWGYbaabeaakiabgkHiTiabeo8aZnaaBaaaleaacqaH4oqCcqaH 4oqCaeqaaOGaaiykaiaac+cacaWGzbaabaGaamizaiabew7aLnaaDa aaleaacqaH4oqCcqaH4oqCaeaacaWGWbaaaOGaeyypa0Jaamizaiab ew7aLnaaDaaaleaacqaHvpGzcqaHvpGzaeaacaWGWbaaaOGaeyypa0 Jaamizaiqbew7aLzaaraWaaWbaaSqabeaacaWGWbaaaOGaaiikaiab eo8aZnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyOeI0Iaeq4Wdm 3aaSbaaSqaaiaadkhacaWGYbaabeaakiaacMcacaGGVaGaaiikaiaa ikdacaWGzbGaaiykaaaaaa@6BA1@

 

 Equilibrium Equations

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

 Boundary Conditions

 

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

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

 

The equilibrium and strain-displacement equations can be derived following the procedure outlined in Section 4.1.2.  The stress-strain relations are derived by substituting the strain components into the general constitutive equation and simplifying the result.

 

Unlike the elastic solution discussed in Sect 4.1, there is no clean, direct and general method for integrating these equations.  Instead, solutions must be found using a combination of physical intuition and some algebraic tricks, as illustrated in the sections below.

 

 

4.2.3 Elastic-perfectly plastic hollow sphere subjected to monotonically increasing internal pressure

 

Assume that

 The sphere is stress free before it is loaded

 No body forces act on the sphere

 The sphere has uniform temperature

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

 The outer surface r=b is traction free

 Strains are infinitesimal

 

Solution:

 

(i) Preliminaries:

 The sphere first reaches yield (at r=a) at an internal pressure p a /Y=2(1 a 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH9aqpcaaIYaGaaiikaiaaigdacqGHsislcaWG HbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqaai aaiodaaaGccaGGPaGaai4laiaaiodaaaa@4103@

 For pressures in the range 2(1 a 3 / b 3 )/3< p a /Y<2log(b/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIYaGaaiikaiaaigdacqGHsislca WGHbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqa aiaaiodaaaGccaGGPaGaai4laiaaiodacqGH8aapcaWGWbWaaSbaaS qaaiaadggaaeqaaOGaai4laiaadMfacqGH8aapcaaIYaGaciiBaiaa c+gacaGGNbGaaiikaiaadkgacaGGVaGaamyyaiaacMcaaaa@496A@  the region between r=a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0Jaamyyaaaa@35B9@  and r=c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0Jaam4yaaaa@35BB@  deforms plastically; while the region between c<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbGaeyipaWJaamOCaiabgYda8i aadkgaaaa@3794@  remains elastic, where c satisfies the equation p a /Y=2log(c/a)+ 2 3 (1 c 3 / b 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH9aqpcaaIYaGaciiBaiaac+gacaGGNbGaaiik aiaadogacaGGVaGaamyyaiaacMcacqGHRaWkdaWcaaqaaiaaikdaae aacaaIZaaaaiaacIcacaaIXaGaeyOeI0Iaam4yamaaCaaaleqabaGa aG4maaaakiaac+cacaWGIbWaaWbaaSqabeaacaaIZaaaaOGaaiykaa aa@48AA@

 At a pressure p a /Y=2log(b/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH9aqpcaaIYaGaciiBaiaac+gacaGGNbGaaiik aiaadkgacaGGVaGaamyyaiaacMcaaaa@3ED3@  the entire cylinder is plastic.  At this point the sphere collapses MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the displacements become infinitely large.

 

(ii) Solution in the plastic region a<r<c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbGaeyipaWJaamOCaiabgYda8i aadogaaaa@37A3@

u= ( 12ν ) E r{ 2Ylog(r/a) p a } e r + Y( 1ν ) c 3 E r 2 e r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwhacqGH9aqpdaWcaaqaamaabmaaba GaaGymaiabgkHiTiaaikdacqaH9oGBaiaawIcacaGLPaaaaeaacaWG fbaaaiaadkhadaGadaqaaiaaikdacaWGzbGaciiBaiaac+gacaGGNb GaaiikaiaadkhacaGGVaGaamyyaiaacMcacqGHsislcaWGWbWaaSba aSqaaiaadggaaeqaaaGccaGL7bGaayzFaaGaaCyzamaaBaaaleaaca WGYbaabeaakiabgUcaRmaalaaabaGaamywamaabmaabaGaaGymaiab gkHiTiabe27aUbGaayjkaiaawMcaaiaadogadaahaaWcbeqaaiaaio daaaaakeaacaWGfbGaamOCamaaCaaaleqabaGaaGOmaaaaaaGccaWH LbWaaSbaaSqaaiaadkhaaeqaaaaa@56D9@

σ rr =2Ylog(r/a) p a σ θθ = σ ϕϕ =2Ylog(r/a) p a +Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcaaIYaGaamywaiGacYgacaGGVbGaai4zaiaacIca caWGYbGaai4laiaadggacaGGPaGaeyOeI0IaamiCamaaBaaaleaaca WGHbaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0JaaGOmaiaadMfaciGGSbGaai4BaiaacEgacaGGOa GaamOCaiaac+cacaWGHbGaaiykaiabgkHiTiaadchadaWgaaWcbaGa amyyaaqabaGccqGHRaWkcaWGzbaaaa@6A1F@     

(iii) Solution in the elastic region c<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbGaeyipaWJaamOCaiabgYda8i aadkgaaaa@37A4@

u= Y c 3 3E b 3 r 2 { 2( 12ν ) r 3 +( 1+ν ) b 3 } e r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0ZaaSaaaeaacaWGzb Gaam4yamaaCaaaleqabaGaaG4maaaaaOqaaiaaiodacaWGfbGaamOy amaaCaaaleqabaGaaG4maaaakiaadkhadaahaaWcbeqaaiaaikdaaa aaaOWaaiWaaeaacaaIYaWaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiab e27aUbGaayjkaiaawMcaaiaadkhadaahaaWcbeqaaiaaiodaaaGccq GHRaWkdaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaa caWGIbWaaWbaaSqabeaacaaIZaaaaaGccaGL7bGaayzFaaGaaCyzam aaBaaaleaacaWGYbaabeaaaaa@5109@

σ rr = 2Y c 3 3 b 3 ( 1 b 3 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaaIYaGaamywaiaadogadaahaaWc beqaaiaaiodaaaaakeaacaaIZaGaamOyamaaCaaaleqabaGaaG4maa aaaaGcdaqadaqaaiaaigdacqGHsisldaWcaaqaaiaadkgadaahaaWc beqaaiaaiodaaaaakeaacaWGYbWaaWbaaSqabeaacaaIZaaaaaaaaO GaayjkaiaawMcaaaaa@44D8@            σ θθ = σ ϕϕ = 2Y c 3 3 b 3 ( 1+ b 3 2 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaacaaIYaGaamywaiaadogadaahaaWcbe qaaiaaiodaaaaakeaacaaIZaGaamOyamaaCaaaleqabaGaaG4maaaa aaGcdaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaadkgadaahaaWcbe qaaiaaiodaaaaakeaacaaIYaGaamOCamaaCaaaleqabaGaaG4maaaa aaaakiaawIcacaGLPaaaaaa@4D96@

These results are plotted in the figures below.

 

  

(a)                                                   (b)                                              (c)

(a) Stress and (b) displacement distributions for a pressurized elastic-perfectly plastic spherical shell; and (c) Displacement at r=a as a function of pressure. Displacements are shown for ν=0.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBcqGH9aqpcaaIWaGaaiOlai aaiodaaaa@37BD@

 

 

 

 

Derivation: By substituting the stresses for the elastic solution given in 4.1.4 into the Von-Mises yield criterion, we see that a pressurized elastic sphere first reaches yield at r=a. If the pressure is increased beyond yield we anticipate that a region a<r<c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacqGH8aapcaWGYbGaeyipaWJaam 4yaaaa@3545@  will deform plastically, while a region c<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogacqGH8aapcaWGYbGaeyipaWJaam Oyaaaa@3546@  remains elastic. We must find separate solutions in the plastic and elastic regimes.

 

In the plastic regime a<r<c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacqGH8aapcaWGYbGaeyipaWJaam 4yaaaa@3545@

(i) We anticipate that σ rr <0 σ θθ = σ ϕϕ >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH8aapcaaIWaGaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSba aSqaaiabeI7aXjabeI7aXbqabaGccqGH9aqpcqaHdpWCdaWgaaWcba Gaeqy1dyMaeqy1dygabeaakiabg6da+iaaicdaaaa@4874@ . The yield criterion then gives σ θθ σ rr =Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4o qCaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadkhacaWGYbaabeaa kiabg2da9iaadMfaaaa@3C95@ .

