Chapter 5

 

Analytical techniques and solutions for linear elastic solids

 

 

 

5.4 Solutions to 3D static problems in linear elasticity

 

The field equations of linear elasticity are much more difficult to solve in 3D than in 2D.  Nevertheless, several important problems have been solved.  In this section, we outline a common representation for 3D problems, and give solutions to selected 3D problems.

 

 

5.4.1 Papkovich-Neuber Potential representations for 3D solutions for isotropic solids

 

In this section we outline a general technique for solving 3D static linear elasticity problems.  The technique is similar to the 2D Airy function method, in that the solution is derived by differentiating a potential, which is governed by a PDE.  Many other potential representations are used in 3D elasticity, but most are simply special cases of the general Papkovich-Neuber representation.

Assume that

 The solid has Young’s modulus E, mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaa@356C@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@ .

 The solid is subjected to body force distribution b i ( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkgadaWgaaWcbaGaamyAaaqabaGcca GGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSba aSqaaiaaikdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaG4maaqaba GccaGGPaaaaa@3AF9@  (per unit mass)

 Part of the boundary S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGymaaqabaaaaa@3227@  is subjected to prescribed displacements u i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaqhaaWcbaGaamyAaaqaaiaacQ caaaaaaa@332B@

 A second part of the boundary S 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGOmaaqabaaaaa@3228@  is subjected to prescribed tractions t i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshadaqhaaWcbaGaamyAaaqaaiaacQ caaaaaaa@332A@

 

The Papkovich-Neuber procedure can be summarized as follows:

1.      Begin by finding a vector function Ψ i ( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfI6aznaaBaaaleaacaWGPbaabeaaki aacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWg aaWcbaGaaGOmaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIZaaabe aakiaacMcaaaa@3BA1@  and scalar function ϕ( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjaacIcacaWG4bWaaSbaaSqaai aaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGG SaGaamiEamaaBaaaleaacaaIZaaabeaakiaacMcaaaa@3AB5@  which satisfy

2 Ψ i x j x j = ρ 0 b i 2 ϕ x k x k = ρ 0 b i x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaeuiQdK1aaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRa amiEamaaBaaaleaacaWGQbaabeaakiabgkGi2kaadIhadaWgaaWcba GaamOAaaqabaaaaOGaeyypa0JaeyOeI0IaeqyWdi3aaSbaaSqaaiaa icdaaeqaaOGaamOyamaaBaaaleaacaWGPbaabeaakiaaygW6caaMbS +aaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaa cqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaeyOaIyRaamiEam aaBaaaleaacaWGRbaabeaaaaGccqGH9aqpcqGHsislcqaHbpGCdaWg aaWcbaGaaGimaaqabaGccaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaam iEamaaBaaaleaacaWGPbaabeaaaaa@5A55@

as well as boundary conditions

2(1+ν) E ( Ψ i + 1 4(1ν) x i ( ϕ x k Ψ k ) )= u i * on  S 1 2ν Ψ k x k n i +(12ν)( Ψ i x j + Ψ j x i ) n j x k 2 Ψ k x i x j n j + 2 ϕ x i x j n j =2(1ν) t i * on  S 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacaaIYaGaaiikaiaaig dacqGHRaWkcqaH9oGBcaGGPaaabaGaamyraaaadaqadaqaaiabfI6a znaaBaaaleaacaWGPbaabeaakiabgUcaRmaalaaabaGaaGymaaqaai aaisdacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcaaaWaaSaaaeaa cqGHciITaeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaakm aabmaabaGaeqy1dyMaeyOeI0IaamiEamaaBaaaleaacaWGRbaabeaa kiabfI6aznaaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMcaaaGaay jkaiaawMcaaiabg2da9iaadwhadaqhaaWcbaGaamyAaaqaaiaacQca aaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaab+gacaqGUbGaaeiiaiaadofadaWgaaWcba GaaGymaaqabaaakeaacaaIYaGaeqyVd42aaSaaaeaacqGHciITcqqH OoqwdaWgaaWcbaGaam4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaS qaaiaadUgaaeqaaaaakiaad6gadaWgaaWcbaGaamyAaaqabaGccqGH RaWkcaGGOaGaaGymaiabgkHiTiaaikdacqaH9oGBcaGGPaWaaeWaae aadaWcaaqaaiabgkGi2kabfI6aznaaBaaaleaacaWGPbaabeaaaOqa aiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOGaey4kaSYaaS aaaeaacqGHciITcqqHOoqwdaWgaaWcbaGaamOAaaqabaaakeaacqGH ciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaaaOGaayjkaiaawMcaai aad6gadaWgaaWcbaGaamOAaaqabaGccqGHsislcaWG4bWaaSbaaSqa aiaadUgaaeqaaOWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaa GccqqHOoqwdaWgaaWcbaGaam4AaaqabaaakeaacqGHciITcaWG4bWa aSbaaSqaaiaadMgaaeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGQb aabeaaaaGccaWGUbWaaSbaaSqaaiaadQgaaeqaaOGaey4kaSYaaSaa aeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaacqGHci ITcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOaIyRaamiEamaaBaaa leaacaWGQbaabeaaaaGccaWGUbWaaSbaaSqaaiaadQgaaeqaaOGaey ypa0JaaGOmaiaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiykaiaadsha daqhaaWcbaGaamyAaaqaaiaacQcaaaGccaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caqGVbGaaeOBaiaabccacaWGtbWaaSbaaSqa aiaaikdaaeqaaaaaaa@3531@

2.      Calculate displacements from the formula

u i = 2(1+ν) E ( Ψ i + 1 4(1ν) x i ( ϕ x k Ψ k ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiaaikdacaGGOaGaaGymaiabgUcaRiabe27aUjaa cMcaaeaacaWGfbaaamaabmaabaGaeuiQdK1aaSbaaSqaaiaadMgaae qaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGinaiaacIcacaaIXaGa eyOeI0IaeqyVd4MaaiykaaaadaWcaaqaaiabgkGi2cqaaiabgkGi2k aadIhadaWgaaWcbaGaamyAaaqabaaaaOWaaeWaaeaacqaHvpGzcqGH sislcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaeuiQdK1aaSbaaSqaai aadUgaaeqaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaGaaGPaRdaa @54C9@

3.      Calculate stresses from the formula

2(1ν) σ ij =2ν Ψ k x k δ ij +(12ν)( Ψ i x j + Ψ j x i ) x k 2 Ψ k x i x j + 2 ϕ x i x j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacaGGOaGaaGymaiabgkHiTiabe2 7aUjaacMcacqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyyp a0JaaGOmaiabe27aUnaalaaabaGaeyOaIyRaeuiQdK1aaSbaaSqaai aadUgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGRbaabeaa aaGccqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIaai ikaiaaigdacqGHsislcaaIYaGaeqyVd4MaaiykamaabmaabaWaaSaa aeaacqGHciITcqqHOoqwdaWgaaWcbaGaamyAaaqabaaakeaacqGHci ITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaakiabgUcaRmaalaaabaGa eyOaIyRaeuiQdK1aaSbaaSqaaiaadQgaaeqaaaGcbaGaeyOaIyRaam iEamaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaacqGHsisl caWG4bWaaSbaaSqaaiaadUgaaeqaaOWaaSaaaeaacqGHciITdaahaa WcbeqaaiaaikdaaaGccqqHOoqwdaWgaaWcbaGaam4Aaaqabaaakeaa cqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOaIyRaamiEam aaBaaaleaacaWGQbaabeaaaaGccqGHRaWkdaWcaaqaaiabgkGi2oaa CaaaleqabaGaaGOmaaaakiabew9aMbqaaiabgkGi2kaadIhadaWgaa WcbaGaamyAaaqabaGccqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqa aaaaaaa@7C11@

 

HEALTH WARNING: Although the displacements and stresses that solve a linear elasticity problem are unique, the Papkovich-Neuber potentials that generate a particular solution are not.   Consequently, if you find several different sets of potentials in the literature that claim to solve the same problem, don’t panic.   It is likely that they really do solve the same problem. 

 

 

 

5.4.2 Demonstration that the Papkovich-Neuber solution satisfies the governing equations

 

We need to show two things:

1.      That the displacement field satisfies the equilibrium equation (See sect 5.1.2)

1 12ν 2 u k x k x i + 2 u i x k x k =2(1+ν) ρ 0 b i E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaG ymaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gaaiaaykW6daWcaaqa aiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaam 4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGa eyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaGccqGHRaWkcaaMcS +aaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWG1bWaaSba aSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGRb aabeaakiabgkGi2kaadIhadaWgaaWcbaGaam4AaaqabaaaaOGaeyyp a0JaeyOeI0IaaGOmaiaacIcacaaIXaGaey4kaSIaeqyVd4Maaiykai abeg8aYnaaBaaaleaacaaIWaaabeaakmaalaaabaGaamOyamaaBaaa leaacaWGPbaabeaaaOqaaiaadweaaaaaaa@6212@

2.      That the stresses are related to the displacements by the elastic stress-strain equations

To show the first result, differentiate the formula relating potentials to the displacement to see that

2 u k x i x j = (1+ν) 2E(1ν) ( (34ν) 2 Ψ k x i x j 2 Ψ j x k x i 2 Ψ i x k x j x m 3 Ψ m x k x i x j + 3 ϕ x k x i x j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaamyDamaaBaaaleaacaWGRbaabeaaaOqaaiabgkGi2kaa dIhadaWgaaWcbaGaamyAaaqabaGccqGHciITcaWG4bWaaSbaaSqaai aadQgaaeqaaaaakiabg2da9maalaaabaGaaiikaiaaigdacqGHRaWk cqaH9oGBcaGGPaaabaGaaGOmaiaadweacaGGOaGaaGymaiabgkHiTi abe27aUjaacMcaaaWaaeWaaeaacaGGOaGaaG4maiabgkHiTiaaisda cqaH9oGBcaGGPaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaa GccqqHOoqwdaWgaaWcbaGaam4AaaqabaaakeaacqGHciITcaWG4bWa aSbaaSqaaiaadMgaaeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGQb aabeaaaaGccqGHsisldaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOm aaaakiabfI6aznaaBaaaleaacaWGQbaabeaaaOqaaiabgkGi2kaadI hadaWgaaWcbaGaam4AaaqabaGccqGHciITcaWG4bWaaSbaaSqaaiaa dMgaaeqaaaaakiabgkHiTmaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaeuiQdK1aaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRa amiEamaaBaaaleaacaWGRbaabeaakiabgkGi2kaadIhadaWgaaWcba GaamOAaaqabaaaaOGaeyOeI0IaamiEamaaBaaaleaacaWGTbaabeaa kmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIZaaaaOGaeuiQdK1aaS baaSqaaiaad2gaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWG RbaabeaakiabgkGi2kaadIhadaWgaaWcbaGaamyAaaqabaGccqGHci ITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaakiabgUcaRmaalaaabaGa eyOaIy7aaWbaaSqabeaacaaIZaaaaOGaeqy1dygabaGaeyOaIyRaam iEamaaBaaaleaacaWGRbaabeaakiabgkGi2kaadIhadaWgaaWcbaGa amyAaaqabaGccqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaaaO GaayjkaiaawMcaaiaaykW6aaa@9963@

Substitute this result into the governing equation to see that

1 12ν 2 u k x k x i + 2 u i x k x k = (1+ν) 2E(1ν)(12ν) ( 2(12ν) 2 Ψ k x i x k 2 Ψ i x k x k x m 3 Ψ m x k x i x k + 3 ϕ x k x i x k ) + (1+ν) 2E(1ν) ( (34ν) 2 Ψ i x k x k 2 2 Ψ k x k x i x m 3 Ψ m x k x i x k + 3 ϕ x k x i x k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaWaaSaaae aacaaIXaaabaGaaGymaiabgkHiTiaaikdacqaH9oGBaaGaaGPaRpaa laaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyDamaaBaaale aacaWGRbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaam4Aaaqa baGccqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaakiabgUcaRi aaykW6daWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadwha daWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaai aadUgaaeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGRbaabeaaaaGc cqGH9aqpdaWcaaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4Maaiykaa qaaiaaikdacaWGfbGaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaGa aiikaiaaigdacqGHsislcaaIYaGaeqyVd4Maaiykaaaadaqadaqaai aaikdacaGGOaGaaGymaiabgkHiTiaaikdacqaH9oGBcaGGPaWaaSaa aeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqqHOoqwdaWgaaWcba Gaam4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqa aOGaeyOaIyRaamiEamaaBaaaleaacaWGRbaabeaaaaGccqGHsislda WcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiabfI6aznaaBaaa leaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaam4Aaa qabaGccqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaaaakiabgkHi TiaadIhadaWgaaWcbaGaamyBaaqabaGcdaWcaaqaaiabgkGi2oaaCa aaleqabaGaaG4maaaakiabfI6aznaaBaaaleaacaWGTbaabeaaaOqa aiabgkGi2kaadIhadaWgaaWcbaGaam4AaaqabaGccqGHciITcaWG4b WaaSbaaSqaaiaadMgaaeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWG RbaabeaaaaGccqGHRaWkdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaG 4maaaakiabew9aMbqaaiabgkGi2kaadIhadaWgaaWcbaGaam4Aaaqa baGccqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOaIyRaam iEamaaBaaaleaacaWGRbaabeaaaaaakiaawIcacaGLPaaaaeaacaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabgUcaRm aalaaabaGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaaabaGaaGOm aiaadweacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcaaaWaaeWaae aacaGGOaGaaG4maiabgkHiTiaaisdacqaH9oGBcaGGPaWaaSaaaeaa cqGHciITdaahaaWcbeqaaiaaikdaaaGccqqHOoqwdaWgaaWcbaGaam yAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGa eyOaIyRaamiEamaaBaaaleaacaWGRbaabeaaaaGccqGHsislcaaIYa WaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqqHOoqwdaWg aaWcbaGaam4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadU gaaeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaGccqGH sislcaWG4bWaaSbaaSqaaiaad2gaaeqaaOWaaSaaaeaacqGHciITda ahaaWcbeqaaiaaiodaaaGccqqHOoqwdaWgaaWcbaGaamyBaaqabaaa keaacqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaeyOaIyRaam iEamaaBaaaleaacaWGPbaabeaakiabgkGi2kaadIhadaWgaaWcbaGa am4AaaqabaaaaOGaey4kaSYaaSaaaeaacqGHciITdaahaaWcbeqaai aaiodaaaGccqaHvpGzaeaacqGHciITcaWG4bWaaSbaaSqaaiaadUga aeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaakiabgkGi2k aadIhadaWgaaWcbaGaam4AaaqabaaaaaGccaGLOaGaayzkaaGaaGPa Rdaaaa@5169@

Finally, substitute the governing equations for the potentials

2 Ψ i x j x j = ρ 0 b i 2 ϕ x k x k = ρ 0 b i x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaeuiQdK1aaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRa amiEamaaBaaaleaacaWGQbaabeaakiabgkGi2kaadIhadaWgaaWcba GaamOAaaqabaaaaOGaeyypa0JaeyOeI0IaeqyWdi3aaSbaaSqaaiaa icdaaeqaaOGaamOyamaaBaaaleaacaWGPbaabeaakiaaygW6caaMbS +aaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaa cqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaeyOaIyRaamiEam aaBaaaleaacaWGRbaabeaaaaGccqGH9aqpcqGHsislcqaHbpGCdaWg aaWcbaGaaGimaaqabaGccaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaam iEamaaBaaaleaacaWGPbaabeaaaaa@5A55@

and simplify the result to verify that the governing equation is indeed satisfied. The second result can be derived by substituting the formula for displacement into the elastic stress-strain equations and simplifying.

 

 

 

 

5.4.3 Point force in an infinite solid

 

The displacements and stresses induced by a point force P= P 1 e 1 + P 2 e 2 + P 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHqbGaeyypa0JaamiuamaaBaaale aacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWk caWGqbWaaSbaaSqaaiaaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYa aabeaakiabgUcaRiaadcfadaWgaaWcbaGaaG4maaqabaGccaWHLbWa aSbaaSqaaiaaiodaaeqaaaaa@4154@  acting at the origin of a large (infinite) elastic solid with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@  are generated by the Papkovich-Neuber potentials

Ψ i = P i 4πR ϕ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfI6aznaaBaaaleaacaWGPbaabeaaki abg2da9maalaaabaGaamiuamaaBaaaleaacaWGPbaabeaaaOqaaiaa isdacqaHapaCcaWGsbaaaiaaykW6caaMcSUaaGPaRlaaykW6caaMcS UaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6 caaMcSUaeqy1dyMaeyypa0JaaGimaaaa@5265@

where R= x i x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacqGH9aqpdaGcaaqaaiaadIhada WgaaWcbaGaamyAaaqabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaaqa baaaaa@368D@ . The displacements, strains and stresses follow as

u i = (1+ν) 8πE(1ν)R { P k x k x i R 2 +(34ν) P i } ε ij = (1+ν) 8πE(1ν) R 2 { 3 P k x k x i x j R 3 P k x k δ ij R +(12ν) P i x j + P j x i R } σ ij = 1 8π(1ν) R 2 { 3 P k x k x i x j R 3 +(12ν) P i x j + P j x i δ ij P k x k R } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyDamaaBaaaleaacaWGPbaabe aakiabg2da9maalaaabaGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGG PaaabaGaaGioaiabec8aWjaadweacaGGOaGaaGymaiabgkHiTiabe2 7aUjaacMcacaWGsbaaamaacmaabaWaaSaaaeaacaWGqbWaaSbaaSqa aiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaakiaadIhada WgaaWcbaGaamyAaaqabaaakeaacaWGsbWaaWbaaSqabeaacaaIYaaa aaaakiabgUcaRiaacIcacaaIZaGaeyOeI0IaaGinaiabe27aUjaacM cacaWGqbWaaSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzFaaaabaGa eqyTdu2aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9maalaaaba GaeyOeI0IaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaaabaGaaGio aiabec8aWjaadweacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcaca WGsbWaaWbaaSqabeaacaaIYaaaaaaakmaaceaabaWaaSaaaeaacaaI ZaGaamiuamaaBaaaleaacaWGRbaabeaakiaadIhadaWgaaWcbaGaam 4AaaqabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaamiEamaaBaaa leaacaWGQbaabeaaaOqaaiaadkfadaahaaWcbeqaaiaaiodaaaaaaO GaeyOeI0YaaSaaaeaacaWGqbWaaSbaaSqaaiaadUgaaeqaaOGaamiE amaaBaaaleaacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGPbGaam OAaaqabaaakeaacaWGsbaaaaGaay5EaaGaey4kaSIaaiikaiaaigda cqGHsislcaaIYaGaeqyVd4MaaiykamaaciaabaWaaSaaaeaacaWGqb WaaSbaaSqaaiaadMgaaeqaaOGaamiEamaaBaaaleaacaWGQbaabeaa kiabgUcaRiaadcfadaWgaaWcbaGaamOAaaqabaGccaWG4bWaaSbaaS qaaiaadMgaaeqaaaGcbaGaamOuaaaaaiaaw2haaaqaaiabeo8aZnaa DaaaleaacaWGPbGaamOAaaqaaaaakiabg2da9maalaaabaGaeyOeI0 IaaGymaaqaaiaaiIdacqaHapaCcaGGOaGaaGymaiabgkHiTiabe27a UjaacMcacaWGsbWaaWbaaSqabeaacaaIYaaaaaaakmaaceaabaWaaS aaaeaacaaIZaGaamiuamaaBaaaleaacaWGRbaabeaakiaadIhadaWg aaWcbaGaam4AaaqabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaam iEamaaBaaaleaacaWGQbaabeaaaOqaaiaadkfadaahaaWcbeqaaiaa iodaaaaaaaGccaGL7baadaGacaqaaiabgUcaRiaacIcacaaIXaGaey OeI0IaaGOmaiabe27aUjaacMcadaWcaaqaaiaadcfadaWgaaWcbaGa amyAaaqabaGccaWG4bWaaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaam iuamaaBaaaleaacaWGQbaabeaakiaadIhadaWgaaWcbaGaamyAaaqa baGccqGHsislcqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaam iuamaaBaaaleaacaWGRbaabeaakiaadIhadaWgaaWcbaGaam4Aaaqa baaakeaacaWGsbaaaaGaayzFaaaaaaa@C50F@

 

 

 

 

5.4.4 Point force normal to the surface of an infinite half-space

 

The displacements and stresses induced by a point force P=P e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHqbGaeyypa0Jaamiuaiaahwgada WgaaWcbaGaaG4maaqabaaaaa@3751@  acting normal to the surface of a semi-infinite solid with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@  are generated by the Papkovich-Neuber potentials

Ψ i = (1ν) δ i3 πR ϕ= (12ν)(1ν) π log(R+ x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfI6aznaaBaaaleaacaWGPbaabeaaki abg2da9maalaaabaGaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaGa eqiTdq2aaSbaaSqaaiaadMgacaaIZaaabeaaaOqaaiabec8aWjaadk faaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabew9aMjabg2da9i abgkHiTmaalaaabaGaaiikaiaaigdacqGHsislcaaIYaGaeqyVd4Ma aiykaiaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiykaaqaaiabec8aWb aaciGGSbGaai4BaiaacEgacaGGOaGaamOuaiabgUcaRiaadIhadaWg aaWcbaGaaG4maaqabaGccaGGPaaaaa@6A9B@

where R= x k x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacqGH9aqpdaGcaaqaaiaadIhada WgaaWcbaGaam4AaaqabaGccaWG4bWaaSbaaSqaaiaadUgaaeqaaaqa baaaaa@3691@

 

The displacements and stresses follow as

u i = (1+ν)P 2πE { x 3 x i R 3 +(34ν) δ i3 R (12ν) R+ x 3 ( δ 3i + x i R ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4Maaiykaiaa dcfaaeaacaaIYaGaeqiWdaNaamyraaaadaGadaqaamaalaaabaGaam iEamaaBaaaleaacaaIZaaabeaakiaadIhadaWgaaWcbaGaamyAaaqa baaakeaacaWGsbWaaWbaaSqabeaacaaIZaaaaaaakiabgUcaRiaacI cacaaIZaGaeyOeI0IaaGinaiabe27aUjaacMcadaWcaaqaaiabes7a KnaaBaaaleaacaWGPbGaaG4maaqabaaakeaacaWGsbaaaiabgkHiTm aalaaabaGaaiikaiaaigdacqGHsislcaaIYaGaeqyVd4Maaiykaaqa aiaadkfacqGHRaWkcaWG4bWaaSbaaSqaaiaaiodaaeqaaaaakmaabm aabaGaeqiTdq2aaSbaaSqaaiaaiodacaWGPbaabeaakiabgUcaRmaa laaabaGaamiEamaaBaaaleaacaWGPbaabeaaaOqaaiaadkfaaaaaca GLOaGaayzkaaaacaGL7bGaayzFaaaaaa@625A@

