Chapter 5

 

Analytical techniques and solutions for linear elastic solids

 

 

 

5.2 Airy Function Solution to Plane Stress and Strain Static Linear Elastic Problems

 

In this section we outline a general technique for solving 2D static linear elasticity problems.  The technique is known as the `Airy Stress Function’ method.

 

A typical plane elasticity problem is illustrated in the picture.  The solid is two dimensional, which means either that

1.      The solid is a thin sheet, with small thickness h, and is loaded only in the { e 1 , e 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGG9baaaa@36D7@  plane.  In this case the plane stress solution is applicable

2.      The solid is very long in the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@323F@  direction, is prevented from stretching parallel to the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@323F@  axis, and every cross section is loaded identically and only in the { e 1 , e 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGG9baaaa@36D7@  plane.  In this case, the plane strain solution is applicable.

 

Some additional basic assumptions and restrictions are:

 The Airy stress function is applicable only to isotropic solids.  We will assume that the solid has Young’s modulus E, Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@  and mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaaa a@330E@

 The Airy Stress function can only be used if the body force has a special form. Specifically, the requirement is

ρ 0 b 1 = Ω x 1 ρ 0 b 2 = Ω x 2 b 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaki aadkgadaWgaaWcbaGaaGymaaqabaGccqGH9aqpdaWcaaqaaiabgkGi 2kabfM6axbqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaaaO GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaeqyWdi3aaSbaaSqaaiaaicdaae qaaOGaamOyamaaBaaaleaacaaIYaaabeaakiabg2da9maalaaabaGa eyOaIyRaeuyQdCfabaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabe aaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caWGIbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaG imaaaa@6C36@

where Ω( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axjaacIcacaWG4bWaaSbaaSqaai aaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGG Paaaaa@37EB@  is a scalar function of position.  Fortunately, most practical body forces can be expressed in this form, including gravity.

 The Airy Stress Function approach works best for problems where a solid is subjected to prescribed tractions on its boundary, rather than prescribed displacements.  Specifically, we will assume that the solid is loaded by boundary tractions t 1 ( x 1 , x 2 ) t 2 ( x 1 , x 2 ) t 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshadaWgaaWcbaGaaGymaaqabaGcca GGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSba aSqaaiaaikdaaeqaaOGaaiykaiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaadshadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiE amaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaik daaeqaaOGaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaamiDamaaBaaaleaacaaIZaaabeaakiabg2da9i aaicdaaaa@594F@ .

 

 

5.2.1 The Airy solution in rectangular coordinates

 

The Airy function procedure can then be summarized as follows:

1.      Begin by finding a scalar function ϕ( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjaacIcacaWG4bWaaSbaaSqaai aaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGG Paaaaa@3815@  (known as the Airy potential) which satisfies:

4 ϕ 4 ϕ x 1 4 +2 4 ϕ x 1 2 x 2 2 + 4 ϕ x 1 4 =C( ν )( b 1 x 1 + b 2 x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgEGirpaaCaaaleqabaGaaGinaaaaki abew9aMjabggMi6oaalaaabaGaeyOaIy7aaWbaaSqabeaacaaI0aaa aOGaeqy1dygabaGaeyOaIyRaamiEamaaDaaaleaacaaIXaaabaGaaG inaaaaaaGccqGHRaWkcaaIYaWaaSaaaeaacqGHciITdaahaaWcbeqa aiaaisdaaaGccqaHvpGzaeaacqGHciITcaWG4bWaa0baaSqaaiaaig daaeaacaaIYaaaaOGaeyOaIyRaamiEamaaDaaaleaacaaIYaaabaGa aGOmaaaaaaGccqGHRaWkdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaG inaaaakiabew9aMbqaaiabgkGi2kaadIhadaqhaaWcbaGaaGymaaqa aiaaisdaaaaaaOGaeyypa0Jaam4qamaabmaabaGaeqyVd4gacaGLOa GaayzkaaWaaeWaaeaadaWcaaqaaiabgkGi2kaadkgadaWgaaWcbaGa aGymaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaa aakiabgUcaRmaalaaabaGaeyOaIyRaamOyamaaBaaaleaacaaIYaaa beaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaaGcca GLOaGaayzkaaaaaa@6A95@

where

C( ν )={ 1ν 12ν (Plane Strain) 1 1ν (Plane Stress) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaqadaqaaiabe27aUbGaayjkai aawMcaaiabg2da9maaceaaeaqabeaadaWcaaqaaiaaigdacqGHsisl cqaH9oGBaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbaacaaMc8UaaG PaVlaaykW7caaMc8UaaeikaiaabcfacaqGSbGaaeyyaiaab6gacaqG LbGaaeiiaiaabofacaqG0bGaaeOCaiaabggacaqGPbGaaeOBaiaabM caaeaadaWcaaqaaiaabgdaaeaacaqGXaGaeyOeI0IaeqyVd4gaaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaabIcacaqGqbGaaeiBaiaabggacaqGUbGaaeyzaiaabccacaqG tbGaaeiDaiaabkhacaqGLbGaae4CaiaabohacaqGPaaaaiaawUhaai aaykW7aaa@6FE5@

In addition ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@  must satisfy the following traction boundary conditions on the surface of the solid

2 ϕ x 2 2 n 1 2 ϕ x 1 x 2 n 2 = t 1 2 ϕ x 1 2 n 2 2 ϕ x 1 x 2 n 1 = t 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaeqy1dygabaGaeyOaIyRaamiEamaaDaaaleaacaaIYaaa baGaaGOmaaaaaaGccaWGUbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0 YaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaa cqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyOaIyRaamiEam aaBaaaleaacaaIYaaabeaaaaGccaWGUbWaaSbaaSqaaiaaikdaaeqa aOGaeyypa0JaamiDamaaBaaaleaacaaIXaaabeaakiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8+aaSaaaeaacqGHciITdaahaaWcbe qaaiaaikdaaaGccqaHvpGzaeaacqGHciITcaWG4bWaa0baaSqaaiaa igdaaeaacaaIYaaaaaaakiaad6gadaWgaaWcbaGaaGOmaaqabaGccq GHsisldaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiabew9a MbqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaGccqGHciITca WG4bWaaSbaaSqaaiaaikdaaeqaaaaakiaad6gadaWgaaWcbaGaaGym aaqabaGccqGH9aqpcaWG0bWaaSbaaSqaaiaaikdaaeqaaaaa@7B7D@

where ( n 1 , n 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWGUbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaad6gadaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@363A@  are the components of a unit vector normal to the boundary.

2.      Given ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@ , the stress field within the region of interest can be calculated from the formulas

σ 11 = 2 ϕ x 2 2 Ω σ 22 = 2 ϕ x 1 2 Ω σ 12 = σ 21 = 2 ϕ x 1 x 2 σ 33 =0(Plane Stress) σ 33 =ν( σ 11 + σ 22 )(Plane Strain) σ 23 = σ 13 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aaSbaaSqaaiaaigdaca aIXaaabeaakiabg2da9maalaaabaGaeyOaIy7aaWbaaSqabeaacaaI YaaaaOGaeqy1dygabaGaeyOaIyRaamiEamaaDaaaleaacaaIYaaaba GaaGOmaaaaaaGccaaMc8UaeyOeI0IaeuyQdCLaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWg aaWcbaGaaGOmaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacqGHciITda ahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaacqGHciITcaWG4bWaa0ba aSqaaiaaigdaaeaacaaIYaaaaaaakiaaykW7cqGHsislcqqHPoWvca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq 4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iabeo8aZnaa BaaaleaacaaIYaGaaGymaaqabaGccqGH9aqpcqGHsisldaWcaaqaai abgkGi2oaaCaaaleqabaGaaGOmaaaakiabew9aMbqaaiabgkGi2kaa dIhadaWgaaWcbaGaaGymaaqabaGccqGHciITcaWG4bWaaSbaaSqaai aaikdaaeqaaaaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaaG4maaqa baGccqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaiikaiaabcfacaqGSbGaaeyy aiaab6gacaqGLbGaaeiiaiaabofacaqG0bGaaeOCaiaabwgacaqGZb Gaae4CaiaabMcaaeaacqaHdpWCdaWgaaWcbaGaae4maiaabodaaeqa aOGaeyypa0JaeqyVd42aaeWaaeaacqaHdpWCdaWgaaWcbaGaaGymai aaigdaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaa beaaaOGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaiikaiaabcfacaqGSbGaaeyyaiaab6gacaqGLbGaaeiiai aabofacaqG0bGaaeOCaiaabggacaqGPbGaaeOBaiaabMcaaeaacqaH dpWCdaWgaaWcbaGaaeOmaiaabodaaeqaaOGaeyypa0Jaeq4Wdm3aaS baaSqaaiaaigdacaaIZaaabeaakiabg2da9iaaicdaaaaa@263C@

3.      If the strains are needed, they may be computed from the stresses using the elastic stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ strain relations.

4.      If the displacement field is needed, it may be computed by integrating the strains, following the procedure described in Section 2.1.20.  An example (in polar coordinates) is given in Section 5.2.4 below.

 

Although it is easier to solve for ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@  than it is to solve for stress directly, this is still not a trivial exercise.  Usually, one guesses a suitable form for ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@ , as illustrated below.  This may seem highly unsatisfactory, but remember that we are essentially integrating a system of PDEs.  The general procedure to evaluate any integral is to guess a solution, differentiate it, and see if the guess was correct. 

 

 

5.2.2 Demonstration that the Airy solution satisfies the governing equations

 

Recall that to solve a linear elasticity problem, we need to satisfy the following equations:

 Displacement MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E8@ strain relation ε ij = 1 2 ( u i x j + u j x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaabaWa aSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaey OaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaGccqGHRaWkdaWcaaqa aiabgkGi2kaadwhadaWgaaWcbaGaamOAaaqabaaakeaacqGHciITca WG4bWaaSbaaSqaaiaadMgaaeqaaaaaaOGaayjkaiaawMcaaaaa@4751@

 Stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E8@ strain relation ε ij = 1+ν E σ ij ν E σ kk δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaaigdacqGHRaWkcqaH9oGBaeaacaWG fbaaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsislda Wcaaqaaiabe27aUbqaaiaadweaaaGaeq4Wdm3aaSbaaSqaaiaadUga caWGRbaabeaakiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaaaaa@4832@

 Equilibrium Equation σ ij x i + ρ 0 b j =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadMgacaWGQbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGa amyAaaqabaaaaOGaey4kaSIaeqyWdi3aaSbaaSqaaiaaicdaaeqaaO GaamOyamaaBaaaleaacaWGQbaabeaakiabg2da9iaaicdaaaa@4098@  

 where we have neglected thermal expansion, for simplicity.