(ii) Substituting this result into the equilibrium equation given in Sect 4.2.2 shows that

d σ rr dr + 1 r ( 2 σ rr σ θθ σ ϕϕ )=0 d σ rr dr 2 Y r =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacqaHdpWCdaWgaa WcbaGaamOCaiaadkhaaeqaaaGcbaGaamizaiaadkhaaaGaey4kaSYa aSaaaeaacaaIXaaabaGaamOCaaaadaqadaqaaiaaikdacqaHdpWCda WgaaWcbaGaamOCaiaadkhaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqa aiabeI7aXjabeI7aXbqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaeq y1dyMaeqy1dygabeaaaOGaayjkaiaawMcaaiaaykW7cqGH9aqpcaaI WaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7cqGHshI3caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8+aaSaaaeaacaWGKbGaeq4Wdm3aaSbaaS qaaiaadkhacaWGYbaabeaaaOqaaiaadsgacaWGYbaaaiabgkHiTiaa ikdadaWcaaqaaiaadMfaaeaacaWGYbaaaiabg2da9iaaicdaaaa@7C99@

(iii) Integrating, and using the boundary condition σ rr = p a r=a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcqGHsislcaWGWbWaaSbaaSqaaiaadggaaeqaaOGa aGPaVlaaykW7caWGYbGaeyypa0Jaamyyaaaa@3E5C@  together with the yield condition (i) gives

σ rr =2Ylog(r/a) p a σ θθ = σ ϕϕ =2Ylog(r/a) p a +Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcaaIYaGaamywaiGacYgacaGGVbGaai4zaiaacIca caWGYbGaai4laiaadggacaGGPaGaeyOeI0IaamiCamaaBaaaleaaca WGHbaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0JaaGOmaiaadMfaciGGSbGaai4BaiaacEgacaGGOa GaamOCaiaac+cacaWGHbGaaiykaiabgkHiTiaadchadaWgaaWcbaGa amyyaaqabaGccqGHRaWkcaWGzbaaaa@6A1F@

(iv) Since the pressure is monotonically increasing, the incremental stress-strain relations for the elastic-plastic region given in 4.2.2 can be integrated. The elastic strains follow as

ε rr e =( σ rr 2ν σ θθ )/E ε ϕϕ e = ε θθ e =( (1ν) σ θθ ν σ rr )/E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaWGYbGaamOCaa qaaiaadwgaaaGccqGH9aqpdaqadaqaaiabeo8aZnaaBaaaleaacaWG YbGaamOCaaqabaGccqGHsislcaaIYaGaeqyVd4Maeq4Wdm3aaSbaaS qaaiabeI7aXjabeI7aXbqabaaakiaawIcacaGLPaaacaGGVaGaamyr aiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqyTdu2aa0baaSqaai abew9aMjabew9aMbqaaiaadwgaaaGccqGH9aqpcaaMc8UaeqyTdu2a a0baaSqaaiabeI7aXjabeI7aXbqaaiaadwgaaaGccqGH9aqpdaqada qaaiaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiykaiabeo8aZnaaBaaa leaacqaH4oqCcqaH4oqCaeqaaOGaeyOeI0IaeqyVd4Maeq4Wdm3aaS baaSqaaiaadkhacaWGYbaabeaaaOGaayjkaiaawMcaaiaac+cacaWG fbaaaa@7161@

(v) The plastic strains satisfy ε rr p +2 ε θθ p =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaWGYbGaamOCaa qaaiaadchaaaGccqGHRaWkcaaIYaGaeqyTdu2aa0baaSqaaiabeI7a XjabeI7aXbqaaiaadchaaaGccqGH9aqpcaaIWaaaaa@3EC6@ .  Consequently, using the strain partition formula, the results of (iv), and the strain-displacement relation shows that

ε rr +2 ε θθ = ε rr e +2 ε θθ e = (12ν) E ( σ rr +2 σ θθ ) du dr + 2u r = 1 r 2 d dr ( r 2 u )= (12ν) E ( 6Ylog(r/a)3 p a +2Y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeqyTdu2aaSbaaSqaaiaadkhaca WGYbaabeaakiabgUcaRiaaikdacqaH1oqzdaWgaaWcbaGaeqiUdeNa eqiUdehabeaakiabg2da9iabew7aLnaaDaaaleaacaWGYbGaamOCaa qaaiaadwgaaaGccqGHRaWkcaaIYaGaeqyTdu2aa0baaSqaaiabeI7a XjabeI7aXbqaaiaadwgaaaGccqGH9aqpdaWcaaqaaiaacIcacaaIXa GaeyOeI0IaaGOmaiabe27aUjaacMcaaeaacaWGfbaaamaabmaabaGa eq4Wdm3aaSbaaSqaaiaadkhacaWGYbaabeaakiabgUcaRiaaikdacq aHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaaaOGaayjkaiaawMca aiaaykW7aeaacaaMc8UaeyO0H49aaSaaaeaacaWGKbGaamyDaaqaai aadsgacaWGYbaaaiabgUcaRmaalaaabaGaaGOmaiaadwhaaeaacaWG Ybaaaiabg2da9maalaaabaGaaGymaaqaaiaadkhadaahaaWcbeqaai aaikdaaaaaaOWaaSaaaeaacaWGKbaabaGaamizaiaadkhaaaWaaeWa aeaacaWGYbWaaWbaaSqabeaacaaIYaaaaOGaamyDaaGaayjkaiaawM caaiabg2da9maalaaabaGaaiikaiaaigdacqGHsislcaaIYaGaeqyV d4MaaiykaaqaaiaadweaaaWaaeWaaeaacaaI2aGaamywaiGacYgaca GGVbGaai4zaiaacIcacaWGYbGaai4laiaadggacaGGPaGaeyOeI0Ia aG4maiaadchadaWgaaWcbaGaamyyaaqabaGccqGHRaWkcaaIYaGaam ywaaGaayjkaiaawMcaaaaaaa@8BC9@

(vi) Integrating gives

u= ( 12ν ) E r{ 2Ylog(r/a) p a }+C/ r 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhacqGH9aqpdaWcaaqaamaabmaaba GaaGymaiabgkHiTiaaikdacqaH9oGBaiaawIcacaGLPaaaaeaacaWG fbaaaiaadkhadaGadaqaaiaaikdacaWGzbGaciiBaiaac+gacaGGNb GaaiikaiaadkhacaGGVaGaamyyaiaacMcacqGHsislcaWGWbWaaSba aSqaaiaadggaaeqaaaGccaGL7bGaayzFaaGaey4kaSIaam4qaiaac+ cacaWGYbWaaWbaaSqabeaacaaIYaaaaaaa@4B9D@

where C is a constant of integration

(vii) The constant of integration can be found by noting that the radial displacements in the elastic and plastic regimes must be equal at r=c.  Using the expression for the elastic displacement field below and solving for C gives

C= 3 2 ( 1ν ) c 3 b 3 E( b 3 c 3 ) { p a 2Ylog(c/a) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbGaeyypa0ZaaSaaaeaacaaIZa aabaGaaGOmaaaadaWcaaqaamaabmaabaGaaGymaiabgkHiTiabe27a UbGaayjkaiaawMcaaiaadogadaahaaWcbeqaaiaaiodaaaGccaWGIb WaaWbaaSqabeaacaaIZaaaaaGcbaGaamyraiaacIcacaWGIbWaaWba aSqabeaacaaIZaaaaOGaeyOeI0Iaam4yamaaCaaaleqabaGaaG4maa aakiaacMcaaaWaaiWaaeaacaWGWbWaaSbaaSqaaiaadggaaeqaaOGa eyOeI0IaaGOmaiaadMfaciGGSbGaai4BaiaacEgacaGGOaGaam4yai aac+cacaWGHbGaaiykaaGaay5Eaiaaw2haaaaa@5307@

This result can be simplified by noting that p a 2Ylog(c/a)=2Y(1 c 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO GaeyOeI0IaaGOmaiaadMfaciGGSbGaai4BaiaacEgacaGGOaGaam4y aiaac+cacaWGHbGaaiykaiabg2da9iaaikdacaWGzbGaaiikaiaaig dacqGHsislcaWGJbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkga daahaaWcbeqaaiaaiodaaaGccaGGPaGaai4laiaaiodaaaa@4983@  from the expression for the location of the elastic-plastic boundary given below.

 

In the elastic regime

The solution can be found directly from the solution to the internally pressurized elastic sphere given in Sect 4.1.4.  From step (iii) in the solution for the plastic regime we see that the radial pressure at r=c is p c = σ rr = p a 2Ylog(c/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaam4yaaqabaGccq GH9aqpcqGHsislcqaHdpWCdaWgaaWcbaGaamOCaiaadkhaaeqaaOGa eyypa0JaamiCamaaBaaaleaacaWGHbaabeaakiabgkHiTiaaikdaca WGzbGaciiBaiaac+gacaGGNbGaaiikaiaadogacaGGVaGaamyyaiaa cMcaaaa@449D@ . We can simplify the solution by noting p a 2Ylog(c/a)=2Y(1 c 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO GaeyOeI0IaaGOmaiaadMfaciGGSbGaai4BaiaacEgacaGGOaGaam4y aiaac+cacaWGHbGaaiykaiabg2da9iaaikdacaWGzbGaaiikaiaaig dacqGHsislcaWGJbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkga daahaaWcbeqaaiaaiodaaaGccaGGPaGaai4laiaaiodaaaa@4983@  from the expression for the location of the elastic-plastic boundary.   Substituting into the expressions for stress and displacement shows that

σ rr = p c c 3 ( b 3 c 3 ) ( 1 b 3 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaWGWbWaaSbaaSqaaiaadogaaeqa aOGaam4yamaaCaaaleqabaGaaG4maaaaaOqaamaabmaabaGaamOyam aaCaaaleqabaGaaG4maaaakiabgkHiTiaadogadaahaaWcbeqaaiaa iodaaaaakiaawIcacaGLPaaaaaWaaeWaaeaacaaIXaGaeyOeI0YaaS aaaeaacaWGIbWaaWbaaSqabeaacaaIZaaaaaGcbaGaamOCamaaCaaa leqabaGaaG4maaaaaaaakiaawIcacaGLPaaaaaa@48E6@            σ θθ = σ ϕϕ = p c c 3 ( b 3 c 3 ) ( 1+ b 3 2 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaacaWGWbWaaSbaaSqaaiaadogaaeqaaO Gaam4yamaaCaaaleqabaGaaG4maaaaaOqaamaabmaabaGaamOyamaa CaaaleqabaGaaG4maaaakiabgkHiTiaadogadaahaaWcbeqaaiaaio daaaaakiaawIcacaGLPaaaaaWaaeWaaeaacaaIXaGaey4kaSYaaSaa aeaacaWGIbWaaWbaaSqabeaacaaIZaaaaaGcbaGaaGOmaiaadkhada ahaaWcbeqaaiaaiodaaaaaaaGccaGLOaGaayzkaaaaaa@51A4@

u= p c c 3 2E( b 3 c 3 ) r 2 { 2( 12ν ) r 3 +( 1+ν ) b 3 } e r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0ZaaSaaaeaacaWGWb WaaSbaaSqaaiaadogaaeqaaOGaam4yamaaCaaaleqabaGaaG4maaaa aOqaaiaaikdacaWGfbWaaeWaaeaacaWGIbWaaWbaaSqabeaacaaIZa aaaOGaeyOeI0Iaam4yamaaCaaaleqabaGaaG4maaaaaOGaayjkaiaa wMcaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaOWaaiWaaeaacaaIYa WaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbGaayjkaiaawMca aiaadkhadaahaaWcbeqaaiaaiodaaaGccqGHRaWkdaqadaqaaiaaig dacqGHRaWkcqaH9oGBaiaawIcacaGLPaaacaWGIbWaaWbaaSqabeaa caaIZaaaaaGccaGL7bGaayzFaaGaaCyzamaaBaaaleaacaWGYbaabe aaaaa@568F@