σ ij = P 2π R 2 { 3 x i x j x 3 R 3 + (12ν)(2R+ x 3 ) R (R+ x 3 ) 2 ( x i x j + δ ij x 3 2 x 3 ( δ i3 x j + δ j3 x i ) ) + (12ν) R 2 (R+ x 3 ) 2 ( δ i3 δ j3 δ ij ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaadcfaaeaacaaIYaGaeqiWdaNaamOu amaaCaaaleqabaGaaGOmaaaaaaGcdaGabaqaaiabgkHiTiaaiodada WcaaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccaWG4bWaaSbaaSqa aiaadQgaaeqaaOGaamiEamaaBaaaleaacaaIZaaabeaaaOqaaiaadk fadaahaaWcbeqaaiaaiodaaaaaaaGccaGL7baacqGHRaWkdaWcaaqa aiaacIcacaaIXaGaeyOeI0IaaGOmaiabe27aUjaacMcacaGGOaGaaG OmaiaadkfacqGHRaWkcaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiyk aaqaaiaadkfacaGGOaGaamOuaiabgUcaRiaadIhadaWgaaWcbaGaaG 4maaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaaaakmaabmaabaGa amiEamaaBaaaleaacaWGPbaabeaakiaadIhadaWgaaWcbaGaamOAaa qabaGccqGHRaWkcqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaOGa amiEamaaDaaaleaacaaIZaaabaGaaGOmaaaakiabgkHiTiaadIhada WgaaWcbaGaaG4maaqabaGcdaqadaqaaiabes7aKnaaBaaaleaacaWG PbGaaG4maaqabaGccaWG4bWaaSbaaSqaaiaadQgaaeqaaOGaey4kaS IaeqiTdq2aaSbaaSqaaiaadQgacaaIZaaabeaakiaadIhadaWgaaWc baGaamyAaaqabaaakiaawIcacaGLPaaaaiaawIcacaGLPaaadaGaca qaaiabgUcaRmaalaaabaGaaiikaiaaigdacqGHsislcaaIYaGaeqyV d4MaaiykaiaadkfadaahaaWcbeqaaiaaikdaaaaakeaacaGGOaGaam OuaiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaWaaWba aSqabeaacaaIYaaaaaaakmaabmaabaGaeqiTdq2aaSbaaSqaaiaadM gacaaIZaaabeaakiabes7aKnaaBaaaleaacaWGQbGaaG4maaqabaGc cqGHsislcqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOa GaayzkaaaacaGL9baaaaa@9258@

 

 

 

5.4.5 Point force tangent to the surface of an infinite half-space

 

The displacements and stresses induced by a point force P=P e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHqbGaeyypa0Jaamiuaiaahwgada WgaaWcbaGaaGymaaqabaaaaa@374F@  acting tangent to the surface of a semi-infinite solid with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@  are generated by the Papkovich-Neuber potentials

 

Ψ i = P 2π(R+ x 3 ) ( δ i1 x 2 2 δ i1 R(R+ x 3 ) + x 1 x 2 δ i2 R(R+ x 3 ) +2(1ν) x 1 δ i3 R ) ϕ= P 2π ( 1+4 (1ν) 2 ) x 1 R+ x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeuiQdK1aaSbaaSqaaiaadMgaae qaaOGaeyypa0ZaaSaaaeaacaWGqbaabaGaaGOmaiabec8aWjaacIca caWGsbGaey4kaSIaamiEamaaBaaaleaacaaIZaaabeaakiaacMcaaa WaaeWaaeaacqaH0oazdaWgaaWcbaGaamyAaiaaigdaaeqaaOGaeyOe I0YaaSaaaeaacaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaOGaeq iTdq2aaSbaaSqaaiaadMgacaaIXaaabeaaaOqaaiaadkfacaGGOaGa amOuaiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaaaai abgUcaRmaalaaabaGaamiEamaaBaaaleaacaaIXaaabeaakiaadIha daWgaaWcbaGaaGOmaaqabaGccqaH0oazdaWgaaWcbaGaamyAaiaaik daaeqaaaGcbaGaamOuaiaacIcacaWGsbGaey4kaSIaamiEamaaBaaa leaacaaIZaaabeaakiaacMcaaaGaey4kaSIaaGOmaiaacIcacaaIXa GaeyOeI0IaeqyVd4MaaiykamaalaaabaGaamiEamaaBaaaleaacaaI Xaaabeaakiabes7aKnaaBaaaleaacaWGPbGaaG4maaqabaaakeaaca WGsbaaaaGaayjkaiaawMcaaiaaykW7aeaacaaMc8Uaeqy1dyMaeyyp a0ZaaSaaaeaacaWGqbaabaGaaGOmaiabec8aWbaadaqadaqaaiaaig dacqGHRaWkcaaI0aGaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaWa aWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaSaaaeaacaWG4b WaaSbaaSqaaiaaigdaaeqaaaGcbaGaamOuaiabgUcaRiaadIhadaWg aaWcbaGaaG4maaqabaaaaaaaaa@81E9@

The displacements and stresses can be calculated from these potentials as

u i = P(1+ν) 2πER { δ i1 + x 1 x i R 2 +(12ν)( R δ i1 R+ x 3 R x 1 δ i3 (R+ x 3 ) 2 x 1 x i (R+ x 3 ) 2 ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiaadcfacaGGOaGaaGymaiabgUcaRiabe27aUjaa cMcaaeaacaaIYaGaeqiWdaNaamyraiaadkfaaaWaaiWaaeaacqaH0o azdaWgaaWcbaGaamyAaiaaigdaaeqaaOGaey4kaSYaaSaaaeaacaWG 4bWaaSbaaSqaaiaaigdaaeqaaOGaamiEamaaBaaaleaacaWGPbaabe aaaOqaaiaadkfadaahaaWcbeqaaiaaikdaaaaaaOGaey4kaSIaaiik aiaaigdacqGHsislcaaIYaGaeqyVd4MaaiykamaabmaabaWaaSaaae aacaWGsbGaeqiTdq2aaSbaaSqaaiaadMgacaaIXaaabeaaaOqaaiaa dkfacqGHRaWkcaWG4bWaaSbaaSqaaiaaiodaaeqaaaaakiabgkHiTm aalaaabaGaamOuaiaadIhadaWgaaWcbaGaaGymaaqabaGccqaH0oaz daWgaaWcbaGaamyAaiaaiodaaeqaaaGcbaGaaiikaiaadkfacqGHRa WkcaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykamaaCaaaleqabaGa aGOmaaaaaaGccqGHsisldaWcaaqaaiaadIhadaWgaaWcbaGaaGymaa qabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaaiikaiaadkfa cqGHRaWkcaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykamaaCaaale qabaGaaGOmaaaaaaaakiaawIcacaGLPaaaaiaawUhacaGL9baaaaa@71EB@

σ ij = P 2π [ 3 x 1 x i x j R 5 +(12ν){ x 1 x i x j R 3 (R+ x 3 ) 2 + 2 x 1 x i x j R 2 (R+ x 3 ) 3 }(1 δ i3 )(1 δ j3 ) +(12ν) { x 1 x 3 (2R+ x 3 ) R 3 (R+ x 3 ) 2 ( δ i1 δ j1 + δ i2 δ j2 ) 2 x 1 δ i1 δ j1 R (R+ x 3 ) 2 x 2 ( δ i1 δ j2 + δ i2 δ j1 ) R (R+ x 3 ) 2 } ](no sum on i or j) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aaSbaaSqaaiaadMgaca WGQbaabeaakiabg2da9maalaaabaGaamiuaaqaaiaaikdacqaHapaC aaWaamqaaeaacqGHsisldaWcaaqaaiaaiodacaWG4bWaaSbaaSqaai aaigdaaeqaaOGaamiEamaaBaaaleaacaWGPbaabeaakiaadIhadaWg aaWcbaGaamOAaaqabaaakeaacaWGsbWaaWbaaSqabeaacaaI1aaaaa aakiabgUcaRiaacIcacaaIXaGaeyOeI0IaaGOmaiabe27aUjaacMca daGadaqaamaalaaabaGaamiEamaaBaaaleaacaaIXaaabeaakiaadI hadaWgaaWcbaGaamyAaaqabaGccaWG4bWaaSbaaSqaaiaadQgaaeqa aaGcbaGaamOuamaaCaaaleqabaGaaG4maaaakiaacIcacaWGsbGaey 4kaSIaamiEamaaBaaaleaacaaIZaaabeaakiaacMcadaahaaWcbeqa aiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaaIYaGaamiEamaaBaaale aacaaIXaaabeaakiaadIhadaWgaaWcbaGaamyAaaqabaGccaWG4bWa aSbaaSqaaiaadQgaaeqaaaGcbaGaamOuamaaCaaaleqabaGaaGOmaa aakiaacIcacaWGsbGaey4kaSIaamiEamaaBaaaleaacaaIZaaabeaa kiaacMcadaahaaWcbeqaaiaaiodaaaaaaaGccaGL7bGaayzFaaGaai ikaiaaigdacqGHsislcqaH0oazdaWgaaWcbaGaamyAaiaaiodaaeqa aOGaaiykaiaacIcacaaIXaGaeyOeI0IaeqiTdq2aaSbaaSqaaiaadQ gacaaIZaaabeaakiaacMcaaiaawUfaaaqaaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaey4kaSIaaiikaiaaig dacqGHsislcaaIYaGaeqyVd4MaaiykamaadiaabaWaaiWaaeaadaWc aaqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaWG4bWaaSbaaSqaai aaiodaaeqaaOGaaiikaiaaikdacaWGsbGaey4kaSIaamiEamaaBaaa leaacaaIZaaabeaakiaacMcaaeaacaWGsbWaaWbaaSqabeaacaaIZa aaaOGaaiikaiaadkfacqGHRaWkcaWG4bWaaSbaaSqaaiaaiodaaeqa aOGaaiykamaaCaaaleqabaGaaGOmaaaaaaGcdaqadaqaaiabes7aKn aaBaaaleaacaWGPbGaaGymaaqabaGccqaH0oazdaWgaaWcbaGaamOA aiaaigdaaeqaaOGaey4kaSIaeqiTdq2aaSbaaSqaaiaadMgacaaIYa aabeaakiabes7aKnaaBaaaleaacaWGQbGaaGOmaaqabaaakiaawIca caGLPaaacqGHsisldaWcaaqaaiaaikdacaWG4bWaaSbaaSqaaiaaig daaeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaaIXaaabeaakiabes7a KnaaBaaaleaacaWGQbGaaGymaaqabaaakeaacaWGsbGaaiikaiaadk facqGHRaWkcaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykamaaCaaa leqabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaadIhadaWgaaWcba GaaGOmaaqabaGcdaqadaqaaiabes7aKnaaBaaaleaacaWGPbGaaGym aaqabaGccqaH0oazdaWgaaWcbaGaamOAaiaaikdaaeqaaOGaey4kaS IaeqiTdq2aaSbaaSqaaiaadMgacaaIYaaabeaakiabes7aKnaaBaaa leaacaWGQbGaaGymaaqabaaakiaawIcacaGLPaaaaeaacaWGsbGaai ikaiaadkfacqGHRaWkcaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiyk amaaCaaaleqabaGaaGOmaaaaaaaakiaawUhacaGL9baacaaMc8oaca GLDbaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caGGOaGa aeOBaiaab+gacaqGGaGaae4CaiaabwhacaqGTbGaaeiiaiaab+gaca qGUbGaaeiiaiaadMgacaqGGaGaae4BaiaabkhacaqGGaGaamOAaiaa cMcaaaaa@F3CB@

 

 

 

5.4.6 The Eshelby Inclusion Problem

 

The Eshelby problem (Eshelby, Proc Roy Soc A 241 p. 376) is posed as follows:

1.      Consider an infinite, isotropic, linear elastic solid, with (homogeneous) Young’s modulus and Poisson’s ratio E,ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaGGSaGaeqyVd4gaaa@339A@ .

2.      The solid is initially stress free, with displacements, strains and stresses u i = ε ij = σ ij =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0Ja eq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iaaicdaaa a@3DE2@ .

3.      Some unspecified external agency then induces a uniform `transformation strain ε ij T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaWGPbGaamOAaa qaaiaadsfaaaaaaa@34F2@  inside an ellipsoidal region, with semi-axes ( a 1 , a 2 , a 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWGHbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaadggadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamyy amaaBaaaleaacaaIZaaabeaakiaacMcaaaa@38A9@  centered at the origin.  The `transformation strain’ can be visualized as an anisotropic thermal expansion MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  if the ellipsoidal region were separated from the surrounding elastic solid, it would be stress free, and would change its shape according to the strain tensor ε ij T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaWGPbGaamOAaa qaaiaadsfaaaaaaa@34F2@ .

4.      Because the ellipsoid is encapsulated within the surrounding elastic solid, stress, strain and displacement fields are induced throughout the elastic solid.  These fields must be defined carefully because the initial configuration for the solid could be chosen in a number of different ways.  In the following, u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaaaaa@327C@  will denote the displacement of a material particle from the initial, unstressed configuration, as the transformation strain is introduced.   The total strain is defined as

ε ij =( u i / x j + u j / x i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaqadaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyA aaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaki abgUcaRiabgkGi2kaadwhadaWgaaWcbaGaamOAaaqabaGccaGGVaGa eyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaai aac+cacaaIYaaaaa@4880@

Inside the ellipsoid, the total strain consists of the transformation strain  (which does not induce stress); together with an additional elastic strain ε ij = ε ij T + ε ij e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWG ubaaaOGaey4kaSIaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaam yzaaaaaaa@3F39@ . Outside the ellipsoid, ε ij = ε ij e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWG Lbaaaaaa@39C3@ . The stress in the solid is related to the elastic part of the strain by the usual linear elastic equations

σ ij = E 1+ν { ε ij e + ν 12ν ε kk e δ ij } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaadweaaeaacaaIXaGaey4kaSIaeqyV d4gaamaacmaabaGaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaam yzaaaakiabgUcaRmaalaaabaGaeqyVd4gabaGaaGymaiabgkHiTiaa ikdacqaH9oGBaaGaeqyTdu2aa0baaSqaaiaadUgacaWGRbaabaGaam yzaaaakiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaaakiaawUha caGL9baaaaa@4F6E@

 

The Eshelby solution gives full expressions for these fields.  It has proved to be one of the most important solutions in all of linear elasticity: it is of some interest in its own right, because it provides some insight into the mechanics of phase transformations in crystals.  More importantly, a number of very important boundary value problems can be solved by manipulating the Eshelby solution.  These include (i) the solution for an ellipsoidal inclusion embedded within an elastically mismatched matrix; (ii) the solution for an ellipsoidal cavity in an elastic solid; (iii) solutions for circular and elliptical cracks in an elastic solid.  In addition, the Eshelby solution is used extensively in theories that provide estimates of elastic properties of composite materials.

 

The displacement field is generated by Papkovich-Neuber potentials

Ψ i ( x k )= S p ji T n j (ξ) 4πR(x,ξ) dA(ξ)ϕ( x k )= S ξ i p ji T n j (ξ) 4πR(x,ξ) dA(ξ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfI6aznaaBaaaleaacaWGPbaabeaaki aacIcacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiykaiabg2da9maa pefabaWaaSaaaeaacaWGWbWaa0baaSqaaiaadQgacaWGPbaabaGaam ivaaaakiaad6gadaWgaaWcbaGaamOAaaqabaGccaGGOaGaaCOVdiaa cMcaaeaacaaI0aGaeqiWdaNaamOuaiaacIcacaWH4bGaaiilaiaah6 7acaGGPaaaaaWcbaGaam4uaaqab0Gaey4kIipakiaadsgacaWGbbGa aiikaiaah67acaGGPaGaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6ca aMcSUaaGPaRlaaykW6caaMcSUaeqy1dyMaaiikaiaadIhadaWgaaWc baGaam4AaaqabaGccaGGPaGaeyypa0Zaa8quaeaadaWcaaqaaiabe6 7a4naaBaaaleaacaWGPbaabeaakiaadchadaqhaaWcbaGaamOAaiaa dMgaaeaacaWGubaaaOGaamOBamaaBaaaleaacaWGQbaabeaakiaacI cacaWH+oGaaiykaaqaaiaaisdacqaHapaCcaWGsbGaaiikaiaahIha caGGSaGaaCOVdiaacMcaaaaaleaacaWGtbaabeqdcqGHRiI8aOGaam izaiaadgeacaGGOaGaaCOVdiaacMcaaaa@7E2E@

where the integral is taken over the surface of the ellipsoid, n j (ξ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gadaWgaaWcbaGaamOAaaqabaGcca GGOaGaaCOVdiaacMcaaaa@3523@  denotes the components of a unit vector perpendicular to the surface of the ellipsoid (pointing outwards); R= ( x k ξ k )( x k ξ k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacqGH9aqpdaGcaaqaamaabmaaba GaamiEamaaBaaaleaacaWGRbaabeaakiabgkHiTiabe67a4naaBaaa leaacaWGRbaabeaaaOGaayjkaiaawMcaamaabmaabaGaamiEamaaBa aaleaacaWGRbaabeaakiabgkHiTiabe67a4naaBaaaleaacaWGRbaa beaaaOGaayjkaiaawMcaaaWcbeaaaaa@4164@ , and

p ij T = E 1+ν { ε ij T + ν 12ν ε kk T δ ij } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaqhaaWcbaGaamyAaiaadQgaae aacaWGubaaaOGaeyypa0ZaaSaaaeaacaWGfbaabaGaaGymaiabgUca Riabe27aUbaadaGadaqaaiabew7aLnaaDaaaleaacaWGPbGaamOAaa qaaiaadsfaaaGccqGHRaWkdaWcaaqaaiabe27aUbqaaiaaigdacqGH sislcaaIYaGaeqyVd4gaaiabew7aLnaaDaaaleaacaWGRbGaam4Aaa qaaiaadsfaaaGccqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaaGc caGL7bGaayzFaaaaaa@4F58@

is the transformation stress (i.e. the stress that would be induced by applying an elastic strain to the inclusion that is equal to the transformation strain).  The stresses outside the inclusion can be calculated using the standard Papkovich-Neuber representation given in Section 5.4.1.  To calculate stresses inside the inclusion, the formula must be modified to account for the transformation strain, which gives

2(1ν) σ ij =2(1ν) p ij T +2ν Ψ k x k δ ij +(12ν)( Ψ i x j + Ψ j x i ) x k 2 Ψ k x i x j + 2 ϕ x i x j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacaGGOaGaaGymaiabgkHiTiabe2 7aUjaacMcacqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyyp a0JaeyOeI0IaaGOmaiaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiykai aadchadaqhaaWcbaGaamyAaiaadQgaaeaacaWGubaaaOGaey4kaSIa aGOmaiabe27aUnaalaaabaGaeyOaIyRaeuiQdK1aaSbaaSqaaiaadU gaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGRbaabeaaaaGc cqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIaaiikai aaigdacqGHsislcaaIYaGaeqyVd4MaaiykamaabmaabaWaaSaaaeaa cqGHciITcqqHOoqwdaWgaaWcbaGaamyAaaqabaaakeaacqGHciITca WG4bWaaSbaaSqaaiaadQgaaeqaaaaakiabgUcaRmaalaaabaGaeyOa IyRaeuiQdK1aaSbaaSqaaiaadQgaaeqaaaGcbaGaeyOaIyRaamiEam aaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaacqGHsislcaWG 4bWaaSbaaSqaaiaadUgaaeqaaOWaaSaaaeaacqGHciITdaahaaWcbe qaaiaaikdaaaGccqqHOoqwdaWgaaWcbaGaam4AaaqabaaakeaacqGH ciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOaIyRaamiEamaaBa aaleaacaWGQbaabeaaaaGccqGHRaWkdaWcaaqaaiabgkGi2oaaCaaa leqabaGaaGOmaaaakiabew9aMbqaaiabgkGi2kaadIhadaWgaaWcba GaamyAaaqabaGccqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaa aaa@8737@

For the general ellipsoid, the expressions for displacement and stress can be reduced to elliptic integrals, with the results

 

Solution inside the ellipsoid: Remarkably, it turns out that the stresses and strains are uniform inside the ellipsoid.  The displacements, strains and stresses can be expressed as:

(i) Displacement u i =( S ijkl + Π ijkl ) ε kl T x j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaqadaqaaiaadofadaqhaaWcbaGaamyAaiaadQgacaWGRbGa amiBaaqaaiabgEHiQaaakiabgUcaRiabfc6aqnaaBaaaleaacaWGPb GaamOAaiaadUgacaWGSbaabeaaaOGaayjkaiaawMcaaiabew7aLnaa DaaaleaacaWGRbGaamiBaaqaaiaadsfaaaGccaWG4bWaaSbaaSqaai aadQgaaeqaaaaa@47D4@

(ii) Strain ε ij = ε ij e + ε ij T = S ijkl ε kl T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWG LbaaaOGaey4kaSIaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaam ivaaaakiabg2da9iaadofadaqhaaWcbaGaamyAaiaadQgacaWGRbGa amiBaaqaaiabgEHiQaaakiabew7aLnaaDaaaleaacaWGRbGaamiBaa qaaiaadsfaaaaaaa@4A92@

(iii) Stress σ ij = E 1+ν ( S ijkl * ε kl T + ν 12ν δ ij S ppkl * ε kl T ) p ij T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaadweaaeaacaaIXaGaey4kaSIaeqyV d4gaamaabmaabaGaam4uamaaDaaaleaacaWGPbGaamOAaiaadUgaca WGSbaabaGaaiOkaaaakiabew7aLnaaDaaaleaacaWGRbGaamiBaaqa aiaadsfaaaGccqGHRaWkdaWcaaqaaiabe27aUbqaaiaaigdacqGHsi slcaaIYaGaeqyVd4gaaiabes7aKnaaBaaaleaacaWGPbGaamOAaaqa baGccaWGtbWaa0baaSqaaiaadchacaWGWbGaam4AaiaadYgaaeaaca GGQaaaaOGaeqyTdu2aa0baaSqaaiaadUgacaWGSbaabaGaamivaaaa aOGaayjkaiaawMcaaiabgkHiTiaadchadaqhaaWcbaGaamyAaiaadQ gaaeaacaWGubaaaaaa@5E71@

Here, S ijkl * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaqhaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqaaiaacQcaaaaaaa@35D9@  is a constant called the `Eshelby Tensor,’ and Π ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHGoaudaWgaaWcbaGaamyAaiaadQ gacaWGRbGaamiBaaqabaaaaa@382E@  is a second (anonymous) constant tensor. These tensors can be calculated as follows. Choose the coordinate system so that a 1 > a 2 > a 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaaGymaaqabaGccq GH+aGpcaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaeyOpa4Jaamyyamaa BaaaleaacaaIZaaabeaaaaa@37F5@ . Define