 

The Airy function is chosen so as to satisfy the equilibrium equations automatically.  For plane stress or plane strain conditions, the equilibrium equations reduce to

σ 11 x 1 + σ 12 x 2 + ρ 0 b 1 =0 σ 12 x 1 + σ 22 x 2 + ρ 0 b 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaaigdacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGa aGymaaqabaaaaOGaey4kaSYaaSaaaeaacqGHciITcqaHdpWCdaWgaa WcbaGaaGymaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaa caaIYaaabeaaaaGccqGHRaWkcqaHbpGCdaWgaaWcbaGaaGimaaqaba GccaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7daWcaaqaaiabgkGi2kabeo8aZnaaBaaaleaacaaIXa GaaGOmaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqa aaaakiabgUcaRmaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaaik dacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqa baaaaOGaey4kaSIaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaamOyam aaBaaaleaacaaIYaaabeaakiabg2da9iaaicdaaaa@7EBB@

Substitute for the stresses in terms of ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@  to see that

x 1 ( 2 ϕ x 2 2 Ω )+ x 2 ( 2 ϕ x 1 x 2 )+ ρ 0 b 1 =0 x 1 ( 2 ϕ x 1 x 2 )+ x 2 ( 2 ϕ x 1 2 Ω )+ ρ 0 b 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacqGHciITaeaacqGHci ITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakmaabmaabaWaaSaaaeaa cqGHciITdaahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaacqGHciITca WG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaaakiabgkHiTiabfM6a xbGaayjkaiaawMcaaiabgUcaRmaalaaabaGaeyOaIylabaGaeyOaIy RaamiEamaaBaaaleaacaaIYaaabeaaaaGcdaqadaqaaiabgkHiTmaa laaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqy1dygabaGaey OaIyRaamiEamaaBaaaleaacaaIXaaabeaakiabgkGi2kaadIhadaWg aaWcbaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaGaey4kaSIaeqyWdi 3aaSbaaSqaaiaaicdaaeqaaOGaamOyamaaBaaaleaacaaIXaaabeaa kiabg2da9iaaicdaaeaadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadI hadaWgaaWcbaGaaGymaaqabaaaaOWaaeWaaeaacqGHsisldaWcaaqa aiabgkGi2oaaCaaaleqabaGaaGOmaaaakiabew9aMbqaaiabgkGi2k aadIhadaWgaaWcbaGaaGymaaqabaGccqGHciITcaWG4bWaaWbaaSqa beaacaaIYaaaaaaaaOGaayjkaiaawMcaaiabgUcaRmaalaaabaGaey OaIylabaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaGcdaqa daqaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqy1dy gabaGaeyOaIyRaamiEamaaDaaaleaacaaIXaaabaGaaGOmaaaaaaGc cqGHsislcqqHPoWvaiaawIcacaGLPaaacqGHRaWkcqaHbpGCdaWgaa WcbaGaaGimaaqabaGccaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaeyyp a0JaaGimaaaaaa@86DF@

so that the equilibrium equations are satisfied automatically for any choice of ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@ .  To ensure that the other two equations are satisfied, we first compute the strains using the elastic stress-strain equations.  Recall that

σ 33 =βν( σ 11 + σ 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaaG4maa qabaGccqGH9aqpcqaHYoGycqaH9oGBdaqadaqaaiabeo8aZnaaBaaa leaacaaIXaGaaGymaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaG OmaiaaikdaaeqaaaGccaGLOaGaayzkaaaaaa@4184@

with β=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjabg2da9iaaicdaaaa@33C8@  for plane stress and β=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjabg2da9iaaigdaaaa@33C9@  for plane strain.  Hence

ε ij = 1+ν E σ ij ν E σ kk δ ij ε 11 = 1+ν E σ 11 ν E ( 1+βν )( σ 11 + σ 22 ) ε 22 = 1+ν E σ 22 ν E ( 1+βν )( σ 11 + σ 22 ) ε 12 = 1+ν E σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeqyTdu2aaSbaaSqaaiaadMgaca WGQbaabeaakiabg2da9maalaaabaGaaGymaiabgUcaRiabe27aUbqa aiaadweaaaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiabgk HiTmaalaaabaGaeqyVd4gabaGaamyraaaacqaHdpWCdaWgaaWcbaGa am4AaiaadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabe aaaOqaaiabgkDiElabew7aLnaaBaaaleaacaaIXaGaaGymaaqabaGc cqGH9aqpdaWcaaqaaiaaigdacqGHRaWkcqaH9oGBaeaacaWGfbaaai abeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGHsisldaWcaaqa aiabe27aUbqaaiaadweaaaWaaeWaaeaacaaIXaGaey4kaSIaeqOSdi MaeqyVd4gacaGLOaGaayzkaaWaaeWaaeaacqaHdpWCdaWgaaWcbaGa aGymaiaaigdaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiaaikdaca aIYaaabeaaaOGaayjkaiaawMcaaaqaaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqyTdu2aaSbaaSqaaiaaik dacaaIYaaabeaakiabg2da9maalaaabaGaaGymaiabgUcaRiabe27a UbqaaiaadweaaaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaaki abgkHiTmaalaaabaGaeqyVd4gabaGaamyraaaadaqadaqaaiaaigda cqGHRaWkcqaHYoGycqaH9oGBaiaawIcacaGLPaaadaqadaqaaiabeo 8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGHRaWkcqaHdpWCdaWg aaWcbaGaaGOmaiaaikdaaeqaaaGccaGLOaGaayzkaaaabaGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH1oqz daWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaaIXa Gaey4kaSIaeqyVd4gabaGaamyraaaacqaHdpWCdaWgaaWcbaGaaGym aiaaikdaaeqaaaaaaa@ADC5@

Next, recall that the strain MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ displacement relation is satisfied provided that the strains obey the compatibility conditions

2 ε 11 x 2 2 + 2 ε 22 x 1 2 2 2 ε 12 x 1 x 2 =0 2 ε 11 x 3 2 + 2 ε 33 x 1 2 2 2 ε 13 x 1 x 3 =0 2 ε 22 x 3 2 + 2 ε 33 x 2 2 2 2 ε 23 x 2 x 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaalaaabaGaeyOaIy7aaWbaaS qabeaacaaIYaaaaOGaeqyTdu2aaSbaaSqaaiaaigdacaaIXaaabeaa aOqaaiabgkGi2kaadIhadaqhaaWcbaGaaGOmaaqaaiaaikdaaaaaaO Gaey4kaSYaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaH 1oqzdaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEam aaDaaaleaacaaIXaaabaGaaGOmaaaaaaGccqGHsislcaaIYaWaaSaa aeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaH1oqzdaWgaaWcba GaaGymaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaI XaaabeaakiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaOGaey ypa0JaaGimaaqaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaa aOGaeqyTdu2aaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiabgkGi2k aadIhadaqhaaWcbaGaaG4maaqaaiaaikdaaaaaaOGaey4kaSYaaSaa aeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaH1oqzdaWgaaWcba GaaG4maiaaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaI XaaabaGaaGOmaaaaaaGccqGHsislcaaIYaWaaSaaaeaacqGHciITda ahaaWcbeqaaiaaikdaaaGccqaH1oqzdaWgaaWcbaGaaGymaiaaioda aeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaakiabgk Gi2kaadIhadaWgaaWcbaGaaG4maaqabaaaaOGaeyypa0JaaGimaaqa amaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyTdu2aaS baaSqaaiaaikdacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaqhaaWc baGaaG4maaqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacqGHciITda ahaaWcbeqaaiaaikdaaaGccqaH1oqzdaWgaaWcbaGaaG4maiaaioda aeqaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaIYaaabaGaaGOmaa aaaaGccqGHsislcaaIYaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaa ikdaaaGccqaH1oqzdaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcbaGaey OaIyRaamiEamaaBaaaleaacaaIYaaabeaakiabgkGi2kaadIhadaWg aaWcbaGaaG4maaqabaaaaOGaeyypa0JaaGimaaaaaa@9FC6@       2 ε 11 x 2 x 3 x 1 ( ε 23 x 1 + ε 31 x 2 + ε 12 x 3 )=0 2 ε 22 x 3 x 1 x 2 ( ε 31 x 2 + ε 12 x 3 + ε 23 x 1 )=0 2 ε 33 x 1 x 2 x 3 ( ε 12 x 3 + ε 23 x 1 + ε 31 x 2 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaalaaabaGaeyOaIy7aaWbaaS qabeaacaaIYaaaaOGaeqyTdu2aaSbaaSqaaiaaigdacaaIXaaabeaa aOqaaiabgkGi2kaadIhadaqhaaWcbaGaaGOmaaqaaaaakiabgkGi2k aadIhadaqhaaWcbaGaaG4maaqaaaaaaaGccqGHsisldaWcaaqaaiab gkGi2kaaykW7caaMc8oabaGaeyOaIyRaamiEamaaBaaaleaacaaIXa aabeaaaaGcdaqadaqaaiabgkHiTmaalaaabaGaeyOaIyRaeqyTdu2a aSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiabgkGi2kaadIhadaqhaa WcbaGaaGymaaqaaaaaaaGccqGHRaWkdaWcaaqaaiabgkGi2kabew7a LnaaBaaaleaacaaIZaGaaGymaaqabaaakeaacqGHciITcaWG4bWaa0 baaSqaaiaaikdaaeaaaaaaaOGaey4kaSYaaSaaaeaacqGHciITcqaH 1oqzdaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEam aaDaaaleaacaaIZaaabaaaaaaaaOGaayjkaiaawMcaaiabg2da9iaa icdaaeaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiabew 7aLnaaBaaaleaacaaIYaGaaGOmaaqabaaakeaacqGHciITcaWG4bWa a0baaSqaaiaaiodaaeaaaaGccqGHciITcaWG4bWaa0baaSqaaiaaig daaeaaaaaaaOGaeyOeI0YaaSaaaeaacqGHciITcaaMc8UaaGPaVdqa aiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaOWaaeWaaeaacq GHsisldaWcaaqaaiabgkGi2kabew7aLnaaBaaaleaacaaIZaGaaGym aaqabaaakeaacqGHciITcaWG4bWaa0baaSqaaiaaikdaaeaaaaaaaO Gaey4kaSYaaSaaaeaacqGHciITcqaH1oqzdaWgaaWcbaGaaGymaiaa ikdaaeqaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaIZaaabaaaaa aakiabgUcaRmaalaaabaGaeyOaIyRaeqyTdu2aaSbaaSqaaiaaikda caaIZaaabeaaaOqaaiabgkGi2kaadIhadaqhaaWcbaGaaGymaaqaaa aaaaaakiaawIcacaGLPaaacqGH9aqpcaaIWaaabaWaaSaaaeaacqGH ciITdaahaaWcbeqaaiaaikdaaaGccqaH1oqzdaWgaaWcbaGaaG4mai aaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaIXaaabaaa aOGaeyOaIyRaamiEamaaDaaaleaacaaIYaaabaaaaaaakiabgkHiTm aalaaabaGaeyOaIyRaaGPaVlaaykW7aeaacqGHciITcaWG4bWaaSba aSqaaiaaiodaaeqaaaaakmaabmaabaGaeyOeI0YaaSaaaeaacqGHci ITcqaH1oqzdaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaeyOaIyRa amiEamaaDaaaleaacaaIZaaabaaaaaaakiabgUcaRmaalaaabaGaey OaIyRaeqyTdu2aaSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiabgkGi 2kaadIhadaqhaaWcbaGaaGymaaqaaaaaaaGccqGHRaWkdaWcaaqaai abgkGi2kabew7aLnaaBaaaleaacaaIZaGaaGymaaqabaaakeaacqGH ciITcaWG4bWaa0baaSqaaiaaikdaaeaaaaaaaaGccaGLOaGaayzkaa Gaeyypa0JaaGimaaaaaa@CD8C@

All but the first of these equations are satisfied automatically by any plane strain or plane stress field. Substitute into the first equation in terms of stress to see that