 

Location of the elastic-plastic boundary

Finally, the elastic-platsic boundary is located by the condition that the stress in the elastic region must just reach yield at r=c (so there is a smooth transition into the plastic region).  The yield condition is σ θθ σ rr =Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4o qCaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadkhacaWGYbaabeaa kiabg2da9iaadMfaaaa@3C95@ , so substituting the expressions for stress in the elastic region and simplifying yields

σ θθ σ rr = 3( p a 2Ylog(c/a) ) b 3 2( b 3 c 3 ) =Y p a Y =2log(c/a)+ 2 3 (1 c 3 / b 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabgkHiTiabeo8aZnaaBaaaleaacaWGYbGaamOCaaqa baGccqGH9aqpdaWcaaqaaiaaiodadaqadaqaaiaadchadaWgaaWcba GaamyyaaqabaGccqGHsislcaaIYaGaamywaiGacYgacaGGVbGaai4z aiaacIcacaWGJbGaai4laiaadggacaGGPaaacaGLOaGaayzkaaGaam OyamaaCaaaleqabaGaaG4maaaaaOqaaiaaikdadaqadaqaaiaadkga daahaaWcbeqaaiaaiodaaaGccqGHsislcaWGJbWaaWbaaSqabeaaca aIZaaaaaGccaGLOaGaayzkaaaaaiabg2da9iaadMfacaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 Uaeyi1HSTaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7daWcaaqaaiaadchadaWgaaWcbaGaamyyaaqabaaakeaaca WGzbaaaiabg2da9iaaikdaciGGSbGaai4BaiaacEgacaGGOaGaam4y aiaac+cacaWGHbGaaiykaiabgUcaRmaalaaabaGaaGOmaaqaaiaaio daaaGaaiikaiaaigdacqGHsislcaWGJbWaaWbaaSqabeaacaaIZaaa aOGaai4laiaadkgadaahaaWcbeqaaiaaiodaaaGccaGGPaaaaa@89A8@

If p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaa aa@34DD@ , Y, a and b are specified this equation can be solved (numerically) for c.  For graphing purposes it is preferable to choose c and then calculate the corresponding value of p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaa aa@34DD@  

 

 

4.2.4 Elastic-perfectly plastic hollow sphere subjected to cyclic internal pressure

 

Assume that

 The sphere is stress free before it is loaded

 No body forces act on the sphere

 The sphere has uniform temperature

 The outer surface r=b is traction free

 

Suppose that the inner surface of the sphere r=a is repeatedly subjected to pressure p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaa aa@34CD@  and then unloaded to zero pressure.

 

Solution:

 

(i) Preliminaries:

 If the maximum pressure p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaa aa@34CD@  applied to the sphere does not exceed the elastic limit (i.e. p a /Y<2(1 a 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH8aapcaaIYaGaaiikaiaaigdacqGHsislcaWG HbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqaai aaiodaaaGccaGGPaGaai4laiaaiodaaaa@4101@  ) the solid remains elastic throughout the loading cycle.  In this case, the sphere is stress free after unloading, and remains elastic throughout all subsequent load cycles.

 For pressures in the range 2(1 a 3 / b 3 )/3< p a /Y<2log(b/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIYaGaaiikaiaaigdacqGHsislca WGHbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqa aiaaiodaaaGccaGGPaGaai4laiaaiodacqGH8aapcaWGWbWaaSbaaS qaaiaadggaaeqaaOGaai4laiaadMfacqGH8aapcaaIYaGaciiBaiaa c+gacaGGNbGaaiikaiaadkgacaGGVaGaamyyaiaacMcaaaa@496A@  the region between r=a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0Jaamyyaaaa@35B9@  and r=c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0Jaam4yaaaa@35BB@  deforms plastically during the first application of pressure; while the region between c<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbGaeyipaWJaamOCaiabgYda8i aadkgaaaa@37A4@  remains elastic, where c satisfies the equation p a /Y=2log(c/a)+ 2 3 (1 c 3 / b 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH9aqpcaaIYaGaciiBaiaac+gacaGGNbGaaiik aiaadogacaGGVaGaamyyaiaacMcacqGHRaWkdaWcaaqaaiaaikdaae aacaaIZaaaaiaacIcacaaIXaGaeyOeI0Iaam4yamaaCaaaleqabaGa aG4maaaakiaac+cacaWGIbWaaWbaaSqabeaacaaIZaaaaOGaaiykaa aa@48AA@ .  In this case, the solid is permanently deformed.  After unloading, its internal and external radii are slightly increased, and the sphere is in a state of residual stress

 If the maximum internal pressure satisfies 2(1 a 3 / b 3 )/3< p a /Y<4(1 a 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIYaGaaiikaiaaigdacqGHsislca WGHbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqa aiaaiodaaaGccaGGPaGaai4laiaaiodacqGH8aapcaWGWbWaaSbaaS qaaiaadggaaeqaaOGaai4laiaadMfacqGH8aapcaaI0aGaaiikaiaa igdacqGHsislcaWGHbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadk gadaahaaWcbeqaaiaaiodaaaGccaGGPaGaai4laiaaiodaaaa@4B9C@ , the cylinder deforms plastically during the first application of pressure.  It then deforms elastically (no yield) while the pressure is removed.  During subsequent pressure cycles between zero and the maximum pressure, the cylinder deforms elastically.  Residual stresses introduced during the first loading cycle are protective, and prevent further plasticity.  This behavior is known as `shakedown’ and the maximum load for which it can occur ( p a /Y=4(1 a 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH9aqpcaaI0aGaaiikaiaaigdacqGHsislcaWG HbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqaai aaiodaaaGccaGGPaGaai4laiaaiodaaaa@4105@  ) is known as the `shakedown limit’

 If the maximum internal pressure reaches the shakedown limit p a /Y=4(1 a 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH9aqpcaaI0aGaaiikaiaaigdacqGHsislcaWG HbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqaai aaiodaaaGccaGGPaGaai4laiaaiodaaaa@4105@ , the residual stress just reaches yield at r=a when the pressure is reduced to zero after the first loading cycle.

 For internal pressures p a /Y>4(1 a 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH+aGpcaaI0aGaaiikaiaaigdacqGHsislcaWG HbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqaai aaiodaaaGccaGGPaGaai4laiaaiodaaaa@4107@ , a plastic zone forms between a<r<d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbGaeyipaWJaamOCaiabgYda8i aadsgaaaa@3794@  as the pressure is reduced to zero, where d satisfies the equation p a =4Y(1 d 3 / b 3 )/3+4Ylog(d/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaeyypa0JaaGinaiaadMfacaGGOaGaaGymaiabgkHiTiaadsgadaah aaWcbeqaaiaaiodaaaGccaGGVaGaamOyamaaCaaaleqabaGaaG4maa aakiaacMcacaGGVaGaaG4maiabgUcaRiaaisdacaWGzbGaciiBaiaa c+gacaGGNbGaaiikaiaadsgacaGGVaGaamyyaiaacMcaaaa@498E@ .   During subsequent cycles of loading, the region a<r<d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbGaeyipaWJaamOCaiabgYda8i aadsgaaaa@37A4@  is repeatedly plastically deformed, stretching in the hoop direction during increasing pressure, and compressing as the pressure is reduced to zero.  The region between d<r<c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeyipaWJaamOCaiabgYda8i aadogaaaa@37A6@  deforms plastically during the first cycle of pressure, but remains elastic for all subsequent cycles.  This is a `shakedown region.’ The remainder of the sphere experiences elastic cycles of strain. 

 

In the preceding discussion, we have assumed that the cylinder is thick enough to support an arbitrarily large pressure.   The internal pressure cannot exceed the collapse load p a /Y=2log(b/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH9aqpcaaIYaGaciiBaiaac+gacaGGNbGaaiik aiaadkgacaGGVaGaamyyaiaacMcaaaa@3ED3@ , so some regimes are inaccessible for thinner walled spheres.

 

The stress fields at maximum and minimum load for these various ranges of applied load are listed below.  The displacements can be computed, but the formulas are too long to record here.

The residual stress distributions (after unloading to zero pressure) are shown in the figure on the right, for a sphere with b/a=3. The solution for c/a=1.25 is below the shakedown limit; the other two solutions are for pressures exceeding the shakedown limit.  The region of cyclic plasticity can be seen from the discontinuity in the hoop stress curves. Note that the residual stresses are predominantly compressive MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  for this reason, bolt holes, pressure vessels and gun barrels are often purposely pressurized above the elastic limit so as to introduce a compressive stress near the loaded surface.  This protects the component against fatigue, since fatigue cracks do not propagate under compressive loading.

 

 

Solution for pressures below the elastic limit p a /Y<2(1 a 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH8aapcaaIYaGaaiikaiaaigdacqGHsislcaWG HbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqaai aaiodaaaGccaGGPaGaai4laiaaiodaaaa@4101@

 

The displacement, strain and stress field at maximum load are given by the elastic solution in Section 4.1.4

 

Solution for pressures between the elastic and shakedown limits p a /Y<4(1 a 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH8aapcaaI0aGaaiikaiaaigdacqGHsislcaWG HbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqaai aaiodaaaGccaGGPaGaai4laiaaiodaaaa@4103@

 

 At maximum pressure, the displacement and stress fields are given by the elastic-plastic solution in Section 4.2.3.