I 1 = 4π a 1 a 2 a 3 ( a 1 2 a 2 2 ) ( a 1 2 a 3 2 ) ( F(θ,k)E(θ,k) ) I 3 = 4π a 1 a 2 a 3 ( a 2 2 a 3 2 ) ( a 1 2 a 3 2 ) { a 2 ( a 1 2 a 3 2 ) 1/2 a 1 a 3 E(θ,k) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGymaaqabaGccq GH9aqpdaWcaaqaaiaaisdacqaHapaCcaaMcSUaamyyamaaBaaaleaa caaIXaaabeaakiaadggadaWgaaWcbaGaaGOmaaqabaGccaWGHbWaaS baaSqaaiaaiodaaeqaaaGcbaGaaiikaiaadggadaqhaaWcbaGaaGym aaqaaiaaikdaaaGccqGHsislcaWGHbWaa0baaSqaaiaaikdaaeaaca aIYaaaaOGaaiykamaakaaabaGaaiikaiaadggadaqhaaWcbaGaaGym aaqaaiaaikdaaaGccqGHsislcaWGHbWaa0baaSqaaiaaiodaaeaaca aIYaaaaOGaaiykaaWcbeaaaaGcdaqadaqaaiaadAeacaGGOaGaeqiU deNaaiilaiaadUgacaGGPaGaeyOeI0IaamyraiaacIcacqaH4oqCca GGSaGaam4AaiaacMcaaiaawIcacaGLPaaacaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaamysamaaBaaaleaacaaIZaaabe aakiabg2da9maalaaabaGaaGinaiabec8aWjaaykW6caWGHbWaaSba aSqaaiaaigdaaeqaaOGaamyyamaaBaaaleaacaaIYaaabeaakiaadg gadaWgaaWcbaGaaG4maaqabaaakeaacaGGOaGaamyyamaaDaaaleaa caaIYaaabaGaaGOmaaaakiabgkHiTiaadggadaqhaaWcbaGaaG4maa qaaiaaikdaaaGccaGGPaWaaOaaaeaacaGGOaGaamyyamaaDaaaleaa caaIXaaabaGaaGOmaaaakiabgkHiTiaadggadaqhaaWcbaGaaG4maa qaaiaaikdaaaGccaGGPaaaleqaaaaakmaacmaabaWaaSaaaeaacaWG HbWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaadggadaqhaaWcbaGaaG ymaaqaaiaaikdaaaGccqGHsislcaWGHbWaa0baaSqaaiaaiodaaeaa caaIYaaaaOGaaiykamaaCaaaleqabaGaaGymaiaac+cacaaIYaaaaa GcbaGaamyyamaaBaaaleaacaaIXaaabeaakiaadggadaWgaaWcbaGa aG4maaqabaaaaOGaeyOeI0IaamyraiaacIcacqaH4oqCcaGGSaGaam 4AaiaacMcaaiaawUhacaGL9baaaaa@9767@

I 2 =4π I 1 I 3 I ij =( I j I i )/3( a i 2 a j 2 )(ij,no sum on i,j) I 11 =4π/3 a 2 I 12 I 13 I 22 =4π/3 a 2 I 23 I 21 I 33 =4π/3 a 2 I 31 I 32 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadMeadaWgaaWcbaGaaGOmaa qabaGccqGH9aqpcaaI0aGaeqiWdaNaeyOeI0IaamysamaaBaaaleaa caaIXaaabeaakiabgkHiTiaadMeadaWgaaWcbaGaaG4maaqabaGcca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadMeadaWgaaWcba GaamyAaiaadQgaaeqaaOGaeyypa0JaaiikaiaadMeadaWgaaWcbaGa amOAaaqabaGccqGHsislcaWGjbWaaSbaaSqaaiaadMgaaeqaaOGaai ykaiaac+cacaaIZaGaaiikaiaadggadaqhaaWcbaGaamyAaaqaaiaa ikdaaaGccqGHsislcaWGHbWaa0baaSqaaiaadQgaaeaacaaIYaaaaO GaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaacIcacaWGPbGaeyiyIK RaamOAaiaacYcacaaMc8UaaGPaVlaab6gacaqGVbGaaeiiaiaaboha caqG1bGaaeyBaiaabccacaqGVbGaaeOBaiaabccacaWGPbGaaiilai aadQgacaGGPaaabaGaamysamaaBaaaleaacaaIXaGaaGymaaqabaGc cqGH9aqpcaaI0aGaeqiWdaNaai4laiaaiodacaWGHbWaaWbaaSqabe aacaaIYaaaaOGaeyOeI0IaamysamaaBaaaleaacaaIXaGaaGOmaaqa baGccqGHsislcaWGjbWaaSbaaSqaaiaaigdacaaIZaaabeaakiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaamys amaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH9aqpcaaI0aGaeqiWda Naai4laiaaiodacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Ia amysamaaBaaaleaacaaIYaGaaG4maaqabaGccqGHsislcaWGjbWaaS baaSqaaiaaikdacaaIXaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaamysamaaBaaaleaacaaIZaGaaG 4maaqabaGccqGH9aqpcaaI0aGaeqiWdaNaai4laiaaiodacaWGHbWa aWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamysamaaBaaaleaacaaIZa GaaGymaaqabaGccqGHsislcaWGjbWaaSbaaSqaaiaaiodacaaIYaaa beaaaaaa@D5A7@

where θ= sin 1 (1 a 3 2 / a 1 2 ) k 2 =( a 1 2 a 2 2 )/( a 1 2 a 3 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iGacohacaGGPbGaai OBamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaakaaabaGaaiikaiaa igdacqGHsislcaWGHbWaa0baaSqaaiaaiodaaeaacaaIYaaaaOGaai 4laiaadggadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccaGGPaaaleqa aOGaaGPaRlaaykW6caaMcSUaaGPaVlaaykW6caWGRbWaaWbaaSqabe aacaaIYaaaaOGaeyypa0JaaiikaiaadggadaqhaaWcbaGaaGymaaqa aiaaikdaaaGccqGHsislcaWGHbWaa0baaSqaaiaaikdaaeaacaaIYa aaaOGaaiykaiaac+cacaGGOaGaamyyamaaDaaaleaacaaIXaaabaGa aGOmaaaakiabgkHiTiaadggadaqhaaWcbaGaaG4maaqaaiaaikdaaa GccaGGPaaaaa@5AFF@  and

F(θ,k)= 0 θ dw (1 k 2 sin 2 w) 1/2 E(θ,k)= 0 θ (1 k 2 sin 2 w) 1/2 dw MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeacaGGOaGaeqiUdeNaaiilaiaadU gacaGGPaGaeyypa0Zaa8qCaeaadaWcaaqaaiaadsgacaWG3baabaGa aiikaiaaigdacqGHsislcaWGRbWaaWbaaSqabeaacaaIYaaaaOGaci 4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaam4DaiaacMca daahaaWcbeqaaiaaigdacaGGVaGaaGOmaaaaaaaabaGaaGimaaqaai abeI7aXbqdcqGHRiI8aOGaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6 caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRl aaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUa amyraiaacIcacqaH4oqCcaGGSaGaam4AaiaacMcacqGH9aqpdaWdXb qaaiaacIcacaaIXaGaeyOeI0Iaam4AamaaCaaaleqabaGaaGOmaaaa kiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiaadEhaca GGPaWaaWbaaSqabeaacaaIXaGaai4laiaaikdaaaGccaWGKbGaam4D aaWcbaGaaGimaaqaaiabeI7aXbqdcqGHRiI8aaaa@8312@

are elliptic integrals of the first and second kinds.  Then

S 1111 = 3 8π(1ν) a 1 2 I 11 + 12ν 8π(1ν) I 1 S 1122 = 3 8π(1ν) a 2 2 I 12 12ν 8π(1ν) I 1 S 1133 = 3 8π(1ν) a 3 2 I 13 12ν 8π(1ν) I 1 S 1212 = a 1 2 + a 2 2 16π(1ν) I 12 + 12ν 16π(1ν) ( I 1 + I 2 ) Π ijij =( I i I j )/8πij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaCaaaleqabaGaey4fIO caaOWaaSbaaSqaaiaaigdacaaIXaGaaGymaiaaigdaaeqaaOGaeyyp a0ZaaSaaaeaacaaIZaaabaGaaGioaiabec8aWjaacIcacaaIXaGaey OeI0IaeqyVd4MaaiykaaaacaWGHbWaa0baaSqaaiaaigdaaeaacaaI YaaaaOGaamysamaaBaaaleaacaaIXaGaaGymaaqabaGccqGHRaWkda WcaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gabaGaaGioaiabec8a WjaacIcacaaIXaGaeyOeI0IaeqyVd4MaaiykaaaacaWGjbWaaSbaaS qaaiaaigdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGtbWaaWbaaSqabe aacqGHxiIkaaGcdaWgaaWcbaGaaGymaiaaigdacaaIYaGaaGOmaaqa baGccqGH9aqpdaWcaaqaaiaaiodaaeaacaaI4aGaeqiWdaNaaiikai aaigdacqGHsislcqaH9oGBcaGGPaaaaiaadggadaqhaaWcbaGaaGOm aaqaaiaaikdaaaGccaWGjbWaaSbaaSqaaiaaigdacaaIYaaabeaaki abgkHiTmaalaaabaGaaGymaiabgkHiTiaaikdacqaH9oGBaeaacaaI 4aGaeqiWdaNaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaaaaiaadM eadaWgaaWcbaGaaGymaaqabaaakeaacaWGtbWaaWbaaSqabeaacqGH xiIkaaGcdaWgaaWcbaGaaGymaiaaigdacaaIZaGaaG4maaqabaGccq GH9aqpdaWcaaqaaiaaiodaaeaacaaI4aGaeqiWdaNaaiikaiaaigda cqGHsislcqaH9oGBcaGGPaaaaiaadggadaqhaaWcbaGaaG4maaqaai aaikdaaaGccaWGjbWaaSbaaSqaaiaaigdacaaIZaaabeaakiabgkHi TmaalaaabaGaaGymaiabgkHiTiaaikdacqaH9oGBaeaacaaI4aGaeq iWdaNaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaaaaiaadMeadaWg aaWcbaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadofadaahaaWc beqaaiabgEHiQaaakmaaBaaaleaacaaIXaGaaGOmaiaaigdacaaIYa aabeaakiabg2da9maalaaabaGaamyyamaaDaaaleaacaaIXaaabaGa aGOmaaaakiabgUcaRiaadggadaqhaaWcbaGaaGOmaaqaaiaaikdaaa aakeaacaaIXaGaaGOnaiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyV d4MaaiykaaaacaWGjbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgU caRmaalaaabaGaaGymaiabgkHiTiaaikdacqaH9oGBaeaacaaIXaGa aGOnaiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiykaaaaca GGOaGaamysamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadMeadaWg aaWcbaGaaGOmaaqabaGccaGGPaaabaGaeuiOda1aaSbaaSqaaiaadM gacaWGQbGaamyAaiaadQgaaeqaaOGaeyypa0JaaiikaiaadMeadaWg aaWcbaGaamyAaaqabaGccqGHsislcaWGjbWaaSbaaSqaaiaadQgaae qaaOGaaiykaiaac+cacaaI4aGaeqiWdaNaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaamyAaiabgcMi5kaadQgaaaaa@0A3F@

The remaining components of S ijkl * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaqhaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqaaiaacQcaaaaaaa@35D9@  can be calculated by cyclic permutations of (1,2,3).  Any components that cannot be obtained from these formulas are zero: thus S 1112 = S 1223 = S 1232 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaqhaaWcbaGaaGymaiaaigdaca aIXaGaaGOmaaqaaiabgEHiQaaakiabg2da9iaadofadaqhaaWcbaGa aGymaiaaikdacaaIYaGaaG4maaqaaiabgEHiQaaakiabg2da9iaado fadaqhaaWcbaGaaGymaiaaikdacaaIZaGaaGOmaaqaaiabgEHiQaaa kiabg2da9iaaicdaaaa@42FA@ , Π 1112 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHGoaudaWgaaWcbaGaaGymaiaaig dacaaIXaGaaGOmaaqabaGccqGH9aqpcaaIWaaaaa@3927@ , etc.  Note that S ijkl * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaqhaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqaaiaacQcaaaaaaa@35D9@  has many of the symmetries of the elastic compliance tensor (e.g S ijkl * = S jikl * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaqhaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqaaiaacQcaaaGccqGH9aqpcaWGtbWaa0baaSqaaiaa dQgacaWGPbGaam4AaiaadYgaaeaacaGGQaaaaaaa@3C5A@  ), but does not have major symmetry S ijkl * S klij * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaqhaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqaaiaacQcaaaGccqGHGjsUcaWGtbWaa0baaSqaaiaa dUgacaWGSbGaamyAaiaadQgaaeaacaGGQaaaaaaa@3D1B@

 

For certain special shapes the expressions given for I k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaam4Aaaqabaaaaa@3252@  break down and simplified formulas must be used

 

Oblate spheroid a 1 > a 2 = a 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaaGymaaqabaGccq GH+aGpcaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jaamyyamaa BaaaleaacaaIZaaabeaaaaa@37F4@  

I 1 = I 2 = 2π a 1 a 2 a 3 ( a 1 2 a 3 2 ) 3/2 { cos 1 a 3 a 1 a 3 a 1 ( 1 a 3 2 a 1 2 ) 1/2 } I 12 = I 21 =π/3 a 2 I 13 /4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaa caaIYaGaeqiWdaNaamyyamaaBaaaleaacaaIXaaabeaakiaadggada WgaaWcbaGaaGOmaaqabaGccaWGHbWaaSbaaSqaaiaaiodaaeqaaaGc baWaaeWaaeaacaWGHbWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey OeI0IaamyyamaaDaaaleaacaaIZaaabaGaaGOmaaaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaaG4maiaac+cacaaIYaaaaaaakmaacmaaba Gaci4yaiaac+gacaGGZbWaaWbaaSqabeaacqGHsislcaaIXaaaaOWa aSaaaeaacaWGHbWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamyyamaaBa aaleaacaaIXaaabeaaaaGccqGHsisldaWcaaqaaiaadggadaWgaaWc baGaaG4maaqabaaakeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaaaakm aabmaabaGaaGymaiabgkHiTmaalaaabaGaamyyamaaDaaaleaacaaI ZaaabaGaaGOmaaaaaOqaaiaadggadaqhaaWcbaGaaGymaaqaaiaaik daaaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4laiaa ikdaaaaakiaawUhacaGL9baacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadMeadaWg aaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0JaamysamaaBaaaleaaca aIYaGaaGymaaqabaGccqGH9aqpcqaHapaCcaGGVaGaaG4maiaadgga daahaaWcbeqaaiaaikdaaaGccqGHsislcaWGjbWaaSbaaSqaaiaaig dacaaIZaaabeaakiaac+cacaaI0aGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7aaa@9056@

 

Prolate spheroid a 1 = a 2 > a 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaeyOpa4Jaamyyamaa BaaaleaacaaIZaaabeaaaaa@37F4@  

I 2 = I 3 = 2π a 1 a 2 a 3 ( a 1 2 a 3 2 ) 3/2 { a 1 a 3 ( a 1 2 a 3 2 1 ) 1/2 cosh 1 a 1 a 3 } I 13 = I 31 =π/3 a 2 I 12 /4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaWGjbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0ZaaSaaaeaa caaIYaGaeqiWdaNaamyyamaaBaaaleaacaaIXaaabeaakiaadggada WgaaWcbaGaaGOmaaqabaGccaWGHbWaaSbaaSqaaiaaiodaaeqaaaGc baWaaeWaaeaacaWGHbWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey OeI0IaamyyamaaDaaaleaacaaIZaaabaGaaGOmaaaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaaG4maiaac+cacaaIYaaaaaaakmaacmaaba WaaSaaaeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamyyamaa BaaaleaacaaIZaaabeaaaaGcdaqadaqaamaalaaabaGaamyyamaaDa aaleaacaaIXaaabaGaaGOmaaaaaOqaaiaadggadaqhaaWcbaGaaG4m aaqaaiaaikdaaaaaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCa aaleqabaGaaGymaiaac+cacaaIYaaaaOGaeyOeI0Iaci4yaiaac+ga caGGZbGaaiiAamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaalaaaba GaamyyamaaBaaaleaacaaIXaaabeaaaOqaaiaadggadaWgaaWcbaGa aG4maaqabaaaaaGccaGL7bGaayzFaaGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG jbWaaSbaaSqaaiaaigdacaaIZaaabeaakiabg2da9iaadMeadaWgaa WcbaGaaG4maiaaigdaaeqaaOGaeyypa0JaeqiWdaNaai4laiaaioda caWGHbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamysamaaBaaale aacaaIXaGaaGOmaaqabaGccaGGVaGaaGinaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8oaaa@9145@

 

Sphere a 1 = a 2 = a 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jaamyyamaa BaaaleaacaaIZaaabeaaaaa@37F2@ . In this case the Eshelby tensor can be calculated analytically

S 1111 = S 2222 = S 3333 = 75ν 15(1ν) S 1212 = S 2323 = S 3131 = 45ν 15(1ν) S 1122 = S 2233 = S 3311 = S 1133 = S 2211 = S 3322 = 5ν1 15(1ν) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadofadaqhaaWcbaGaaGymai aaigdacaaIXaGaaGymaaqaaiabgEHiQaaakiabg2da9iaadofadaqh aaWcbaGaaGOmaiaaikdacaaIYaGaaGOmaaqaaiabgEHiQaaakiabg2 da9iaadofadaqhaaWcbaGaaG4maiaaiodacaaIZaGaaG4maaqaaiab gEHiQaaakiabg2da9maalaaabaGaaG4naiabgkHiTiaaiwdacqaH9o GBaeaacaaIXaGaaGynaiaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiyk aaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8Uaam4uamaaDaaaleaacaaIXaGaaGOmaiaaigda caaIYaaabaGaey4fIOcaaOGaeyypa0Jaam4uamaaDaaaleaacaaIYa GaaG4maiaaikdacaaIZaaabaGaey4fIOcaaOGaeyypa0Jaam4uamaa DaaaleaacaaIZaGaaGymaiaaiodacaaIXaaabaGaey4fIOcaaOGaey ypa0ZaaSaaaeaacaaI0aGaeyOeI0IaaGynaiabe27aUbqaaiaaigda caaI1aGaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaaaaaqaaiaado fadaqhaaWcbaGaaGymaiaaigdacaaIYaGaaGOmaaqaaiabgEHiQaaa kiabg2da9iaadofadaqhaaWcbaGaaGOmaiaaikdacaaIZaGaaG4maa qaaiabgEHiQaaakiabg2da9iaadofadaqhaaWcbaGaaG4maiaaioda caaIXaGaaGymaaqaaiabgEHiQaaakiabg2da9iaadofadaqhaaWcba GaaGymaiaaigdacaaIZaGaaG4maaqaaiabgEHiQaaakiabg2da9iaa dofadaqhaaWcbaGaaGOmaiaaikdacaaIXaGaaGymaaqaaiabgEHiQa aakiabg2da9iaadofadaqhaaWcbaGaaG4maiaaiodacaaIYaGaaGOm aaqaaiabgEHiQaaakiabg2da9maalaaabaGaaGynaiabe27aUjabgk HiTiaaigdaaeaacaaIXaGaaGynaiaacIcacaaIXaGaeyOeI0IaeqyV d4Maaiykaaaaaeaaaaaa@A8DA@

Additional terms follow from the symmetry conditions S ijkl = S jikl = S ijlk = S jilk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpcaWGtbWaaSbaaSqaaiaadQgacaWG PbGaam4AaiaadYgaaeqaaOGaeyypa0Jaam4uamaaBaaaleaacaWGPb GaamOAaiaadYgacaWGRbaabeaakiabg2da9iaadofadaWgaaWcbaGa amOAaiaadMgacaWGSbGaam4Aaaqabaaaaa@46A0@ .  The remaining terms are zero.