1+ν E ( 2 σ 11 x 2 2 + 2 σ 22 x 1 2 ) ν E ( 1+βν )( 2 x 1 2 + 2 x 2 2 )( σ 11 + σ 22 )2 1+ν E 2 σ 12 x 1 x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaGymaiabgUcaRiabe27aUb qaaiaadweaaaWaaeWaaeaadaWcaaqaaiabgkGi2oaaCaaaleqabaGa aGOmaaaakiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaaakeaacq GHciITcaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaaakiabgUca RmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeq4Wdm3aaS baaSqaaiaaikdacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaqhaaWc baGaaGymaaqaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaeyOeI0YaaS aaaeaacqaH9oGBaeaacaWGfbaaamaabmaabaGaaGymaiabgUcaRiab ek7aIjabe27aUbGaayjkaiaawMcaamaabmaabaWaaSaaaeaacqGHci ITdaahaaWcbeqaaiaaikdaaaaakeaacqGHciITcaWG4bWaa0baaSqa aiaaigdaaeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaeyOaIy7aaW baaSqabeaacaaIYaaaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaI YaaabaGaaGOmaaaaaaaakiaawIcacaGLPaaadaqadaqaaiabeo8aZn aaBaaaleaacaaIXaGaaGymaaqabaGccqGHRaWkcqaHdpWCdaWgaaWc baGaaGOmaiaaikdaaeqaaaGccaGLOaGaayzkaaGaeyOeI0IaaGOmam aalaaabaGaaGymaiabgUcaRiabe27aUbqaaiaadweaaaWaaSaaaeaa cqGHciITdaahaaWcbeqaaiaaikdaaaGccqaHdpWCdaWgaaWcbaGaaG ymaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaa beaakiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaOGaeyypa0 JaaGimaaaa@8199@

Finally, substitute into this horrible looking equation for stress in terms of ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@  and rearrange to see that

4 ϕ x 2 4 2 Ω x 2 2 + 4 ϕ x 1 4 2 Ω x 1 2 ν( 1+βν ) 1+ν ( 2 x 1 2 + 2 x 2 2 )( 2 ϕ x 1 2 + 2 ϕ x 2 2 2Ω )+2 4 ϕ x 1 2 x 2 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aI0aaaaOGaeqy1dygabaGaeyOaIyRaamiEamaaDaaaleaacaaIYaaa baGaaGinaaaaaaGccqGHsisldaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiabfM6axbqaaiabgkGi2kaadIhadaqhaaWcbaGaaGOm aaqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacqGHciITdaahaaWcbe qaaiaaisdaaaGccqaHvpGzaeaacqGHciITcaWG4bWaa0baaSqaaiaa igdaaeaacaaI0aaaaaaakiabgkHiTmaalaaabaGaeyOaIy7aaWbaaS qabeaacaaIYaaaaOGaeuyQdCfabaGaeyOaIyRaamiEamaaDaaaleaa caaIXaaabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiabe27aUnaabm aabaGaaGymaiabgUcaRiabek7aIjabe27aUbGaayjkaiaawMcaaaqa aiaaigdacqGHRaWkcqaH9oGBaaWaaeWaaeaadaWcaaqaaiabgkGi2o aaCaaaleqabaGaaGOmaaaaaOqaaiabgkGi2kaadIhadaqhaaWcbaGa aGymaaqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacqGHciITdaahaa WcbeqaaiaaikdaaaaakeaacqGHciITcaWG4bWaa0baaSqaaiaaikda aeaacaaIYaaaaaaaaOGaayjkaiaawMcaamaabmaabaWaaSaaaeaacq GHciITdaahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaacqGHciITcaWG 4bWaa0baaSqaaiaaigdaaeaacaaIYaaaaaaakiabgUcaRmaalaaaba GaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqy1dygabaGaeyOaIyRa amiEamaaDaaaleaacaaIYaaabaGaaGOmaaaaaaGccqGHsislcaaIYa GaeuyQdCfacaGLOaGaayzkaaGaey4kaSIaaGOmamaalaaabaGaeyOa Iy7aaWbaaSqabeaacaaI0aaaaOGaeqy1dygabaGaeyOaIyRaamiEam aaDaaaleaacaaIXaaabaGaaGOmaaaakiabgkGi2kaadIhadaqhaaWc baGaaGOmaaqaaiaaikdaaaaaaOGaeyypa0JaaGimaaaa@9585@

A few more weeks of algebra reduces this to

4 ϕ x 1 4 +2 4 ϕ x 1 2 x 2 2 + 4 ϕ x 1 4 = 1β ν 2 1ν2β ν 2 ( 2 Ω x 1 2 + 2 Ω x 2 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aI0aaaaOGaeqy1dygabaGaeyOaIyRaamiEamaaDaaaleaacaaIXaaa baGaaGinaaaaaaGccqGHRaWkcaaIYaWaaSaaaeaacqGHciITdaahaa WcbeqaaiaaisdaaaGccqaHvpGzaeaacqGHciITcaWG4bWaa0baaSqa aiaaigdaaeaacaaIYaaaaOGaeyOaIyRaamiEamaaDaaaleaacaaIYa aabaGaaGOmaaaaaaGccqGHRaWkdaWcaaqaaiabgkGi2oaaCaaaleqa baGaaGinaaaakiabew9aMbqaaiabgkGi2kaadIhadaqhaaWcbaGaaG ymaaqaaiaaisdaaaaaaOGaeyypa0ZaaSaaaeaacaaIXaGaeyOeI0Ia eqOSdiMaeqyVd42aaWbaaSqabeaacaaIYaaaaaGcbaGaaGymaiabgk HiTiabe27aUjabgkHiTiaaikdacqaHYoGycqaH9oGBdaahaaWcbeqa aiaaikdaaaaaaOWaaeWaaeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiabfM6axbqaaiabgkGi2kaadIhadaqhaaWcbaGaaGym aaqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacqGHciITdaahaaWcbe qaaiaaikdaaaGccqqHPoWvaeaacqGHciITcaWG4bWaa0baaSqaaiaa ikdaaeaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@72A4@

which is the result we were looking for.

 

This proves that the Airy representation satisfies the governing equations.  A second important question is MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  is it possible to find an Airy function for all 2D plane stress and plane strain problems?  If not, the method would be useless, because you couldn’t tell ahead of time whether ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMbaa@3240@  existed for the problem you were trying to solve.  Fortunately it is possible to prove that all properly posed 2D elasticity problems do have an Airy representation.

 

 

 

5.2.3 The Airy solution in cylindrical-polar coordinates

 

Boundary value problems involving cylindrical regions are best solved using Cylindrical-polar coordinates.  It is worth recording the Airy function equations for this coordinate system.

 

In a 2D cylindrical-polar coordinate system, a point in the solid is specified by its radial distance r= x 1 2 + x 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhacqGH9aqpdaGcaaqaaiaadIhada qhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHRaWkcaWG4bWaa0baaSqa aiaaikdaaeaacaaIYaaaaaqabaaaaa@38A4@  from the origin and the angle θ= tan 1 x 2 / x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iGacshacaGGHbGaai OBamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaadIhadaWgaaWcbaGa aGOmaaqabaGccaGGVaGaamiEamaaBaaaleaacaaIXaaabeaaaaa@3C5A@ .  The solution is independent of z.  The Airy function is written as a function of the coordinates as ϕ(r,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjaacIcacaWGYbGaaiilaiabeI 7aXjaacMcaaaa@36E6@ .  Vector quantities (displacement, body force) and tensor quantities (strain, stress) are expressed as components in the basis { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaam OCaaqabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiil aiaahwgadaWgaaWcbaGaamOEaaqabaaakiaawUhacaGL9baaaaa@3D6F@  shown in the picture.

 

The governing equation for the Airy function in this coordinate system is

( 2 r 2 + 1 r r + 1 r 2 2 θ 2 ) 2 ϕ=C( ν )( b r r + 1 r b θ θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaWaaeWaaeaadaWcaaqaaiabgkGi2oaaCa aaleqabaGaaGOmaaaaaOqaaiabgkGi2kaadkhadaahaaWcbeqaaiaa ikdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaamOCaaaadaWcaa qaaiabgkGi2cqaaiabgkGi2kaadkhaaaGaey4kaSYaaSaaaeaacaaI XaaabaGaamOCamaaCaaaleqabaGaaGOmaaaaaaGcdaWcaaqaaiabgk Gi2oaaCaaaleqabaGaaGOmaaaaaOqaaiabgkGi2kabeI7aXnaaCaaa leqabaGaaGOmaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaik daaaGccqaHvpGzcqGH9aqpcaWGdbWaaeWaaeaacqaH9oGBaiaawIca caGLPaaadaqadaqaamaalaaabaGaeyOaIyRaamOyamaaBaaaleaaca WGYbaabeaaaOqaaiabgkGi2kaadkhaaaGaey4kaSYaaSaaaeaacaaI XaaabaGaamOCaaaadaWcaaqaaiabgkGi2kaadkgadaWgaaWcbaGaeq iUdehabeaaaOqaaiabgkGi2kabeI7aXbaaaiaawIcacaGLPaaaaaa@61D9@

C( ν )={ 1ν 12ν (Plane Strain) 1 1ν (Plane Stress) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaqadaqaaiabe27aUbGaayjkai aawMcaaiabg2da9maaceaaeaqabeaadaWcaaqaaiaaigdacqGHsisl cqaH9oGBaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbaacaaMc8UaaG PaVlaaykW7caaMc8UaaeikaiaabcfacaqGSbGaaeyyaiaab6gacaqG LbGaaeiiaiaabofacaqG0bGaaeOCaiaabggacaqGPbGaaeOBaiaabM caaeaadaWcaaqaaiaabgdaaeaacaqGXaGaeyOeI0IaeqyVd4gaaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaabIcacaqGqbGaaeiBaiaabggacaqGUbGaaeyzaiaabccacaqG tbGaaeiDaiaabkhacaqGLbGaae4CaiaabohacaqGPaaaaiaawUhaai aaykW7aaa@6FE5@

The state of stress is related to the Airy function by

σ rr = 1 r ϕ r + 1 r 2 2 ϕ θ 2 Ω σ θθ = 2 ϕ r 2 Ω σ rθ = r ( 1 r ϕ θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadkhacaWGYb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaadkhaaaWaaSaaaeaa cqGHciITcqaHvpGzaeaacqGHciITcaWGYbaaaiabgUcaRmaalaaaba GaaGymaaqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaOWaaSaaaeaa cqGHciITdaahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaacqGHciITcq aH4oqCdaahaaWcbeqaaiaaikdaaaaaaOGaeyOeI0IaeuyQdCLaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaai abeI7aXjabeI7aXbqabaGccqGH9aqpdaWcaaqaaiabgkGi2oaaCaaa leqabaGaaGOmaaaakiabew9aMbqaaiabgkGi2kaadkhadaahaaWcbe qaaiaaikdaaaaaaOGaeyOeI0IaeuyQdCLaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaamOCaiabeI7aXbqabaGccq GH9aqpcqGHsisldaWcaaqaaiabgkGi2cqaaiabgkGi2kaadkhaaaWa aeWaaeaadaWcaaqaaiaaigdaaeaacaWGYbaaamaalaaabaGaeyOaIy Raeqy1dygabaGaeyOaIyRaeqiUdehaaaGaayjkaiaawMcaaaaa@80DB@