 At zero pressure, the solution is

 

(i) Solution for  a<r<c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbGaeyipaWJaamOCaiabgYda8i aadogaaaa@3793@

σ rr =2Ylog( r a ) p a p a a 3 ( b 3 a 3 ) ( 1 b 3 r 3 ) σ θθ = σ ϕϕ =2Ylog( r a ) p a +Y p a a 3 ( b 3 a 3 ) ( 1+ b 3 2 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aaSbaaSqaaiaadkhaca WGYbaabeaakiabg2da9iaaikdacaWGzbGaciiBaiaac+gacaGGNbWa aeWaaeaadaWcaaqaaiaadkhaaeaacaWGHbaaaaGaayjkaiaawMcaai abgkHiTiaadchadaWgaaWcbaGaamyyaaqabaGccqGHsisldaWcaaqa aiaadchadaWgaaWcbaGaamyyaaqabaGccaWGHbWaaWbaaSqabeaaca aIZaaaaaGcbaWaaeWaaeaacaWGIbWaaWbaaSqabeaacaaIZaaaaOGa eyOeI0IaamyyamaaCaaaleqabaGaaG4maaaaaOGaayjkaiaawMcaaa aadaqadaqaaiaaigdacqGHsisldaWcaaqaaiaadkgadaahaaWcbeqa aiaaiodaaaaakeaacaWGYbWaaWbaaSqabeaacaaIZaaaaaaaaOGaay jkaiaawMcaaaqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4oqCaeqa aOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiabew9aMjabew9aMbqabaGccq GH9aqpcaaIYaGaamywaiGacYgacaGGVbGaai4zamaabmaabaWaaSaa aeaacaWGYbaabaGaamyyaaaaaiaawIcacaGLPaaacqGHsislcaWGWb WaaSbaaSqaaiaadggaaeqaaOGaey4kaSIaamywaiabgkHiTmaalaaa baGaamiCamaaBaaaleaacaWGHbaabeaakiaadggadaahaaWcbeqaai aaiodaaaaakeaadaqadaqaaiaadkgadaahaaWcbeqaaiaaiodaaaGc cqGHsislcaWGHbWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaayzkaa aaamaabmaabaGaaGymaiabgUcaRmaalaaabaGaamOyamaaCaaaleqa baGaaG4maaaaaOqaaiaaikdacaWGYbWaaWbaaSqabeaacaaIZaaaaa aaaOGaayjkaiaawMcaaaaaaa@7EB7@     

 

(ii) Solution for c<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbGaeyipaWJaamOCaiabgYda8i aadkgaaaa@3794@

σ rr = 2Y c 3 3 b 3 ( 1 b 3 r 3 ) p a a 3 ( b 3 a 3 ) ( 1 b 3 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaaIYaGaamywaiaadogadaahaaWc beqaaiaaiodaaaaakeaacaaIZaGaamOyamaaCaaaleqabaGaaG4maa aaaaGcdaqadaqaaiaaigdacqGHsisldaWcaaqaaiaadkgadaahaaWc beqaaiaaiodaaaaakeaacaWGYbWaaWbaaSqabeaacaaIZaaaaaaaaO GaayjkaiaawMcaaiabgkHiTmaalaaabaGaamiCamaaBaaaleaacaWG HbaabeaakiaadggadaahaaWcbeqaaiaaiodaaaaakeaadaqadaqaai aadkgadaahaaWcbeqaaiaaiodaaaGccqGHsislcaWGHbWaaWbaaSqa beaacaaIZaaaaaGccaGLOaGaayzkaaaaamaabmaabaGaaGymaiabgk HiTmaalaaabaGaamOyamaaCaaaleqabaGaaG4maaaaaOqaaiaadkha daahaaWcbeqaaiaaiodaaaaaaaGccaGLOaGaayzkaaaaaa@56F2@

           σ θθ = σ ϕϕ = 2Y c 3 3 b 3 ( 1+ b 3 2 r 3 ) p a a 3 ( b 3 a 3 ) ( 1+ b 3 2 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaacaaIYaGaamywaiaadogadaahaaWcbe qaaiaaiodaaaaakeaacaaIZaGaamOyamaaCaaaleqabaGaaG4maaaa aaGcdaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaadkgadaahaaWcbe qaaiaaiodaaaaakeaacaaIYaGaamOCamaaCaaaleqabaGaaG4maaaa aaaakiaawIcacaGLPaaacqGHsisldaWcaaqaaiaadchadaWgaaWcba GaamyyaaqabaGccaWGHbWaaWbaaSqabeaacaaIZaaaaaGcbaWaaeWa aeaacaWGIbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamyyamaaCa aaleqabaGaaG4maaaaaOGaayjkaiaawMcaaaaadaqadaqaaiaaigda cqGHRaWkdaWcaaqaaiaadkgadaahaaWcbeqaaiaaiodaaaaakeaaca aIYaGaamOCamaaCaaaleqabaGaaG4maaaaaaaakiaawIcacaGLPaaa aaa@6061@

 

 

Solution for pressures exceeding the shakedown limit p a /Y>4(1 a 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH+aGpcaaI0aGaaiikaiaaigdacqGHsislcaWG HbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqaai aaiodaaaGccaGGPaGaai4laiaaiodaaaa@4107@

 

 At maximum pressure, the displacement, strain and stress fields are given in Section 4.2.3.

 At zero pressure, the solution is

 

(i) Solution for cyclic plastic region  a<r<d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbGaeyipaWJaamOCaiabgYda8i aadsgaaaa@3794@

σ rr =2Ylog(r/a) σ θθ = σ ϕϕ =2Ylog(r/a)Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcqGHsislcaaIYaGaamywaiGacYgacaGGVbGaai4z aiaacIcacaWGYbGaai4laiaadggacaGGPaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8a ZnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyypa0Jaeq4Wdm3aaS baaSqaaiabew9aMjabew9aMbqabaGccqGH9aqpcqGHsislcaaIYaGa amywaiGacYgacaGGVbGaai4zaiaacIcacaWGYbGaai4laiaadggaca GGPaGaeyOeI0Iaamywaaaa@6608@

 

(ii) Solution for shakedown region d<r<c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqGH8aapcaWGYbGaeyipaWJaam 4yaaaa@3538@

σ rr =2Ylog(r/a) p a 4Y d 3 3 b 3 ( 1 b 3 r 3 ) σ θθ = σ ϕϕ =2Ylog(r/a) p a +Y 4Y d 3 3 b 3 ( 1+ b 3 2 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aaSbaaSqaaiaadkhaca WGYbaabeaakiabg2da9iaaikdacaWGzbGaciiBaiaac+gacaGGNbGa aiikaiaadkhacaGGVaGaamyyaiaacMcacqGHsislcaWGWbWaaSbaaS qaaiaadggaaeqaaOGaeyOeI0YaaSaaaeaacaaI0aGaamywaiaadsga daahaaWcbeqaaiaaiodaaaaakeaacaaIZaGaamOyamaaCaaaleqaba GaaG4maaaaaaGcdaqadaqaaiaaigdacqGHsisldaWcaaqaaiaadkga daahaaWcbeqaaiaaiodaaaaakeaacaWGYbWaaWbaaSqabeaacaaIZa aaaaaaaOGaayjkaiaawMcaaaqaaiabeo8aZnaaBaaaleaacqaH4oqC cqaH4oqCaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiabew9aMjabew 9aMbqabaGccqGH9aqpcaaIYaGaamywaiGacYgacaGGVbGaai4zaiaa cIcacaWGYbGaai4laiaadggacaGGPaGaeyOeI0IaamiCamaaBaaale aacaWGHbaabeaakiabgUcaRiaadMfacqGHsisldaWcaaqaaiaaisda caWGzbGaamizamaaCaaaleqabaGaaG4maaaaaOqaaiaaiodacaWGIb WaaWbaaSqabeaacaaIZaaaaaaakmaabmaabaGaaGymaiabgUcaRmaa laaabaGaamOyamaaCaaaleqabaGaaG4maaaaaOqaaiaaikdacaWGYb WaaWbaaSqabeaacaaIZaaaaaaaaOGaayjkaiaawMcaaaaaaa@7793@

 

(iii) Solution for the elastic region c<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogacqGH8aapcaWGYbGaeyipaWJaam Oyaaaa@3536@

σ rr = 2Y c 3 3 b 3 ( 1 b 3 r 3 ) 4Y d 3 3 b 3 ( 1 b 3 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaaIYaGaamywaiaadogadaahaaWc beqaaiaaiodaaaaakeaacaaIZaGaamOyamaaCaaaleqabaGaaG4maa aaaaGcdaqadaqaaiaaigdacqGHsisldaWcaaqaaiaadkgadaahaaWc beqaaiaaiodaaaaakeaacaWGYbWaaWbaaSqabeaacaaIZaaaaaaaaO GaayjkaiaawMcaaiabgkHiTmaalaaabaGaaGinaiaadMfacaWGKbWa aWbaaSqabeaacaaIZaaaaaGcbaGaaG4maiaadkgadaahaaWcbeqaai aaiodaaaaaaOWaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGIbWa aWbaaSqabeaacaaIZaaaaaGcbaGaamOCamaaCaaaleqabaGaaG4maa aaaaaakiaawIcacaGLPaaaaaa@52ED@            σ θθ = σ ϕϕ = 2Y c 3 3 b 3 ( 1+ b 3 2 r 3 ) 4Y d 3 3 b 3 ( 1+ b 3 2 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaacaaIYaGaamywaiaadogadaahaaWcbe qaaiaaiodaaaaakeaacaaIZaGaamOyamaaCaaaleqabaGaaG4maaaa aaGcdaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaadkgadaahaaWcbe qaaiaaiodaaaaakeaacaaIYaGaamOCamaaCaaaleqabaGaaG4maaaa aaaakiaawIcacaGLPaaacqGHsisldaWcaaqaaiaaisdacaWGzbGaam izamaaCaaaleqabaGaaG4maaaaaOqaaiaaiodacaWGIbWaaWbaaSqa beaacaaIZaaaaaaakmaabmaabaGaaGymaiabgUcaRmaalaaabaGaam OyamaaCaaaleqabaGaaG4maaaaaOqaaiaaikdacaWGYbWaaWbaaSqa beaacaaIZaaaaaaaaOGaayjkaiaawMcaaaaa@5C5C@

 

Derivation of stress after unloading in the cyclic  plastic regime a<r<d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacqGH8aapcaWGYbGaeyipaWJaam izaaaa@3536@

 

(i) We anticipate that σ rr >0 σ θθ <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH+aGpcaaIWaGaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSba aSqaaiabeI7aXjabeI7aXbqabaGccqGH8aapcaaIWaaaaa@41D5@ . The yield criterion then gives σ θθ σ rr =Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4o qCaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadkhacaWGYbaabeaa kiabg2da9iabgkHiTiaadMfaaaa@3D72@ .