 

Cylinder a 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaaG4maaqabaGccq GHsgIRcqGHEisPaaa@359F@ . For this case the Eshelby tensor reduces to

S 1111 * = a 2 [ 2(1ν)( a 1 + a 2 )+ a 1 ] 2(1ν) ( a 1 + a 2 ) 2 S 2222 * = a 1 [ 2(1ν)( a 1 + a 2 )+ a 2 ] 2(1ν) ( a 1 + a 2 ) 2 S 3333 * =0 S 1122 * = a 2 [ (2ν1) a 1 +2ν a 2 ] 2(1ν) ( a 1 + a 2 ) 2 S 2211 * = a 1 [ (2ν1) a 2 +2ν a 1 ] 2(1ν) ( a 1 + a 2 ) 2 S 1133 * = ν a 2 (1ν)( a 1 + a 2 ) S 3311 * =0 S 2233 * = ν a 1 (1ν)( a 1 + a 2 ) S 3322 * =0 S 1212 * = [ (1ν)( a 1 2 + a 2 2 )+(12ν) a 1 a 2 ] 2(1ν) ( a 1 + a 2 ) 2 S 1313 * = a 2 (2ν) 2(1ν)( a 1 + a 2 ) S 2323 * = a 1 (2ν) 2(1ν)( a 1 + a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadofadaqhaaWcbaGaaGymai aaigdacaaIXaGaaGymaaqaaiaacQcaaaGccqGH9aqpdaWcaaqaaiaa dggadaWgaaWcbaGaaGOmaaqabaGcdaWadaqaaiaaikdacaGGOaGaaG ymaiabgkHiTiabe27aUjaacMcacaGGOaGaamyyamaaBaaaleaacaaI XaaabeaakiabgUcaRiaadggadaWgaaWcbaGaaGOmaaqabaGccaGGPa Gaey4kaSIaamyyamaaBaaaleaacaaIXaaabeaaaOGaay5waiaaw2fa aaqaaiaaikdacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcacaGGOa GaamyyamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadggadaWgaaWc baGaaGOmaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaaaakiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caWGtbWaa0baaSqaaiaaikdaca aIYaGaaGOmaiaaikdaaeaacaGGQaaaaOGaeyypa0ZaaSaaaeaacaWG HbWaaSbaaSqaaiaaigdaaeqaaOWaamWaaeaacaaIYaGaaiikaiaaig dacqGHsislcqaH9oGBcaGGPaGaaiikaiaadggadaWgaaWcbaGaaGym aaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaaiykai abgUcaRiaadggadaWgaaWcbaGaaGOmaaqabaaakiaawUfacaGLDbaa aeaacaaIYaGaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaGaaiikai aadggadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGHbWaaSbaaSqa aiaaikdaaeqaaOGaaiykamaaCaaaleqabaGaaGOmaaaaaaGccaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8Uaam4uamaaDaaaleaacaaIZaGaaG 4maiaaiodacaaIZaaabaGaaiOkaaaakiabg2da9iaaicdaaeaacaWG tbWaa0baaSqaaiaaigdacaaIXaGaaGOmaiaaikdaaeaacaGGQaaaaO Gaeyypa0ZaaSaaaeaacaWGHbWaaSbaaSqaaiaaikdaaeqaaOWaamWa aeaacaGGOaGaaGOmaiabe27aUjabgkHiTiaaigdacaGGPaGaamyyam aaBaaaleaacaaIXaaabeaakiabgUcaRiaaikdacqaH9oGBcaWGHbWa aSbaaSqaaiaaikdaaeqaaaGccaGLBbGaayzxaaaabaGaaGOmaiaacI cacaaIXaGaeyOeI0IaeqyVd4MaaiykaiaacIcacaWGHbWaaSbaaSqa aiaaigdaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaaIYaaabeaaki aacMcadaahaaWcbeqaaiaaikdaaaaaaOGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4uamaaDaaale aacaaIYaGaaGOmaiaaigdacaaIXaaabaGaaiOkaaaakiabg2da9maa laaabaGaamyyamaaBaaaleaacaaIXaaabeaakmaadmaabaGaaiikai aaikdacqaH9oGBcqGHsislcaaIXaGaaiykaiaadggadaWgaaWcbaGa aGOmaaqabaGccqGHRaWkcaaIYaGaeqyVd4MaamyyamaaBaaaleaaca aIXaaabeaaaOGaay5waiaaw2faaaqaaiaaikdacaGGOaGaaGymaiab gkHiTiabe27aUjaacMcacaGGOaGaamyyamaaBaaaleaacaaIXaaabe aakiabgUcaRiaadggadaWgaaWcbaGaaGOmaaqabaGccaGGPaWaaWba aSqabeaacaaIYaaaaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8Uaam4uamaaDaaaleaacaaIXaGaaGymaiaa iodacaaIZaaabaGaaiOkaaaakiabg2da9maalaaabaGaeqyVd4Maam yyamaaBaaaleaacaaIYaaabeaaaOqaaiaacIcacaaIXaGaeyOeI0Ia eqyVd4MaaiykaiaacIcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaey 4kaSIaamyyamaaBaaaleaacaaIYaaabeaakiaacMcaaaGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadofadaqhaaWcba GaaG4maiaaiodacaaIXaGaaGymaaqaaiaacQcaaaGccqGH9aqpcaaI WaaabaGaam4uamaaDaaaleaacaaIYaGaaGOmaiaaiodacaaIZaaaba GaaiOkaaaakiabg2da9maalaaabaGaeqyVd4MaamyyamaaBaaaleaa caaIXaaabeaaaOqaaiaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiykai aacIcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyyamaa BaaaleaacaaIYaaabeaakiaacMcaaaGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaadofadaqhaaWcbaGaaG4maiaaioda caaIYaGaaGOmaaqaaiaacQcaaaGccqGH9aqpcaaIWaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4uamaaDaaaleaaca aIXaGaaGOmaiaaigdacaaIYaaabaGaaiOkaaaakiabg2da9maalaaa baWaamWaaeaacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcacaGGOa GaamyyamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgUcaRiaadgga daqhaaWcbaGaaGOmaaqaaiaaikdaaaGccaGGPaGaey4kaSIaaiikai aaigdacqGHsislcaaIYaGaeqyVd4MaaiykaiaadggadaWgaaWcbaGa aGymaaqabaGccaWGHbWaaSbaaSqaaiaaikdaaeqaaaGccaGLBbGaay zxaaaabaGaaGOmaiaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiykaiaa cIcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyyamaaBa aaleaacaaIYaaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaaaaOGa aGPaVdqaaiaadofadaqhaaWcbaGaaGymaiaaiodacaaIXaGaaG4maa qaaiaacQcaaaGccqGH9aqpdaWcaaqaaiaadggadaWgaaWcbaGaaGOm aaqabaGccaGGOaGaaGOmaiabgkHiTiabe27aUjaacMcaaeaacaaIYa GaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaGaaiikaiaadggadaWg aaWcbaGaaGymaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaikdaae qaaOGaaiykaaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8Uaam4uamaaDaaaleaacaaIYaGaaG 4maiaaikdacaaIZaaabaGaaiOkaaaakiabg2da9maalaaabaGaamyy amaaBaaaleaacaaIXaaabeaakiaacIcacaaIYaGaeyOeI0IaeqyVd4 MaaiykaaqaaiaaikdacaGGOaGaaGymaiabgkHiTiabe27aUjaacMca caGGOaGaamyyamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadggada WgaaWcbaGaaGOmaaqabaGccaGGPaaaaaaaaa@D442@

Additional terms follow from the symmetry conditions S ijkl = S jikl = S ijlk = S jilk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpcaWGtbWaaSbaaSqaaiaadQgacaWG PbGaam4AaiaadYgaaeqaaOGaeyypa0Jaam4uamaaBaaaleaacaWGPb GaamOAaiaadYgacaWGRbaabeaakiabg2da9iaadofadaWgaaWcbaGa amOAaiaadMgacaWGSbGaam4Aaaqabaaaaa@46A0@ .  The remaining terms are zero.

 

Solution outside the ellipsoid: The solution outside the ellipsoid can also be expressed in simplified form:

J. D. Eshelby “The Elastic Field Outside an Ellipsoidal Inclusion,” Proceedings of the Royal Society of London. A252, pp. 561-569, (1959) shows that the displacement can be obtained from a single scalar potential ω( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHjpWDcaGGOaGaamiEamaaBaaale aacaWGPbaabeaakiaacMcaaaa@380D@ .  For actual calculations only the derivatives of the potential are required, which can be reduced to

dω d x 1 = x 1 a 1 a 2 a 3 l 3 k 2 { E(θ,k)F(θ,k) } dω d x 2 = x 2 a 1 a 2 a 3 l 3 k 2 k ^ 2 { k ^ 2 F(θ,k)E(θ,k)+l A 3 k 2 /( A 1 A 2 ) } dω d x 3 = x 3 a 1 a 2 a 3 l 3 k ^ 2 { E(θ,k)l A 2 /( A 1 A 3 ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaalaaabaGaamizaiabeM8a3b qaaiaadsgacaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakiabg2da9maa laaabaGaamiEamaaBaaaleaacaaIXaaabeaakiaadggadaWgaaWcba GaaGymaaqabaGccaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaamyyamaa BaaaleaacaaIZaaabeaaaOqaaiaadYgadaahaaWcbeqaaiaaiodaaa GccaWGRbWaaWbaaSqabeaacaaIYaaaaaaakmaacmaabaGaamyraiaa cIcacqaH4oqCcaGGSaGaam4AaiaacMcacqGHsislcaWGgbGaaiikai abeI7aXjaacYcacaWGRbGaaiykaaGaay5Eaiaaw2haaaqaamaalaaa baGaamizaiabeM8a3bqaaiaadsgacaWG4bWaaSbaaSqaaiaaikdaae qaaaaakiabg2da9maalaaabaGaamiEamaaBaaaleaacaaIYaaabeaa kiaadggadaWgaaWcbaGaaGymaaqabaGccaWGHbWaaSbaaSqaaiaaik daaeqaaOGaamyyamaaBaaaleaacaaIZaaabeaaaOqaaiaadYgadaah aaWcbeqaaiaaiodaaaGccaWGRbWaaWbaaSqabeaacaaIYaaaaOGabm 4AayaajaWaaWbaaSqabeaacaaIYaaaaaaakmaacmaabaGabm4Aayaa jaWaaWbaaSqabeaacaaIYaaaaOGaamOraiaacIcacqaH4oqCcaGGSa Gaam4AaiaacMcacqGHsislcaWGfbGaaiikaiabeI7aXjaacYcacaWG RbGaaiykaiabgUcaRiaadYgacaWGbbWaaSbaaSqaaiaaiodaaeqaaO Gaam4AamaaCaaaleqabaGaaGOmaaaakiaac+cacaGGOaGaamyqamaa BaaaleaacaaIXaaabeaakiaadgeadaWgaaWcbaGaaGOmaaqabaGcca GGPaaacaGL7bGaayzFaaaabaWaaSaaaeaacaWGKbGaeqyYdChabaGa amizaiaadIhadaWgaaWcbaGaaG4maaqabaaaaOGaeyypa0ZaaSaaae aacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaamyyamaaBaaaleaacaaI XaaabeaakiaadggadaWgaaWcbaGaaGOmaaqabaGccaWGHbWaaSbaaS qaaiaaiodaaeqaaaGcbaGaamiBamaaCaaaleqabaGaaG4maaaakiqa dUgagaqcamaaCaaaleqabaGaaGOmaaaaaaGcdaGadaqaaiaadweaca GGOaGaeqiUdeNaaiilaiaadUgacaGGPaGaeyOeI0IaamiBaiaadgea daWgaaWcbaGaaGOmaaqabaGccaGGVaGaaiikaiaadgeadaWgaaWcba GaaGymaaqabaGccaWGbbWaaSbaaSqaaiaaiodaaeqaaOGaaiykaaGa ay5Eaiaaw2haaaaaaa@A465@

A i = a i 2 +λ l= a 1 2 a 3 2 k= ( a 1 2 a 2 2 )/( a 1 2 a 3 2 ) k ^ = ( a 2 2 a 3 2 )/( a 1 2 a 3 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0ZaaOaaaeaacaWGHbWaa0baaSqaaiaadMgaaeaacaaIYaaa aOGaey4kaSIaeq4UdWgaleqaaOGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGSbGa eyypa0ZaaOaaaeaacaWGHbWaa0baaSqaaiaaigdaaeaacaaIYaaaaO GaeyOeI0IaamyyamaaDaaaleaacaaIZaaabaGaaGOmaaaaaeqaaOGa aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadUgacq GH9aqpdaGcaaqaaiaacIcacaWGHbWaa0baaSqaaiaaigdaaeaacaaI YaaaaOGaeyOeI0IaamyyamaaDaaaleaacaaIYaaabaGaaGOmaaaaki aacMcacaGGVaGaaiikaiaadggadaqhaaWcbaGaaGymaaqaaiaaikda aaGccqGHsislcaWGHbWaa0baaSqaaiaaiodaaeaacaaIYaaaaOGaai ykaaWcbeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlqadUgagaqcaiabg2da9maakaaabaGaaiikai aadggadaqhaaWcbaGaaGOmaaqaaiaaikdaaaGccqGHsislcaWGHbWa a0baaSqaaiaaiodaaeaacaaIYaaaaOGaaiykaiaac+cacaGGOaGaam yyamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgkHiTiaadggadaqh aaWcbaGaaG4maaqaaiaaikdaaaGccaGGPaaaleqaaaaa@9061@

where λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH7oaBaaa@347A@  is the greatest positive root of λ 3 L λ 2 +MλN=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH7oaBdaahaaWcbeqaaiaaiodaaa GccqGHsislcaWGmbGaeq4UdW2aaWbaaSqabeaacaaIYaaaaOGaey4k aSIaamytaiabeU7aSjabgkHiTiaad6eacqGH9aqpcaaIWaaaaa@40BB@ , with

L= r 2 R 2 M= i=1 3 a i 2 x i a 1 2 a 2 2 a 2 2 a 3 2 a 1 2 a 3 2 + r 2 R 2 N= a 1 2 a 2 2 a 3 2 ( i=1 3 x i 2 a i 2 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGmbGaeyypa0JaamOCamaaCaaale qabaGaaGOmaaaakiabgkHiTiaadkfadaahaaWcbeqaaiaaikdaaaGc caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGnbGaeyypa0 ZaaabCaeaacaWGHbWaa0baaSqaaiaadMgaaeaacaaIYaaaaOGaamiE amaaBaaaleaacaWGPbaabeaaaeaacaWGPbGaeyypa0JaaGymaaqaai aaiodaa0GaeyyeIuoakiabgkHiTiaadggadaqhaaWcbaGaaGymaaqa aiaaikdaaaGccaWGHbWaa0baaSqaaiaaikdaaeaacaaIYaaaaOGaey OeI0IaamyyamaaDaaaleaacaaIYaaabaGaaGOmaaaakiaadggadaqh aaWcbaGaaG4maaqaaiaaikdaaaGccqGHsislcaWGHbWaa0baaSqaai aaigdaaeaacaaIYaaaaOGaamyyamaaDaaaleaacaaIZaaabaGaaGOm aaaakiabgUcaRiaadkhadaahaaWcbeqaaiaaikdaaaGccaWGsbWaaW baaSqabeaacaaIYaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caWGobGaeyypa0JaamyyamaaDaaaleaaca aIXaaabaGaaGOmaaaakiaadggadaqhaaWcbaGaaGOmaaqaaiaaikda aaGccaWGHbWaa0baaSqaaiaaiodaaeaacaaIYaaaaOWaaeWaaeaada aeWbqaamaalaaabaGaamiEamaaDaaaleaacaWGPbaabaGaaGOmaaaa aOqaaiaadggadaqhaaWcbaGaamyAaaqaaiaaikdaaaaaaOGaeyOeI0 IaaGymaaWcbaGaamyAaiabg2da9iaaigdaaeaacaaIZaaaniabggHi LdaakiaawIcacaGLPaaaaaa@8A4C@

and r= x k x k R= a k a k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0ZaaOaaaeaacaWG4b WaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaa aeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaadkfacqGH9aqpdaGcaaqaaiaadggadaWg aaWcbaGaam4AaaqabaGccaWGHbWaaSbaaSqaaiaadUgaaeqaaaqaba aaaa@4E82@ .  Additional derivatives can be computed using the relations

dF/dλ=l/(2 A 1 A 2 A 3 )dE/dλ=l A 2 /( A 1 3 A 3 )dλ/d x i =2 x i /( A i h) h 2 = i=1 3 x i 2 / A i 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamOraiaac+cacaWGKbGaeq 4UdWMaeyypa0JaeyOeI0IaamiBaiaac+cacaGGOaGaaGOmaiaadgea daWgaaWcbaGaaGymaaqabaGccaWGbbWaaSbaaSqaaiaaikdaaeqaaO GaamyqamaaBaaaleaacaaIZaaabeaakiaacMcacaaMc8UaaGPaVlaa ykW7caaMc8UaamizaiaadweacaGGVaGaamizaiabeU7aSjabg2da9i abgkHiTiaadYgacaWGbbWaaSbaaSqaaiaaikdaaeqaaOGaai4laiaa cIcacaWGbbWaa0baaSqaaiaaigdaaeaacaaIZaaaaOGaamyqamaaBa aaleaacaaIZaaabeaakiaacMcacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadsgacq aH7oaBcaGGVaGaamizaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGH 9aqpcaaIYaGaamiEamaaBaaaleaacaWGPbaabeaakiaac+cacaGGOa GaamyqamaaBaaaleaacaWGPbaabeaakiaadIgacaGGPaGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWGObWaaWbaaSqabeaacaaIYaaaaOGaeyypa0ZaaabC aeaacaWG4bWaa0baaSqaaiaadMgaaeaacaaIYaaaaOGaai4laiaadg eadaqhaaWcbaGaamyAaaqaaiaaisdaaaaabaGaamyAaiabg2da9iaa igdaaeaacaaIZaaaniabggHiLdaaaa@986E@

The displacements follow as

2(1ν) u 1 = ε 11 T ε 22 T a 1 2 a 2 2 x 2 ( a 1 2 x 2 ω x 1 a 2 2 x 1 ω x 2 )+ ε 33 T ε 11 T a 3 2 a 1 2 x 3 ( a 3 2 x 1 ω x 3 a 1 2 x 3 ω x 3 ) 2{ (1ν) ε 11 T +ν( ε 11 T + ε 22 T ) } ω x 1 4(1ν)( ε 12 T ω x 2 + ε 13 T ω x 3 )+ β x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaaikdacaGGOaGaaGymaiabgk HiTiabe27aUjaacMcacaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaeyyp a0ZaaSaaaeaacqaH1oqzdaqhaaWcbaGaaGymaiaaigdaaeaacaWGub aaaOGaeyOeI0IaeqyTdu2aa0baaSqaaiaaikdacaaIYaaabaGaamiv aaaaaOqaaiaadggadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHsi slcaWGHbWaa0baaSqaaiaaikdaaeaacaaIYaaaaaaakmaalaaabaGa eyOaIylabaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaGcda qadaqaaiaadggadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccaWG4bWa aSbaaSqaaiaaikdaaeqaaOWaaSaaaeaacqGHciITcqaHjpWDaeaacq GHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakiabgkHiTiaadgga daqhaaWcbaGaaGOmaaqaaiaaikdaaaGccaWG4bWaaSbaaSqaaiaaig daaeqaaOWaaSaaaeaacqGHciITcqaHjpWDaeaacqGHciITcaWG4bWa aSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaiabgUcaRmaala aabaGaeqyTdu2aa0baaSqaaiaaiodacaaIZaaabaGaamivaaaakiab gkHiTiabew7aLnaaDaaaleaacaaIXaGaaGymaaqaaiaadsfaaaaake aacaWGHbWaa0baaSqaaiaaiodaaeaacaaIYaaaaOGaeyOeI0Iaamyy amaaDaaaleaacaaIXaaabaGaaGOmaaaaaaGcdaWcaaqaaiabgkGi2c qaaiabgkGi2kaadIhadaWgaaWcbaGaaG4maaqabaaaaOWaaeWaaeaa caWGHbWaa0baaSqaaiaaiodaaeaacaaIYaaaaOGaamiEamaaBaaale aacaaIXaaabeaakmaalaaabaGaeyOaIyRaeqyYdChabaGaeyOaIyRa amiEamaaBaaaleaacaaIZaaabeaaaaGccqGHsislcaWGHbWaa0baaS qaaiaaigdaaeaacaaIYaaaaOGaamiEamaaBaaaleaacaaIZaaabeaa kmaalaaabaGaeyOaIyRaeqyYdChabaGaeyOaIyRaamiEamaaBaaale aacaaIZaaabeaaaaaakiaawIcacaGLPaaaaeaacaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHsislcaaIYaWaai WaaeaacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcacqaH1oqzdaqh aaWcbaGaaGymaiaaigdaaeaacaWGubaaaOGaey4kaSIaeqyVd4Maai ikaiabew7aLnaaDaaaleaacaaIXaGaaGymaaqaaiaadsfaaaGccqGH RaWkcqaH1oqzdaqhaaWcbaGaaGOmaiaaikdaaeaacaWGubaaaOGaai ykaaGaay5Eaiaaw2haamaalaaabaGaeyOaIyRaeqyYdChabaGaeyOa IyRaamiEamaaBaaaleaacaaIXaaabeaaaaGccqGHsislcaaI0aGaai ikaiaaigdacqGHsislcqaH9oGBcaGGPaWaaeWaaeaacqaH1oqzdaqh aaWcbaGaaGymaiaaikdaaeaacaWGubaaaOWaaSaaaeaacqGHciITcq aHjpWDaeaacqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaakiab gUcaRiabew7aLnaaDaaaleaacaaIXaGaaG4maaqaaiaadsfaaaGcda WcaaqaaiabgkGi2kabeM8a3bqaaiabgkGi2kaadIhadaWgaaWcbaGa aG4maaqabaaaaaGccaGLOaGaayzkaaGaey4kaSYaaSaaaeaacqGHci ITcqaHYoGyaeaacqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaa aaaa@0117@

where

β= 2 ε 12 T a 1 2 a 2 2 ( a 1 2 x 2 ω x 1 a 2 2 x 1 ω x 2 )+ 2 ε 23 T a 2 2 a 3 2 ( a 2 2 x 3 ω x 2 a 3 2 x 2 ω x 3 )+ 2 ε 31 T a 3 2 a 1 2 ( a 3 2 x 1 ω x 3 a 1 2 x 3 ω x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHYoGycqGH9aqpdaWcaaqaaiaaik dacqaH1oqzdaqhaaWcbaGaaGymaiaaikdaaeaacaWGubaaaaGcbaGa amyyamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgkHiTiaadggada qhaaWcbaGaaGOmaaqaaiaaikdaaaaaaOWaaeWaaeaacaWGHbWaa0ba aSqaaiaaigdaaeaacaaIYaaaaOGaamiEamaaBaaaleaacaaIYaaabe aakmaalaaabaGaeyOaIyRaeqyYdChabaGaeyOaIyRaamiEamaaBaaa leaacaaIXaaabeaaaaGccqGHsislcaWGHbWaa0baaSqaaiaaikdaae aacaaIYaaaaOGaamiEamaaBaaaleaacaaIXaaabeaakmaalaaabaGa eyOaIyRaeqyYdChabaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabe aaaaaakiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaaikdacqaH1oqz daqhaaWcbaGaaGOmaiaaiodaaeaacaWGubaaaaGcbaGaamyyamaaDa aaleaacaaIYaaabaGaaGOmaaaakiabgkHiTiaadggadaqhaaWcbaGa aG4maaqaaiaaikdaaaaaaOWaaeWaaeaacaWGHbWaa0baaSqaaiaaik daaeaacaaIYaaaaOGaamiEamaaBaaaleaacaaIZaaabeaakmaalaaa baGaeyOaIyRaeqyYdChabaGaeyOaIyRaamiEamaaBaaaleaacaaIYa aabeaaaaGccqGHsislcaWGHbWaa0baaSqaaiaaiodaaeaacaaIYaaa aOGaamiEamaaBaaaleaacaaIYaaabeaakmaalaaabaGaeyOaIyRaeq yYdChabaGaeyOaIyRaamiEamaaBaaaleaacaaIZaaabeaaaaaakiaa wIcacaGLPaaacqGHRaWkdaWcaaqaaiaaikdacqaH1oqzdaqhaaWcba GaaG4maiaaigdaaeaacaWGubaaaaGcbaGaamyyamaaDaaaleaacaaI ZaaabaGaaGOmaaaakiabgkHiTiaadggadaqhaaWcbaGaaGymaaqaai aaikdaaaaaaOWaaeWaaeaacaWGHbWaa0baaSqaaiaaiodaaeaacaaI YaaaaOGaamiEamaaBaaaleaacaaIXaaabeaakmaalaaabaGaeyOaIy RaeqyYdChabaGaeyOaIyRaamiEamaaBaaaleaacaaIZaaabeaaaaGc cqGHsislcaWGHbWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaamiEam aaBaaaleaacaaIZaaabeaakmaalaaabaGaeyOaIyRaeqyYdChabaGa eyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaaaakiaawIcacaGLPa aaaaa@A270@

The remaining displacement components can be calculated by cyclic permutations of (1,2,3), and strains and stresses can be calculated by differentiating the displacements appropriately.  The results are far too complicated to write out in full, and in practice the algebra can only be done with the aid of a symbolic manipulation program.  Some special results can be reduced to a tractable form, however:

 