In polar coordinates the strains are related to the stresses by

[ ε rr ε θθ 2 ε rθ ]= (1+ν) E [ 1ν ν 0 ν 1ν 0 0 0 2 ][ σ rr σ θθ σ rθ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabmqaaaqaaiabew7aLn aaBaaaleaacaWGYbGaamOCaaqabaaakeaacqaH1oqzdaWgaaWcbaGa eqiUdeNaeqiUdehabeaaaOqaaiaaikdacqaH1oqzdaWgaaWcbaGaam OCaiabeI7aXbqabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaaSaaaeaa caGGOaGaaGymaiabgUcaRiabe27aUjaacMcaaeaacaWGfbaaamaadm aabaqbaeqabmWaaaqaaiaaigdacqGHsislcqaH9oGBaeaacqGHsisl cqaH9oGBaeaacaaIWaaabaGaeyOeI0IaeqyVd4gabaGaaGymaiabgk HiTiabe27aUbqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaikda aaaacaGLBbGaayzxaaWaamWaaeaafaqabeWabaaabaGaeq4Wdm3aaS baaSqaaiaadkhacaWGYbaabeaaaOqaaiabeo8aZnaaBaaaleaacqaH 4oqCcqaH4oqCaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadkhacqaH4o qCaeqaaaaaaOGaay5waiaaw2faaaaa@68FB@

for plane strain, while

[ ε rr ε θθ 2 ε rθ ]= 1 E [ 1 ν 0 ν 1 0 0 0 2(1+ν) ][ σ rr σ θθ σ rθ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabmqaaaqaaiabew7aLn aaBaaaleaacaWGYbGaamOCaaqabaaakeaacqaH1oqzdaWgaaWcbaGa eqiUdeNaeqiUdehabeaaaOqaaiaaikdacqaH1oqzdaWgaaWcbaGaam OCaiabeI7aXbqabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaaSaaaeaa caaIXaaabaGaamyraaaadaWadaqaauaabeqadmaaaeaacaaIXaaaba GaeyOeI0IaeqyVd4gabaGaaGimaaqaaiabgkHiTiabe27aUbqaaiaa igdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIYaGaaiikai aaigdacqGHRaWkcqaH9oGBcaGGPaaaaaGaay5waiaaw2faamaadmaa baqbaeqabmqaaaqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaaqaba aakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaaaOqaaiab eo8aZnaaBaaaleaacaWGYbGaeqiUdehabeaaaaaakiaawUfacaGLDb aaaaa@646C@

for plane stress.  The displacements must be determined by integrating these strains following the procedure similar to that outlined in Section 2.1.20.  To this end, let u= u r e r + u θ e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0JaamyDamaaBaaale aacaWGYbaabeaakiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHRaWk caWG1bWaaSbaaSqaaiabeI7aXbqabaGccaWHLbWaaSbaaSqaaiabeI 7aXbqabaaaaa@3FA4@  denote the displacement vector.  The strain-displacement relations in polar coordinates are:

ε rr = u r r ε θθ = u r r + 1 r u θ θ ε rθ = 1 2 ( 1 r u r θ + u θ r u θ r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaa dkhaaeqaaaGcbaGaeyOaIyRaamOCaaaacaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaeqyTdu2aaSbaaSqaaiabeI7aXjab eI7aXbqabaGccqGH9aqpdaWcaaqaaiaadwhadaWgaaWcbaGaamOCaa qabaaakeaacaWGYbaaaiabgUcaRmaalaaabaGaaGymaaqaaiaadkha aaWaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiabeI7aXbqabaaake aacqGHciITcqaH4oqCaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabew7aLnaaBaaaleaaca WGYbGaeqiUdehabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikda aaWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWGYbaaamaalaaabaGaey OaIyRaamyDamaaBaaaleaacaWGYbaabeaaaOqaaiabgkGi2kabeI7a XbaacqGHRaWkdaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaeqiUde habeaaaOqaaiabgkGi2kaadkhaaaGaeyOeI0YaaSaaaeaacaWG1bWa aSbaaSqaaiabeI7aXbqabaaakeaacaWGYbaaaaGaayjkaiaawMcaaa aa@88F4@

These can be integrated using a procedure analogous to that outlined in Section 2.1.20.  An example is given in Section 5.2.5.

 

In the following sections, we give several examples of Airy function solutions to boundary value problems.

 

 

 

 

5.2.4 Airy function solution to the end loaded cantilever

 

Consider a cantilever beam, with length L, height 2a and out-of-plane thickness b, as shown in the figure. The beam is made from an isotropic linear elastic solid with Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbaaaa@3279@  and Poisson ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@3367@ . The top and bottom of the beam x 2 =±a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaeyySaeRaamyyaaaa@388F@  are traction free, the left hand end is subjected to a resultant force P, and the right hand end is clamped.  Assume that b<<a, so that a state of plane stress is developed in the beam. An approximate solution to the stress in the beam can be calculated from the Airy function

ϕ= 3P 4ab x 1 x 2 + P 4 a 3 b x 1 x 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcqGH9aqpcqGHsisldaWcaa qaaiaaiodacaWGqbaabaGaaGinaiaadggacaWGIbaaaiaadIhadaWg aaWcbaGaaGymaaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaey 4kaSYaaSaaaeaacaWGqbaabaGaaGinaiaadggadaahaaWcbeqaaiaa iodaaaGccaWGIbaaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaWG4b Waa0baaSqaaiaaikdaaeaacaaIZaaaaaaa@4862@

You can easily show that this function satisfies the governing equation for the Airy function. The stresses follow as

σ 11 = 2 ϕ x 2 2 Ω= 3P 2 a 3 b x 1 x 2 σ 22 = 2 ϕ x 1 2 Ω=0 σ 12 = σ 21 = 2 ϕ x 1 x 2 = 3P 4ab ( 1 x 2 2 a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaa kiabew9aMbqaaiabgkGi2kaadIhadaqhaaWcbaGaaGOmaaqaaiaaik daaaaaaOGaaGPaVlabgkHiTiabfM6axjabg2da9maalaaabaGaaG4m aiaadcfaaeaacaaIYaGaamyyamaaCaaaleqabaGaaG4maaaakiaadk gaaaGaamiEamaaBaaaleaacaaIXaaabeaakiaadIhadaWgaaWcbaGa aGOmaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab eo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH9aqpdaWcaaqaai abgkGi2oaaCaaaleqabaGaaGOmaaaakiabew9aMbqaaiabgkGi2kaa dIhadaqhaaWcbaGaaGymaaqaaiaaikdaaaaaaOGaaGPaVlabgkHiTi abfM6axjabg2da9iaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdp WCdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0Jaeq4Wdm3aaSba aSqaaiaaikdacaaIXaaabeaakiabg2da9iabgkHiTmaalaaabaGaey OaIy7aaWbaaSqabeaacaaIYaaaaOGaeqy1dygabaGaeyOaIyRaamiE amaaBaaaleaacaaIXaaabeaakiabgkGi2kaadIhadaWgaaWcbaGaaG OmaaqabaaaaOGaeyypa0ZaaSaaaeaacaaIZaGaamiuaaqaaiaaisda caWGHbGaamOyaaaadaqadaqaaiaaigdacqGHsisldaWcaaqaaiaadI hadaqhaaWcbaGaaGOmaaqaaiaaikdaaaaakeaacaWGHbWaaWbaaSqa beaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@A770@

 

To see that this solution satisfies the boundary conditions, note that

1.      The top and bottom surfaces of the beam x 2 =±a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaeyySaeRaamyyaaaa@388F@  are traction free ( σ ij n i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaamOBamaaBaaaleaacaWGPbaabeaakiabg2da9iaaicda aaa@3A73@  ).  Since the normal is in the e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaikdaaeqaaa aa@34AC@  direction on these surfaces, this requires that σ 22 = σ 21 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaikdacaaIXaaabeaa kiabg2da9iaaicdaaaa@3C6D@ .  The stress field clearly satisfies this condition.

2.      The plane stress assumption automatically satisfies boundary conditions on x 3 =±b/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaO Gaeyypa0JaeyySaeRaamOyaiaac+cacaaIYaaaaa@3A00@ .

3.      The traction boundary condition on the left hand end of the beam ( x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaaGimaaaa@3674@  ) was not specified in detail: instead, we only required that the resultant of the traction acting on the surface is P e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaWGqbGaaCyzamaaBaaale aacaaIYaaabeaaaaa@365E@ .  The normal to the surface at the left hand end of the beam is in the e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaWHLbWaaSbaaSqaaiaaig daaeqaaaaa@3588@  direction, so the traction vector is

t i = σ ij n i = σ 12 δ i2 = 3P 4ab ( 1 x 2 2 a 2 ) δ i2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG0bWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaad6ga daWgaaWcbaGaamyAaaqabaGccqGH9aqpcqGHsislcqaHdpWCdaWgaa WcbaGaaGymaiaaikdaaeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaaI Yaaabeaakiabg2da9iabgkHiTmaalaaabaGaaG4maiaadcfaaeaaca aI0aGaamyyaiaadkgaaaWaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaa caWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaGcbaGaamyyamaaCa aaleqabaGaaGOmaaaaaaaakiaawIcacaGLPaaacqaH0oazdaWgaaWc baGaamyAaiaaikdaaeqaaaaa@561F@

The resultant force can be calculated by integrating the traction over the end of the beam:

F i =b a a 3P 4ab ( 1 x 2 2 a 2 ) δ i2 d x 2 =P δ i2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaamOyamaapehabaGaeyOeI0YaaSaaaeaacaaIZaGaamiu aaqaaiaaisdacaWGHbGaamOyaaaadaqadaqaaiaaigdacqGHsislda WcaaqaaiaadIhadaqhaaWcbaGaaGOmaaqaaiaaikdaaaaakeaacaWG HbWaaWbaaSqabeaacaaIYaaaaaaaaOGaayjkaiaawMcaaiabes7aKn aaBaaaleaacaWGPbGaaGOmaaqabaGccaWGKbGaamiEamaaBaaaleaa caaIYaaabeaaaeaacqGHsislcaWGHbaabaGaamyyaaqdcqGHRiI8aO Gaeyypa0JaeyOeI0Iaamiuaiabes7aKnaaBaaaleaacaWGPbGaaGOm aaqabaaaaa@5539@

The stresses thus satisfy the boundary condition.  Note that by Saint-Venant’s principle, other distributions of traction with the same resultant will induce the same stresses sufficiently far ( x 1 >3a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaO GaeyOpa4JaaG4maiaadggaaaa@375F@  ) from the end of the beam.