(ii) Substituting this result into the equilibrium equation shows that

d σ rr dr +2 Y r =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacqaHdpWCdaWgaa WcbaGaamOCaiaadkhaaeqaaaGcbaGaamizaiaadkhaaaGaey4kaSIa aGOmamaalaaabaGaamywaaqaaiaadkhaaaGaeyypa0JaaGimaaaa@3EC8@

(iii) Integrating, and using the boundary condition σ rr =0r=a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcaaIWaGaaGPaVlaaykW7caWGYbGaeyypa0Jaamyy aaaa@3C08@  together with the yield condition in step (i) gives

σ rr =2Ylog(r/a) σ θθ = σ ϕϕ =2Ylog(r/a)Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcqGHsislcaaIYaGaamywaiGacYgacaGGVbGaai4z aiaacIcacaWGYbGaai4laiaadggacaGGPaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8a ZnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyypa0Jaeq4Wdm3aaS baaSqaaiabew9aMjabew9aMbqabaGccqGH9aqpcqGHsislcaaIYaGa amywaiGacYgacaGGVbGaai4zaiaacIcacaWGYbGaai4laiaadggaca GGPaGaeyOeI0Iaamywaaaa@6608@

 

Derivation of stress after unloading in the shakedown regime d<r<c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqGH8aapcaWGYbGaeyipaWJaam 4yaaaa@3538@

 

(i) In this region, the stress at maximum load are given by the expressions for r<c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyipaWJaam4yaaaa@35B9@  in 4.2.3, i.e.

σ rr =2Ylog(r/a) p a σ θθ = σ ϕϕ =2Ylog(r/a) p a +Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcaaIYaGaamywaiGacYgacaGGVbGaai4zaiaacIca caWGYbGaai4laiaadggacaGGPaGaeyOeI0IaamiCamaaBaaaleaaca WGHbaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0JaaGOmaiaadMfaciGGSbGaai4BaiaacEgacaGGOa GaamOCaiaac+cacaWGHbGaaiykaiabgkHiTiaadchadaWgaaWcbaGa amyyaaqabaGccqGHRaWkcaWGzbaaaa@6A1F@

The solid then unloads elastically while the pressure is removed. 

(ii) The change in stress during unloading can be calculated quickly by regarding the region d<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeyipaWJaamOCaiabgYda8i aadkgaaaa@3795@  as a spherical shell with internal radius d and external radius b, subjected to radial pressure at r=d.  At maximum load, the pressure at r=d is p a 2Ylog(d/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO GaeyOeI0IaaGOmaiaadMfaciGGSbGaai4BaiaacEgacaGGOaGaamiz aiaac+cacaWGHbGaaiykaaaa@3E09@ ; after unloading the pressure follows from the solution for the cyclic plastic regime as 2Ylog(d/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIYaGaamywaiGacYgacaGGVbGaai 4zaiaacIcacaWGKbGaai4laiaadggacaGGPaaaaa@3B0B@ .  The change in pressure at r=d during unloading is thus Δp=4Ylog(d/a) p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcaWGWbGaeyypa0JaaGinai aadMfaciGGSbGaai4BaiaacEgacaGGOaGaamizaiaac+cacaWGHbGa aiykaiabgkHiTiaadchadaWgaaWcbaGaamyyaaqabaaaaa@4162@ .

(iii) The change in pressure during unloading can also be expressed as Δp=4Y(1 b 3 / d 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcaWGWbGaeyypa0JaeyOeI0 IaaGinaiaadMfacaGGOaGaaGymaiabgkHiTiaadkgadaahaaWcbeqa aiaaiodaaaGccaGGVaGaamizamaaCaaaleqabaGaaG4maaaakiaacM cacaGGVaGaaG4maaaa@418C@  using the governing equation for d shown below.

(iv) We then can simply add the (elastic) stress and displacement induced by this pressure change to the displacement and stress at maximum load, to obtain the solution given above.

 

Boundary of the cyclic plastic zone

 

The boundary of the cyclic plastic zone is determined by the condition that the stress in the shakedown regime must just reach yield at r=d when the pressure reaches zero.  This gives

σ θθ σ rr =Y p a =4Y(1 d 3 / b 3 )/3+4Ylog(d/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabgkHiTiabeo8aZnaaBaaaleaacaWGYbGaamOCaaqa baGccqGH9aqpcqGHsislcaWGzbGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaeyO0H4TaamiCamaaBaaaleaacaWGHbaabeaakiab g2da9iaaisdacaWGzbGaaiikaiaaigdacqGHsislcaWGKbWaaWbaaS qabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqaaiaaiodaaaGc caGGPaGaai4laiaaiodacqGHRaWkcaaI0aGaamywaiGacYgacaGGVb Gaai4zaiaacIcacaWGKbGaai4laiaadggacaGGPaaaaa@6227@

 

Derivation of solution in the elastic region c<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbGaeyipaWJaamOCaiabgYda8i aadkgaaaa@3794@

 

The solution in this region is derived in the same way as the solution for the shakedown region, except that the displacement and stress at maximum load are given by solutions for c<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbGaeyipaWJaamOCaiabgYda8i aadkgaaaa@37A4@

 

 

 

 

4.2.5 Simplified equations for plane strain axially symmetric elastic-perfectly plastic solids

 

An axially symmetric solid is illustrated in the picture.  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).  Temperature changes will be neglected, to simplify calculations.  However, 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.

 

We will assume that 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, so that a state of plane strain exists in the solid. 

 

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0JaamyDaiaacIcaca WGYbGaaiykaiaahwgadaWgaaWcbaGaamOCaaqabaaaaa@3A24@

 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 reduce to

 

 Strain Displacement Relations ε rr = du dr ε θθ = u r ε zz =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaWGKbGaamyDaaqaaiaadsgacaWG YbaaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlabew7aLnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGa eyypa0ZaaSaaaeaacaWG1baabaGaamOCaaaacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqyT du2aaSbaaSqaaiaadQhacaWG6baabeaakiabg2da9iaaicdaaaa@66A6@

 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)

 

In elastic region(s)

[ 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 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadeaaaeaacqaHdp WCdaWgaaWcbaGaamOCaiaadkhaaeqaaaGcbaGaeq4Wdm3aaSbaaSqa aiabeI7aXjabeI7aXbqabaaakeaacqaHdpWCdaWgaaWcbaGaamOEai aadQhaaeqaaaaaaOGaay5waiaaw2faaiabg2da9maalaaabaGaamyr aaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4MaaiykaiaacIcacaaIXa GaeyOeI0IaaGOmaiabe27aUjaacMcaaaWaamWaaeaafaqabeWadaaa baGaaGymaiabgkHiTiabe27aUbqaaiabe27aUbqaaiabe27aUbqaai abe27aUbqaaiaaigdacqGHsislcqaH9oGBaeaacqaH9oGBaeaacqaH 9oGBaeaacqaH9oGBaeaacaaIXaGaeyOeI0IaeqyVd4gaaaGaay5wai aaw2faamaadmaabaqbaeqabmqaaaqaaiabew7aLnaaBaaaleaacaWG YbGaamOCaaqabaaakeaacqaH1oqzdaWgaaWcbaGaeqiUdeNaeqiUde habeaaaOqaaiabew7aLnaaBaaaleaacaWG6bGaamOEaaqabaaaaaGc caGLBbGaayzxaaaaaa@7376@

{ ( σ θθ σ rr ) 2 + ( σ θθ σ zz ) 2 + ( σ rr σ zz ) 2 }/2 <Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaakaaabaWaaiWaaeaadaqadaqaaiabeo 8aZnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyOeI0Iaeq4Wdm3a aSbaaSqaaiaadkhacaWGYbaabeaaaOGaayjkaiaawMcaamaaCaaale qabaGaaGOmaaaakiabgUcaRmaabmaabaGaeq4Wdm3aaSbaaSqaaiab eI7aXjabeI7aXbqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaamOEai aadQhaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa ey4kaSYaaeWaaeaacqaHdpWCdaWgaaWcbaGaamOCaiaadkhaaeqaaO GaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadQhacaWG6baabeaaaOGaayjk aiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaay5Eaiaaw2haaiaac+ cacaaIYaaaleqaaOGaeyipaWJaamywaaaa@5C94@

In plastic region(s)

Yield criterion:   { ( σ θθ σ rr ) 2 + ( σ θθ σ zz ) 2 + ( σ rr σ zz ) 2 }/2 =Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaakaaabaWaaiWaaeaadaqadaqaaiabeo 8aZnaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyOeI0Iaeq4Wdm3a aSbaaSqaaiaadkhacaWGYbaabeaaaOGaayjkaiaawMcaamaaCaaale qabaGaaGOmaaaakiabgUcaRmaabmaabaGaeq4Wdm3aaSbaaSqaaiab eI7aXjabeI7aXbqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaamOEai aadQhaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa ey4kaSYaaeWaaeaacqaHdpWCdaWgaaWcbaGaamOCaiaadkhaaeqaaO GaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadQhacaWG6baabeaaaOGaayjk aiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaay5Eaiaaw2haaiaac+ cacaaIYaaaleqaaOGaeyypa0Jaamywaaaa@5C96@