Displacements far from the ellipsoid R= x k x k >> a 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbGaeyypa0ZaaOaaaeaacaWG4b WaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaa aeqaaOGaeyOpa4JaeyOpa4JaamyyamaaBaaaleaacaaIXaaabeaaaa a@3CD6@

u i = a 1 a 2 a 3 8(1ν) R 2 { 3 ε jk T x i x j x k R 3 +(12ν) 2 ε ik T x k ε kk T x i R } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0ZaaSaaaeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaamyy amaaBaaaleaacaaIYaaabeaakiaadggadaWgaaWcbaGaaG4maaqaba aakeaacaaI4aGaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaGaamOu amaaCaaaleqabaGaaGOmaaaaaaGcdaGadaqaamaalaaabaGaaG4mai abew7aLnaaDaaaleaacaWGQbGaam4AaaqaaiaadsfaaaGccaWG4bWa aSbaaSqaaiaadMgaaeqaaOGaamiEamaaBaaaleaacaWGQbaabeaaki aadIhadaWgaaWcbaGaam4AaaqabaaakeaacaWGsbWaaWbaaSqabeaa caaIZaaaaaaakiabgUcaRiaacIcacaaIXaGaeyOeI0IaaGOmaiabe2 7aUjaacMcadaWcaaqaaiaaikdacqaH1oqzdaqhaaWcbaGaamyAaiaa dUgaaeaacaWGubaaaOGaamiEamaaBaaaleaacaWGRbaabeaakiabgk HiTiabew7aLnaaDaaaleaacaWGRbGaam4AaaqaaiaadsfaaaGccaWG 4bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaamOuaaaaaiaawUhacaGL9b aaaaa@68E3@

 

Solution outside a spherical inclusion: For this case the Papkovich-Neuber potentials can be reduced to

Ψ i = a 3 p ik T x k 3 R 3 ϕ= a 3 p ij T 15 R 3 ( (5 R 2 a 2 ) δ ij +3 a 2 x i x j R 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHOoqwdaWgaaWcbaGaamyAaaqaba GccqGH9aqpdaWcaaqaaiaadggadaahaaWcbeqaaiaaiodaaaGccaWG WbWaa0baaSqaaiaadMgacaWGRbaabaGaamivaaaakiaadIhadaWgaa WcbaGaam4AaaqabaaakeaacaaIZaGaamOuamaaCaaaleqabaGaaG4m aaaaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHvpGzcqGH9aqpdaWc aaqaaiaadggadaahaaWcbeqaaiaaiodaaaGccaWGWbWaa0baaSqaai aadMgacaWGQbaabaGaamivaaaaaOqaaiaaigdacaaI1aGaamOuamaa CaaaleqabaGaaG4maaaaaaGcdaqadaqaaiaacIcacaaI1aGaamOuam aaCaaaleqabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqaaiaa ikdaaaGccaGGPaGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaki abgUcaRiaaiodacaWGHbWaaWbaaSqabeaacaaIYaaaaOWaaSaaaeaa caWG4bWaaSbaaSqaaiaadMgaaeqaaOGaamiEamaaBaaaleaacaWGQb aabeaaaOqaaiaadkfadaahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGa ayzkaaaaaa@74D8@

The displacements and stresses follow as

u i = (1+ν) a 3 2(1ν)E { (2 p ik T x k + p kk T x i ) 15 R 5 (3 a 2 5 R 2 )+ p jk T x j x k x i R 7 ( R 2 a 2 )+ 4(1ν) p ik T x k 3 R 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0ZaaSaaaeaacaGGOaGaaGymaiabgUcaRiabe27aUjaacMca caWGHbWaaWbaaSqabeaacaaIZaaaaaGcbaGaaGOmaiaacIcacaaIXa GaeyOeI0IaeqyVd4MaaiykaiaadweaaaWaaiWaaeaadaWcaaqaaiaa cIcacaaIYaGaamiCamaaDaaaleaacaWGPbGaam4Aaaqaaiaadsfaaa GccaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaey4kaSIaamiCamaaDaaa leaacaWGRbGaam4AaaqaaiaadsfaaaGccaWG4bWaaSbaaSqaaiaadM gaaeqaaOGaaiykaaqaaiaaigdacaaI1aGaamOuamaaCaaaleqabaGa aGynaaaaaaGccaGGOaGaaG4maiaadggadaahaaWcbeqaaiaaikdaaa GccqGHsislcaaI1aGaamOuamaaCaaaleqabaGaaGOmaaaakiaacMca cqGHRaWkdaWcaaqaaiaadchadaqhaaWcbaGaamOAaiaadUgaaeaaca WGubaaaOGaamiEamaaBaaaleaacaWGQbaabeaakiaadIhadaWgaaWc baGaam4AaaqabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaam OuamaaCaaaleqabaGaaG4naaaaaaGccaGGOaGaamOuamaaCaaaleqa baGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqaaiaaikdaaaGcca GGPaGaey4kaSYaaSaaaeaacaaI0aGaaiikaiaaigdacqGHsislcqaH 9oGBcaGGPaGaamiCamaaDaaaleaacaWGPbGaam4Aaaqaaiaadsfaaa GccaWG4bWaaSbaaSqaaiaadUgaaeqaaaGcbaGaaG4maiaadkfadaah aaWcbeqaaiaaiodaaaaaaaGccaGL7bGaayzFaaaaaa@809C@

σ ij = a 3 2(1ν) R 3 { p ij T 15 ( 10(12ν)+6 a 2 R 2 ) + p ik T x k x j + p jk T x k x i R 2 ( 2ν2 a 2 R 2 )+ δ ij p kk T 15 ( 3 a 2 R 2 5(12ν) ) + δ ij p kl T x k x l R 2 ( (12ν) a 2 R 2 ) x i x j p kl T x k x l R 4 ( 57 a 2 R 2 )+ x i x j p kk T R 2 ( 1 a 2 R 2 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aaSbaaSqaaiaadMgaca WGQbaabeaakiabg2da9maalaaabaGaamyyamaaCaaaleqabaGaaG4m aaaaaOqaaiaaikdacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcaca WGsbWaaWbaaSqabeaacaaIZaaaaaaakmaaceaabaWaaSaaaeaacaWG WbWaa0baaSqaaiaadMgacaWGQbaabaGaamivaaaaaOqaaiaaigdaca aI1aaaamaabmaabaGaaGymaiaaicdacaGGOaGaaGymaiabgkHiTiaa ikdacqaH9oGBcaGGPaGaey4kaSIaaGOnamaalaaabaGaamyyamaaCa aaleqabaGaaGOmaaaaaOqaaiaadkfadaahaaWcbeqaaiaaikdaaaaa aaGccaGLOaGaayzkaaaacaGL7baacqGHRaWkdaWcaaqaaiaadchada qhaaWcbaGaamyAaiaadUgaaeaacaWGubaaaOGaamiEamaaBaaaleaa caWGRbaabeaakiaadIhadaWgaaWcbaGaamOAaaqabaGccqGHRaWkca WGWbWaa0baaSqaaiaadQgacaWGRbaabaGaamivaaaakiaadIhadaWg aaWcbaGaam4AaaqabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaaGcba GaamOuamaaCaaaleqabaGaaGOmaaaaaaGcdaqadaqaaiaaikdacqaH 9oGBcqGHsislcaaIYaWaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIYa aaaaGcbaGaamOuamaaCaaaleqabaGaaGOmaaaaaaaakiaawIcacaGL PaaacqGHRaWkdaWcaaqaaiabes7aKnaaBaaaleaacaWGPbGaamOAaa qabaGccaWGWbWaa0baaSqaaiaadUgacaWGRbaabaGaamivaaaaaOqa aiaaigdacaaI1aaaamaabmaabaGaaG4mamaalaaabaGaamyyamaaCa aaleqabaGaaGOmaaaaaOqaaiaadkfadaahaaWcbeqaaiaaikdaaaaa aOGaeyOeI0IaaGynaiaacIcacaaIXaGaeyOeI0IaaGOmaiabe27aUj aacMcaaiaawIcacaGLPaaaaeaacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7cqGHRaWkdaWcaaqaaiabes7aKnaaBaaaleaacaWG PbGaamOAaaqabaGccaWGWbWaa0baaSqaaiaadUgacaWGSbaabaGaam ivaaaakiaadIhadaWgaaWcbaGaam4AaaqabaGccaWG4bWaaSbaaSqa aiaadYgaaeqaaaGcbaGaamOuamaaCaaaleqabaGaaGOmaaaaaaGcda qadaqaaiaacIcacaaIXaGaeyOeI0IaaGOmaiabe27aUjaacMcacqGH sisldaWcaaqaaiaadggadaahaaWcbeqaaiaaikdaaaaakeaacaWGsb WaaWbaaSqabeaacaaIYaaaaaaaaOGaayjkaiaawMcaaiabgkHiTmaa ciaabaWaaSaaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaamiEam aaBaaaleaacaWGQbaabeaakiaadchadaqhaaWcbaGaam4AaiaadYga aeaacaWGubaaaOGaamiEamaaBaaaleaacaWGRbaabeaakiaadIhada WgaaWcbaGaamiBaaqabaaakeaacaWGsbWaaWbaaSqabeaacaaI0aaa aaaakmaabmaabaGaaGynaiabgkHiTiaaiEdadaWcaaqaaiaadggada ahaaWcbeqaaiaaikdaaaaakeaacaWGsbWaaWbaaSqabeaacaaIYaaa aaaaaOGaayjkaiaawMcaaiabgUcaRmaalaaabaGaamiEamaaBaaale aacaWGPbaabeaakiaadIhadaWgaaWcbaGaamOAaaqabaGccaWGWbWa a0baaSqaaiaadUgacaWGRbaabaGaamivaaaaaOqaaiaadkfadaahaa WcbeqaaiaaikdaaaaaaOWaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaa caWGHbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOuamaaCaaaleqaba GaaGOmaaaaaaaakiaawIcacaGLPaaaaiaaw2haaaaaaa@2C67@

where R= x k x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbGaeyypa0ZaaOaaaeaacaWG4b WaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaa aeqaaaaa@38EF@  and p ij T ={ E/(1+ν) }{ ε ij T +ν ε kk T δ ij /(12ν) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaqhaaWcbaGaamyAaiaadQgaae aacaWGubaaaOGaeyypa0ZaaiWaaeaacaWGfbGaai4laiaacIcacaaI XaGaey4kaSIaeqyVd4MaaiykaaGaay5Eaiaaw2haamaacmaabaGaeq yTdu2aa0baaSqaaiaadMgacaWGQbaabaGaamivaaaakiabgUcaRiab e27aUjabew7aLnaaDaaaleaacaWGRbGaam4AaaqaaiaadsfaaaGccq aH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaai4laiaacIcacaaI XaGaeyOeI0IaaGOmaiabe27aUjaacMcaaiaawUhacaGL9baaaaa@5581@

 

 

 

5.4.7 Elastically mismatched ellipsoidal inclusion in an infinite solid subjected to remote stress

 

The figure shows an ellipsoidal inclusion, with semi-axes ( a 1 , a 2 , a 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamyyamaaBaaaleaacaaIXa aabeaakiaacYcacaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa dggadaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@3B07@ .  The inclusion is made from an isotropic, elastic solid with Young’s modulus E I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbWaaWbaaSqabeaacaWGjbaaaa aa@348B@  and Poisson’s ratio ν I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBdaahaaWcbeqaaiaadMeaaa aaaa@3579@ .  It is embedded in an infinite, isotropic elastic matrix with Young’s modulus E M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbWaaWbaaSqabeaacaWGnbaaaa aa@348F@  and Poisson’s ratio ν M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBdaahaaWcbeqaaiaad2eaaa aaaa@357D@ .  The solid is loaded at infinity by a uniform stress state σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacqGHEisPaaaaaa@3804@ , strains ε ij =((1+ ν M ) σ ij ν M σ kk δ ij )/ E M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaqhaaWcbaGaamyAaiaadQ gaaeaacqGHEisPaaGccqGH9aqpcaGGOaGaaiikaiaaigdacqGHRaWk cqaH9oGBdaahaaWcbeqaaiaad2eaaaGccaGGPaGaeq4Wdm3aa0baaS qaaiaadMgacaWGQbaabaGaeyOhIukaaOGaeyOeI0IaeqyVd42aaWba aSqabeaacaWGnbaaaOGaeq4Wdm3aa0baaSqaaiaadUgacaWGRbaaba GaeyOhIukaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiaa cMcacaGGVaGaamyramaaCaaaleqabaGaamytaaaaaaa@547D@  and displacements u i = ε ij x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaa0baaSqaaiaadMgaaeaacq GHEisPaaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaa cqGHEisPaaGccaWG4bWaaSbaaSqaaiaadQgaaeqaaaaa@3EA0@ .

 

The solution is constructed by superposing the Eshelby solution to the uniform stress state.  To represent the Eshelby solution, we introduce:

1.      The Eshelby transformation strain ε ij T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGubaaaaaa@3750@

2.      The Eshelby tensor S ijkl * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaa0baaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeaacaGGQaaaaaaa@3837@

3.      The displacement induced by the Eshelby transformation u i = U ikl ( x m ) ε kl T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaamyvamaaBaaaleaacaWGPbGaam4AaiaadYgaaeqaaOGa aiikaiaadIhadaWgaaWcbaGaamyBaaqabaGccaGGPaGaeqyTdu2aa0 baaSqaaiaadUgacaWGSbaabaGaamivaaaaaaa@41D5@

4.      The stresses induced by the Eshelby transformation σ ij = Σ ijkl ( x m ) ε kl T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaeu4Odm1aaSbaaSqaaiaadMgacaWGQbGaam4A aiaadYgaaeqaaOGaaiikaiaadIhadaWgaaWcbaGaamyBaaqabaGcca GGPaGaeqyTdu2aa0baaSqaaiaadUgacaWGSbaabaGaamivaaaaaaa@4526@

The functions S ijkl * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaa0baaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeaacaGGQaaaaaaa@3837@   U ikl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGvbWaaSbaaSqaaiaadMgacaWGRb GaamiBaaqabaaaaa@369B@  and Σ ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHJoWudaWgaaWcbaGaamyAaiaadQ gacaWGRbGaamiBaaqabaaaaa@3834@  can be calculated using the results given in Section 5.4.6 (the elastic properties of the matrix should be used when evaluating the formulas).

 

The solution for the solid containing the inclusion follows as

u i = U ikl ( x m ) ε kl T + u i σ ij = Σ ijkl ( x m ) ε kl T + σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaamyvamaaBaaaleaacaWGPbGaam4AaiaadYgaaeqaaOGa aiikaiaadIhadaWgaaWcbaGaamyBaaqabaGccaGGPaGaeqyTdu2aa0 baaSqaaiaadUgacaWGSbaabaGaamivaaaakiabgUcaRiaadwhadaqh aaWcbaGaamyAaaqaaiabg6HiLcaakiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa Vlabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcqqHJo WudaWgaaWcbaGaamyAaiaadQgacaWGRbGaamiBaaqabaGccaGGOaGa amiEamaaBaaaleaacaWGTbaabeaakiaacMcacqaH1oqzdaqhaaWcba Gaam4AaiaadYgaaeaacaWGubaaaOGaey4kaSIaeq4Wdm3aa0baaSqa aiaadMgacaWGQbaabaGaeyOhIukaaaaa@715F@

where the transformation strain is calculated by solving

( C ijkl M C ijkl I ) ε k , l =( C ijpq M ( C ijkl M C ijkl I ) S klpq ) ε pq T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaam4qamaaDaaaleaacaWGPb GaamOAaiaadUgacaWGSbaabaGaamytaaaakiabgkHiTiaadoeadaqh aaWcbaGaamyAaiaadQgacaWGRbGaamiBaaqaaiaadMeaaaGccaGGPa GaeqyTdu2aa0baaSqaaiaadUgaaeaacqGHEisPaaGccaGGSaWaaSba aSqaaiaadYgaaeqaaOGaeyypa0JaaiikaiaadoeadaqhaaWcbaGaam yAaiaadQgacaWGWbGaamyCaaqaaiaad2eaaaGccqGHsislcaGGOaGa am4qamaaDaaaleaacaWGPbGaamOAaiaadUgacaWGSbaabaGaamytaa aakiabgkHiTiaadoeadaqhaaWcbaGaamyAaiaadQgacaWGRbGaamiB aaqaaiaadMeaaaGccaGGPaGaam4uamaaDaaaleaacaWGRbGaamiBai aadchacaWGXbaabaGaey4fIOcaaOGaaiykaiabew7aLnaaDaaaleaa caWGWbGaamyCaaqaaiaadsfaaaaaaa@66E2@

for ε ij T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGubaaaaaa@3750@ .  Here,

C ijkl M = E M 2( 1+ ν M ) ( δ il δ jk + δ ik δ jl )+ E M ν M ( 1+ ν M )( 12 ν M ) δ ij δ kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaa0baaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeaacaWGnbaaaOGaeyypa0ZaaSaaaeaacaWGfbWa aWbaaSqabeaacaWGnbaaaaGcbaGaaGOmamaabmaabaGaaGymaiabgU caRiabe27aUnaaCaaaleqabaGaamytaaaaaOGaayjkaiaawMcaaaaa daqadaqaaiabes7aKnaaBaaaleaacaWGPbGaamiBaaqabaGccqaH0o azdaWgaaWcbaGaamOAaiaadUgaaeqaaOGaey4kaSIaeqiTdq2aaSba aSqaaiaadMgacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGQbGaam iBaaqabaaakiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaadweadaah aaWcbeqaaiaad2eaaaGccqaH9oGBdaahaaWcbeqaaiaad2eaaaaake aadaqadaqaaiaaigdacqGHRaWkcqaH9oGBdaahaaWcbeqaaiaad2ea aaaakiaawIcacaGLPaaadaqadaqaaiaaigdacqGHsislcaaIYaGaeq yVd42aaWbaaSqabeaacaWGnbaaaaGccaGLOaGaayzkaaaaaiabes7a KnaaBaaaleaacaWGPbGaamOAaaqabaGccqaH0oazdaWgaaWcbaGaam 4AaiaadYgaaeqaaaaa@6CB9@

is the stiffness of the matrix, with a similar expression for the stiffness of the inclusion.

 

 

 

5.4.8 Spherical cavity in an infinite solid subjected to remote stress

 

The figure shows a spherical cavity with radius a in an infinite, isotropic linear elastic solid. Far from the cavity, the solid is subjected to a tensile stress σ 33 = σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaaG4maa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaa@378A@ , with all other stress components zero.

 

The solution is generated by potentials

Ψ i = (1ν) σ 0 (1+ν) x 3 δ i3 + a 3 (1ν) σ 0 R 3 (75ν) ( (5ν1) 2(12ν) x i +5 x 3 δ i3 ) ϕ= ν(1ν) σ 0 (1+ν) (3 x 3 2 R 2 )+ a 3 (1ν) σ 0 R 3 (75ν) ( (75ν) 2(12ν) R 2 a 2 + 3 x 3 2 a 2 R 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabfI6aznaaBaaaleaacaWGPb aabeaakiabg2da9maalaaabaGaaiikaiaaigdacqGHsislcqaH9oGB caGGPaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaGcbaGaaiikaiaaig dacqGHRaWkcqaH9oGBcaGGPaaaaiaadIhadaWgaaWcbaGaaG4maaqa baGccqaH0oazdaWgaaWcbaGaamyAaiaaiodaaeqaaOGaey4kaSYaaS aaaeaacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaaiikaiaaigdacqGH sislcqaH9oGBcaGGPaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaGcba GaamOuamaaCaaaleqabaGaaG4maaaakiaacIcacaaI3aGaeyOeI0Ia aGynaiabe27aUjaacMcaaaWaaeWaaeaadaWcaaqaaiaacIcacaaI1a GaeqyVd4MaeyOeI0IaaGymaiaacMcaaeaacaaIYaGaaiikaiaaigda cqGHsislcaaIYaGaeqyVd4MaaiykaaaacaWG4bWaaSbaaSqaaiaadM gaaeqaaOGaey4kaSIaaGynaiaadIhadaWgaaWcbaGaaG4maaqabaGc cqaH0oazdaWgaaWcbaGaamyAaiaaiodaaeqaaaGccaGLOaGaayzkaa aabaGaaGPaVlabew9aMjabg2da9maalaaabaGaeqyVd4Maaiikaiaa igdacqGHsislcqaH9oGBcaGGPaGaeq4Wdm3aaSbaaSqaaiaaicdaae qaaaGcbaGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaaaaiaacIca caaIZaGaamiEamaaDaaaleaacaaIZaaabaGaaGOmaaaakiabgkHiTi aadkfadaahaaWcbeqaaiaaikdaaaGccaGGPaGaey4kaSYaaSaaaeaa caWGHbWaaWbaaSqabeaacaaIZaaaaOGaaiikaiaaigdacqGHsislcq aH9oGBcaGGPaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaGcbaGaamOu amaaCaaaleqabaGaaG4maaaakiaacIcacaaI3aGaeyOeI0IaaGynai abe27aUjaacMcaaaWaaeWaaeaadaWcaaqaaiaacIcacaaI3aGaeyOe I0IaaGynaiabe27aUjaacMcaaeaacaaIYaGaaiikaiaaigdacqGHsi slcaaIYaGaeqyVd4MaaiykaaaacaWGsbWaaWbaaSqabeaacaaIYaaa aOGaeyOeI0IaamyyamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaala aabaGaaG4maiaadIhadaqhaaWcbaGaaG4maaqaaiaaikdaaaGccaWG HbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOuamaaCaaaleqabaGaaG OmaaaaaaaakiaawIcacaGLPaaaaaaa@B52E@