4.      The boundary conditions on the right hand end of the beam are not satisfied exactly.  The exact solution should satisfy both u 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaaGimaaaa@3671@  and u 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaaGimaaaa@3672@  on x 1 =L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0Jaamitaaaa@368B@ .  The displacement field corresponding to the stress distribution was calculated in the example problem in Sect 2.1.20, where we found that

u 1 = 3P 4E a 3 b x 1 2 x 2 P 4E a 3 b (2+ν) x 2 3 + 3P 2E a 3 b (1+ν) a 2 x 2 ω x 2 +c u 2 =ν 3P 4E a 3 b x 1 x 2 2 P 4E a 3 b x 1 3 +ω x 1 +d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadwhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpdaWcaaqaaiaaiodacaWGqbaabaGaaGinaiaadwea caWGHbWaaWbaaSqabeaacaaIZaaaaOGaamOyaaaacaWG4bWaa0baaS qaaiaaigdaaeaacaaIYaaaaOGaamiEamaaBaaaleaacaaIYaaabeaa kiabgkHiTmaalaaabaGaamiuaaqaaiaaisdacaWGfbGaamyyamaaCa aaleqabaGaaG4maaaakiaadkgaaaGaaiikaiaaikdacqGHRaWkcqaH 9oGBcaGGPaGaamiEamaaDaaaleaacaaIYaaabaGaaG4maaaakiabgU caRmaalaaabaGaaG4maiaadcfaaeaacaaIYaGaamyraiaadggadaah aaWcbeqaaiaaiodaaaGccaWGIbaaaiaacIcacaaIXaGaey4kaSIaeq yVd4MaaiykaiaadggadaahaaWcbeqaaiaaikdaaaGccaWG4bWaaSba aSqaaiaaikdaaeqaaOGaeyOeI0IaeqyYdCNaamiEamaaBaaaleaaca aIYaaabeaakiabgUcaRiaadogaaeaacaaMc8UaamyDamaaBaaaleaa caaIYaaabeaakiabg2da9iabgkHiTiabe27aUnaalaaabaGaaG4mai aadcfaaeaacaaI0aGaamyraiaadggadaahaaWcbeqaaiaaiodaaaGc caWGIbaaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaWG4bWaa0baaS qaaiaaikdaaeaacaaIYaaaaOGaeyOeI0YaaSaaaeaacaWGqbaabaGa aGinaiaadweacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaamOyaaaaca WG4bWaa0baaSqaaiaaigdaaeaacaaIZaaaaOGaey4kaSIaeqyYdCNa amiEamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadsgaaaaa@8443@

where c,d,ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbGaaiilaiaadsgacaGGSaGaeq yYdChaaa@37C4@  are constants that may be selected to satisfy the boundary condition as far as possible.  We can satisfy u 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaaGimaaaa@3681@  and u 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaaGimaaaa@3682@  at some, but not all, points on x 1 =L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0Jaamitaaaa@369B@ .  The choice is arbitrary.  Usually the boundary condition is approximated by requiring u 1 = u 2 = u 2 / x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaamyDamaaBaaaleaacaaIYaaabeaakiabg2da9iabgkGi 2kaadwhadaWgaaWcbaGaaGOmaaqabaGccaGGVaGaeyOaIyRaamiEam aaBaaaleaacaaIXaaabeaakiabg2da9iaaicdaaaa@41C2@  at x 1 =L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0Jaamitaaaa@368B@ , x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaaGimaaaa@3675@ .  This gives c=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbGaeyypa0JaaGimaaaa@356E@ , d=P L 3 /2E a 3 b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeyypa0JaeyOeI0Iaamiuai aadYeadaahaaWcbeqaaiaaiodaaaGccaGGVaGaaGOmaiaadweacaWG HbWaaWbaaSqabeaacaaIZaaaaOGaamOyaaaa@3D36@  and ω=3P L 2 /4E a 3 b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHjpWDcqGH9aqpcaaIZaGaamiuai aadYeadaahaaWcbeqaaiaaikdaaaGccaGGVaGaaGinaiaadweacaWG HbWaaWbaaSqabeaacaaIZaaaaOGaamOyaaaa@3DEB@ .   By Saint-Venant’s principle, applying other boundary conditions (including the exact boundary condition) will not influence the stresses and displacements sufficiently far from the end.

 

 

 

5.2.5 2D Line load acting perpendicular to the surface of an infinite solid

 

As a second example, the stress fields due to a line load magnitude P per unit out-of-plane length acting on the surface of a homogeneous, isotropic half-space can be generated from the Airy function

ϕ= P π rθsinθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaey ypa0JaeyOeI0YaaSaaaeaacaWGqbaabaGaeqiWdahaaiaadkhacqaH 4oqCciGGZbGaaiyAaiaac6gacqaH4oqCaaa@437E@

The formulas in the preceding section yield

σ rr = 2P π cosθ r σ θθ = σ rθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadkhacaWGYbaabeaakiabg2da9iabgkHiTmaalaaabaGa aGOmaiaadcfaaeaacqaHapaCaaWaaSaaaeaaciGGJbGaai4Baiaaco hacqaH4oqCaeaacaWGYbaaaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8Uaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7aXbqabaGccqGH 9aqpcqaHdpWCdaWgaaWcbaGaamOCaiabeI7aXbqabaGccqGH9aqpca aIWaaaaa@6830@

The stresses in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BC6@  basis are

σ 11 = 2P π x 1 3 ( x 1 2 + x 2 2 ) 2 σ 22 = 2P π x 1 x 2 2 ( x 1 2 + x 2 2 ) 2 σ 12 = 2P π x 1 2 x 2 ( x 1 2 + x 2 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaacaaIYaGaamiuaaqaaiab ec8aWbaadaWcaaqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaaiodaaa aakeaadaqadaqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikdaaaGc cqGHRaWkcaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaGccaGLOa GaayzkaaWaaWbaaSqabeaacaaIYaaaaaaakiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBa aaleaacaaIYaGaaGOmaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaa ikdacaWGqbaabaGaeqiWdahaamaalaaabaGaamiEamaaDaaaleaaca aIXaaabaaaaOGaamiEamaaDaaaleaacaaIYaaabaGaaGOmaaaaaOqa amaabmaabaGaamiEamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgU caRiaadIhadaqhaaWcbaGaaGOmaaqaaiaaikdaaaaakiaawIcacaGL PaaadaahaaWcbeqaaiaaikdaaaaaaOGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH dpWCdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0JaeyOeI0YaaS aaaeaacaaIYaGaamiuaaqaaiabec8aWbaadaWcaaqaaiaadIhadaqh aaWcbaGaaGymaaqaaiaaikdaaaGccaWG4bWaa0baaSqaaiaaikdaae aaaaaakeaadaqadaqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikda aaGccqGHRaWkcaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaGcca GLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaaaaa@920C@

 

The method outlined in section 5.2.3 can be used to calculate the displacements: the procedure is described in detail below to provide a representative example.  For plane strain deformation, we find

u r = 2( 1 ν 2 ) πE Pcosθlogr ( 1+ν )( 12ν ) πE Pθsinθ u θ = 2( 1 ν 2 ) πE Psinθlogr+ 1+ν πE Psinθ 2( 12ν )( 1+ν ) πE Pθcosθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG1b WaaSbaaSqaaiaadkhaaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaacaaI YaWaaeWaaeaacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYa aaaaGccaGLOaGaayzkaaaabaGaeqiWdaNaamyraaaacaWGqbGaci4y aiaac+gacaGGZbGaeqiUdeNaciiBaiaac+gacaGGNbGaamOCaiabgk HiTmaalaaabaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGa ayzkaaWaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbGaayjkai aawMcaaaqaaiabec8aWjaadweaaaGaamiuaiabeI7aXjGacohacaGG PbGaaiOBaiabeI7aXbqaaiaadwhadaWgaaWcbaGaeqiUdehabeaaki abg2da9maalaaabaGaaGOmamaabmaabaGaaGymaiabgkHiTiabe27a UnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaqaaiabec8aWj aadweaaaGaamiuaiGacohacaGGPbGaaiOBaiabeI7aXjGacYgacaGG VbGaai4zaiaadkhacqGHRaWkdaWcaaqaaiaaigdacqGHRaWkcqaH9o GBaeaacqaHapaCcaWGfbaaaiaadcfaciGGZbGaaiyAaiaac6gacqaH 4oqCcqGHsisldaWcaaqaaiaaikdadaqadaqaaiaaigdacqGHsislca aIYaGaeqyVd4gacaGLOaGaayzkaaWaaeWaaeaacaaIXaGaey4kaSIa eqyVd4gacaGLOaGaayzkaaaabaGaeqiWdaNaamyraaaacaWGqbGaeq iUdeNaci4yaiaac+gacaGGZbGaeqiUdehaaaa@9ADF@

to within an arbitrary rigid motion.  Note that the displacements vary as log(r) so they are unbounded both at the origin and at infinity.  Moreover, the displacements due to any distribution of traction that exerts a nonzero resultant force on the surface will also be unbounded at infinity. 

 

It is easy to see that this solution satisfies all the relevant boundary conditions.  The surface is traction free ( σ 22 = σ 12 =0 x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaa kiabg2da9iaaicdacaaMc8UaaGPaVlaaykW7caWG4bWaaSbaaSqaai aaigdaaeqaaOGaeyypa0JaaGimaaaa@44BC@  ) except at r=0.  To see that the stresses are consistent with a vertical point force, note that the resultant vertical force exerted by the tractions acting on the dashed curve shown in the picture can be calculated as

F 1 = π/2 π/2 σ rr cosθrdθ = π/2 π/2 2P π cosθ r cosθrdθ =P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaaGymaaqabaGccq GH9aqpdaWdXbqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaaqabaGc ciGGJbGaai4BaiaacohacqaH4oqCcaaMc8UaaGPaVlaadkhacaWGKb GaeqiUdehaleaacqGHsislcqaHapaCcaGGVaGaaGOmaaqaaiabec8a Wjaac+cacaaIYaaaniabgUIiYdGccqGH9aqpdaWdXbqaaiabgkHiTm aalaaabaGaaGOmaiaadcfaaeaacqaHapaCaaWaaSaaaeaaciGGJbGa ai4BaiaacohacqaH4oqCaeaacaWGYbaaaiGacogacaGGVbGaai4Cai abeI7aXjaaykW7caaMc8UaamOCaiaadsgacqaH4oqCaSqaaiabgkHi Tiabec8aWjaac+cacaaIYaaabaGaeqiWdaNaai4laiaaikdaa0Gaey 4kIipakiabg2da9iabgkHiTiaadcfaaaa@6E94@

 

The expressions for displacement can be derived as follows.  Substituting the expression for stress into the stress-strain laws and using the strain-displacement relations yields

ε rr = u r r = ( 1+ν ) E [ (1ν) σ rr ν σ θθ ]= 2P( 1 ν 2 ) πE cosθ r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaa dkhaaeqaaaGcbaGaeyOaIyRaamOCaaaacqGH9aqpdaWcaaqaamaabm aabaGaaGymaiabgUcaRiabe27aUbGaayjkaiaawMcaaaqaaiaadwea aaWaamWaaeaacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcacqaHdp WCdaWgaaWcbaGaamOCaiaadkhaaeqaaOGaeyOeI0IaeqyVd4Maeq4W dm3aaSbaaSqaaiabeI7aXjabeI7aXbqabaaakiaawUfacaGLDbaacq GH9aqpcqGHsisldaWcaaqaaiaaikdacaWGqbWaaeWaaeaacaaIXaGa eyOeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaa aabaGaeqiWdaNaamyraaaadaWcaaqaaiGacogacaGGVbGaai4Caiab eI7aXbqaaiaadkhaaaaaaa@6872@

Integrating

u r = 2P( 1 ν 2 ) πE cosθlog(r)+ f r (θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadkhaaeqaaO Gaeyypa0JaeyOeI0YaaSaaaeaacaaIYaGaamiuamaabmaabaGaaGym aiabgkHiTiabe27aUnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawM caaaqaaiabec8aWjaadweaaaGaci4yaiaac+gacaGGZbGaeqiUdeNa ciiBaiaac+gacaGGNbGaaiikaiaadkhacaGGPaGaey4kaSIaamOzam aaBaaaleaacaWGYbaabeaakiaacIcacqaH4oqCcaGGPaaaaa@5096@

where f r (θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiaadkhaaeqaaO GaaiikaiabeI7aXjaacMcaaaa@37ED@  is a function of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@347C@  to be determined.  Similarly, considering the hoop stresses gives