Strain partition: d ε rr =d ε rr p +d ε rr e d ε ϕϕ =d ε ϕϕ p +d ε ϕϕ e d ε θθ =d ε θθ p +d ε θθ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH1oqzdaWgaaWcbaGaamOCai aadkhaaeqaaOGaeyypa0Jaamizaiabew7aLnaaDaaaleaacaWGYbGa amOCaaqaaiaadchaaaGccqGHRaWkcaWGKbGaeqyTdu2aa0baaSqaai aadkhacaWGYbaabaGaamyzaaaakiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaadsgacqaH1oqzdaWgaaWcbaGaeqy1dyMaeqy1dy gabeaakiabg2da9iaadsgacqaH1oqzdaqhaaWcbaGaeqy1dyMaeqy1 dygabaGaamiCaaaakiabgUcaRiaadsgacqaH1oqzdaqhaaWcbaGaeq y1dyMaeqy1dygabaGaamyzaaaakiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaadsgacqaH1oqzdaWgaaWcbaGaeqiUdeNaeqiUde habeaakiabg2da9iaadsgacqaH1oqzdaqhaaWcbaGaeqiUdeNaeqiU dehabaGaamiCaaaakiabgUcaRiaadsgacqaH1oqzdaqhaaWcbaGaeq iUdeNaeqiUdehabaGaamyzaaaaaaa@81F1@

Elastic strain:    d ε rr e =d σ rr /Eν(d σ θθ +d σ zz )/E d ε θθ e =d σ θθ /Eν(d σ rr +d σ zz )/E d ε zz e =d σ zz /Eν(d σ rr +d σ θθ )/E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamizaiabew7aLnaaDaaaleaaca WGYbGaamOCaaqaaiaadwgaaaGccqGH9aqpcaWGKbGaeq4Wdm3aaSba aSqaaiaadkhacaWGYbaabeaakiaac+cacaWGfbGaeyOeI0IaeqyVd4 MaaiikaiaadsgacqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaa kiabgUcaRiaadsgacqaHdpWCdaWgaaWcbaGaamOEaiaadQhaaeqaaO Gaaiykaiaac+cacaWGfbaabaGaamizaiabew7aLnaaDaaaleaacqaH 4oqCcqaH4oqCaeaacaWGLbaaaOGaeyypa0Jaamizaiabeo8aZnaaBa aaleaacqaH4oqCcqaH4oqCaeqaaOGaai4laiaadweacqGHsislcqaH 9oGBcaGGOaGaamizaiabeo8aZnaaBaaaleaacaWGYbGaamOCaaqaba GccqGHRaWkcaWGKbGaeq4Wdm3aaSbaaSqaaiaadQhacaWG6baabeaa kiaacMcacaGGVaGaamyraaqaaiaadsgacqaH1oqzdaqhaaWcbaGaam OEaiaadQhaaeaacaWGLbaaaOGaeyypa0Jaamizaiabeo8aZnaaBaaa leaacaWG6bGaamOEaaqabaGccaGGVaGaamyraiabgkHiTiabe27aUj aacIcacaWGKbGaeq4Wdm3aaSbaaSqaaiaadkhacaWGYbaabeaakiab gUcaRiaadsgacqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaaki aacMcacaGGVaGaamyraaaaaa@8D74@

Flow rule:          d ε rr p =d ε ¯ p σ rr ( σ θθ + σ zz )/2 Y d ε θθ p =d ε ¯ p σ θθ ( σ rr + σ zz )/2 Y d ε zz p =d ε ¯ p σ zz ( σ rr + σ θθ )/2 Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamizaiabew7aLnaaDaaaleaaca WGYbGaamOCaaqaaiaadchaaaGccqGH9aqpcaWGKbGafqyTduMbaeba daahaaWcbeqaaiaadchaaaGcdaWcaaqaaiabeo8aZnaaBaaaleaaca WGYbGaamOCaaqabaGccqGHsisldaqadaqaaiabeo8aZnaaBaaaleaa cqaH4oqCcqaH4oqCaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiaadQ hacaWG6baabeaaaOGaayjkaiaawMcaaiaac+cacaaIYaaabaGaamyw aaaaaeaacaaMc8Uaamizaiabew7aLnaaDaaaleaacqaH4oqCcqaH4o qCaeaacaWGWbaaaOGaeyypa0Jaamizaiqbew7aLzaaraWaaWbaaSqa beaacaWGWbaaaOWaaSaaaeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabgkHiTmaabmaabaGaeq4Wdm3aaSbaaSqaaiaadkha caWGYbaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaWG6bGaamOEaa qabaaakiaawIcacaGLPaaacaGGVaGaaGOmaaqaaiaadMfaaaaabaGa amizaiabew7aLnaaDaaaleaacaWG6bGaamOEaaqaaiaadchaaaGccq GH9aqpcaWGKbGafqyTduMbaebadaahaaWcbeqaaiaadchaaaGcdaWc aaqaaiabeo8aZnaaBaaaleaacaWG6bGaamOEaaqabaGccqGHsislda qadaqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaaqabaGccqGHRaWk cqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaaaOGaayjkaiaawM caaiaac+cacaaIYaaabaGaamywaaaaaaaa@8BFC@

 

 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@

 

The equilibrium and strain-displacement equations can be derived following the procedure outlined in Section 4.1.2.  The stress-strain relations are derived by substituting the strain components into the general constitutive equation and simplifying the result.

 

Unlike the elastic solution in Sect 4.1, there is no clean, direct and general method for integrating these equations.  Instead, solutions must be found using a combination of physical intuition and some algebraic tricks, as illustrated in the sections below.

 

 

4.2.6 Long (plane strain) cylinder subjected to internal 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 free of pressure

 The cylinder does not stretch parallel to its axis

 

The solution given below is approximate, because it assumes that both elastic and plastic axial strains vanish separately (when in fact only the sum of elastic and plastic strains should be zero).

 

 

Solution:

 

(i) Preliminaries:

 The cylinder first reaches yield (at r=a) at an internal pressure 3 p a /Y(1 a 2 / b 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGcaaqaaiaaiodaaSqabaGccaWGWb WaaSbaaSqaaiaadggaaeqaaOGaai4laiaadMfacqGHijYUcaGGOaGa aGymaiabgkHiTiaadggadaahaaWcbeqaaiaaikdaaaGccaGGVaGaam OyamaaCaaaleqabaGaaGOmaaaakiaacMcaaaa@4062@

 For pressures in the range (1 a 2 / b 2 )< 3 p a /Y<2log(b/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaaGymaiabgkHiTiaadggada ahaaWcbeqaaiaaikdaaaGccaGGVaGaamOyamaaCaaaleqabaGaaGOm aaaakiaacMcacqGH8aapdaGcaaqaaiaaiodaaSqabaGccaWGWbWaaS baaSqaaiaadggaaeqaaOGaai4laiaadMfacqGH8aapcaaIYaGaciiB aiaac+gacaGGNbGaaiikaiaadkgacaGGVaGaamyyaiaacMcaaaa@481E@  the region between r=a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0Jaamyyaaaa@35A9@  and r=c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0Jaam4yaaaa@35BB@  deforms plastically; while the region between c<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbGaeyipaWJaamOCaiabgYda8i aadkgaaaa@37A4@  remains elastic, where c satisfies the equation 3 p a /Y=2log(c/a)+1 c 2 / b 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGcaaqaaiaaiodaaSqabaGccaWGWb WaaSbaaSqaaiaadggaaeqaaOGaai4laiaadMfacqGH9aqpcaaIYaGa ciiBaiaac+gacaGGNbGaaiikaiaadogacaGGVaGaamyyaiaacMcacq GHRaWkcaaIXaGaeyOeI0Iaam4yamaaCaaaleqabaGaaGOmaaaakiaa c+cacaWGIbWaaWbaaSqabeaacaaIYaaaaaaa@469E@

 At a pressure p a /Y=( 2/ 3 )log(b/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH9aqpdaqadaqaaiaaikdacaGGVaWaaOaaaeaa caaIZaaaleqaaaGccaGLOaGaayzkaaGaciiBaiaac+gacaGGNbGaai ikaiaadkgacaGGVaGaamyyaiaacMcaaaa@41F1@  the entire cylinder is plastic.  At this point the sphere collapses MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the displacements become infinitely large.

 

(ii) Solution in the plastic region a<r<c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbGaeyipaWJaamOCaiabgYda8i aadogaaaa@37A3@

u= ( 12ν )( 1+ν ) E 2r{ ( 2Y/ 3 )log(r/a) p a }+ ( 1+ν )Y c 2 (2(1ν) b 2 +(12ν)( b 2 c 2 )) 3 E b 2 r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhacqGH9aqpdaWcaaqaamaabmaaba GaaGymaiabgkHiTiaaikdacqaH9oGBaiaawIcacaGLPaaadaqadaqa aiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaaaeaacaWGfbaaai aaikdacaWGYbWaaiWaaeaadaqadaqaaiaaikdacaWGzbGaai4lamaa kaaabaGaaG4maaWcbeaaaOGaayjkaiaawMcaaiGacYgacaGGVbGaai 4zaiaacIcacaWGYbGaai4laiaadggacaGGPaGaeyOeI0IaamiCamaa BaaaleaacaWGHbaabeaaaOGaay5Eaiaaw2haaiabgUcaRmaalaaaba WaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzkaaGaamyw aiaadogadaahaaWcbeqaaiaaikdaaaGccaGGOaGaaGOmaiaacIcaca aIXaGaeyOeI0IaeqyVd4MaaiykaiaadkgadaahaaWcbeqaaiaaikda aaGccqGHRaWkcaGGOaGaaGymaiabgkHiTiaaikdacqaH9oGBcaGGPa GaaiikaiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGJbWa aWbaaSqabeaacaaIYaaaaOGaaiykaiaacMcaaeaadaGcaaqaaiaaio daaSqabaGccaWGfbGaamOyamaaCaaaleqabaGaaGOmaaaakiaadkha aaaaaa@7218@