The displacements and stresses follow as

u i = (1+ν) σ 0 2E ( [ 2+ 5(54ν) (75ν) a 3 R 3 + 6 (75ν) a 5 R 5 ] x 3 δ i3 +[ 2ν (1+ν) + (5ν6) (75ν) a 3 R 3 + 3 (75ν) a 5 R 5 ( 1 5 x 3 2 R 2 ) ] x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyDamaaBaaaleaacaWGPbaabe aakiabg2da9maalaaabaGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGG PaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaGcbaGaaGOmaiaadweaaa WaaeqaaeaadaWadaqaaiaaikdacqGHRaWkdaWcaaqaaiaaiwdacaGG OaGaaGynaiabgkHiTiaaisdacqaH9oGBcaGGPaaabaGaaiikaiaaiE dacqGHsislcaaI1aGaeqyVd4MaaiykaaaadaWcaaqaaiaadggadaah aaWcbeqaaiaaiodaaaaakeaacaWGsbWaaWbaaSqabeaacaaIZaaaaa aakiabgUcaRmaalaaabaGaaGOnaaqaaiaacIcacaaI3aGaeyOeI0Ia aGynaiabe27aUjaacMcaaaWaaSaaaeaacaWGHbWaaWbaaSqabeaaca aI1aaaaaGcbaGaamOuamaaCaaaleqabaGaaGynaaaaaaaakiaawUfa caGLDbaaaiaawIcaaiaadIhadaWgaaWcbaGaaG4maaqabaGccqaH0o azdaWgaaWcbaGaamyAaiaaiodaaeqaaaGcbaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daqaca qaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7cqGHRaWkdaWadaqaamaalaaabaGaeyOeI0 IaaGOmaiabe27aUbqaaiaacIcacaaIXaGaey4kaSIaeqyVd4Maaiyk aaaacqGHRaWkdaWcaaqaaiaacIcacaaI1aGaeqyVd4MaeyOeI0IaaG OnaiaacMcaaeaacaGGOaGaaG4naiabgkHiTiaaiwdacqaH9oGBcaGG PaaaamaalaaabaGaamyyamaaCaaaleqabaGaaG4maaaaaOqaaiaadk fadaahaaWcbeqaaiaaiodaaaaaaOGaey4kaSYaaSaaaeaacaaIZaaa baGaaiikaiaaiEdacqGHsislcaaI1aGaeqyVd4MaaiykaaaadaWcaa qaaiaadggadaahaaWcbeqaaiaaiwdaaaaakeaacaWGsbWaaWbaaSqa beaacaaI1aaaaaaakmaabmaabaGaaGymaiabgkHiTmaalaaabaGaaG ynaiaadIhadaqhaaWcbaGaaG4maaqaaiaaikdaaaaakeaacaWGsbWa aWbaaSqabeaacaaIYaaaaaaaaOGaayjkaiaawMcaaaGaay5waiaaw2 faaiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawMcaaiaaykW6aaaa @FD94@
σ ij σ 0 = 3 a 3 2(75ν) R 3 ( 35ν4 a 2 R 2 ) δ ij + 3 a 3 x i x j 2(75ν) R 5 ( 65ν5 a 2 R 2 +10 x 3 2 R 2 ) + δ i3 δ j3 (75ν) ( (75ν)+5(12ν) a 3 R 3 +3 a 5 R 5 ) 15 a 3 x 3 ( x j δ i3 + x i δ j3 ) (75ν) R 5 ( a 2 R 2 ν ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacqaHdpWCdaWgaaWcba GaamyAaiaadQgaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqa aaaakiabg2da9maalaaabaGaaG4maiaadggadaahaaWcbeqaaiaaio daaaaakeaacaaIYaGaaiikaiaaiEdacqGHsislcaaI1aGaeqyVd4Ma aiykaiaadkfadaahaaWcbeqaaiaaiodaaaaaaOWaaeWaaeaacaaIZa GaeyOeI0IaaGynaiabe27aUjabgkHiTiaaisdadaWcaaqaaiaadgga daahaaWcbeqaaiaaikdaaaaakeaacaWGsbWaaWbaaSqabeaacaaIYa aaaaaaaOGaayjkaiaawMcaaiabes7aKnaaBaaaleaacaWGPbGaamOA aaqabaGccqGHRaWkdaWcaaqaaiaaiodacaWGHbWaaWbaaSqabeaaca aIZaaaaOGaamiEamaaBaaaleaacaWGPbaabeaakiaadIhadaWgaaWc baGaamOAaaqabaaakeaacaaIYaGaaiikaiaaiEdacqGHsislcaaI1a GaeqyVd4MaaiykaiaadkfadaahaaWcbeqaaiaaiwdaaaaaaOWaaeWa aeaacaaI2aGaeyOeI0IaaGynaiabe27aUjabgkHiTiaaiwdadaWcaa qaaiaadggadaahaaWcbeqaaiaaikdaaaaakeaacaWGsbWaaWbaaSqa beaacaaIYaaaaaaakiabgUcaRiaaigdacaaIWaWaaSaaaeaacaWG4b Waa0baaSqaaiaaiodaaeaacaaIYaaaaaGcbaGaamOuamaaCaaaleqa baGaaGOmaaaaaaaakiaawIcacaGLPaaaaeaacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlabgUcaRmaalaaabaGaeqiTdq2aaSbaaSqaaiaadMgacaaIZaaabe aakiabes7aKnaaBaaaleaacaWGQbGaaG4maaqabaaakeaacaGGOaGa aG4naiabgkHiTiaaiwdacqaH9oGBcaGGPaaaamaabmaabaGaaiikai aaiEdacqGHsislcaaI1aGaeqyVd4MaaiykaiabgUcaRiaaiwdacaGG OaGaaGymaiabgkHiTiaaikdacqaH9oGBcaGGPaWaaSaaaeaacaWGHb WaaWbaaSqabeaacaaIZaaaaaGcbaGaamOuamaaCaaaleqabaGaaG4m aaaaaaGccqGHRaWkcaaIZaWaaSaaaeaacaWGHbWaaWbaaSqabeaaca aI1aaaaaGcbaGaamOuamaaCaaaleqabaGaaGynaaaaaaaakiaawIca caGLPaaacqGHsisldaWcaaqaaiaaigdacaaI1aGaamyyamaaCaaale qabaGaaG4maaaakiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGOaGa amiEamaaBaaaleaacaWGQbaabeaakiabes7aKnaaBaaaleaacaWGPb GaaG4maaqabaGccqGHRaWkcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGa eqiTdq2aaSbaaSqaaiaadQgacaaIZaaabeaakiaacMcaaeaacaGGOa GaaG4naiabgkHiTiaaiwdacqaH9oGBcaGGPaGaamOuamaaCaaaleqa baGaaGynaaaaaaGcdaqadaqaamaalaaabaGaamyyamaaCaaaleqaba GaaGOmaaaaaOqaaiaadkfadaahaaWcbeqaaiaaikdaaaaaaOGaeyOe I0IaeqyVd4gacaGLOaGaayzkaaaaaaa@CBCE@
 

Derivation: This solution can be derived by superposing two solutions:

  1. A uniform state of stress σ ij = σ 0 δ i3 δ j3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOGaeqiT dq2aaSbaaSqaaiaadMgacaaIZaaabeaakiabes7aKnaaBaaaleaaca WGQbGaaG4maaqabaaaaa@4158@ , which can be generated from potentials Ψ i =(1ν) σ 0 x 3 δ i3 /(1+ν) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHOoqwdaWgaaWcbaGaamyAaaqaba GccqGH9aqpcaGGOaGaaGymaiabgkHiTiabe27aUjaacMcacqaHdpWC daWgaaWcbaGaaGimaaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaO GaeqiTdq2aaSbaaSqaaiaadMgacaaIZaaabeaakiaac+cacaGGOaGa aGymaiabgUcaRiabe27aUjaacMcaaaa@48C2@ , ϕ=ν(1ν) σ 0 (3 x 3 2 R 2 )/(1+ν) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcqGH9aqpcqaH9oGBcaGGOa GaaGymaiabgkHiTiabe27aUjaacMcacqaHdpWCdaWgaaWcbaGaaGim aaqabaGccaGGOaGaaG4maiaadIhadaqhaaWcbaGaaG4maaqaaiaaik daaaGccqGHsislcaWGsbWaaWbaaSqabeaacaaIYaaaaOGaaiykaiaa c+cacaGGOaGaaGymaiabgUcaRiabe27aUjaacMcaaaa@4B93@
  2. The Eshelby solution for a sphere with transformation stress p ij T =A δ ij +B δ i3 δ j3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaa0baaSqaaiaadMgacaWGQb aabaGaamivaaaakiabg2da9iaadgeacqaH0oazdaWgaaWcbaGaamyA aiaadQgaaeqaaOGaey4kaSIaamOqaiabes7aKnaaBaaaleaacaWGPb GaaG4maaqabaGccqaH0oazdaWgaaWcbaGaamOAaiaaiodaaeqaaaaa @44D8@ .

 The unknown coefficients A and B must be chosen to satisfy the traction free boundary condition σ ij n j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaamOBamaaBaaaleaacaWGQbaabeaakiabg2da9iaaicda aaa@3A74@  on the surface of the hole R=a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbGaeyypa0Jaamyyaaaa@3589@ .  Noting that n j = x j /a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadQgaaeqaaO Gaeyypa0JaeyOeI0IaamiEamaaBaaaleaacaWGQbaabeaakiaac+ca caWGHbaaaa@3A8C@  and working through some tedious algebra shows that

A= 3 σ 0 (1ν)(5ν1) 2(75ν)(12ν) B= 15 σ 0 (1ν) (75ν) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbGaeyypa0ZaaSaaaeaacaaIZa Gaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOGaaiikaiaaigdacqGHsisl cqaH9oGBcaGGPaGaaiikaiaaiwdacqaH9oGBcqGHsislcaaIXaGaai ykaaqaaiaaikdacaGGOaGaaG4naiabgkHiTiaaiwdacqaH9oGBcaGG PaGaaiikaiaaigdacqGHsislcaaIYaGaeqyVd4MaaiykaaaacaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caWGcbGaeyypa0ZaaSaaaeaacaaIXaGaaG ynaiabeo8aZnaaBaaaleaacaaIWaaabeaakiaacIcacaaIXaGaeyOe I0IaeqyVd4MaaiykaaqaaiaacIcacaaI3aGaeyOeI0IaaGynaiabe2 7aUjaacMcaaaaaaa@70B7@

Substituting back into the Eshelby potentials and simplifying yields the results given.  The same approach can be used to derive the solution for a rigid inclusion in an infinite solid subjected to remote stress, as well as the solution to an elastically mismatched spherical inclusion in an infinite solid.

 

 

 

5.4.9 Flat ended cylindrical indenter in contact with an elastic half-space

 

A rigid flat ended cylindrical punch with radius a is pushed into the surface of an elastic half-space with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@  by a force P.  The indenter sinks into the surface by a depth h. The interface between the contacting surfaces is frictionless

 

The load is related to the displacement of the punch by

P= 2Ea (1 ν 2 ) h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGqbGaeyypa0ZaaSaaaeaacaaIYa GaamyraiaadggaaeaacaGGOaGaaGymaiabgkHiTiabe27aUnaaCaaa leqabaGaaGOmaaaakiaacMcaaaGaamiAaaaa@3DB6@

The solution can be generated from Papkovich-Neuber potentials

Ψ k = 2Eh δ k3 (1+ν)π Im{ log( R * + x 3 +ia ) }ϕ= 2(12ν)Eh (1+ν)π Im{ ( x 3 +ia)log( R * + x 3 +ia ) R * } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHOoqwdaWgaaWcbaGaam4Aaaqaba GccqGH9aqpdaWcaaqaaiaaikdacaWGfbGaamiAaiabes7aKnaaBaaa leaacaWGRbGaaG4maaqabaaakeaacaGGOaGaaGymaiabgUcaRiabe2 7aUjaacMcacqaHapaCaaGaciysaiaac2gadaGadaqaaiGacYgacaGG VbGaai4zamaabmaabaGaamOuamaaCaaaleqabaGaaiOkaaaakiabgU caRiaadIhadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcaWGPbGaamyy aaGaayjkaiaawMcaaaGaay5Eaiaaw2haaiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua eqy1dyMaeyypa0JaeyOeI0YaaSaaaeaacaaIYaGaaiikaiaaigdacq GHsislcaaIYaGaeqyVd4MaaiykaiaadweacaWGObaabaGaaiikaiaa igdacqGHRaWkcqaH9oGBcaGGPaGaeqiWdahaaiGacMeacaGGTbWaai WaaeaacaGGOaGaamiEamaaBaaaleaacaaIZaaabeaakiabgUcaRiaa dMgacaWGHbGaaiykaiGacYgacaGGVbGaai4zamaabmaabaGaamOuam aaCaaaleqabaGaaiOkaaaakiabgUcaRiaadIhadaWgaaWcbaGaaG4m aaqabaGccqGHRaWkcaWGPbGaamyyaaGaayjkaiaawMcaaiabgkHiTi aadkfadaahaaWcbeqaaiaacQcaaaaakiaawUhacaGL9baaaaa@8DD2@

where R * = x 1 2 + x 2 2 + ( x 3 +ia) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaWbaaSqabeaacaGGQaaaaO Gaeyypa0ZaaOaaaeaacaWG4bWaa0baaSqaaiaaigdaaeaacaaIYaaa aOGaey4kaSIaamiEamaaDaaaleaacaaIYaaabaGaaGOmaaaakiabgU caRiaacIcacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamyA aiaadggacaGGPaWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@439B@ , i= 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGPbGaeyypa0ZaaOaaaeaacqGHsi slcaaIXaaaleqaaaaa@367D@  and Im{ z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGjbGaaiyBamaacmaabaGaamOEaa Gaay5Eaiaaw2haaaaa@37B6@  denotes the imaginary part of z.  The displacements and stresses follow as

u k = h π(1ν) Im{ 2(1ν) δ k3 log( R * + x 3 +ia ) x 3 δ k3 R * + x 1 δ k1 + x 2 δ k2 R * + x 3 +ia ( 12ν x 3 R * ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadUgaaeqaaO Gaeyypa0ZaaSaaaeaacaWGObaabaGaeqiWdaNaaiikaiaaigdacqGH sislcqaH9oGBcaGGPaaaaiGacMeacaGGTbWaaiWaaeaacaaIYaGaai ikaiaaigdacqGHsislcqaH9oGBcaGGPaGaeqiTdq2aaSbaaSqaaiaa dUgacaaIZaaabeaakiGacYgacaGGVbGaai4zamaabmaabaGaamOuam aaCaaaleqabaGaaiOkaaaakiabgUcaRiaadIhadaWgaaWcbaGaaG4m aaqabaGccqGHRaWkcaWGPbGaamyyaaGaayjkaiaawMcaaiabgkHiTm aalaaabaGaamiEamaaBaaaleaacaaIZaaabeaakiabes7aKnaaBaaa leaacaWGRbGaaG4maaqabaaakeaacaWGsbWaaWbaaSqabeaacaGGQa aaaaaakiabgUcaRmaalaaabaGaamiEamaaBaaaleaacaaIXaaabeaa kiabes7aKnaaBaaaleaacaWGRbGaaGymaaqabaGccqGHRaWkcaWG4b WaaSbaaSqaaiaaikdaaeqaaOGaeqiTdq2aaSbaaSqaaiaadUgacaaI YaaabeaaaOqaaiaadkfadaahaaWcbeqaaiaacQcaaaGccqGHRaWkca WG4bWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamyAaiaadggaaaWa aeWaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUjabgkHiTmaalaaaba GaamiEamaaBaaaleaacaaIZaaabeaaaOqaaiaadkfadaahaaWcbeqa aiaacQcaaaaaaaGccaGLOaGaayzkaaaacaGL7bGaayzFaaaaaa@7C6B@

σ 11 = Eh (1 ν 2 )π Im{ 2ν R * + 12ν x 3 / R * R * + x 3 +ia x 1 2 R * ( R * + x 3 +ia) 2 ( 1+ x 3 (1+2 R * + x 3 +ia) R *2 ) } σ 22 = Eh (1 ν 2 )π Im{ 2ν R * + 12ν x 3 / R * R * + x 3 +ia x 2 2 R * ( R * + x 3 +ia) 2 ( 1+ x 3 (1+2 R * + x 3 +ia) R *2 ) } σ 33 = Eh (1 ν 2 )π Im{ 1 R * + ( x 3 +ia) x 3 R *3 } σ 13 = Eh (1 ν 2 )π Im{ x 1 x 3 R *3 } σ 23 = Eh (1 ν 2 )π Im{ x 2 x 3 R *3 } σ 12 = Eh (1 ν 2 )π Im{ x 1 x 2 R * ( R * + x 3 +ia) ( (12ν)+ x 3 (2 R * + x 3 +ia) R * ( R * + x 3 +ia) ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH9aqpdaWcaaqaaiaadweacaWGObaabaGaaiik aiaaigdacqGHsislcqaH9oGBdaahaaWcbeqaaiaaikdaaaGccaGGPa GaeqiWdahaaiGacMeacaGGTbWaaiWaaeaadaWcaaqaaiaaikdacqaH 9oGBaeaacaWGsbWaaWbaaSqabeaacaGGQaaaaaaakiabgUcaRmaala aabaGaaGymaiabgkHiTiaaikdacqaH9oGBcqGHsislcaWG4bWaaSba aSqaaiaaiodaaeqaaOGaai4laiaadkfadaahaaWcbeqaaiaacQcaaa aakeaacaWGsbWaaWbaaSqabeaacaGGQaaaaOGaey4kaSIaamiEamaa BaaaleaacaaIZaaabeaakiabgUcaRiaadMgacaWGHbaaaiabgkHiTm aalaaabaGaamiEamaaDaaaleaacaaIXaaabaGaaGOmaaaaaOqaaiaa dkfadaahaaWcbeqaaiaacQcaaaGccaGGOaGaamOuamaaCaaaleqaba GaaiOkaaaakiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGccqGH RaWkcaWGPbGaamyyaiaacMcadaahaaWcbeqaaiaaikdaaaaaaOWaae WaaeaacaaIXaGaey4kaSYaaSaaaeaacaWG4bWaaSbaaSqaaiaaioda aeqaaOGaaiikaiaaigdacqGHRaWkcaaIYaGaamOuamaaCaaaleqaba GaaiOkaaaakiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGccqGH RaWkcaWGPbGaamyyaiaacMcaaeaacaWGsbWaaWbaaSqabeaacaGGQa GaaGOmaaaaaaaakiaawIcacaGLPaaaaiaawUhacaGL9baaaeaacqaH dpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaaca WGfbGaamiAaaqaaiaacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqa beaacaaIYaaaaOGaaiykaiabec8aWbaaciGGjbGaaiyBamaacmaaba WaaSaaaeaacaaIYaGaeqyVd4gabaGaamOuamaaCaaaleqabaGaaiOk aaaaaaGccqGHRaWkdaWcaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4 MaeyOeI0IaamiEamaaBaaaleaacaaIZaaabeaakiaac+cacaWGsbWa aWbaaSqabeaacaGGQaaaaaGcbaGaamOuamaaCaaaleqabaGaaiOkaa aakiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcaWG PbGaamyyaaaacqGHsisldaWcaaqaaiaadIhadaqhaaWcbaGaaGOmaa qaaiaaikdaaaaakeaacaWGsbWaaWbaaSqabeaacaGGQaaaaOGaaiik aiaadkfadaahaaWcbeqaaiaacQcaaaGccqGHRaWkcaWG4bWaaSbaaS qaaiaaiodaaeqaaOGaey4kaSIaamyAaiaadggacaGGPaWaaWbaaSqa beaacaaIYaaaaaaakmaabmaabaGaaGymaiabgUcaRmaalaaabaGaam iEamaaBaaaleaacaaIZaaabeaakiaacIcacaaIXaGaey4kaSIaaGOm aiaadkfadaahaaWcbeqaaiaacQcaaaGccqGHRaWkcaWG4bWaaSbaaS qaaiaaiodaaeqaaOGaey4kaSIaamyAaiaadggacaGGPaaabaGaamOu amaaCaaaleqabaGaaiOkaiaaikdaaaaaaaGccaGLOaGaayzkaaaaca GL7bGaayzFaaaabaGaeq4Wdm3aaSbaaSqaaiaaiodacaaIZaaabeaa kiabg2da9maalaaabaGaamyraiaadIgaaeaacaGGOaGaaGymaiabgk HiTiabe27aUnaaCaaaleqabaGaaGOmaaaakiaacMcacqaHapaCaaGa ciysaiaac2gadaGadaqaamaalaaabaGaaGymaaqaaiaadkfadaahaa WcbeqaaiaacQcaaaaaaOGaey4kaSYaaSaaaeaacaGGOaGaamiEamaa BaaaleaacaaIZaaabeaakiabgUcaRiaadMgacaWGHbGaaiykaiaadI hadaWgaaWcbaGaaG4maaqabaaakeaacaWGsbWaaWbaaSqabeaacaGG QaGaaG4maaaaaaaakiaawUhacaGL9baacaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaaigdacaaIZaaabeaakiabg2 da9maalaaabaGaamyraiaadIgaaeaacaGGOaGaaGymaiabgkHiTiab e27aUnaaCaaaleqabaGaaGOmaaaakiaacMcacqaHapaCaaGaciysai aac2gadaGadaqaamaalaaabaGaamiEamaaBaaaleaacaaIXaaabeaa kiaadIhadaWgaaWcbaGaaG4maaqabaaakeaacaWGsbWaaWbaaSqabe aacaGGQaGaaG4maaaaaaaakiaawUhacaGL9baacaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIYaGaaG4m aaqabaGccqGH9aqpdaWcaaqaaiaadweacaWGObaabaGaaiikaiaaig dacqGHsislcqaH9oGBdaahaaWcbeqaaiaaikdaaaGccaGGPaGaeqiW dahaaiGacMeacaGGTbWaaiWaaeaadaWcaaqaaiaadIhadaWgaaWcba GaaGOmaaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamOu amaaCaaaleqabaGaaiOkaiaaiodaaaaaaaGccaGL7bGaayzFaaaaba Gaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9maalaaa baGaamyraiaadIgaaeaacaGGOaGaaGymaiabgkHiTiabe27aUnaaCa aaleqabaGaaGOmaaaakiaacMcacqaHapaCaaGaciysaiaac2gadaGa daqaamaalaaabaGaamiEamaaBaaaleaacaaIXaaabeaakiaadIhada WgaaWcbaGaaGOmaaqabaaakeaacaWGsbWaaWbaaSqabeaacaGGQaaa aOGaaiikaiaadkfadaahaaWcbeqaaiaacQcaaaGccqGHRaWkcaWG4b WaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamyAaiaadggacaGGPaaa amaabmaabaGaeyOeI0IaaiikaiaaigdacqGHsislcaaIYaGaeqyVd4 MaaiykaiabgUcaRmaalaaabaGaamiEamaaBaaaleaacaaIZaaabeaa kiaacIcacaaIYaGaamOuamaaCaaaleqabaGaaiOkaaaakiabgUcaRi aadIhadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcaWGPbGaamyyaiaa cMcaaeaacaWGsbWaaWbaaSqabeaacaGGQaaaaOGaaiikaiaadkfada ahaaWcbeqaaiaacQcaaaGccqGHRaWkcaWG4bWaaSbaaSqaaiaaioda aeqaaOGaey4kaSIaamyAaiaadggacaGGPaaaaaGaayjkaiaawMcaaa Gaay5Eaiaaw2haaaaaaa@7C49@

A symbolic manipulation program can handle the complex arithmetic in these formulas without difficulty.  If you wish to find analytical formulas for the displacement or stress, the following expressions are helpful