ε θθ = u r r + 1 r u θ θ = ( 1+ν ) E [ (1ν) σ θθ ν σ rr ]= 2Pν(1+ν) πE cosθ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9maalaaabaGaamyDamaaBaaaleaacaWGYbaa beaaaOqaaiaadkhaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOCaa aadaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaeqiUdehabeaaaOqa aiabgkGi2kabeI7aXbaacqGH9aqpdaWcaaqaamaabmaabaGaaGymai abgUcaRiabe27aUbGaayjkaiaawMcaaaqaaiaadweaaaWaamWaaeaa caGGOaGaaGymaiabgkHiTiabe27aUjaacMcacqaHdpWCdaWgaaWcba GaeqiUdeNaeqiUdehabeaakiabgkHiTiabe27aUjabeo8aZnaaBaaa leaacaWGYbGaamOCaaqabaaakiaawUfacaGLDbaacqGH9aqpdaWcaa qaaiaaikdacaWGqbGaeqyVd4MaaiikaiaaigdacqGHRaWkcqaH9oGB caGGPaaabaGaeqiWdaNaamyraaaadaWcaaqaaiGacogacaGGVbGaai 4CaiabeI7aXbqaaiaadkhaaaaaaa@70DC@

Rearrange and integrate with respect to θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@348C@

u θ = 2P( 1+ν ) πE sinθ( ν+(1ν)log(r) ) f r (θ)dθ + f θ (r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiabeI7aXbqaba GccqGH9aqpdaWcaaqaaiaaikdacaWGqbWaaeWaaeaacaaIXaGaey4k aSIaeqyVd4gacaGLOaGaayzkaaaabaGaeqiWdaNaamyraaaaciGGZb GaaiyAaiaac6gacqaH4oqCdaqadaqaaiabe27aUjabgUcaRiaacIca caaIXaGaeyOeI0IaeqyVd4MaaiykaiGacYgacaGGVbGaai4zaiaacI cacaWGYbGaaiykaaGaayjkaiaawMcaaiabgkHiTmaapeaabaGaamOz amaaBaaaleaacaWGYbaabeaakiaacIcacqaH4oqCcaGGPaGaamizai abeI7aXbWcbeqab0Gaey4kIipakiabgUcaRiaadAgadaWgaaWcbaGa eqiUdehabeaakiaacIcacaWGYbGaaiykaaaa@6303@

where f θ (r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiabeI7aXbqaba GccaGGOaGaamOCaiaacMcaaaa@37FD@  is a function of r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbaaaa@33CD@  to be determined.  Finally, substituting for stresses into the expression for shear strain shows that

ε rθ = 1 2 ( 1 r u r θ + u θ r u θ r )= ( 1+ν ) E σ rθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOCaiabeI 7aXbqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaa baWaaSaaaeaacaaIXaaabaGaamOCaaaadaWcaaqaaiabgkGi2kaadw hadaWgaaWcbaGaamOCaaqabaaakeaacqGHciITcqaH4oqCaaGaey4k aSYaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiabeI7aXbqabaaake aacqGHciITcaWGYbaaaiabgkHiTmaalaaabaGaamyDamaaBaaaleaa cqaH4oqCaeqaaaGcbaGaamOCaaaaaiaawIcacaGLPaaacqGH9aqpda WcaaqaamaabmaabaGaaGymaiabgUcaRiabe27aUbGaayjkaiaawMca aaqaaiaadweaaaGaeq4Wdm3aaSbaaSqaaiaadkhacqaH4oqCaeqaaO Gaeyypa0JaaGimaaaa@5D79@

Inserting the expressions for displacement and simplifying gives

1 r { f r (θ) θ + f r (θ)dθ+ 2P(1+ν)(12ν) πE sinθ }+{ f θ (r) r f θ (r) r }=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaaigdaaeaacaWGYbaaam aacmaabaWaaSaaaeaacqGHciITcaWGMbWaaSbaaSqaaiaadkhaaeqa aOGaaiikaiabeI7aXjaacMcaaeaacqGHciITcqaH4oqCaaGaey4kaS Yaa8qaaeaacaWGMbWaaSbaaSqaaiaadkhaaeqaaOGaaiikaiabeI7a XjaacMcacaWGKbGaeqiUdeNaey4kaSYaaSaaaeaacaaIYaGaamiuai aacIcacaaIXaGaey4kaSIaeqyVd4MaaiykaiaacIcacaaIXaGaeyOe I0IaaGOmaiabe27aUjaacMcaaeaacqaHapaCcaWGfbaaaiGacohaca GGPbGaaiOBaiabeI7aXbWcbeqab0Gaey4kIipaaOGaay5Eaiaaw2ha aiabgUcaRmaacmaabaWaaSaaaeaacqGHciITcaWGMbWaaSbaaSqaai abeI7aXbqabaGccaGGOaGaamOCaiaacMcaaeaacqGHciITcaWGYbaa aiabgkHiTmaalaaabaGaamOzamaaBaaaleaacqaH4oqCaeqaaOGaai ikaiaadkhacaGGPaaabaGaamOCaaaaaiaawUhacaGL9baacqGH9aqp caaIWaaaaa@73C2@

The two terms in parentheses are functions of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@348C@  and r, respectively, and so must both be separately equal to zero to satisfy this expression for all possible values of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@348C@  and r. Therefore

2 f r (θ) θ 2 + f r (θ)= 2P(1+ν)(12ν) πE cosθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadAgadaWgaaWcbaGaamOCaaqabaGccaGGOaGaeqiU deNaaiykaaqaaiabgkGi2kabeI7aXnaaCaaaleqabaGaaGOmaaaaaa GccqGHRaWkcaWGMbWaaSbaaSqaaiaadkhaaeqaaOGaaiikaiabeI7a XjaacMcacqGH9aqpcqGHsisldaWcaaqaaiaaikdacaWGqbGaaiikai aaigdacqGHRaWkcqaH9oGBcaGGPaGaaiikaiaaigdacqGHsislcaaI YaGaeqyVd4Maaiykaaqaaiabec8aWjaadweaaaGaci4yaiaac+gaca GGZbGaeqiUdehaaa@5935@

This ODE has solution

f r (θ)= P(1+ν)(12ν) πE θsinθ+Asinθ+Bcosθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiaadkhaaeqaaO GaaiikaiabeI7aXjaacMcacqGH9aqpcqGHsisldaWcaaqaaiaadcfa caGGOaGaaGymaiabgUcaRiabe27aUjaacMcacaGGOaGaaGymaiabgk HiTiaaikdacqaH9oGBcaGGPaaabaGaeqiWdaNaamyraaaacqaH4oqC ciGGZbGaaiyAaiaac6gacqaH4oqCcqGHRaWkcaWGbbGaci4CaiaacM gacaGGUbGaeqiUdeNaey4kaSIaamOqaiGacogacaGGVbGaai4Caiab eI7aXbaa@5A1B@

The second equation gives

f θ (r) r f θ (r) r =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadAgadaWgaa WcbaGaeqiUdehabeaakiaacIcacaWGYbGaaiykaaqaaiabgkGi2kaa dkhaaaGaeyOeI0YaaSaaaeaacaWGMbWaaSbaaSqaaiabeI7aXbqaba GccaGGOaGaamOCaiaacMcaaeaacaWGYbaaaiabg2da9iaaicdaaaa@449B@

which has solution f θ (r)=Cr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiabeI7aXbqaba GccaGGOaGaamOCaiaacMcacqGH9aqpcaWGdbGaamOCaaaa@3AB2@ .  The constants A,B,C represent an arbitrary rigid displacement, and can be taken to be zero.  This gives the required answer.

 

 

5.2.6 2D Line load acting parallel to the surface of an infinite solid

 

Similarly, the stress fields due to a line load magnitude P per unit out-of-plane length acting tangent to the surface of a homogeneous, isotropic half-space can be generated from the Airy function

ϕ= P π rθcosθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKe9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaey ypa0JaeyOeI0YaaSaaaeaacaWGqbaabaGaeqiWdahaaiaadkhacqaH 4oqCciGGJbGaai4BaiaacohacqaH4oqCaaa@44B0@

The formulas in the preceding section yield

σ rr = 2P π sinθ r σ θθ = σ rθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadkhacaWGYbaabeaakiabg2da9iabgkHiTmaalaaabaGa aGOmaiaadcfaaeaacqaHapaCaaWaaSaaaeaaciGGZbGaaiyAaiaac6 gacqaH4oqCaeaacaWGYbaaaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8Uaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7aXbqabaGccqGH 9aqpcqaHdpWCdaWgaaWcbaGaamOCaiabeI7aXbqabaGccqGH9aqpca aIWaaaaa@6835@

The method outlined in the preceding section can be used to calculate the displacements. The procedure gives

u r = 2( 1 ν 2 ) πE Psinθlogr ( 1+ν )( 12ν ) πE Pθcosθ u θ = 2( 1 ν 2 ) πE Pcosθlogr+ 1+ν πE Pcosθ 2( 12ν )( 1+ν ) πE Pθsinθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG1b WaaSbaaSqaaiaadkhaaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaacaaI YaWaaeWaaeaacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYa aaaaGccaGLOaGaayzkaaaabaGaeqiWdaNaamyraaaacaWGqbGaci4C aiaacMgacaGGUbGaeqiUdeNaciiBaiaac+gacaGGNbGaamOCaiabgk HiTmaalaaabaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGa ayzkaaWaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbGaayjkai aawMcaaaqaaiabec8aWjaadweaaaGaamiuaiabeI7aXjGacogacaGG VbGaai4CaiabeI7aXbqaaiaadwhadaWgaaWcbaGaeqiUdehabeaaki abg2da9maalaaabaGaaGOmamaabmaabaGaaGymaiabgkHiTiabe27a UnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaqaaiabec8aWj aadweaaaGaamiuaiGacogacaGGVbGaai4CaiabeI7aXjGacYgacaGG VbGaai4zaiaadkhacqGHRaWkdaWcaaqaaiaaigdacqGHRaWkcqaH9o GBaeaacqaHapaCcaWGfbaaaiaadcfaciGGJbGaai4BaiaacohacqaH 4oqCcqGHsisldaWcaaqaaiaaikdadaqadaqaaiaaigdacqGHsislca aIYaGaeqyVd4gacaGLOaGaayzkaaWaaeWaaeaacaaIXaGaey4kaSIa eqyVd4gacaGLOaGaayzkaaaabaGaeqiWdaNaamyraaaacaWGqbGaeq iUdeNaci4CaiaacMgacaGGUbGaeqiUdehaaaa@9ADA@

to within an arbitrary rigid motion. 