σ rr =(2Y/ 3 )log(r/a) p a σ θθ =(2Y/ 3 )log(r/a) p a +2Y/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcaGGOaGaaGOmaiaadMfacaGGVaWaaOaaaeaacaaI ZaaaleqaaOGaaiykaiGacYgacaGGVbGaai4zaiaacIcacaWGYbGaai 4laiaadggacaGGPaGaeyOeI0IaamiCamaaBaaaleaacaWGHbaabeaa kiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaa kiabg2da9iaacIcacaaIYaGaamywaiaac+cadaGcaaqaaiaaiodaaS qabaGccaGGPaGaciiBaiaac+gacaGGNbGaaiikaiaadkhacaGGVaGa amyyaiaacMcacqGHsislcaWGWbWaaSbaaSqaaiaadggaaeqaaOGaey 4kaSIaaGOmaiaadMfacaGGVaWaaOaaaeaacaaIZaaaleqaaaaa@6BB3@     

(iii) Solution in the elastic region c<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbGaeyipaWJaamOCaiabgYda8i aadkgaaaa@37A4@

u r = ( 1+ν ) c 2 Y E 3 { 1 r +( 12ν ) r b 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadkhaaeqaaO Gaeyypa0ZaaSaaaeaadaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaa wIcacaGLPaaacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaamywaaqaai aadweadaGcaaqaaiaaiodaaSqabaaaaOWaaiWaaeaadaWcaaqaaiaa igdaaeaacaWGYbaaaiabgUcaRmaabmaabaGaaGymaiabgkHiTiaaik dacqaH9oGBaiaawIcacaGLPaaadaWcaaqaaiaadkhaaeaacaWGIbWa aWbaaSqabeaacaaIYaaaaaaaaOGaay5Eaiaaw2haaaaa@4CA0@

σ rr = Y c 2 3 b 2 { 1 b 2 r 2 } σ θθ = Y c 2 3 b 2 { 1+ b 2 r 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaWGzbGaam4yamaaCaaaleqabaGa aGOmaaaaaOqaamaakaaabaGaaG4maaWcbeaakiaadkgadaahaaWcbe qaaiaaikdaaaaaaOWaaiWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWG IbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOCamaaCaaaleqabaGaaG OmaaaaaaaakiaawUhacaGL9baacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cq aHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaakiabg2da9maalaaa baGaamywaiaadogadaahaaWcbeqaaiaaikdaaaaakeaadaGcaaqaai aaiodaaSqabaGccaWGIbWaaWbaaSqabeaacaaIYaaaaaaakmaacmaa baGaaGymaiabgUcaRmaalaaabaGaamOyamaaCaaaleqabaGaaGOmaa aaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaaGccaGL7bGaayzF aaaaaa@6AFA@

The stress and displacement fields are plotted in the figure below, for various positions of the elastic-plastic boundary.  The results are for b/a=3, and the displacement is shown for a solid with ν=0.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBcqGH9aqpcaaIWaGaaiOlai aaiodaaaa@37BD@

 

       

Stress distribution                                          Radial displacement

 

Derivation: By substituting the stresses for the elastic solution given in 4.1.4 into the Von-Mises yield criterion, we see that a pressurized elastic cylinder first reaches yield at r=a. If the pressure is increased beyond yield, a region a<r<c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacqGH8aapcaWGYbGaeyipaWJaam 4yaaaa@3535@  will deform plastically, while a region c<r<b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogacqGH8aapcaWGYbGaeyipaWJaam Oyaaaa@3536@  remains elastic. We must find separate solutions in the plastic and elastic regimes.

 

In the plastic regime a<r<c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacqGH8aapcaWGYbGaeyipaWJaam 4yaaaa@3545@

(i) To simplify the calculation we assume that d ε zz p =d ε zz e =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeqyTdu2aa0baaSqaaiaadQ hacaWG6baabaGaamiCaaaakiabg2da9iaadsgacqaH1oqzdaqhaaWc baGaamOEaiaadQhaaeaacaWGLbaaaOGaeyypa0JaaGimaaaa@40F5@ .  This turns out to be exact for ν=1/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBcqGH9aqpcaaIXaGaai4lai aaikdaaaa@37AE@ , but is approximate for other values of Poisson ratio.  The plastic flow rule shows that d ε zz p =d ε ¯ p σ zz ( σ rr + σ θθ )/2 Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH1oqzdaqhaaWcbaGaamOEai aadQhaaeaacaWGWbaaaOGaeyypa0Jaamizaiqbew7aLzaaraWaaWba aSqabeaacaWGWbaaaOWaaSaaaeaacqaHdpWCdaWgaaWcbaGaamOEai aadQhaaeqaaOGaeyOeI0YaaeWaaeaacqaHdpWCdaWgaaWcbaGaamOC aiaadkhaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI 7aXbqabaaakiaawIcacaGLPaaacaGGVaGaaGOmaaqaaiaadMfaaaaa aa@4DF3@ , in which case d ε zz p =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeqyTdu2aa0baaSqaaiaadQ hacaWG6baabaGaamiCaaaakiabg2da9iaaicdaaaa@3A40@  requires that σ zz =( σ rr + σ θθ )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWG6bGaamOEaa qabaGccqGH9aqpdaqadaqaaiabeo8aZnaaBaaaleaacaWGYbGaamOC aaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabe aaaOGaayjkaiaawMcaaiaac+cacaaIYaaaaa@428A@

(ii) We anticipate that σ rr <0 σ θθ >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH8aapcaaIWaGaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSba aSqaaiabeI7aXjabeI7aXbqabaGccqGH+aGpcaaIWaaaaa@41D5@ . Substituting the result of (i) into the yield criterion then gives σ θθ σ rr =2Y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4o qCaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadkhacaWGYbaabeaa kiabg2da9iaaikdacaWGzbWaaOaaaeaacaaIZaaaleqaaaaa@3E19@ .

(iii) Substituting this result into the equilibrium equation shows that

d σ rr dr 2Y 3 r =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacqaHdpWCdaWgaa WcbaGaamOCaiaadkhaaeqaaaGcbaGaamizaiaadkhaaaGaeyOeI0Ya aSaaaeaacaaIYaGaamywaaqaamaakaaabaGaaG4maaWcbeaakiaadk haaaGaeyypa0JaaGimaaaa@3FB5@

(iv) Integrating, and using the boundary condition σ rr = p a r=a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpcqGHsislcaWGWbWaaSbaaSqaaiaadggaaeqaaOGa aGPaVlaaykW7caWGYbGaeyypa0Jaamyyaaaa@3E4C@  together with the yield condition (i) gives

σ rr =( 2Y/ 3 )log(r/a) p a σ θθ =( 2Y/ 3 )log(r/a) p a +2Y/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccqGH9aqpdaqadaqaaiaaikdacaWGzbGaai4lamaakaaabaGa aG4maaWcbeaaaOGaayjkaiaawMcaaiGacYgacaGGVbGaai4zaiaacI cacaWGYbGaai4laiaadggacaGGPaGaeyOeI0IaamiCamaaBaaaleaa caWGHbaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaeqiUdeNa eqiUdehabeaakiabg2da9maabmaabaGaaGOmaiaadMfacaGGVaWaaO aaaeaacaaIZaaaleqaaaGccaGLOaGaayzkaaGaciiBaiaac+gacaGG NbGaaiikaiaadkhacaGGVaGaamyyaiaacMcacqGHsislcaWGWbWaaS baaSqaaiaadggaaeqaaOGaey4kaSIaaGOmaiaadMfacaGGVaWaaOaa aeaacaaIZaaaleqaaaaa@6C13@

(v) The elastic strains follow as

ε rr e =( σ rr ν σ θθ ν σ zz )/E ε θθ e =( σ θθ ν σ rr ν σ zz )/E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaWGYbGaamOCaa qaaiaadwgaaaGccqGH9aqpdaqadaqaaiabeo8aZnaaBaaaleaacaWG YbGaamOCaaqabaGccqGHsislcqaH9oGBcqaHdpWCdaWgaaWcbaGaeq iUdeNaeqiUdehabeaakiabgkHiTiabe27aUjabeo8aZnaaBaaaleaa caWG6bGaamOEaaqabaaakiaawIcacaGLPaaacaGGVaGaamyraiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabew7aLnaaDaaaleaa cqaH4oqCcqaH4oqCaeaacaWGLbaaaOGaeyypa0ZaaeWaaeaacqaHdp WCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaakiabgkHiTiabe27aUjab eo8aZnaaBaaaleaacaWGYbGaamOCaaqabaGccqGHsislcqaH9oGBcq aHdpWCdaWgaaWcbaGaamOEaiaadQhaaeqaaaGccaGLOaGaayzkaaGa ai4laiaadweaaaa@71C6@

(vi) With assumption (i), the flow rule shows that plastic strains satisfy ε rr p + ε θθ p =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaWGYbGaamOCaa qaaiaadchaaaGccqGHRaWkcqaH1oqzdaqhaaWcbaGaeqiUdeNaeqiU dehabaGaamiCaaaakiabg2da9iaaicdaaaa@3E0A@ .  Consequently, using the strain partition formula and the strain-displacement relation shows that

ε rr + ε θθ = du dr + u r = 1 r d dr ( ru )= ( 12ν )( 1+ν ) E { σ rr + σ θθ } = ( 12ν )( 1+ν ) E 2( 2Y[log(r/a)+1]/ 3 p a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeqyTdu2aaSbaaSqaaiaadkhaca WGYbaabeaakiabgUcaRiabew7aLnaaBaaaleaacqaH4oqCcqaH4oqC aeqaaOGaeyypa0ZaaSaaaeaacaWGKbGaamyDaaqaaiaadsgacaWGYb aaaiabgUcaRmaalaaabaGaamyDaaqaaiaadkhaaaGaeyypa0ZaaSaa aeaacaaIXaaabaGaamOCaaaadaWcaaqaaiaadsgaaeaacaWGKbGaam OCaaaadaqadaqaaiaadkhacaWG1baacaGLOaGaayzkaaGaeyypa0Za aSaaaeaadaqadaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gacaGLOa GaayzkaaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzk aaaabaGaamyraaaadaGadaqaaiabeo8aZnaaBaaaleaacaWGYbGaam OCaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdeha beaaaOGaay5Eaiaaw2haaaqaaiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlabg2da9maalaaabaWaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiab e27aUbGaayjkaiaawMcaamaabmaabaGaaGymaiabgUcaRiabe27aUb GaayjkaiaawMcaaaqaaiaadweaaaGaaGOmamaabmaabaGaaGOmaiaa dMfacaGGBbGaciiBaiaac+gacaGGNbGaaiikaiaadkhacaGGVaGaam yyaiaacMcacqGHRaWkcaaIXaGaaiyxaiaac+cadaGcaaqaaiaaioda aSqabaGccqGHsislcaWGWbWaaSbaaSqaaiaadggaaeqaaaGccaGLOa Gaayzkaaaaaaa@A350@