R * = 1 2 ( ρ 2 + R 2 a 2 +i ρ 2 + a 2 R 2 )Im{ log( R * + x 3 +ia) }= tan 1 ( ρ 2 + a 2 R 2 +a 2 ρ 2 + R 2 a 2 + x 3 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaWbaaSqabeaacaGGQaaaaO Gaeyypa0ZaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIYaaaleqaaaaa kmaabmaabaWaaOaaaeaacqaHbpGCdaahaaWcbeqaaiaaikdaaaGccq GHRaWkcaWGsbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaamyyamaa CaaaleqabaGaaGOmaaaaaeqaaOGaey4kaSIaamyAamaakaaabaGaeq yWdi3aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyyamaaCaaaleqa baGaaGOmaaaakiabgkHiTiaadkfadaahaaWcbeqaaiaaikdaaaaabe aaaOGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7ciGGjbGaaiyBamaacmaabaGaciiBaiaac+gacaGGNb GaaiikaiaadkfadaahaaWcbeqaaiaacQcaaaGccqGHRaWkcaWG4bWa aSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamyAaiaadggacaGGPaaaca GL7bGaayzFaaGaeyypa0JaciiDaiaacggacaGGUbWaaWbaaSqabeaa cqGHsislcaaIXaaaaOWaaeWaaeaadaWcaaqaamaakaaabaGaeqyWdi 3aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyyamaaCaaaleqabaGa aGOmaaaakiabgkHiTiaadkfadaahaaWcbeqaaiaaikdaaaaabeaaki abgUcaRiaadggadaGcaaqaaiaaikdaaSqabaaakeaadaGcaaqaaiab eg8aYnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadkfadaahaaWcbe qaaiaaikdaaaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIYaaaaaqa baGccqGHRaWkcaWG4bWaaSbaaSqaaiaaiodaaeqaaOWaaOaaaeaaca aIYaaaleqaaaaaaOGaayjkaiaawMcaaaaa@8336@

where R= x k x k ρ= ( [ R 2 a 2 ] 2 +4 x 3 2 a 2 ) 1/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbGaeyypa0ZaaOaaaeaacaWG4b WaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaa aeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaeqyWdiNaeyypa0ZaaeWaaeaadaWadaqaaiaadkfa daahaaWcbeqaaiaaikdaaaGccqGHsislcaWGHbWaaWbaaSqabeaaca aIYaaaaaGccaGLBbGaayzxaaWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaaGinaiaadIhadaqhaaWcbaGaaG4maaqaaiaaikdaaaGccaWGHb WaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaa caaIXaGaai4laiaaisdaaaaaaa@5B1F@

 

Important features of these results include:

1.      Contact pressure: The pressure exerted by the indenter on the elastic solid follows as

p( x 1 )= σ 33 (r, x 3 =0)= P 2πa a 2 r 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGPaGaeyypa0JaeyOeI0Iaeq4Wdm3aaSbaaSqa aiaaiodacaaIZaaabeaakiaacIcacaWGYbGaaiilaiaadIhadaWgaa WcbaGaaG4maaqabaGccqGH9aqpcaaIWaGaaiykaiabg2da9maalaaa baGaamiuaaqaaiaaikdacqaHapaCcaWGHbWaaOaaaeaacaWGHbWaaW baaSqabeaacaaIYaaaaOGaeyOeI0IaamOCamaaCaaaleqabaGaaGOm aaaaaeqaaaaaaaa@4D18@

2.      Surface displacement: The vertical displacement of the surface is

u 3 ={ 2h π tan 1 a r 2 a 2 r>a hr<a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaiodaaeqaaO Gaeyypa0ZaaiqaaeaafaqabeGabaaabaWaaSaaaeaacaaIYaGaamiA aaqaaiabec8aWbaaciGG0bGaaiyyaiaac6gadaahaaWcbeqaaiabgk HiTiaaigdaaaGcdaWcaaqaaiaadggaaeaadaGcaaqaaiaadkhadaah aaWcbeqaaiaaikdaaaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIYa aaaaqabaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamOCaiabg6da+i aadggaaeaacaWGObGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGYbGa eyipaWJaamyyaaaaaiaawUhaaaaa@AFBA@

3.      Contact stiffness: the stiffness of a contact is defined as the ratio of the force acting on the indenter to its displacement k c =P/h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGRbWaaSbaaSqaaiaadogaaeqaaO Gaeyypa0Jaamiuaiaac+cacaWGObaaaa@384F@ , and is of interest in practical applications.  The stiffness of a 3D contact is well defined (unlike 2D contacts discussed in Section 5.4) and is given by k c =2Ea/(1 ν 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGRbWaaSbaaSqaaiaadogaaeqaaO Gaeyypa0JaaGOmaiaadweacaWGHbGaai4laiaacIcacaaIXaGaeyOe I0IaeqyVd42aaWbaaSqabeaacaaIYaaaaOGaaiykaaaa@3EA5@ .  This turns out to be a universal relation for any axisymmetric contact with contact radius a.

 

 

5.4.10 Frictionless contact between two elastic spheres

 
This solution is known as the `Hertz contact problem’ after its author.  The figure illustrates the problem to be solved.  Two elastic spheres, with radii R A , R B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaamyqaaqabaGcca GGSaGaamOuamaaBaaaleaacaWGcbaabeaaaaa@34B5@  and elastic constants E A , ν A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaamyqaaqabaGcca GGSaGaeqyVd42aaSbaaSqaaiaadgeaaeqaaaaa@3588@ , E B , ν B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaamOqaaqabaGcca GGSaGaeqyVd42aaSbaaSqaaiaadkeaaeqaaaaa@358A@  initially meet at a point, and are pushed into contact by a force P.  The two spheres deform so as to make contact over a small circular patch with radius a<< R B , R B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacqGH8aapcqGH8aapcaWGsbWaaS baaSqaaiaadkeaaeqaaOGaaiilaiaadkfadaWgaaWcbaGaamOqaaqa baaaaa@37A4@ , and the centers of the two spheres approach one another by a distance h.
 
The solution is conveniently expressed in terms of an effective modulus and radius for the contact pair:

E * = E A E B (1 ν A 2 ) E B +(1 ν B 2 ) E A R * = R A R B R A + R B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaahaaWcbeqaaiaacQcaaaGccq GH9aqpdaWcaaqaaiaadweadaWgaaWcbaGaamyqaaqabaGccaWGfbWa aSbaaSqaaiaadkeaaeqaaaGcbaGaaiikaiaaigdacqGHsislcqaH9o GBdaqhaaWcbaGaamyqaaqaaiaaikdaaaGccaGGPaGaamyramaaBaaa leaacaWGcbaabeaakiabgUcaRiaacIcacaaIXaGaeyOeI0IaeqyVd4 2aa0baaSqaaiaadkeaaeaacaaIYaaaaOGaaiykaiaadweadaWgaaWc baGaamyqaaqabaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaadkfada ahaaWcbeqaaiaacQcaaaGccqGH9aqpdaWcaaqaaiaadkfadaWgaaWc baGaamyqaaqabaGccaWGsbWaaSbaaSqaaiaadkeaaeqaaaGcbaGaam OuamaaBaaaleaacaWGbbaabeaakiabgUcaRiaadkfadaWgaaWcbaGa amOqaaqabaaaaaaa@5932@

 
Relations between P,h,a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfacaGGSaGaamiAaiaacYcacaWGHb aaaa@3470@ The force P, approach of distant points h and contact area a are related by

a= ( 3PR 4 E * ) 1/3 h= a 2 R = ( 9 P 2 16R E *2 ) 1/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacqGH9aqpdaqadaqaamaalaaaba GaaG4maiaadcfacaWGsbaabaGaaGinaiaadweadaahaaWcbeqaaiaa cQcaaaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4lai aaiodaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaamiAaiabg2da9maalaaabaGaamyyam aaCaaaleqabaGaaGOmaaaaaOqaaiaadkfaaaGaeyypa0ZaaeWaaeaa daWcaaqaaiaaiMdacaWGqbWaaWbaaSqabeaacaaIYaaaaaGcbaGaaG ymaiaaiAdacaWGsbGaamyramaaCaaaleqabaGaaiOkaiaaikdaaaaa aaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4laiaaiodaaa GccaaMc8oaaa@5D1A@

 

Contact pressure:  The two solids are subjected to a repulsive pressure p(r)= p 0 1 r 2 / a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacaGGOaGaamOCaiaacMcacqGH9a qpcaWGWbWaaSbaaSqaaiaaicdaaeqaaOWaaOaaaeaacaaIXaGaeyOe I0IaamOCamaaCaaaleqabaGaaGOmaaaakiaac+cacaWGHbWaaWbaaS qabeaacaaIYaaaaaqabaaaaa@3CBC@  within the contact area.  The maximum contact pressure is related to the load applied to the spheres by

p 0 =( 3P 2π a 2 )= ( 6P E *2 π 3 R 2 ) 1/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaaGimaaqabaGccq GH9aqpdaqadaqaamaalaaabaGaaG4maiaadcfaaeaacaaIYaGaeqiW daNaamyyamaaCaaaleqabaGaaGOmaaaaaaaakiaawIcacaGLPaaacq GH9aqpdaqadaqaamaalaaabaGaaGOnaiaadcfacaWGfbWaaWbaaSqa beaacaGGQaGaaGOmaaaaaOqaaiabec8aWnaaCaaaleqabaGaaG4maa aakiaadkfadaahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGaayzkaaWa aWbaaSqabeaacaaIXaGaai4laiaaiodaaaaaaa@4842@

 

Contact stiffness: the stiffness of a contact is defined as the ratio of the force acting on the indenter to its displacement k c =dP/dh MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGRbWaaSbaaSqaaiaadogaaeqaaO Gaeyypa0JaamizaiaadcfacaGGVaGaamizaiaadIgaaaa@3A21@ , and is of interest in practical applications.  The stiffness of a 3D contact is well defined (unlike 2D contacts discussed in Section 5.4) and is given by k c =2 E * a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGRbWaaSbaaSqaaiaadogaaeqaaO Gaeyypa0JaaGOmaiaadweadaahaaWcbeqaaiaacQcaaaGccaWGHbaa aa@392B@ .  This turns out to be a universal relation for any axisymmetric contact with contact radius a.

 

Stress field The two spheres are subjected to the same contact pressure, and are both assumed to deform like a half-space (with a flat surface).  Consequently, the stress field is identical inside both spheres, and can be calculated from formulas derived by Hamilton, Proc I. Mech E. 197C (1983) p. 53.

σ 11 = p 0 a [ ϕ+ 1 r 2 { x 1 2 x 2 2 r 2 ( (1ν)N 12ν 3 ( NS+2AN+ a 3 )νM x 3 a )N( x 1 2 +2ν x 2 2 ) M x 1 2 x 3 a S } ] σ 22 = p 0 a [ ϕ+ 1 r 2 { x 2 2 x 1 2 r 2 ( (1ν)N 12ν 3 ( NS+2AN+ a 3 )νM x 3 a )N( x 2 2 +2ν x 1 2 ) M x 2 2 x 3 a S } ] σ 33 = p 0 a ( N+ a x 3 M S ) σ 13 = x 3 x 1 p 0 a ( N S x 3 H G 2 + H 2 ) σ 23 = x 3 x 2 p 0 a ( N S x 3 H G 2 + H 2 ) σ 12 = p 0 x 1 x 2 a r 4 [ (12ν){ N r 2 + 2 3 N(S+2A) x 3 ( x 3 N+aM)+ 2 3 a 3 }+ x 3 { aM r 2 S x 3 N+aM } ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aaSbaaSqaaiaaigdaca aIXaaabeaakiabg2da9maalaaabaGaamiCamaaBaaaleaacaaIWaaa beaaaOqaaiaadggaaaWaamWaaeaacqaHvpGzcqGHRaWkdaWcaaqaai aaigdaaeaacaWGYbWaaWbaaSqabeaacaaIYaaaaaaakmaacmaabaWa aSaaaeaacaWG4bWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaeyOeI0 IaamiEamaaDaaaleaacaaIYaaabaGaaGOmaaaaaOqaaiaadkhadaah aaWcbeqaaiaaikdaaaaaaOWaaeWaaeaacaGGOaGaaGymaiabgkHiTi abe27aUjaacMcacaWGobGaeyOeI0YaaSaaaeaacaaIXaGaeyOeI0Ia aGOmaiabe27aUbqaaiaaiodaaaWaaeWaaeaacaWGobGaam4uaiabgU caRiaaikdacaWGbbGaamOtaiabgUcaRiaadggadaahaaWcbeqaaiaa iodaaaaakiaawIcacaGLPaaacqGHsislcqaH9oGBcaWGnbGaamiEam aaBaaaleaacaaIZaaabeaakiaadggaaiaawIcacaGLPaaacqGHsisl caWGobGaaiikaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccq GHRaWkcaaIYaGaeqyVd4MaamiEamaaDaaaleaacaaIYaaabaGaaGOm aaaakiaacMcacqGHsisldaWcaaqaaiaad2eacaWG4bWaa0baaSqaai aaigdaaeaacaaIYaaaaOGaamiEamaaBaaaleaacaaIZaaabeaakiaa dggaaeaacaWGtbaaaaGaay5Eaiaaw2haaaGaay5waiaaw2faaaqaai abeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH9aqpdaWcaaqa aiaadchadaWgaaWcbaGaaGimaaqabaaakeaacaWGHbaaamaadmaaba Gaeqy1dyMaey4kaSYaaSaaaeaacaaIXaaabaGaamOCamaaCaaaleqa baGaaGOmaaaaaaGcdaGadaqaamaalaaabaGaamiEamaaDaaaleaaca aIYaaabaGaaGOmaaaakiabgkHiTiaadIhadaqhaaWcbaGaaGymaaqa aiaaikdaaaaakeaacaWGYbWaaWbaaSqabeaacaaIYaaaaaaakmaabm aabaGaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaGaamOtaiabgkHi TmaalaaabaGaaGymaiabgkHiTiaaikdacqaH9oGBaeaacaaIZaaaam aabmaabaGaamOtaiaadofacqGHRaWkcaaIYaGaamyqaiaad6eacqGH RaWkcaWGHbWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaayzkaaGaey OeI0IaeqyVd4MaamytaiaadIhadaWgaaWcbaGaaG4maaqabaGccaWG HbaacaGLOaGaayzkaaGaeyOeI0IaamOtaiaacIcacaWG4bWaa0baaS qaaiaaikdaaeaacaaIYaaaaOGaey4kaSIaaGOmaiabe27aUjaadIha daqhaaWcbaGaaGymaaqaaiaaikdaaaGccaGGPaGaeyOeI0YaaSaaae aacaWGnbGaamiEamaaDaaaleaacaaIYaaabaGaaGOmaaaakiaadIha daWgaaWcbaGaaG4maaqabaGccaWGHbaabaGaam4uaaaaaiaawUhaca GL9baaaiaawUfacaGLDbaaaeaacqaHdpWCdaWgaaWcbaGaaG4maiaa iodaaeqaaOGaeyypa0ZaaSaaaeaacaWGWbWaaSbaaSqaaiaaicdaae qaaaGcbaGaamyyaaaadaqadaqaaiabgkHiTiaad6eacqGHRaWkdaWc aaqaaiaadggacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaamytaaqaai aadofaaaaacaGLOaGaayzkaaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBaaale aacaaIXaGaaG4maaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaadIha daWgaaWcbaGaaG4maaqabaGccaWG4bWaaSbaaSqaaiaaigdaaeqaaO GaamiCamaaBaaaleaacaaIWaaabeaaaOqaaiaadggaaaWaaeWaaeaa daWcaaqaaiaad6eaaeaacaWGtbaaaiabgkHiTmaalaaabaGaamiEam aaBaaaleaacaaIZaaabeaakiaadIeaaeaacaWGhbWaaWbaaSqabeaa caaIYaaaaOGaey4kaSIaamisamaaCaaaleqabaGaaGOmaaaaaaaaki aawIcacaGLPaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIYaGaaG4maaqabaGccq GH9aqpcqGHsisldaWcaaqaaiaadIhadaWgaaWcbaGaaG4maaqabaGc caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaamiCamaaBaaaleaacaaIWa aabeaaaOqaaiaadggaaaWaaeWaaeaadaWcaaqaaiaad6eaaeaacaWG tbaaaiabgkHiTmaalaaabaGaamiEamaaBaaaleaacaaIZaaabeaaki aadIeaaeaacaWGhbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamis amaaCaaaleqabaGaaGOmaaaaaaaakiaawIcacaGLPaaaaeaacqaHdp WCdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaWG WbWaaSbaaSqaaiaaicdaaeqaaOGaamiEamaaBaaaleaacaaIXaaabe aakiaadIhadaWgaaWcbaGaaGOmaaqabaaakeaacaWGHbGaamOCamaa CaaaleqabaGaaGinaaaaaaGcdaWadaqaaiaacIcacaaIXaGaeyOeI0 IaaGOmaiabe27aUjaacMcadaGadaqaaiabgkHiTiaad6eacaWGYbWa aWbaaSqabeaacaaIYaaaaOGaey4kaSYaaSaaaeaacaaIYaaabaGaaG 4maaaacaWGobGaaiikaiaadofacqGHRaWkcaaIYaGaamyqaiaacMca cqGHsislcaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiikaiaadIhada WgaaWcbaGaaG4maaqabaGccaWGobGaey4kaSIaamyyaiaad2eacaGG PaGaey4kaSYaaSaaaeaacaaIYaaabaGaaG4maaaacaWGHbWaaWbaaS qabeaacaaIZaaaaaGccaGL7bGaayzFaaGaey4kaSIaamiEamaaBaaa leaacaaIZaaabeaakmaacmaabaGaeyOeI0YaaSaaaeaacaWGHbGaam ytaiaadkhadaahaaWcbeqaaiaaikdaaaaakeaacaWGtbaaaiabgkHi TiaadIhadaWgaaWcbaGaaG4maaqabaGccaWGobGaey4kaSIaamyyai aad2eaaiaawUhacaGL9baaaiaawUfacaGLDbaaaaaa@5EC7@

where

r= x 1 2 + x 2 2 A= r 2 + x 3 2 a 2 S= A 2 +4 a 2 x 3 2 M= (S+A)/2 N= (SA)/2 G= M 2 N 2 + x 3 MaNH=2MN+aM+ x 3 Nϕ=(1+ν) x 3 tan 1 (a/M) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamOCaiabg2da9maakaaabaGaam iEamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgUcaRiaadIhadaqh aaWcbaGaaGOmaaqaaiaaikdaaaaabeaakiaaykW7caaMc8UaaGPaVl aaykW7caaMc8Uaamyqaiabg2da9iaadkhadaahaaWcbeqaaiaaikda aaGccqGHRaWkcaWG4bWaa0baaSqaaiaaiodaaeaacaaIYaaaaOGaey OeI0IaamyyamaaCaaaleqabaGaaGOmaaaakiaaykW7caaMc8UaaGPa VlaadofacqGH9aqpdaGcaaqaaiaadgeadaahaaWcbeqaaiaaikdaaa GccqGHRaWkcaaI0aGaamyyamaaCaaaleqabaGaaGOmaaaakiaadIha daqhaaWcbaGaaG4maaqaaiaaikdaaaaabeaakiaaykW7caaMc8UaaG PaVlaaykW7caWGnbGaeyypa0ZaaOaaaeaacaGGOaGaam4uaiabgUca RiaadgeacaGGPaGaai4laiaaikdaaSqabaGccaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaamOtaiabg2da9maakaaabaGa aiikaiaadofacqGHsislcaWGbbGaaiykaiaac+cacaaIYaaaleqaaa GcbaGaam4raiabg2da9iaad2eadaahaaWcbeqaaiaaikdaaaGccqGH sislcaWGobWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamiEamaaBa aaleaacaaIZaaabeaakiaad2eacqGHsislcaWGHbGaamOtaiaaykW7 caaMc8UaaGPaVlaadIeacqGH9aqpcaaIYaGaamytaiaad6eacqGHRa WkcaWGHbGaamytaiabgUcaRiaadIhadaWgaaWcbaGaaG4maaqabaGc caWGobGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl abew9aMjabg2da9iaacIcacaaIXaGaey4kaSIaeqyVd4Maaiykaiaa dIhadaWgaaWcbaGaaG4maaqabaGcciGG0bGaaiyyaiaac6gadaahaa WcbeqaaiabgkHiTiaaigdaaaGccaGGOaGaamyyaiaac+cacaWGnbGa aiykaaaaaa@B0C9@

The stresses on r=0 must be computed using a limiting process, with the result

σ 11 = σ 22 = p 0 a [ ( 1+ν )( x 3 tan 1 (a/ x 3 )a )+ a 3 2( a 2 + x 3 2 ) ] σ 33 = p 0 a 2 a 2 + x 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGa eyypa0ZaaSaaaeaacaWGWbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaam yyaaaadaWadaqaamaabmaabaGaaGymaiabgUcaRiabe27aUbGaayjk aiaawMcaamaabmaabaGaamiEamaaBaaaleaacaaIZaaabeaakiGacs hacaGGHbGaaiOBamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaacIca caWGHbGaai4laiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaey OeI0IaamyyaaGaayjkaiaawMcaaiabgUcaRmaalaaabaGaamyyamaa CaaaleqabaGaaG4maaaaaOqaaiaaikdadaqadaqaaiaadggadaahaa WcbeqaaiaaikdaaaGccqGHRaWkcaWG4bWaa0baaSqaaiaaiodaaeaa caaIYaaaaaGccaGLOaGaayzkaaaaaaGaay5waiaaw2faaiaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7cqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaOGaeyypa0Jaey OeI0YaaSaaaeaacaWGWbWaaSbaaSqaaiaaicdaaeqaaOGaamyyamaa CaaaleqabaGaaGOmaaaaaOqaaiaadggadaahaaWcbeqaaiaaikdaaa GccqGHRaWkcaWG4bWaa0baaSqaaiaaiodaaeaacaaIYaaaaaaaaaa@7A56@

 

Conditions to initiate yield: The material under the contact yields when the maximum von-Mises effective stress σ e = 3 S ij S ij /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyzaaqaba GccqGH9aqpdaGcaaqaaiaaiodacaWGtbWaaSbaaSqaaiaadMgacaWG QbaabeaakiaadofadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaai4lai aaikdaaSqabaaaaa@3ECC@  reaches the uniaxial tensile yield stress Y.  The location of the maximum von-Mises stress can be found by plotting contours of σ e / p 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyzaaqaba GccaGGVaGaamiCamaaBaaaleaacaaIWaaabeaaaaa@3837@  as a function of x 1 /a, x 3 /a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaO Gaai4laiaadggacaGGSaGaamiEamaaBaaaleaacaaIZaaabeaakiaa c+cacaWGHbaaaa@3A86@ .  For ν=0.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBcqGH9aqpcaaIWaGaaiOlai aaiodaaaa@37AD@  the maximum value occurs at x 1 = x 2 =0, x 3 =0.481a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaamiEamaaBaaaleaacaaIYaaabeaakiabg2da9iaaicda caGGSaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG4bWaaSbaaS qaaiaaiodaaeqaaOGaeyypa0JaaGimaiaac6cacaaI0aGaaGioaiaa igdacaWGHbaaaa@4953@  and has value σ e / p 0 =0.6200 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyzaaqaba GccaGGVaGaamiCamaaBaaaleaacaaIWaaabeaakiabg2da9iaaicda caGGUaGaaGOnaiaaikdacaaIWaGaaGimaaaa@3DA3@ .  Yield occurs when p 0 =1.61Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaaicdaaeqaaO Gaeyypa0JaaGymaiaac6cacaaI2aGaaGymaiaadMfaaaa@3977@ .