 

The stresses and displacements in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BC6@  basis are

σ 11 = 2P π x 1 2 x 2 ( x 1 2 + x 2 2 ) 2 σ 22 = 2P π x 2 3 ( x 1 2 + x 2 2 ) 2 σ 12 = 2P π x 1 x 2 2 ( x 1 2 + x 2 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaacaaIYaGaamiuaaqaaiab ec8aWbaadaWcaaqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikdaaa GccaWG4bWaaSbaaSqaaiaaikdaaeqaaaGcbaWaaeWaaeaacaWG4bWa a0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaSIaamiEamaaDaaale aacaaIYaaabaGaaGOmaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGa aGOmaaaaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqa aOGaeyypa0JaeyOeI0YaaSaaaeaacaaIYaGaamiuaaqaaiabec8aWb aadaWcaaqaaiaadIhadaqhaaWcbaGaaGOmaaqaaiaaiodaaaaakeaa daqadaqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHRa WkcaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaGccaGLOaGaayzk aaWaaWbaaSqabeaacaaIYaaaaaaakiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4W dm3aaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iabgkHiTmaala aabaGaaGOmaiaadcfaaeaacqaHapaCaaWaaSaaaeaacaWG4bWaa0ba aSqaaiaaigdaaeaaaaGccaWG4bWaa0baaSqaaiaaikdaaeaacaaIYa aaaaGcbaWaaeWaaeaacaWG4bWaa0baaSqaaiaaigdaaeaacaaIYaaa aOGaey4kaSIaamiEamaaDaaaleaacaaIYaaabaGaaGOmaaaaaOGaay jkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaaaaa@920C@

 

 

5.2.7 Arbitrary pressure acting on a flat surface

 

The principle of superposition can be used to extend the point force solutions to arbitrary pressures acting on a surface. For example, we can find the (plane strain) solution for a uniform pressure acting on the strip of width 2a on the surface of a half-space by distributing the point force solution appropriately.

 

Distributing point forces with magnitude p(s)ds e 1 +q(s)ds e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacaGGOaGaam4CaiaacMcacaWGKb Gaam4CaiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGXbGa aiikaiaadohacaGGPaGaamizaiaadohacaWHLbWaaSbaaSqaaiaaik daaeqaaaaa@3F4E@  over the loaded region shows that

σ 11 = 2 π A x 1 2 ( x 1 p(s)+( x 2 s )q(s)) ( x 1 2 + ( x 2 s ) 2 ) 2 ds σ 22 = 2 π A ( x 2 s) 2 ( x 1 p(s)+( x 2 s)q(s) ) ( x 1 2 + ( x 2 s ) 2 ) 2 ds σ 12 = 2 π A x 1 ( x 2 s)( x 1 p(s)+( x 2 s)q(s) ) ( x 1 2 + ( x 2 s ) 2 ) 2 ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaaikdaaeaacqaH apaCaaWaa8quaeaadaWcaaqaaiaadIhadaqhaaWcbaGaaGymaaqaai aaikdaaaGccaGGOaGaamiEamaaDaaaleaacaaIXaaabaaaaOGaamiC aiaacIcacaWGZbGaaiykaiabgUcaRmaabmaabaGaamiEamaaBaaale aacaaIYaaabeaakiabgkHiTiaadohaaiaawIcacaGLPaaacaWGXbGa aiikaiaadohacaGGPaGaaiykaaqaamaabmaabaGaamiEamaaDaaale aacaaIXaaabaGaaGOmaaaakiabgUcaRmaabmaabaGaamiEamaaBaaa leaacaaIYaaabeaakiabgkHiTiaadohaaiaawIcacaGLPaaadaahaa WcbeqaaiaaikdaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikda aaaaaaqaaiaadgeaaeqaniabgUIiYdGccaWGKbGaam4CaiaaykW7ae aacaaMc8Uaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaakiabg2da 9iabgkHiTmaalaaabaGaaGOmaaqaaiabec8aWbaadaWdrbqaamaala aabaGaaiikaiaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWG ZbGaaiykamaaCaaaleqabaGaaGOmaaaakmaabmaabaGaamiEamaaDa aaleaacaaIXaaabaaaaOGaamiCaiaacIcacaWGZbGaaiykaiabgUca RiaacIcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0Iaam4Cai aacMcacaWGXbGaaiikaiaadohacaGGPaaacaGLOaGaayzkaaaabaWa aeWaaeaacaWG4bWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaS YaaeWaaeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0Iaam4C aaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawM caamaaCaaaleqabaGaaGOmaaaaaaaabaGaamyqaaqab0Gaey4kIipa kiaadsgacaWGZbaabaGaaGPaVlabeo8aZnaaBaaaleaacaaIXaGaaG OmaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaaikdaaeaacqaHapaC aaWaa8quaeaadaWcaaqaaiaadIhadaqhaaWcbaGaaGymaaqaaaaaki aacIcacaWG4bWaa0baaSqaaiaaikdaaeaaaaGccqGHsislcaWGZbGa aiykamaabmaabaGaamiEamaaBaaaleaacaaIXaaabeaakiaadchaca GGOaGaam4CaiaacMcacqGHRaWkcaGGOaGaamiEamaaDaaaleaacaaI YaaabaaaaOGaeyOeI0Iaam4CaiaacMcacaWGXbGaaiikaiaadohaca GGPaaacaGLOaGaayzkaaaabaWaaeWaaeaacaWG4bWaa0baaSqaaiaa igdaaeaacaaIYaaaaOGaey4kaSYaaeWaaeaacaWG4bWaaSbaaSqaai aaikdaaeqaaOGaeyOeI0Iaam4CaaGaayjkaiaawMcaamaaCaaaleqa baGaaGOmaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaa GccaWGKbGaam4CaaWcbaGaamyqaaqab0Gaey4kIipaaaaa@C34F@

 

 

5.2.8 Uniform normal pressure acting on a strip

 

For the particular case of a uniform pressure, the integrals can be evaluated to show that

σ 22 = p 2π ( 2( θ 1 θ 2 )+( sin2 θ 1 sin2 θ 2 ) ) σ 11 = p 2π ( 2( θ 1 θ 2 )( sin2 θ 1 sin2 θ 2 ) ) σ 12 = p 2π ( cos2 θ 1 cos2 θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaeyypa0JaeyOeI0YaaSaa aeaacaWGWbaabaGaaGOmaiabec8aWbaadaqadaqaaiaaikdadaqada qaaiabeI7aXnaaBaaaleaacaaIXaaabeaakiabgkHiTiabeI7aXnaa BaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiabgUcaRmaabmaaba Gaci4CaiaacMgacaGGUbGaaGOmaiabeI7aXnaaBaaaleaacaaIXaaa beaakiabgkHiTiGacohacaGGPbGaaiOBaiaaikdacqaH4oqCdaWgaa WcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaiaawIcacaGLPaaaaeaa cqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyypa0JaeyOeI0 YaaSaaaeaacaWGWbaabaGaaGOmaiabec8aWbaadaqadaqaaiaaikda daqadaqaaiabeI7aXnaaBaaaleaacaaIXaaabeaakiabgkHiTiabeI 7aXnaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiabgkHiTmaa bmaabaGaci4CaiaacMgacaGGUbGaaGOmaiabeI7aXnaaBaaaleaaca aIXaaabeaakiabgkHiTiGacohacaGGPbGaaiOBaiaaikdacqaH4oqC daWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaiaawIcacaGLPa aaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0Za aSaaaeaacaWGWbaabaGaaGOmaiabec8aWbaadaqadaqaaiGacogaca GGVbGaai4CaiaaikdacqaH4oqCdaWgaaWcbaGaaGymaaqabaGccqGH sislciGGJbGaai4BaiaacohacaaIYaGaeqiUde3aaSbaaSqaaiaaik daaeqaaaGccaGLOaGaayzkaaaaaaa@935B@

where 0 θ α π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkabeI7aXnaaBaaaleaacqaHXoqyaeqaaOGaeyizImQaeqiWdaha aa@4069@  and θ 1 = tan 1 x 1 /( x 2 a) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaciiDaiaacggacaGGUbWaaWba aSqabeaacqGHsislcaaIXaaaaOGaamiEamaaBaaaleaacaaIXaaabe aakiaac+cacaGGOaGaamiEamaaBaaaleaacaaIYaaabeaakiabgkHi TiaadggacaGGPaaaaa@4716@   θ 2 = tan 1 x 1 /( x 2 +a) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaaikdaaeqaaOGaeyypa0JaciiDaiaacggacaGGUbWaaWba aSqabeaacqGHsislcaaIXaaaaOGaamiEamaaBaaaleaacaaIXaaabe aakiaac+cacaGGOaGaamiEamaaBaaaleaacaaIYaaabeaakiabgUca RiaadggacaGGPaaaaa@470C@

 

 

 

5.2.9 Stresses near the tip of a crack

 

Consider an infinite solid, which contains a semi-infinite crack on the (x1,x3) plane. Suppose that the solid deforms in plane strain and is subjected to bounded stress at infinity.  The stress field near the tip of the crack can be derived from the Airy function

ϕ= K I 3 2π r 3/2 ( cos3θ/2+3cosθ/2 ) K II 2π r 3/2 ( sin3θ/2+sinθ/2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabew9aMjabg2da9maalaaaba Gaam4samaaBaaaleaacaWGjbaabeaaaOqaaiaaiodadaGcaaqaaiaa ikdacqaHapaCaSqabaaaaOGaamOCamaaCaaaleqabaGaaG4maiaac+ cacaaIYaaaaOWaaeWaaeaaciGGJbGaai4BaiaacohacaaIZaGaeqiU deNaai4laiaaikdacqGHRaWkcaaIZaGaci4yaiaac+gacaGGZbGaeq iUdeNaai4laiaaikdaaiaawIcacaGLPaaaaeaacaaMc8UaaGPaVlaa ykW7caaMc8UaeyOeI0YaaSaaaeaacaWGlbWaaSbaaSqaaiaadMeaca WGjbaabeaaaOqaamaakaaabaGaaGOmaiabec8aWbWcbeaaaaGccaWG YbWaaWbaaSqabeaacaaIZaGaai4laiaaikdaaaGcdaqadaqaaiGaco hacaGGPbGaaiOBaiaaiodacqaH4oqCcaGGVaGaaGOmaiabgUcaRiGa cohacaGGPbGaaiOBaiabeI7aXjaac+cacaaIYaaacaGLOaGaayzkaa aaaaa@6CED@

Here, K I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeaaeqaaa aa@3490@  and K II MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGlbWaaSbaaSqaaiaadMeacaWGjb aabeaaaaa@355E@  are two constants, known as mode I and mode II stress intensity factors, respectively.  They quantify the magnitudes of the stresses near the crack tip, as shown below. Their role will be discussed in more detail when we discuss fracture mechanics. The stresses can be calculated as