(vii) Integrating gives

u= ( 12ν )( 1+ν ) E 2r{ ( 2Y/ 3 )log(r/a) p a }+C/r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhacqGH9aqpdaWcaaqaamaabmaaba GaaGymaiabgkHiTiaaikdacqaH9oGBaiaawIcacaGLPaaadaqadaqa aiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaaaeaacaWGfbaaai aaikdacaWGYbWaaiWaaeaadaqadaqaaiaaikdacaWGzbGaai4lamaa kaaabaGaaG4maaWcbeaaaOGaayjkaiaawMcaaiGacYgacaGGVbGaai 4zaiaacIcacaWGYbGaai4laiaadggacaGGPaGaeyOeI0IaamiCamaa BaaaleaacaWGHbaabeaaaOGaay5Eaiaaw2haaiabgUcaRiaadoeaca GGVaGaamOCaaaa@536C@

where C is a constant of integration

(viii) The constant of integration can be found by noting that the radial displacements in the elastic and plastic regimes must be equal at r=c.  Using the expression for the elastic displacement field below and solving for C gives

C= ( 1+ν ) c 2 (2(1ν) b 2 +(12ν)( b 2 c 2 )) E( b 2 c 2 ) { p a ( 2Y/ 3 )log(c/a) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbGaeyypa0ZaaSaaaeaadaqada qaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaacaWGJbWaaWba aSqabeaacaaIYaaaaOGaaiikaiaaikdacaGGOaGaaGymaiabgkHiTi abe27aUjaacMcacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIa aiikaiaaigdacqGHsislcaaIYaGaeqyVd4MaaiykaiaacIcacaWGIb WaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam4yamaaCaaaleqabaGa aGOmaaaakiaacMcacaGGPaaabaGaamyraiaacIcacaWGIbWaaWbaaS qabeaacaaIYaaaaOGaeyOeI0Iaam4yamaaCaaaleqabaGaaGOmaaaa kiaacMcaaaWaaiWaaeaacaWGWbWaaSbaaSqaaiaadggaaeqaaOGaey OeI0YaaeWaaeaacaaIYaGaamywaiaac+cadaGcaaqaaiaaiodaaSqa baaakiaawIcacaGLPaaaciGGSbGaai4BaiaacEgacaGGOaGaam4yai aac+cacaWGHbGaaiykaaGaay5Eaiaaw2haaaaa@67AC@

This result can be simplified by noting that p a ( 2Y/ 3 )log(c/a)=Y(1 c 2 / b 2 )/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO GaeyOeI0YaaeWaaeaacaaIYaGaamywaiaac+cadaGcaaqaaiaaioda aSqabaaakiaawIcacaGLPaaaciGGSbGaai4BaiaacEgacaGGOaGaam 4yaiaac+cacaWGHbGaaiykaiabg2da9iaadMfacaGGOaGaaGymaiab gkHiTiaadogadaahaaWcbeqaaiaaikdaaaGccaGGVaGaamOyamaaCa aaleqabaGaaGOmaaaakiaacMcacaGGVaWaaOaaaeaacaaIZaaaleqa aaaa@4BFE@  from the expression for the location of the elastic-plastic boundary given below.

 

In the elastic regime

The solution can be found directly from the solution to the internally pressurized elastic cylinder given in Sect 4.1.9.  From step (iv) in the solution for the plastic regime we see that the radial pressure at r=c is p c = σ rr = p a ( 2Y/ 3 )log(c/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaam4yaaqabaGccq GH9aqpcqGHsislcqaHdpWCdaWgaaWcbaGaamOCaiaadkhaaeqaaOGa eyypa0JaamiCamaaBaaaleaacaWGHbaabeaakiabgkHiTmaabmaaba GaaGOmaiaadMfacaGGVaWaaOaaaeaacaaIZaaaleqaaaGccaGLOaGa ayzkaaGaciiBaiaac+gacaGGNbGaaiikaiaadogacaGGVaGaamyyai aacMcaaaa@47BB@ . We can simplify the solution by noting p a ( 2Y/ 3 )log(c/a)=Y(1 c 2 / b 2 )/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO GaeyOeI0YaaeWaaeaacaaIYaGaamywaiaac+cadaGcaaqaaiaaioda aSqabaaakiaawIcacaGLPaaaciGGSbGaai4BaiaacEgacaGGOaGaam 4yaiaac+cacaWGHbGaaiykaiabg2da9iaadMfacaGGOaGaaGymaiab gkHiTiaadogadaahaaWcbeqaaiaaikdaaaGccaGGVaGaamOyamaaCa aaleqabaGaaGOmaaaakiaacMcacaGGVaWaaOaaaeaacaaIZaaaleqa aaaa@4BFE@  from the expression for the location of the elastic-plastic boundary.   Substituting into the expressions for stress and displacement in 4.1.9 shows that

σ rr = p c c 2 b 2 c 2 { 1 b 2 r 2 } σ θθ = p c c 2 b 2 c 2 { 1+ b 2 r 2 } σ zz =2ν p c 2 b 2 c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacaWGWbWaaSbaaSqaaiaadogaaeqa aOGaam4yamaaCaaaleqabaGaaGOmaaaaaOqaaiaadkgadaahaaWcbe qaaiaaikdaaaGccqGHsislcaWGJbWaaWbaaSqabeaacaaIYaaaaaaa kmaacmaabaGaaGymaiabgkHiTmaalaaabaGaamOyamaaCaaaleqaba GaaGOmaaaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaaGccaGL 7bGaayzFaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7aXbqa baGccqGH9aqpdaWcaaqaaiaadchadaWgaaWcbaGaam4yaaqabaGcca WGJbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOyamaaCaaaleqabaGa aGOmaaaakiabgkHiTiaadogadaahaaWcbeqaaiaaikdaaaaaaOWaai WaaeaacaaIXaGaey4kaSYaaSaaaeaacaWGIbWaaWbaaSqabeaacaaI YaaaaaGcbaGaamOCamaaCaaaleqabaGaaGOmaaaaaaaakiaawUhaca GL9baacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaadQhacaWG6baabe aakiabg2da9iaaikdacqaH9oGBdaWcaaqaaiaadchacaWGJbWaaWba aSqabeaacaaIYaaaaaGcbaGaamOyamaaCaaaleqabaGaaGOmaaaaki abgkHiTiaadogadaahaaWcbeqaaiaaikdaaaaaaaaa@8AE6@

u r = ( 1+ν ) c 2 b 2 p c E( b 2 c 2 ) { 1 r +( 12ν ) r b 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadkhaaeqaaO Gaeyypa0ZaaSaaaeaadaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaa wIcacaGLPaaacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaamOyamaaCa aaleqabaGaaGOmaaaakiaadchadaWgaaWcbaGaam4yaaqabaaakeaa caWGfbWaaeWaaeaacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0 Iaam4yamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaaadaGa daqaamaalaaabaGaaGymaaqaaiaadkhaaaGaey4kaSYaaeWaaeaaca aIXaGaeyOeI0IaaGOmaiabe27aUbGaayjkaiaawMcaamaalaaabaGa amOCaaqaaiaadkgadaahaaWcbeqaaiaaikdaaaaaaaGccaGL7bGaay zFaaaaaa@54F8@

 

Location of the elastic-plastic boundary

Finally, the elastic-plastic boundary is located by the condition that the stress in the elastic region must just reach yield at r=c (so there is a smooth transition into the plastic region).  The yield condition is σ θθ σ rr =2Y/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4o qCaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadkhacaWGYbaabeaa kiabg2da9iaaikdacaWGzbGaai4lamaakaaabaGaaG4maaWcbeaaaa a@3ECC@ , so substituting the expressions for stress in the elastic region and simplifying yields

σ θθ σ rr = 2( p a ( 2Y/ 3 )log(c/a) ) b 2 ( b 2 c 2 ) =2Y/ 3 3 p a Y =2log(c/a)+(1 c 2 / b 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacqaH4o qCcqaH4oqCaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadkhacaWG Ybaabeaakiabg2da9maalaaabaGaaGOmamaabmaabaGaamiCamaaBa aaleaacaWGHbaabeaakiabgkHiTmaabmaabaGaaGOmaiaadMfacaGG VaWaaOaaaeaacaaIZaaaleqaaaGccaGLOaGaayzkaaGaciiBaiaac+ gacaGGNbGaaiikaiaadogacaGGVaGaamyyaiaacMcaaiaawIcacaGL PaaacaWGIbWaaWbaaSqabeaacaaIYaaaaaGcbaWaaeWaaeaacaWGIb WaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam4yamaaCaaaleqabaGa aGOmaaaaaOGaayjkaiaawMcaaaaacqGH9aqpcaaIYaGaamywaiaac+ cadaGcaaqaaiaaiodaaSqabaaakeaacqGHuhY2daGcaaqaaiaaioda aSqabaGcdaWcaaqaaiaadchadaWgaaWcbaGaamyyaaqabaaakeaaca WGzbaaaiabg2da9iaaikdaciGGSbGaai4BaiaacEgacaGGOaGaam4y aiaac+cacaWGHbGaaiykaiabgUcaRiaacIcacaaIXaGaeyOeI0Iaam 4yamaaCaaaleqabaGaaGOmaaaakiaac+cacaWGIbWaaWbaaSqabeaa caaIYaaaaOGaaiykaaaaaa@71EF@

If p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaa aa@34CD@ , Y, a and b are specified this equation can be solved (numerically) for c.  For graphing purposes it is preferable to choose c and then calculate the corresponding value of p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaa aa@34CD@