 

 

 

5.4.11 Contact area, pressure, stiffness and elastic limit for general non-conformal contacts

 

A non-conformal contact has the following properties: (i) the two contacting solids initially touch at a point or along a line; (ii) both contacting solids are smooth in the neighborhood of the contact, so that their local geometry can be approximated as ellipsoids, (iii) the size of the contact patch between the two solids is much smaller than either solid.

 

Complete solutions for such contacts can be found in Bryant and Keer Journal of Applied Mechanics, 49, p. 345 (1982) or Sackfield and Hills Journal of Strain Analysis 18 p107, p. 195 (1983).  These papers also account for the effects of friction under sliding contacts.   The results are lengthy.  Here, we give formulas that predict the most important features of frictionless nonconformal contacts.

 

Contact Geometry: The geometry of the contacting solids is characterized as follows:

  1. The principal radii of curvature of the two solids at the point of initial contact are denoted by ρ 1 A , ρ 2 A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaDaaaleaacaaIXaaabaGaam yqaaaakiaacYcacqaHbpGCdaqhaaWcbaGaaGOmaaqaaiaadgeaaaaa aa@37FF@ , ρ 1 B , ρ 2 B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaDaaaleaacaaIXaaabaGaam OqaaaakiaacYcacqaHbpGCdaqhaaWcbaGaaGOmaaqaaiaadkeaaaaa aa@3801@ .  The radii of curvature are positive if convex and negative if concave. 
  2. The angle between the principal directions of curvature of the two solids is denoted by α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3207@ . Note that  while labels 1 and 2 can be assigned to the radii of curvature of the two surfaces arbitrarily, α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3207@  must specify the angle between the two planes containing the radii ρ 1 A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaDaaaleaacaaIXaaabaGaam yqaaaaaaa@33D6@  and ρ 1 B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaDaaaleaacaaIXaaabaGaam Oqaaaaaaa@33D7@ .
  3. Define the principal relative contact radii R 1 , R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaaGymaaqabaGcca GGSaGaamOuamaaBaaaleaacaaIYaaabeaaaaa@349F@  as

1 R 1 = 1 2 ( 1 ρ 1 A + 1 ρ 2 A + 1 ρ 1 B + 1 ρ 2 B )+ 1 2 ( 1 ρ 1 A 1 ρ 2 A ) 2 + ( 1 ρ 1 B 1 ρ 2 B ) 2 +2( 1 ρ 1 A 1 ρ 2 A )( 1 ρ 1 B 1 ρ 2 B )cos2α 1 R 2 = 1 2 ( 1 ρ 1 A + 1 ρ 2 A + 1 ρ 1 B + 1 ρ 2 B ) 1 2 ( 1 ρ 1 A 1 ρ 2 A ) 2 + ( 1 ρ 1 B 1 ρ 2 B ) 2 +2( 1 ρ 1 A 1 ρ 2 A )( 1 ρ 1 B 1 ρ 2 B )cos2α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacaaIXaaabaGaamOuam aaBaaaleaacaaIXaaabeaaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaa caaIYaaaamaabmaabaWaaSaaaeaacaaIXaaabaGaeqyWdi3aa0baaS qaaiaaigdaaeaacaWGbbaaaaaakiabgUcaRmaalaaabaGaaGymaaqa aiabeg8aYnaaDaaaleaacaaIYaaabaGaamyqaaaaaaGccqGHRaWkda WcaaqaaiaaigdaaeaacqaHbpGCdaqhaaWcbaGaaGymaaqaaiaadkea aaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeqyWdi3aa0baaSqaai aaikdaaeaacaWGcbaaaaaaaOGaayjkaiaawMcaaiabgUcaRmaalaaa baGaaGymaaqaaiaaikdaaaWaaOaaaeaadaqadaqaamaalaaabaGaaG ymaaqaaiabeg8aYnaaDaaaleaacaaIXaaabaGaamyqaaaaaaGccqGH sisldaWcaaqaaiaaigdaaeaacqaHbpGCdaqhaaWcbaGaaGOmaaqaai aadgeaaaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa ey4kaSYaaeWaaeaadaWcaaqaaiaaigdaaeaacqaHbpGCdaqhaaWcba GaaGymaaqaaiaadkeaaaaaaOGaeyOeI0YaaSaaaeaacaaIXaaabaGa eqyWdi3aa0baaSqaaiaaikdaaeaacaWGcbaaaaaaaOGaayjkaiaawM caamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaikdadaqadaqaamaa laaabaGaaGymaaqaaiabeg8aYnaaDaaaleaacaaIXaaabaGaamyqaa aaaaGccqGHsisldaWcaaqaaiaaigdaaeaacqaHbpGCdaqhaaWcbaGa aGOmaaqaaiaadgeaaaaaaaGccaGLOaGaayzkaaWaaeWaaeaadaWcaa qaaiaaigdaaeaacqaHbpGCdaqhaaWcbaGaaGymaaqaaiaadkeaaaaa aOGaeyOeI0YaaSaaaeaacaaIXaaabaGaeqyWdi3aa0baaSqaaiaaik daaeaacaWGcbaaaaaaaOGaayjkaiaawMcaaiGacogacaGGVbGaai4C aiaaikdacqaHXoqyaSqabaaakeaadaWcaaqaaiaaigdaaeaacaWGsb WaaSbaaSqaaiaaikdaaeqaaaaakiabg2da9maalaaabaGaaGymaaqa aiaaikdaaaWaaeWaaeaadaWcaaqaaiaaigdaaeaacqaHbpGCdaqhaa WcbaGaaGymaaqaaiaadgeaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaa baGaeqyWdi3aa0baaSqaaiaaikdaaeaacaWGbbaaaaaakiabgUcaRm aalaaabaGaaGymaaqaaiabeg8aYnaaDaaaleaacaaIXaaabaGaamOq aaaaaaGccqGHRaWkdaWcaaqaaiaaigdaaeaacqaHbpGCdaqhaaWcba GaaGOmaaqaaiaadkeaaaaaaaGccaGLOaGaayzkaaGaeyOeI0YaaSaa aeaacaaIXaaabaGaaGOmaaaadaGcaaqaamaabmaabaWaaSaaaeaaca aIXaaabaGaeqyWdi3aa0baaSqaaiaaigdaaeaacaWGbbaaaaaakiab gkHiTmaalaaabaGaaGymaaqaaiabeg8aYnaaDaaaleaacaaIYaaaba GaamyqaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGc cqGHRaWkdaqadaqaamaalaaabaGaaGymaaqaaiabeg8aYnaaDaaale aacaaIXaaabaGaamOqaaaaaaGccqGHsisldaWcaaqaaiaaigdaaeaa cqaHbpGCdaqhaaWcbaGaaGOmaaqaaiaadkeaaaaaaaGccaGLOaGaay zkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOmamaabmaabaWa aSaaaeaacaaIXaaabaGaeqyWdi3aa0baaSqaaiaaigdaaeaacaWGbb aaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiabeg8aYnaaDaaaleaa caaIYaaabaGaamyqaaaaaaaakiaawIcacaGLPaaadaqadaqaamaala aabaGaaGymaaqaaiabeg8aYnaaDaaaleaacaaIXaaabaGaamOqaaaa aaGccqGHsisldaWcaaqaaiaaigdaaeaacqaHbpGCdaqhaaWcbaGaaG OmaaqaaiaadkeaaaaaaaGccaGLOaGaayzkaaGaci4yaiaac+gacaGG ZbGaaGOmaiabeg7aHbWcbeaaaaaa@D563@

  1. Introduce an effective contact radius R e = R 1 R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaamyzaaqabaGccq GH9aqpdaGcaaqaaiaadkfadaWgaaWcbaGaaGymaaqabaGccaWGsbWa aSbaaSqaaiaaikdaaeqaaaqabaaaaa@36FC@

 

Elastic constants: The two contacting solids are isotropic, with Young’s modulus E A , E B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaamyqaaqabaGcca GGSaGaamyramaaBaaaleaacaWGcbaabeaaaaa@349B@  and Poisson’s ratio ν A , ν B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUnaaBaaaleaacaWGbbaabeaaki aacYcacqaH9oGBdaWgaaWcbaGaamOqaaqabaaaaa@3677@ .  Define the effective modulus

E * = E A E B (1 ν A 2 ) E B +(1 ν B 2 ) E A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaahaaWcbeqaaiaacQcaaaGccq GH9aqpdaWcaaqaaiaadweadaWgaaWcbaGaamyqaaqabaGccaWGfbWa aSbaaSqaaiaadkeaaeqaaaGcbaGaaiikaiaaigdacqGHsislcqaH9o GBdaqhaaWcbaGaamyqaaqaaiaaikdaaaGccaGGPaGaamyramaaBaaa leaacaWGcbaabeaakiabgUcaRiaacIcacaaIXaGaeyOeI0IaeqyVd4 2aa0baaSqaaiaadkeaaeaacaaIYaaaaOGaaiykaiaadweadaWgaaWc baGaamyqaaqabaaaaaaa@4804@

 

Contact area: The area of contact between the two solids is elliptical, with semi-axes a,b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacaGGSaGaamOyaaaa@32E5@ .   The dimensions of the contact area may be calculated as follows:

  1. Solve the following equation (numerically) for e= 1 b 2 / a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwgacqGH9aqpdaGcaaqaaiaaigdacq GHsislcaWGIbWaaWbaaSqabeaacaaIYaaaaOGaai4laiaadggadaah aaWcbeqaaiaaikdaaaaabeaaaaa@386C@ , with 0e1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdacqGHKjYOcaWGLbGaeyizImQaaG ymaaaa@3631@

R 2 R 1 = K(e)E(e)/(1 e 2 ) E(e)K(e) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamOuamaaBaaaleaacaaIYa aabeaaaOqaaiaadkfadaWgaaWcbaGaaGymaaqabaaaaOGaeyypa0Za aSaaaeaacaWGlbGaaiikaiaadwgacaGGPaGaeyOeI0IaamyraiaacI cacaWGLbGaaiykaiaac+cacaGGOaGaaGymaiabgkHiTiaadwgadaah aaWcbeqaaiaaikdaaaGccaGGPaaabaGaamyraiaacIcacaWGLbGaai ykaiabgkHiTiaadUeacaGGOaGaamyzaiaacMcaaaaaaa@48CA@

where K(e) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeacaGGOaGaamyzaiaacMcaaaa@337B@  and E(e) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaGGOaGaamyzaiaacMcaaaa@3375@  are complete elliptic integrals of the first and second kind

K(e)= 0 π/2 [ 1 e 2 sin 2 θ ] 1/2 dθE(e)= 0 π/2 [ 1 e 2 sin 2 θ ] 1/2 dθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeacaGGOaGaamyzaiaacMcacqGH9a qpdaWdXbqaamaadmaabaGaaGymaiabgkHiTiaadwgadaahaaWcbeqa aiaaikdaaaGcciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaa GccqaH4oqCaiaawUfacaGLDbaadaahaaWcbeqaaiabgkHiTiaaigda caGGVaGaaGOmaaaaaeaacaaIWaaabaGaeqiWdaNaai4laiaaikdaa0 Gaey4kIipakiaadsgacqaH4oqCcaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaadweacaGGOaGaamyzaiaacMcacqGH9aqpdaWdXbqa amaadmaabaGaaGymaiabgkHiTiaadwgadaahaaWcbeqaaiaaikdaaa GcciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaH4oqC aiaawUfacaGLDbaadaahaaWcbeqaaiaaigdacaGGVaGaaGOmaaaaae aacaaIWaaabaGaeqiWdaNaai4laiaaikdaa0Gaey4kIipakiaadsga cqaH4oqCaaa@7B93@

  1. Calculate the contact area from

A=πab=π ( 3P R e π E * ) 2/3 ( R 2 R 1 ) 1/3 1 e 2 e 4/3 { K(e)E(e) } 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeacqGH9aqpcqaHapaCcaWGHbGaam Oyaiabg2da9iabec8aWnaabmaabaWaaSaaaeaacaaIZaGaamiuaiaa dkfadaWgaaWcbaGaamyzaaqabaaakeaacqaHapaCcaWGfbWaaWbaaS qabeaacaGGQaaaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOm aiaac+cacaaIZaaaaOWaaeWaaeaadaWcaaqaaiaadkfadaWgaaWcba GaaGOmaaqabaaakeaacaWGsbWaaSbaaSqaaiaaigdaaeqaaaaaaOGa ayjkaiaawMcaamaaCaaaleqabaGaaGymaiaac+cacaaIZaaaaOWaaS aaaeaadaGcaaqaaiaaigdacqGHsislcaWGLbWaaWbaaSqabeaacaaI YaaaaaqabaaakeaacaWGLbWaaWbaaSqabeaacaaI0aGaai4laiaaio daaaaaaOWaaiWaaeaacaWGlbGaaiikaiaadwgacaGGPaGaeyOeI0Ia amyraiaacIcacaWGLbGaaiykaaGaay5Eaiaaw2haamaaCaaaleqaba GaaGOmaiaac+cacaaIZaaaaaaa@5D89@

(The limit lim e0 { K(e)E(e) } 2/3 / e 4/3 = (π/4) 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacYgacaGGPbGaaiyBamaaBaaaleaaca WGLbGaeyOKH4QaaGimaaqabaGcdaGadaqaaiaadUeacaGGOaGaamyz aiaacMcacqGHsislcaWGfbGaaiikaiaadwgacaGGPaaacaGL7bGaay zFaaWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccaGGVaGaamyz amaaCaaaleqabaGaaGinaiaac+cacaaIZaaaaOGaeyypa0Jaaiikai abec8aWjaac+cacaaI0aGaaiykamaaCaaaleqabaGaaGOmaiaac+ca caaIZaaaaaaa@4E87@  is helpful )

  1. The dimensions of the contact patch follow as a= A/π / (1 e 2 ) 1/4 b= A/π (1 e 2 ) 1/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacqGH9aqpdaGcaaqaaiaadgeaca GGVaGaeqiWdahaleqaaOGaai4laiaacIcacaaIXaGaeyOeI0Iaamyz amaaCaaaleqabaGaaGOmaaaakiaacMcadaahaaWcbeqaaiaaigdaca GGVaGaaGinaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaamOyaiabg2da9maakaaabaGaamyqaiaac+cacq aHapaCaSqabaGccaGGOaGaaGymaiabgkHiTiaadwgadaahaaWcbeqa aiaaikdaaaGccaGGPaWaaWbaaSqabeaacaaIXaGaai4laiaaisdaaa aaaa@567A@

 

Contact pressure: The contact pressure distribution is ellipsoidal, with the form

p( x 1 , x 2 )=(3P/2A) 1 x 1 2 / a 2 x 2 2 / a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacaGGOaGaamiEamaaBaaaleaaca aIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiyk aiabg2da9iaacIcacaaIZaGaamiuaiaac+cacaaIYaGaamyqaiaacM cadaGcaaqaaiaaigdacqGHsislcaWG4bWaa0baaSqaaiaaigdaaeaa caaIYaaaaOGaai4laiaadggadaahaaWcbeqaaiaaikdaaaGccqGHsi slcaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaOGaai4laiaadgga daahaaWcbeqaaiaaikdaaaaabeaaaaa@4A73@

 

 

Approach of contacting solids: Points distant from the contact in the two solids approach one another by a displacement

δ= 3P 2 E * πA ( 1 e 2 ) 1/4 K(e) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjabg2da9maalaaabaGaaG4mai aadcfaaeaacaaIYaGaamyramaaCaaaleqabaGaaiOkaaaakmaakaaa baGaeqiWdaNaamyqaaWcbeaaaaGcdaqadaqaaiaaigdacqGHsislca WGLbWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIXaGaai4laiaaisdaaaGccaWGlbGaaiikaiaadwgacaGGPa aaaa@444C@

 

Contact stiffness: The contact stiffness is defined as k=dP/dδ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgacqGH9aqpcaWGKbGaamiuaiaac+ cacaWGKbGaeqiTdqgaaa@375D@  and is given by

k= ( 3 π 2 E *2 R e P ) 1/3 ( R 2 R 1 ) 1/6 { K(e)E(e) } 1/3 e 2/3 K(e) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgacqGH9aqpdaqadaqaaiaaiodacq aHapaCdaahaaWcbeqaaiaaikdaaaGccaWGfbWaaWbaaSqabeaacaGG QaGaaGOmaaaakiaadkfadaWgaaWcbaGaamyzaaqabaGccaWGqbaaca GLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4laiaaiodaaaGcdaqa daqaamaalaaabaGaamOuamaaBaaaleaacaaIYaaabeaaaOqaaiaadk fadaWgaaWcbaGaaGymaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIXaGaai4laiaaiAdaaaGcdaWcaaqaamaacmaabaGaam4sai aacIcacaWGLbGaaiykaiabgkHiTiaadweacaGGOaGaamyzaiaacMca aiaawUhacaGL9baadaahaaWcbeqaaiaaigdacaGGVaGaaG4maaaaaO qaaiaadwgadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaakiaadUea caGGOaGaamyzaiaacMcaaaaaaa@588C@

 

Elastic Limit: The stresses in both solids are identical, and therefore yield occurs first in the solid with the lower yield stress.  The figure shows the critical load required to cause yield in a solid with Von-Mises yield criterion, and uniaxial tensile yield stress Y, based on tabular values on page 99 of Johnson “Contact Mechanics” CUP (1985)

 

 

 

5.4.12 Load-displacement-contact area relations for arbitrarily shaped axisymmetric contacts

 

The most important properties of general frictionless axisymmetric contacts can be calculated from simple formulas, even when full expressions for the stress and displacement fields cannot be calculated.

 

Assume that:

  1. The two contacting solids have elastic constants E 1 , ν 1 , E 2 , ν 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaaGymaaqabaGcca GGSaGaeqyVd42aaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadweadaWg aaWcbaGaaGOmaaqabaGccaGGSaGaeqyVd42aaSbaaSqaaiaaikdaae qaaaaa@3B38@ .  Define an effective elastic constant as

E * = { 1 ν 1 2 E 1 + 1 ν 2 2 E 2 } 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbWaaWbaaSqabeaacaGGQaaaaO Gaeyypa0ZaaiWaaeaadaWcaaqaaiaaigdacqGHsislcqaH9oGBdaqh aaWcbaGaaGymaaqaaiaaikdaaaaakeaacaWGfbWaaSbaaSqaaiaaig daaeqaaaaakiabgUcaRmaalaaabaGaaGymaiabgkHiTiabe27aUnaa DaaaleaacaaIYaaabaGaaGOmaaaaaOqaaiaadweadaWgaaWcbaGaaG OmaaqabaaaaaGccaGL7bGaayzFaaWaaWbaaSqabeaacqGHsislcaaI Xaaaaaaa@4817@

  1. The surfaces of the two solids are axisymmetric near the point of initial contact.
  2. When the two solids just touch, the gap between them can be described by a monotonically increasing function g(r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbGaaiikaiaadkhacaGGPaaaaa@3602@ , where r is the distance from the point of initial contact. For example, a cone contacting a flat surface would have g(r)=r/tanβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbGaaiikaiaadkhacaGGPaGaey ypa0JaamOCaiaac+caciGG0bGaaiyyaiaac6gacqaHYoGyaaa@3D24@ , where β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHYoGyaaa@3467@  is the cone angle; a sphere contacting a flat surface could be approximated using g(r)= r 2 /D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbGaaiikaiaadkhacaGGPaGaey ypa0JaamOCamaaCaaaleqabaGaaGOmaaaakiaac+cacaWGebaaaa@3A6E@  where D  is the sphere diameter.  In the following we will use g'(r)dg/dr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbGaai4jaiaacIcacaWGYbGaai ykaiabggMi6kaadsgacaWGNbGaai4laiaadsgacaWGYbaaaa@3CDE@
  3. The two solids are pushed into contact by a force P.   The solids deform so as to make contact over a circular region with radius a, and move together by a distance h as the load is applied.
  4. The relationship between h and the contact radius a will be specified by a functional relationship of the form h=H(a).   The derivative of this function with respect to its argument will be denoted by H'(a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGibGaai4jaiaacIcacaWGHbGaai ykaaaa@367D@

 

 

These quantities are related by the following formulas:

1.      Approach as a function of contact radius  H(a)=a 0 a g (ξ) a 2 ξ 2 dξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGibGaaiikaiaadggacaGGPaGaey ypa0JaamyyamaapehabaWaaSaaaeaaceWGNbGbauaacaGGOaGaeqOV dGNaaiykaaqaamaakaaabaGaamyyamaaCaaaleqabaGaaGOmaaaaki abgkHiTiabe67a4naaCaaaleqabaGaaGOmaaaaaeqaaaaaaeaacaaI WaaabaGaamyyaaqdcqGHRiI8aOGaamizaiabe67a4baa@4818@

2.      Applied force as a function of contact radius P=2 E * ( aH(a) 0 a H(ξ)dξ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGqbGaeyypa0JaaGOmaiaadweada ahaaWcbeqaaiaacQcaaaGcdaqadaqaaiaadggacaWGibGaaiikaiaa dggacaGGPaGaeyOeI0Yaa8qCaeaacaWGibGaaiikaiabe67a4jaacM cacaWGKbGaeqOVdGhaleaacaaIWaaabaGaamyyaaqdcqGHRiI8aaGc caGLOaGaayzkaaaaaa@481C@

3.      Contact stiffness  dP dh =2 E * a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWGqbaabaGaam izaiaadIgaaaGaeyypa0JaaGOmaiaadweadaahaaWcbeqaaiaacQca aaGccaWGHbaaaa@3AC1@

4.      Contact pressure distribution  p(r)= E * π r a H (ξ) ξ 2 r 2 dξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbGaaiikaiaadkhacaGGPaGaey ypa0ZaaSaaaeaacaWGfbWaaWbaaSqabeaacaGGQaaaaaGcbaGaeqiW dahaamaapehabaWaaSaaaeaaceWGibGbauaacaGGOaGaeqOVdGNaai ykaaqaamaakaaabaGaeqOVdG3aaWbaaSqabeaacaaIYaaaaOGaeyOe I0IaamOCamaaCaaaleqabaGaaGOmaaaaaeqaaaaaaeaacaWGYbaaba GaamyyaaqdcqGHRiI8aOGaamizaiabe67a4baa@4B16@

 

Once these formulas have been evaluated for a given contact geometry the results can be combined to determine other relationships, such as contact radius or stiffness as a function of load or approach h.