σ rr = K I 2πr ( 5 4 cos θ 2 1 4 cos 3θ 2 )+ K II 2πr ( 5 4 sin θ 2 + 3 4 sin 3θ 2 ) σ θθ = K I 2πr ( 3 4 cos θ 2 + 1 4 cos 3θ 2 ) K II 2πr ( 3 4 sin θ 2 + 3 4 sin 3θ 2 ) σ rθ = K I 2πr ( 1 4 sin θ 2 + 1 4 sin 3θ 2 )+ K II 2πr ( 1 4 cos θ 2 + 3 4 cos 3θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCdaWgaaWcbaGaamOCaiaadkhaaeqaaOGaeyypa0ZaaSaaaeaacaWG lbWaaSbaaSqaaiaadMeaaeqaaaGcbaWaaOaaaeaacaaIYaGaeqiWda NaamOCaaWcbeaaaaGcdaqadaqaamaalaaabaGaaGynaaqaaiaaisda aaGaci4yaiaac+gacaGGZbWaaSaaaeaacqaH4oqCaeaacaaIYaaaai abgkHiTmaalaaabaGaaGymaaqaaiaaisdaaaGaci4yaiaac+gacaGG ZbWaaSaaaeaacaaIZaGaeqiUdehabaGaaGOmaaaaaiaawIcacaGLPa aacqGHRaWkdaWcaaqaaiaadUeadaWgaaWcbaGaamysaiaadMeaaeqa aaGcbaWaaOaaaeaacaaIYaGaeqiWdaNaamOCaaWcbeaaaaGcdaqada qaaiabgkHiTmaalaaabaGaaGynaaqaaiaaisdaaaGaci4CaiaacMga caGGUbWaaSaaaeaacqaH4oqCaeaacaaIYaaaaiabgUcaRmaalaaaba GaaG4maaqaaiaaisdaaaGaci4CaiaacMgacaGGUbWaaSaaaeaacaaI ZaGaeqiUdehabaGaaGOmaaaaaiaawIcacaGLPaaaaeaacqaHdpWCda WgaaWcbaGaeqiUdeNaeqiUdehabeaakiabg2da9maalaaabaGaam4s amaaBaaaleaacaWGjbaabeaaaOqaamaakaaabaGaaGOmaiabec8aWj aadkhaaSqabaaaaOWaaeWaaeaadaWcaaqaaiaaiodaaeaacaaI0aaa aiGacogacaGGVbGaai4CamaalaaabaGaeqiUdehabaGaaGOmaaaacq GHRaWkdaWcaaqaaiaaigdaaeaacaaI0aaaaiGacogacaGGVbGaai4C amaalaaabaGaaG4maiabeI7aXbqaaiaaikdaaaaacaGLOaGaayzkaa GaeyOeI0YaaSaaaeaacaWGlbWaaSbaaSqaaiaadMeacaWGjbaabeaa aOqaamaakaaabaGaaGOmaiabec8aWjaadkhaaSqabaaaaOWaaeWaae aadaWcaaqaaiaaiodaaeaacaaI0aaaaiGacohacaGGPbGaaiOBamaa laaabaGaeqiUdehabaGaaGOmaaaacqGHRaWkdaWcaaqaaiaaiodaae aacaaI0aaaaiGacohacaGGPbGaaiOBamaalaaabaGaaG4maiabeI7a XbqaaiaaikdaaaaacaGLOaGaayzkaaaabaGaeq4Wdm3aaSbaaSqaai aadkhacqaH4oqCaeqaaOGaeyypa0ZaaSaaaeaacaWGlbWaaSbaaSqa aiaadMeaaeqaaaGcbaWaaOaaaeaacaaIYaGaeqiWdaNaamOCaaWcbe aaaaGcdaqadaqaamaalaaabaGaaGymaaqaaiaaisdaaaGaci4Caiaa cMgacaGGUbWaaSaaaeaacqaH4oqCaeaacaaIYaaaaiabgUcaRmaala aabaGaaGymaaqaaiaaisdaaaGaci4CaiaacMgacaGGUbWaaSaaaeaa caaIZaGaeqiUdehabaGaaGOmaaaaaiaawIcacaGLPaaacqGHRaWkda WcaaqaaiaadUeadaWgaaWcbaGaamysaiaadMeaaeqaaaGcbaWaaOaa aeaacaaIYaGaeqiWdaNaamOCaaWcbeaaaaGcdaqadaqaamaalaaaba GaaGymaaqaaiaaisdaaaGaci4yaiaac+gacaGGZbWaaSaaaeaacqaH 4oqCaeaacaaIYaaaaiabgUcaRmaalaaabaGaaG4maaqaaiaaisdaaa Gaci4yaiaac+gacaGGZbWaaSaaaeaacaaIZaGaeqiUdehabaGaaGOm aaaaaiaawIcacaGLPaaaaaaa@D47A@

Equivalent expressions in rectangular coordinates are

σ 11 = K I 2πr cos θ 2 ( 1sin θ 2 sin 3θ 2 ) K II 2πr sin θ 2 ( 2+cos θ 2 cos 3θ 2 ) σ 22 = K I 2πr cos θ 2 ( 1+sin θ 2 sin 3θ 2 )+ K II 2πr cos θ 2 sin θ 2 cos 3θ 2 σ 12 = K I 2πr cos θ 2 sin θ 2 cos 3θ 2 + K II 2πr cos θ 2 ( 1sin θ 2 sin 3θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacaWG lbWaaSbaaSqaaiaadMeaaeqaaaGcbaWaaOaaaeaacaaIYaGaeqiWda NaamOCaaWcbeaaaaGcciGGJbGaai4BaiaacohadaWcaaqaaiabeI7a XbqaaiaaikdaaaWaaeWaaeaacaaIXaGaeyOeI0Iaci4CaiaacMgaca GGUbWaaSaaaeaacqaH4oqCaeaacaaIYaaaaiGacohacaGGPbGaaiOB amaalaaabaGaaG4maiabeI7aXbqaaiaaikdaaaaacaGLOaGaayzkaa GaeyOeI0YaaSaaaeaacaWGlbWaaSbaaSqaaiaadMeacaWGjbaabeaa aOqaamaakaaabaGaaGOmaiabec8aWjaadkhaaSqabaaaaOGaci4Cai aacMgacaGGUbWaaSaaaeaacqaH4oqCaeaacaaIYaaaamaabmaabaGa aGOmaiabgUcaRiGacogacaGGVbGaai4CamaalaaabaGaeqiUdehaba GaaGOmaaaaciGGJbGaai4BaiaacohadaWcaaqaaiaaiodacqaH4oqC aeaacaaIYaaaaaGaayjkaiaawMcaaaqaaiabeo8aZnaaBaaaleaaca aIYaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiaadUeadaWgaaWcbaGa amysaaqabaaakeaadaGcaaqaaiaaikdacqaHapaCcaWGYbaaleqaaa aakiGacogacaGGVbGaai4CamaalaaabaGaeqiUdehabaGaaGOmaaaa daqadaqaaiaaigdacqGHRaWkciGGZbGaaiyAaiaac6gadaWcaaqaai abeI7aXbqaaiaaikdaaaGaci4CaiaacMgacaGGUbWaaSaaaeaacaaI ZaGaeqiUdehabaGaaGOmaaaaaiaawIcacaGLPaaacqGHRaWkdaWcaa qaaiaadUeadaWgaaWcbaGaamysaiaadMeaaeqaaaGcbaWaaOaaaeaa caaIYaGaeqiWdaNaamOCaaWcbeaaaaGcciGGJbGaai4Baiaacohada WcaaqaaiabeI7aXbqaaiaaikdaaaGaci4CaiaacMgacaGGUbWaaSaa aeaacqaH4oqCaeaacaaIYaaaaiGacogacaGGVbGaai4Camaalaaaba GaaG4maiabeI7aXbqaaiaaikdaaaaabaGaeq4Wdm3aaSbaaSqaaiaa igdacaaIYaaabeaakiabg2da9maalaaabaGaam4samaaBaaaleaaca WGjbaabeaaaOqaamaakaaabaGaaGOmaiabec8aWjaadkhaaSqabaaa aOGaci4yaiaac+gacaGGZbWaaSaaaeaacqaH4oqCaeaacaaIYaaaai GacohacaGGPbGaaiOBamaalaaabaGaeqiUdehabaGaaGOmaaaaciGG JbGaai4BaiaacohadaWcaaqaaiaaiodacqaH4oqCaeaacaaIYaaaai abgUcaRmaalaaabaGaam4samaaBaaaleaacaWGjbGaamysaaqabaaa keaadaGcaaqaaiaaikdacqaHapaCcaWGYbaaleqaaaaakiGacogaca GGVbGaai4CamaalaaabaGaeqiUdehabaGaaGOmaaaadaqadaqaaiaa igdacqGHsislciGGZbGaaiyAaiaac6gadaWcaaqaaiabeI7aXbqaai aaikdaaaGaci4CaiaacMgacaGGUbWaaSaaaeaacaaIZaGaeqiUdeha baGaaGOmaaaaaiaawIcacaGLPaaaaaaa@DB9C@

while the displacements can be calculated by integrating the strains, with the result

u 1 = K I μ r 2π [ 12ν+ sin 2 θ 2 ]cos θ 2 + K II μ r 2π [ 22ν+ cos 2 θ 2 ]sin θ 2 u 2 = K I μ r 2π [ 22ν cos 2 θ 2 ]sin θ 2 + K II μ r 2π [ 1+2ν+ sin 2 θ 2 ]cos θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG1b WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacaWGlbWaaSba aSqaaiaadMeaaeqaaaGcbaGaeqiVd0gaamaakaaabaWaaSaaaeaaca WGYbaabaGaaGOmaiabec8aWbaaaSqabaGcdaWadaqaaiaaigdacqGH sislcaaIYaGaeqyVd4Maey4kaSIaci4CaiaacMgacaGGUbWaaWbaaS qabeaacaaIYaaaaOWaaSaaaeaacqaH4oqCaeaacaaIYaaaaaGaay5w aiaaw2faaiGacogacaGGVbGaai4CamaalaaabaGaeqiUdehabaGaaG OmaaaacqGHRaWkdaWcaaqaaiaadUeadaWgaaWcbaGaamysaiaadMea aeqaaaGcbaGaeqiVd0gaamaakaaabaWaaSaaaeaacaWGYbaabaGaaG Omaiabec8aWbaaaSqabaGcdaWadaqaaiaaikdacqGHsislcaaIYaGa eqyVd4Maey4kaSIaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYa aaaOWaaSaaaeaacqaH4oqCaeaacaaIYaaaaaGaay5waiaaw2faaiGa cohacaGGPbGaaiOBamaalaaabaGaeqiUdehabaGaaGOmaaaaaeaaca WG1bWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaWGlbWa aSbaaSqaaiaadMeaaeqaaaGcbaGaeqiVd0gaamaakaaabaWaaSaaae aacaWGYbaabaGaaGOmaiabec8aWbaaaSqabaGcdaWadaqaaiaaikda cqGHsislcaaIYaGaeqyVd4MaeyOeI0Iaci4yaiaac+gacaGGZbWaaW baaSqabeaacaaIYaaaaOWaaSaaaeaacqaH4oqCaeaacaaIYaaaaaGa ay5waiaaw2faaiGacohacaGGPbGaaiOBamaalaaabaGaeqiUdehaba GaaGOmaaaacqGHRaWkdaWcaaqaaiaadUeadaWgaaWcbaGaamysaiaa dMeaaeqaaaGcbaGaeqiVd0gaamaakaaabaWaaSaaaeaacaWGYbaaba GaaGOmaiabec8aWbaaaSqabaGcdaWadaqaaiabgkHiTiaaigdacqGH RaWkcaaIYaGaeqyVd4Maey4kaSIaci4CaiaacMgacaGGUbWaaWbaaS qabeaacaaIYaaaaOWaaSaaaeaacqaH4oqCaeaacaaIYaaaaaGaay5w aiaaw2faaiGacogacaGGVbGaai4CamaalaaabaGaeqiUdehabaGaaG Omaaaaaaaa@A87A@

Note that this displacement field is valid for plane strain deformation only.

 

Observe that the stress intensity factor has the bizarre units of N m 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaad2 gadaahaaWcbeqaaiabgkHiTiaaiodacaGGVaGaaGOmaaaaaaa@3B01@ .